--- a/fit.py	Fri Aug 19 20:16:56 2022 +0000
+++ b/fit.py	Fri Mar 10 16:02:04 2023 +0000
@@ -7,45 +7,118 @@
 @author: rv43
 """
 
-import sys
-import re
 import logging
-import numpy as np
 
-from asteval import Interpreter
+from asteval import Interpreter, get_ast_names
 from copy import deepcopy
-#from lmfit import Minimizer
 from lmfit import Model, Parameters
+from lmfit.model import ModelResult
 from lmfit.models import ConstantModel, LinearModel, QuadraticModel, PolynomialModel,\
-        StepModel, RectangleModel, GaussianModel, LorentzianModel
+        ExponentialModel, StepModel, RectangleModel, ExpressionModel, GaussianModel,\
+        LorentzianModel
+import numpy as np
+from os import cpu_count, getpid, listdir, mkdir, path
+from re import compile, sub
+from shutil import rmtree
+from sympy import diff, simplify
+try:
+    from joblib import Parallel, delayed
+    have_joblib = True
+except:
+    have_joblib = False
+try:
+    import xarray as xr
+    have_xarray = True
+except:
+    have_xarray = False
 
-from general import is_index, index_nearest, quickPlot
+from .general import illegal_value, is_int, is_dict_series, is_index, index_nearest, \
+        almost_equal, quick_plot #, eval_expr
+#from sys import path as syspath
+#syspath.append(f'/nfs/chess/user/rv43/msnctools/msnctools')
+#from general import illegal_value, is_int, is_dict_series, is_index, index_nearest, \
+#        almost_equal, quick_plot #, eval_expr
+
+from sys import float_info
+float_min = float_info.min
+float_max = float_info.max
 
 # sigma = fwhm_factor*fwhm
 fwhm_factor = {
-    'gaussian' : f'fwhm/(2*sqrt(2*log(2)))',
-    'lorentzian' : f'0.5*fwhm',
-    'splitlorentzian' : f'0.5*fwhm', # sigma = sigma_r
-    'voight' : f'0.2776*fwhm', # sigma = gamma
-    'pseudovoight' : f'0.5*fwhm'} # fraction = 0.5
+    'gaussian': f'fwhm/(2*sqrt(2*log(2)))',
+    'lorentzian': f'0.5*fwhm',
+    'splitlorentzian': f'0.5*fwhm', # sigma = sigma_r
+    'voight': f'0.2776*fwhm', # sigma = gamma
+    'pseudovoight': f'0.5*fwhm'} # fraction = 0.5
 
 # amplitude = height_factor*height*fwhm
 height_factor = {
-    'gaussian' : f'height*fwhm*0.5*sqrt(pi/log(2))',
-    'lorentzian' : f'height*fwhm*0.5*pi',
-    'splitlorentzian' : f'height*fwhm*0.5*pi', # sigma = sigma_r
-    'voight' : f'3.334*height*fwhm', # sigma = gamma
-    'pseudovoight' : f'1.268*height*fwhm'} # fraction = 0.5
+    'gaussian': f'height*fwhm*0.5*sqrt(pi/log(2))',
+    'lorentzian': f'height*fwhm*0.5*pi',
+    'splitlorentzian': f'height*fwhm*0.5*pi', # sigma = sigma_r
+    'voight': f'3.334*height*fwhm', # sigma = gamma
+    'pseudovoight': f'1.268*height*fwhm'} # fraction = 0.5
 
 class Fit:
     """Wrapper class for lmfit
     """
-    def __init__(self, x, y, models=None, **kwargs):
-        self._x = x
-        self._y = y
+    def __init__(self, y, x=None, models=None, normalize=True, **kwargs):
+        if not isinstance(normalize, bool):
+            raise ValueError(f'Invalid parameter normalize ({normalize})')
+        self._mask = None
         self._model = None
+        self._norm = None
+        self._normalized = False
         self._parameters = Parameters()
+        self._parameter_bounds = None
+        self._parameter_norms = {}
+        self._linear_parameters = []
+        self._nonlinear_parameters = []
         self._result = None
+        self._try_linear_fit = True
+        self._y = None
+        self._y_norm = None
+        self._y_range = None
+        if 'try_linear_fit' in kwargs:
+            try_linear_fit = kwargs.pop('try_linear_fit')
+            if not isinstance(try_linear_fit, bool):
+                illegal_value(try_linear_fit, 'try_linear_fit', 'Fit.fit', raise_error=True)
+            self._try_linear_fit = try_linear_fit
+        if y is not None:
+            if isinstance(y, (tuple, list, np.ndarray)):
+                self._x = np.asarray(x)
+            elif have_xarray and isinstance(y, xr.DataArray):
+                if x is not None:
+                    logging.warning('Ignoring superfluous input x ({x}) in Fit.__init__')
+                if y.ndim != 1:
+                    illegal_value(y.ndim, 'DataArray dimensions', 'Fit:__init__', raise_error=True)
+                self._x = np.asarray(y[y.dims[0]])
+            else:
+                illegal_value(y, 'y', 'Fit:__init__', raise_error=True)
+            self._y = y
+            if self._x.ndim != 1:
+                raise ValueError(f'Invalid dimension for input x ({self._x.ndim})')
+            if self._x.size != self._y.size:
+                raise ValueError(f'Inconsistent x and y dimensions ({self._x.size} vs '+
+                        f'{self._y.size})')
+            if 'mask' in kwargs:
+                self._mask = kwargs.pop('mask')
+            if self._mask is None:
+                y_min = float(self._y.min())
+                self._y_range = float(self._y.max())-y_min
+                if normalize and self._y_range > 0.0:
+                    self._norm = (y_min, self._y_range)
+            else:
+                self._mask = np.asarray(self._mask).astype(bool)
+                if self._x.size != self._mask.size:
+                    raise ValueError(f'Inconsistent x and mask dimensions ({self._x.size} vs '+
+                            f'{self._mask.size})')
+                y_masked = np.asarray(self._y)[~self._mask]
+                y_min = float(y_masked.min())
+                self._y_range = float(y_masked.max())-y_min
+                if normalize and self._y_range > 0.0:
+                    if normalize and self._y_range > 0.0:
+                        self._norm = (y_min, self._y_range)
         if models is not None:
             if callable(models) or isinstance(models, str):
                 kwargs = self.add_model(models, **kwargs)
@@ -55,73 +128,173 @@
             self.fit(**kwargs)
 
     @classmethod
-    def fit_data(cls, x, y, models, **kwargs):
-        return cls(x, y, models, **kwargs)
+    def fit_data(cls, y, models, x=None, normalize=True, **kwargs):
+        return(cls(y, x=x, models=models, normalize=normalize, **kwargs))
 
     @property
     def best_errors(self):
         if self._result is None:
-            return None
-        errors = {}
-        names = sorted(self._result.params)
-        for name in names:
-            par = self._result.params[name]
-            errors[name] = par.stderr
-        return errors
+            return(None)
+        return({name:self._result.params[name].stderr for name in sorted(self._result.params)
+                if name != 'tmp_normalization_offset_c'})
 
     @property
     def best_fit(self):
         if self._result is None:
-            return None
-        return self._result.best_fit
+            return(None)
+        return(self._result.best_fit)
 
     @property
     def best_parameters(self):
         if self._result is None:
-            return None
-        parameters = []
-        names = sorted(self._result.params)
-        for name in names:
-            par = self._result.params[name]
-            parameters.append({'name' : par.name, 'value' : par.value, 'error' : par.stderr,
-                    'init_value' : par.init_value, 'min' : par.min, 'max' : par.max,
-                    'vary' : par.vary, 'expr' : par.expr})
-        return parameters
+            return(None)
+        parameters = {}
+        for name in sorted(self._result.params):
+            if name != 'tmp_normalization_offset_c':
+                par = self._result.params[name]
+                parameters[name] = {'value': par.value, 'error': par.stderr,
+                        'init_value': par.init_value, 'min': par.min, 'max': par.max,
+                        'vary': par.vary, 'expr': par.expr}
+        return(parameters)
+
+    @property
+    def best_results(self):
+        """Convert the input data array to a data set and add the fit results.
+        """
+        if self._result is None:
+            return(None)
+        if isinstance(self._y, xr.DataArray):
+            best_results = self._y.to_dataset()
+            dims = self._y.dims
+            fit_name = f'{self._y.name}_fit'
+        else:
+            coords = {'x': (['x'], self._x)}
+            dims = ('x')
+            best_results = xr.Dataset(coords=coords)
+            best_results['y'] = (dims, self._y)
+            fit_name = 'y_fit'
+        best_results[fit_name] = (dims, self.best_fit)
+        if self._mask is not None:
+            best_results['mask'] = self._mask
+        best_results.coords['par_names'] = ('peak', [name for name in self.best_values.keys()])
+        best_results['best_values'] = (['par_names'], [v for v in self.best_values.values()])
+        best_results['best_errors'] = (['par_names'], [v for v in self.best_errors.values()])
+        best_results.attrs['components'] = self.components
+        return(best_results)
 
     @property
     def best_values(self):
         if self._result is None:
-            return None
-        return self._result.params.valuesdict()
+            return(None)
+        return({name:self._result.params[name].value for name in sorted(self._result.params)
+                if name != 'tmp_normalization_offset_c'})
 
     @property
     def chisqr(self):
-        return self._result.chisqr
+        if self._result is None:
+            return(None)
+        return(self._result.chisqr)
+
+    @property
+    def components(self):
+        components = {}
+        if self._result is None:
+            logging.warning('Unable to collect components in Fit.components')
+            return(components)
+        for component in self._result.components:
+            if 'tmp_normalization_offset_c' in component.param_names:
+                continue
+            parameters = {}
+            for name in component.param_names:
+                par = self._parameters[name]
+                parameters[name] = {'free': par.vary, 'value': self._result.params[name].value}
+                if par.expr is not None:
+                    parameters[name]['expr'] = par.expr
+            expr = None
+            if isinstance(component, ExpressionModel):
+                name = component._name
+                if name[-1] == '_':
+                    name = name[:-1]
+                expr = component.expr
+            else:
+                prefix = component.prefix
+                if len(prefix):
+                    if prefix[-1] == '_':
+                        prefix = prefix[:-1]
+                    name = f'{prefix} ({component._name})'
+                else:
+                    name = f'{component._name}'
+            if expr is None:
+                components[name] = {'parameters': parameters}
+            else:
+                components[name] = {'expr': expr, 'parameters': parameters}
+        return(components)
 
     @property
     def covar(self):
-        return self._result.covar
+        if self._result is None:
+            return(None)
+        return(self._result.covar)
+
+    @property
+    def init_parameters(self):
+        if self._result is None or self._result.init_params is None:
+            return(None)
+        parameters = {}
+        for name in sorted(self._result.init_params):
+            if name != 'tmp_normalization_offset_c':
+                par = self._result.init_params[name]
+                parameters[name] = {'value': par.value, 'min': par.min, 'max': par.max,
+                        'vary': par.vary, 'expr': par.expr}
+        return(parameters)
 
     @property
     def init_values(self):
+        if self._result is None or self._result.init_params is None:
+            return(None)
+        return({name:self._result.init_params[name].value for name in
+                sorted(self._result.init_params) if name != 'tmp_normalization_offset_c'})
+
+    @property
+    def normalization_offset(self):
         if self._result is None:
-            return None
-        return self._result.init_values
+            return(None)
+        if self._norm is None:
+            return(0.0)
+        else:
+            if self._result.init_params is not None:
+                normalization_offset = self._result.init_params['tmp_normalization_offset_c']
+            else:
+                normalization_offset = self._result.params['tmp_normalization_offset_c']
+            return(normalization_offset)
 
     @property
     def num_func_eval(self):
-        return self._result.nfev
+        if self._result is None:
+            return(None)
+        return(self._result.nfev)
+
+    @property
+    def parameters(self):
+        return({name:{'min': par.min, 'max': par.max, 'vary': par.vary, 'expr': par.expr}
+                for name, par in self._parameters.items() if name != 'tmp_normalization_offset_c'})
 
     @property
     def redchi(self):
-        return self._result.redchi
+        if self._result is None:
+            return(None)
+        return(self._result.redchi)
 
     @property
     def residual(self):
-        return self._result.residual
+        if self._result is None:
+            return(None)
+        return(self._result.residual)
 
     @property
     def success(self):
+        if self._result is None:
+            return(None)
         if not self._result.success:
 #            print(f'ier = {self._result.ier}')
 #            print(f'lmdif_message = {self._result.lmdif_message}')
@@ -133,183 +306,692 @@
                 logging.warning(f'ier = {self._result.ier}: {self._result.message}')
             else:
                 logging.warning(f'ier = {self._result.ier}: {self._result.message}')
-                return True
+                return(True)
 #            self.print_fit_report()
 #            self.plot()
-        return self._result.success
+        return(self._result.success)
 
     @property
     def var_names(self):
         """Intended to be used with covar
         """
         if self._result is None:
-            return None
-        return self._result.var_names
+            return(None)
+        return(getattr(self._result, 'var_names', None))
+
+    @property
+    def x(self):
+        return(self._x)
 
-    def print_fit_report(self, show_correl=False):
-        if self._result is not None:
-            print(self._result.fit_report(show_correl=show_correl))
+    @property
+    def y(self):
+        return(self._y)
+
+    def print_fit_report(self, result=None, show_correl=False):
+        if result is None:
+            result = self._result
+        if result is not None:
+            print(result.fit_report(show_correl=show_correl))
 
     def add_parameter(self, **parameter):
         if not isinstance(parameter, dict):
-            illegal_value(parameter, 'parameter', 'add_parameter')
-            return
+            raise ValueError(f'Invalid parameter ({parameter})')
+        if parameter.get('expr') is not None:
+            raise KeyError(f'Illegal "expr" key in parameter {parameter}')
+        name = parameter['name']
+        if not isinstance(name, str):
+            raise ValueError(f'Illegal "name" value ({name}) in parameter {parameter}')
+        if parameter.get('norm') is None:
+            self._parameter_norms[name] = False
+        else:
+            norm = parameter.pop('norm')
+            if self._norm is None:
+                logging.warning(f'Ignoring norm in parameter {name} in '+
+                            f'Fit.add_parameter (normalization is turned off)')
+                self._parameter_norms[name] = False
+            else:
+                if not isinstance(norm, bool):
+                    raise ValueError(f'Illegal "norm" value ({norm}) in parameter {parameter}')
+                self._parameter_norms[name] = norm
+        vary = parameter.get('vary')
+        if vary is not None:
+            if not isinstance(vary, bool):
+                raise ValueError(f'Illegal "vary" value ({vary}) in parameter {parameter}')
+            if not vary:
+                if 'min' in parameter:
+                    logging.warning(f'Ignoring min in parameter {name} in '+
+                            f'Fit.add_parameter (vary = {vary})')
+                    parameter.pop('min')
+                if 'max' in parameter:
+                    logging.warning(f'Ignoring max in parameter {name} in '+
+                            f'Fit.add_parameter (vary = {vary})')
+                    parameter.pop('max')
+        if self._norm is not None and name not in self._parameter_norms:
+            raise ValueError(f'Missing parameter normalization type for paremeter {name}')
         self._parameters.add(**parameter)
 
-    def add_model(self, model, prefix=None, parameters=None, **kwargs):
+    def add_model(self, model, prefix=None, parameters=None, parameter_norms=None, **kwargs):
         # Create the new model
-#        print('\nAt start adding model:')
-#        self._parameters.pretty_print()
+#        print(f'at start add_model:\nself._parameters:\n{self._parameters}')
+#        print(f'at start add_model: kwargs = {kwargs}')
+#        print(f'parameters = {parameters}')
+#        print(f'parameter_norms = {parameter_norms}')
+#        if len(self._parameters.keys()):
+#            print('\nAt start adding model:')
+#            self._parameters.pretty_print()
+#            print(f'parameter_norms:\n{self._parameter_norms}')
         if prefix is not None and not isinstance(prefix, str):
             logging.warning('Ignoring illegal prefix: {model} {type(model)}')
             prefix = None
+        if prefix is None:
+            pprefix = ''
+        else:
+            pprefix = prefix
+        if parameters is not None:
+            if isinstance(parameters, dict):
+                parameters = (parameters, )
+            elif not is_dict_series(parameters):
+                illegal_value(parameters, 'parameters', 'Fit.add_model', raise_error=True)
+            parameters = deepcopy(parameters)
+        if parameter_norms is not None:
+            if isinstance(parameter_norms, dict):
+                parameter_norms = (parameter_norms, )
+            if not is_dict_series(parameter_norms):
+                illegal_value(parameter_norms, 'parameter_norms', 'Fit.add_model', raise_error=True)
+        new_parameter_norms = {}
         if callable(model):
+            # Linear fit not yet implemented for callable models
+            self._try_linear_fit = False
+            if parameter_norms is None:
+                if parameters is None:
+                    raise ValueError('Either "parameters" or "parameter_norms" is required in '+
+                            f'{model}')
+                for par in parameters:
+                    name = par['name']
+                    if not isinstance(name, str):
+                        raise ValueError(f'Illegal "name" value ({name}) in input parameters')
+                    if par.get('norm') is not None:
+                        norm = par.pop('norm')
+                        if not isinstance(norm, bool):
+                            raise ValueError(f'Illegal "norm" value ({norm}) in input parameters')
+                        new_parameter_norms[f'{pprefix}{name}'] = norm
+            else:
+                for par in parameter_norms:
+                    name = par['name']
+                    if not isinstance(name, str):
+                        raise ValueError(f'Illegal "name" value ({name}) in input parameters')
+                    norm = par.get('norm')
+                    if norm is None or not isinstance(norm, bool):
+                        raise ValueError(f'Illegal "norm" value ({norm}) in input parameters')
+                    new_parameter_norms[f'{pprefix}{name}'] = norm
+            if parameters is not None:
+                for par in parameters:
+                    if par.get('expr') is not None:
+                        raise KeyError(f'Illegal "expr" key ({par.get("expr")}) in parameter '+
+                                f'{name} for a callable model {model}')
+                    name = par['name']
+                    if not isinstance(name, str):
+                        raise ValueError(f'Illegal "name" value ({name}) in input parameters')
+# RV FIX callable model will need partial deriv functions for any linear pars to get the linearized matrix, so for now skip linear solution option
             newmodel = Model(model, prefix=prefix)
         elif isinstance(model, str):
-            if model == 'constant':
+            if model == 'constant':      # Par: c
                 newmodel = ConstantModel(prefix=prefix)
-            elif model == 'linear':
+                new_parameter_norms[f'{pprefix}c'] = True
+                self._linear_parameters.append(f'{pprefix}c')
+            elif model == 'linear':      # Par: slope, intercept
                 newmodel = LinearModel(prefix=prefix)
-            elif model == 'quadratic':
+                new_parameter_norms[f'{pprefix}slope'] = True
+                new_parameter_norms[f'{pprefix}intercept'] = True
+                self._linear_parameters.append(f'{pprefix}slope')
+                self._linear_parameters.append(f'{pprefix}intercept')
+            elif model == 'quadratic':   # Par: a, b, c
                 newmodel = QuadraticModel(prefix=prefix)
-            elif model == 'gaussian':
+                new_parameter_norms[f'{pprefix}a'] = True
+                new_parameter_norms[f'{pprefix}b'] = True
+                new_parameter_norms[f'{pprefix}c'] = True
+                self._linear_parameters.append(f'{pprefix}a')
+                self._linear_parameters.append(f'{pprefix}b')
+                self._linear_parameters.append(f'{pprefix}c')
+            elif model == 'gaussian':    # Par: amplitude, center, sigma (fwhm, height)
                 newmodel = GaussianModel(prefix=prefix)
-            elif model == 'step':
+                new_parameter_norms[f'{pprefix}amplitude'] = True
+                new_parameter_norms[f'{pprefix}center'] = False
+                new_parameter_norms[f'{pprefix}sigma'] = False
+                self._linear_parameters.append(f'{pprefix}amplitude')
+                self._nonlinear_parameters.append(f'{pprefix}center')
+                self._nonlinear_parameters.append(f'{pprefix}sigma')
+                # parameter norms for height and fwhm are needed to get correct errors
+                new_parameter_norms[f'{pprefix}height'] = True
+                new_parameter_norms[f'{pprefix}fwhm'] = False
+            elif model == 'lorentzian':    # Par: amplitude, center, sigma (fwhm, height)
+                newmodel = LorentzianModel(prefix=prefix)
+                new_parameter_norms[f'{pprefix}amplitude'] = True
+                new_parameter_norms[f'{pprefix}center'] = False
+                new_parameter_norms[f'{pprefix}sigma'] = False
+                self._linear_parameters.append(f'{pprefix}amplitude')
+                self._nonlinear_parameters.append(f'{pprefix}center')
+                self._nonlinear_parameters.append(f'{pprefix}sigma')
+                # parameter norms for height and fwhm are needed to get correct errors
+                new_parameter_norms[f'{pprefix}height'] = True
+                new_parameter_norms[f'{pprefix}fwhm'] = False
+            elif model == 'exponential': # Par: amplitude, decay
+                newmodel = ExponentialModel(prefix=prefix)
+                new_parameter_norms[f'{pprefix}amplitude'] = True
+                new_parameter_norms[f'{pprefix}decay'] = False
+                self._linear_parameters.append(f'{pprefix}amplitude')
+                self._nonlinear_parameters.append(f'{pprefix}decay')
+            elif model == 'step':        # Par: amplitude, center, sigma
                 form = kwargs.get('form')
                 if form is not None:
-                    del kwargs['form']
+                    kwargs.pop('form')
                 if form is None or form not in ('linear', 'atan', 'arctan', 'erf', 'logistic'):
-                    logging.error(f'Illegal form parameter for build-in step model ({form})')
-                    return kwargs
+                    raise ValueError(f'Invalid parameter form for build-in step model ({form})')
                 newmodel = StepModel(prefix=prefix, form=form)
-            elif model == 'rectangle':
+                new_parameter_norms[f'{pprefix}amplitude'] = True
+                new_parameter_norms[f'{pprefix}center'] = False
+                new_parameter_norms[f'{pprefix}sigma'] = False
+                self._linear_parameters.append(f'{pprefix}amplitude')
+                self._nonlinear_parameters.append(f'{pprefix}center')
+                self._nonlinear_parameters.append(f'{pprefix}sigma')
+            elif model == 'rectangle':   # Par: amplitude, center1, center2, sigma1, sigma2
                 form = kwargs.get('form')
                 if form is not None:
-                    del kwargs['form']
+                    kwargs.pop('form')
                 if form is None or form not in ('linear', 'atan', 'arctan', 'erf', 'logistic'):
-                    logging.error(f'Illegal form parameter for build-in rectangle model ({form})')
-                    return kwargs
+                    raise ValueError('Invalid parameter form for build-in rectangle model '+
+                            f'({form})')
                 newmodel = RectangleModel(prefix=prefix, form=form)
+                new_parameter_norms[f'{pprefix}amplitude'] = True
+                new_parameter_norms[f'{pprefix}center1'] = False
+                new_parameter_norms[f'{pprefix}center2'] = False
+                new_parameter_norms[f'{pprefix}sigma1'] = False
+                new_parameter_norms[f'{pprefix}sigma2'] = False
+                self._linear_parameters.append(f'{pprefix}amplitude')
+                self._nonlinear_parameters.append(f'{pprefix}center1')
+                self._nonlinear_parameters.append(f'{pprefix}center2')
+                self._nonlinear_parameters.append(f'{pprefix}sigma1')
+                self._nonlinear_parameters.append(f'{pprefix}sigma2')
+            elif model == 'expression':  # Par: by expression
+                expr = kwargs['expr']
+                if not isinstance(expr, str):
+                    raise ValueError(f'Illegal "expr" value ({expr}) in {model}')
+                kwargs.pop('expr')
+                if parameter_norms is not None:
+                        logging.warning('Ignoring parameter_norms (normalization determined from '+
+                                'linearity)}')
+                if parameters is not None:
+                    for par in parameters:
+                        if par.get('expr') is not None:
+                            raise KeyError(f'Illegal "expr" key ({par.get("expr")}) in parameter '+
+                                    f'({par}) for an expression model')
+                        if par.get('norm') is not None:
+                            logging.warning(f'Ignoring "norm" key in parameter ({par}) '+
+                                '(normalization determined from linearity)}')
+                            par.pop('norm')
+                        name = par['name']
+                        if not isinstance(name, str):
+                            raise ValueError(f'Illegal "name" value ({name}) in input parameters')
+                ast = Interpreter()
+                expr_parameters = [name for name in get_ast_names(ast.parse(expr))
+                        if name != 'x' and name not in self._parameters
+                        and name not in ast.symtable]
+#                print(f'\nexpr_parameters: {expr_parameters}')
+#                print(f'expr = {expr}')
+                if prefix is None:
+                    newmodel = ExpressionModel(expr=expr)
+                else:
+                    for name in expr_parameters:
+                        expr = sub(rf'\b{name}\b', f'{prefix}{name}', expr)
+                    expr_parameters = [f'{prefix}{name}' for name in expr_parameters]
+#                    print(f'\nexpr_parameters: {expr_parameters}')
+#                    print(f'expr = {expr}')
+                    newmodel = ExpressionModel(expr=expr, name=name)
+#                print(f'\nnewmodel = {newmodel.__dict__}')
+#                print(f'params_names = {newmodel._param_names}')
+#                print(f'params_names = {newmodel.param_names}')
+                # Remove already existing names
+                for name in newmodel.param_names.copy():
+                    if name not in expr_parameters:
+                        newmodel._func_allargs.remove(name)
+                        newmodel._param_names.remove(name)
+#                print(f'params_names = {newmodel._param_names}')
+#                print(f'params_names = {newmodel.param_names}')
             else:
-                logging.error('Unknown build-in fit model')
-                return kwargs
+                raise ValueError(f'Unknown build-in fit model ({model})')
         else:
-            illegal_value(model, 'model', 'add_model')
-            return kwargs
+            illegal_value(model, 'model', 'Fit.add_model', raise_error=True)
 
         # Add the new model to the current one
+#        print('\nBefore adding model:')
+#        print(f'\nnewmodel = {newmodel.__dict__}')
+#        if len(self._parameters):
+#            self._parameters.pretty_print()
         if self._model is None:
             self._model = newmodel
         else:
             self._model += newmodel
-        if self._parameters is None:
-            self._parameters = newmodel.make_params()
-        else:
-            self._parameters += newmodel.make_params()
+        new_parameters = newmodel.make_params()
+        self._parameters += new_parameters
 #        print('\nAfter adding model:')
+#        print(f'\nnewmodel = {newmodel.__dict__}')
+#        print(f'\nnew_parameters = {new_parameters}')
 #        self._parameters.pretty_print()
 
-        # Initialize the model parameters
+        # Check linearity of expression model paremeters
+        if isinstance(newmodel, ExpressionModel):
+            for name in newmodel.param_names:
+                if not diff(newmodel.expr, name, name):
+                    if name not in self._linear_parameters:
+                        self._linear_parameters.append(name)
+                        new_parameter_norms[name] = True
+#                        print(f'\nADDING {name} TO LINEAR')
+                else:
+                    if name not in self._nonlinear_parameters:
+                        self._nonlinear_parameters.append(name)
+                        new_parameter_norms[name] = False
+#                        print(f'\nADDING {name} TO NONLINEAR')
+#        print(f'new_parameter_norms:\n{new_parameter_norms}')
+
+        # Scale the default initial model parameters
+        if self._norm is not None:
+            for name, norm in new_parameter_norms.copy().items():
+                par = self._parameters.get(name)
+                if par is None:
+                    new_parameter_norms.pop(name)
+                    continue
+                if par.expr is None and norm:
+                    value = par.value*self._norm[1]
+                    _min = par.min
+                    _max = par.max
+                    if not np.isinf(_min) and abs(_min) != float_min:
+                        _min *= self._norm[1]
+                    if not np.isinf(_max) and abs(_max) != float_min:
+                        _max *= self._norm[1]
+                    par.set(value=value, min=_min, max=_max)
+#        print('\nAfter norm defaults:')
+#        self._parameters.pretty_print()
+#        print(f'parameters:\n{parameters}')
+#        print(f'all_parameters:\n{list(self.parameters)}')
+#        print(f'new_parameter_norms:\n{new_parameter_norms}')
+#        print(f'parameter_norms:\n{self._parameter_norms}')
+
+        # Initialize the model parameters from parameters
         if prefix is None:
             prefix = ""
         if parameters is not None:
-            if not isinstance(parameters, (tuple, list)):
-                illegal_value(parameters, 'parameters', 'add_model')
-                return kwargs
             for parameter in parameters:
-                if not isinstance(parameter, dict):
-                    illegal_value(parameter, 'parameter in parameters', 'add_model')
-                    return kwargs
-                parameter['name']  = prefix+parameter['name']
-                self._parameters.add(**parameter)
-        for name, value in kwargs.items():
-            if isinstance(value, (int, float)):
-                self._parameters.add(prefix+name, value=value)
-#        print('\nAt end add_model:')
+                name = parameter['name']
+                if not isinstance(name, str):
+                    raise ValueError(f'Illegal "name" value ({name}) in input parameters')
+                if name not in new_parameters:
+                    name = prefix+name
+                    parameter['name'] = name
+                if name not in new_parameters:
+                    logging.warning(f'Ignoring superfluous parameter info for {name}')
+                    continue
+                if name in self._parameters:
+                    parameter.pop('name')
+                    if 'norm' in parameter:
+                        if not isinstance(parameter['norm'], bool):
+                            illegal_value(parameter['norm'], 'norm', 'Fit.add_model',
+                                    raise_error=True)
+                        new_parameter_norms[name] = parameter['norm']
+                        parameter.pop('norm')
+                    if parameter.get('expr') is not None:
+                        if 'value' in parameter:
+                            logging.warning(f'Ignoring value in parameter {name} '+
+                                    f'(set by expression: {parameter["expr"]})')
+                            parameter.pop('value')
+                        if 'vary' in parameter:
+                            logging.warning(f'Ignoring vary in parameter {name} '+
+                                    f'(set by expression: {parameter["expr"]})')
+                            parameter.pop('vary')
+                        if 'min' in parameter:
+                            logging.warning(f'Ignoring min in parameter {name} '+
+                                    f'(set by expression: {parameter["expr"]})')
+                            parameter.pop('min')
+                        if 'max' in parameter:
+                            logging.warning(f'Ignoring max in parameter {name} '+
+                                    f'(set by expression: {parameter["expr"]})')
+                            parameter.pop('max')
+                    if 'vary' in parameter:
+                        if not isinstance(parameter['vary'], bool):
+                            illegal_value(parameter['vary'], 'vary', 'Fit.add_model',
+                                    raise_error=True)
+                        if not parameter['vary']:
+                            if 'min' in parameter:
+                                logging.warning(f'Ignoring min in parameter {name} in '+
+                                        f'Fit.add_model (vary = {parameter["vary"]})')
+                                parameter.pop('min')
+                            if 'max' in parameter:
+                                logging.warning(f'Ignoring max in parameter {name} in '+
+                                        f'Fit.add_model (vary = {parameter["vary"]})')
+                                parameter.pop('max')
+                    self._parameters[name].set(**parameter)
+                    parameter['name'] = name
+                else:
+                    illegal_value(parameter, 'parameter name', 'Fit.model', raise_error=True)
+        self._parameter_norms = {**self._parameter_norms, **new_parameter_norms}
+#        print('\nAfter parameter init:')
 #        self._parameters.pretty_print()
+#        print(f'parameters:\n{parameters}')
+#        print(f'new_parameter_norms:\n{new_parameter_norms}')
+#        print(f'parameter_norms:\n{self._parameter_norms}')
+#        print(f'kwargs:\n{kwargs}')
 
-        return kwargs
+        # Initialize the model parameters from kwargs
+        for name, value in {**kwargs}.items():
+            full_name = f'{pprefix}{name}'
+            if full_name in new_parameter_norms and isinstance(value, (int, float)):
+                kwargs.pop(name)
+                if self._parameters[full_name].expr is None:
+                    self._parameters[full_name].set(value=value)
+                else:
+                    logging.warning(f'Ignoring parameter {name} in Fit.fit (set by expression: '+
+                            f'{self._parameters[full_name].expr})')
+#        print('\nAfter kwargs init:')
+#        self._parameters.pretty_print()
+#        print(f'parameter_norms:\n{self._parameter_norms}')
+#        print(f'kwargs:\n{kwargs}')
+
+        # Check parameter norms (also need it for expressions to renormalize the errors)
+        if self._norm is not None and (callable(model) or model == 'expression'):
+            missing_norm = False
+            for name in new_parameters.valuesdict():
+                if name not in self._parameter_norms:
+                    print(f'new_parameters:\n{new_parameters.valuesdict()}')
+                    print(f'self._parameter_norms:\n{self._parameter_norms}')
+                    logging.error(f'Missing parameter normalization type for {name} in {model}')
+                    missing_norm = True
+            if missing_norm:
+                raise ValueError
+
+#        print(f'at end add_model:\nself._parameters:\n{list(self.parameters)}')
+#        print(f'at end add_model: kwargs = {kwargs}')
+#        print(f'\nat end add_model: newmodel:\n{newmodel.__dict__}\n')
+        return(kwargs)
 
     def fit(self, interactive=False, guess=False, **kwargs):
+        # Check inputs
         if self._model is None:
             logging.error('Undefined fit model')
             return
-        # Current parameter values
-        pars = self._parameters.valuesdict()
-        # Apply parameter updates through keyword arguments
-        for par in set(pars) & set(kwargs):
-            pars[par] = kwargs.pop(par)
-            self._parameters[par].set(value=pars[par])
-        # Check for uninitialized parameters
-        for par, value in pars.items():
-            if value is None or np.isinf(value) or np.isnan(value):
-                if interactive:
-                    self._parameters[par].set(value=
-                            input_num(f'Enter an initial value for {par}: '))
-                else:
-                    self._parameters[par].set(value=1.0)
-#        print('\nAt start actual fit:')
-#        print(f'kwargs = {kwargs}')
-#        self._parameters.pretty_print()
-#        print(f'parameters:\n{self._parameters}')
-#        print(f'x = {self._x}')
-#        print(f'len(x) = {len(self._x)}')
-#        print(f'y = {self._y}')
-#        print(f'len(y) = {len(self._y)}')
+        if not isinstance(interactive, bool):
+            illegal_value(interactive, 'interactive', 'Fit.fit', raise_error=True)
+        if not isinstance(guess, bool):
+            illegal_value(guess, 'guess', 'Fit.fit', raise_error=True)
+        if 'try_linear_fit' in kwargs:
+            try_linear_fit = kwargs.pop('try_linear_fit')
+            if not isinstance(try_linear_fit, bool):
+                illegal_value(try_linear_fit, 'try_linear_fit', 'Fit.fit', raise_error=True)
+            if not self._try_linear_fit:
+                logging.warning('Ignore superfluous keyword argument "try_linear_fit" (not '+
+                        'yet supported for callable models)')
+            else:
+                self._try_linear_fit = try_linear_fit
+#        if self._result is None:
+#            if 'parameters' in kwargs:
+#                raise ValueError('Invalid parameter parameters ({kwargs["parameters"]})')
+#        else:
+        if self._result is not None:
+            if guess:
+                logging.warning('Ignoring input parameter guess in Fit.fit during refitting')
+                guess = False
+
+        # Check for circular expressions
+        # FIX TODO
+#        for name1, par1 in self._parameters.items():
+#            if par1.expr is not None:
+
+        # Apply mask if supplied:
+        if 'mask' in kwargs:
+            self._mask = kwargs.pop('mask')
+        if self._mask is not None:
+            self._mask = np.asarray(self._mask).astype(bool)
+            if self._x.size != self._mask.size:
+                raise ValueError(f'Inconsistent x and mask dimensions ({self._x.size} vs '+
+                        f'{self._mask.size})')
+
+        # Estimate initial parameters with build-in lmfit guess method (only for a single model)
+#        print(f'\nat start fit: kwargs = {kwargs}')
+#RV        print('\nAt start of fit:')
+#RV        self._parameters.pretty_print()
+#        print(f'parameter_norms:\n{self._parameter_norms}')
         if guess:
-            self._parameters = self._model.guess(self._y, x=self._x)
-        self._result = self._model.fit(self._y, self._parameters, x=self._x, **kwargs)
-#        print('\nAt end actual fit:')
-#        print(f'var_names:\n{self._result.var_names}')
-#        print(f'stderr:\n{np.sqrt(np.diagonal(self._result.covar))}')
-#        self._parameters.pretty_print()
-#        print(f'parameters:\n{self._parameters}')
+            if self._mask is None:
+                self._parameters = self._model.guess(self._y, x=self._x)
+            else:
+                self._parameters = self._model.guess(np.asarray(self._y)[~self._mask],
+                        x=self._x[~self._mask])
+#            print('\nAfter guess:')
+#            self._parameters.pretty_print()
+
+        # Add constant offset for a normalized model
+        if self._result is None and self._norm is not None and self._norm[0]:
+            self.add_model('constant', prefix='tmp_normalization_offset_', parameters={'name': 'c',
+                    'value': -self._norm[0], 'vary': False, 'norm': True})
+                    #'value': -self._norm[0]/self._norm[1], 'vary': False, 'norm': False})
+
+        # Adjust existing parameters for refit:
+        if 'parameters' in kwargs:
+            parameters = kwargs.pop('parameters')
+            if isinstance(parameters, dict):
+                parameters = (parameters, )
+            elif not is_dict_series(parameters):
+                illegal_value(parameters, 'parameters', 'Fit.fit', raise_error=True)
+            for par in parameters:
+                name = par['name']
+                if name not in self._parameters:
+                    raise ValueError(f'Unable to match {name} parameter {par} to an existing one')
+                if self._parameters[name].expr is not None:
+                    raise ValueError(f'Unable to modify {name} parameter {par} (currently an '+
+                            'expression)')
+                if par.get('expr') is not None:
+                    raise KeyError(f'Illegal "expr" key in {name} parameter {par}')
+                self._parameters[name].set(vary=par.get('vary'))
+                self._parameters[name].set(min=par.get('min'))
+                self._parameters[name].set(max=par.get('max'))
+                self._parameters[name].set(value=par.get('value'))
+#RV            print('\nAfter adjust:')
+#RV            self._parameters.pretty_print()
+
+        # Apply parameter updates through keyword arguments
+#        print(f'kwargs = {kwargs}')
+#        print(f'parameter_norms = {self._parameter_norms}')
+        for name in set(self._parameters) & set(kwargs):
+            value = kwargs.pop(name)
+            if self._parameters[name].expr is None:
+                self._parameters[name].set(value=value)
+            else:
+                logging.warning(f'Ignoring parameter {name} in Fit.fit (set by expression: '+
+                        f'{self._parameters[name].expr})')
+
+        # Check for uninitialized parameters
+        for name, par in self._parameters.items():
+            if par.expr is None:
+                value = par.value
+                if value is None or np.isinf(value) or np.isnan(value):
+                    if interactive:
+                        value = input_num(f'Enter an initial value for {name}', default=1.0)
+                    else:
+                        value = 1.0
+                    if self._norm is None or name not in self._parameter_norms:
+                        self._parameters[name].set(value=value)
+                    elif self._parameter_norms[name]:
+                        self._parameters[name].set(value=value*self._norm[1])
 
-    def plot(self):
-        if self._result is None:
+        # Check if model is linear
+        try:
+            linear_model = self._check_linearity_model()
+        except:
+            linear_model = False
+#        print(f'\n\n--------> linear_model = {linear_model}\n')
+        if kwargs.get('check_only_linearity') is not None:
+            return(linear_model)
+
+        # Normalize the data and initial parameters
+#RV        print('\nBefore normalization:')
+#RV        self._parameters.pretty_print()
+#        print(f'parameter_norms:\n{self._parameter_norms}')
+        self._normalize()
+#        print(f'norm = {self._norm}') 
+#RV        print('\nAfter normalization:')
+#RV        self._parameters.pretty_print()
+#        self.print_fit_report()
+#        print(f'parameter_norms:\n{self._parameter_norms}')
+
+        if linear_model:
+            # Perform a linear fit by direct matrix solution with numpy
+            try:
+                if self._mask is None:
+                    self._fit_linear_model(self._x, self._y_norm)
+                else:
+                    self._fit_linear_model(self._x[~self._mask],
+                            np.asarray(self._y_norm)[~self._mask])
+            except:
+                linear_model = False
+        if not linear_model:
+            # Perform a non-linear fit with lmfit
+            # Prevent initial values from sitting at boundaries
+            self._parameter_bounds = {name:{'min': par.min, 'max': par.max} for name, par in
+                    self._parameters.items() if par.vary}
+            for par in self._parameters.values():
+                if par.vary:
+                    par.set(value=self._reset_par_at_boundary(par, par.value))
+#            print('\nAfter checking boundaries:')
+#            self._parameters.pretty_print()
+
+            # Perform the fit
+#            fit_kws = None
+#            if 'Dfun' in kwargs:
+#                fit_kws = {'Dfun': kwargs.pop('Dfun')}
+#            self._result = self._model.fit(self._y_norm, self._parameters, x=self._x,
+#                    fit_kws=fit_kws, **kwargs)
+            if self._mask is None:
+                self._result = self._model.fit(self._y_norm, self._parameters, x=self._x, **kwargs)
+            else:
+                self._result = self._model.fit(np.asarray(self._y_norm)[~self._mask],
+                        self._parameters, x=self._x[~self._mask], **kwargs)
+#RV        print('\nAfter fit:')
+#        print(f'\nself._result ({self._result}):\n\t{self._result.__dict__}')
+#RV        self._parameters.pretty_print()
+#        self.print_fit_report()
+
+        # Set internal parameter values to fit results upon success
+        if self.success:
+            for name, par in self._parameters.items():
+                if par.expr is None and par.vary:
+                    par.set(value=self._result.params[name].value)
+#            print('\nAfter update parameter values:')
+#            self._parameters.pretty_print()
+
+        # Renormalize the data and results
+        self._renormalize()
+#RV        print('\nAfter renormalization:')
+#RV        self._parameters.pretty_print()
+#        self.print_fit_report()
+
+    def plot(self, y=None, y_title=None, result=None, skip_init=False, plot_comp_legends=False,
+            plot_residual=False, plot_masked_data=True, **kwargs):
+        if result is None:
+            result = self._result
+        if result is None:
             return
-        components = self._result.eval_components()
-        plots = ((self._x, self._y, '.'), (self._x, self._result.best_fit, 'k-'),
-                (self._x, self._result.init_fit, 'g-'))
-        legend = ['data', 'best fit', 'init']
-        if len(components) > 1:
+        plots = []
+        legend = []
+        if self._mask is None:
+            mask = np.zeros(self._x.size).astype(bool)
+            plot_masked_data = False
+        else:
+            mask = self._mask
+        if y is not None:
+            if not isinstance(y, (tuple, list, np.ndarray)):
+                illegal_value(y, 'y', 'Fit.plot')
+            if len(y) != len(self._x):
+                logging.warning('Ignoring parameter y in Fit.plot (wrong dimension)')
+                y = None
+        if y is not None:
+            if y_title is None or not isinstance(y_title, str):
+                y_title = 'data'
+            plots += [(self._x, y, '.')]
+            legend += [y_title]
+        if self._y is not None:
+            plots += [(self._x, np.asarray(self._y), 'b.')]
+            legend += ['data']
+            if plot_masked_data:
+                plots += [(self._x[mask], np.asarray(self._y)[mask], 'bx')]
+                legend += ['masked data']
+        if isinstance(plot_residual, bool) and plot_residual:
+            plots += [(self._x[~mask], result.residual, 'k-')]
+            legend += ['residual']
+        plots += [(self._x[~mask], result.best_fit, 'k-')]
+        legend += ['best fit']
+        if not skip_init and hasattr(result, 'init_fit'):
+            plots += [(self._x[~mask], result.init_fit, 'g-')]
+            legend += ['init']
+        components = result.eval_components(x=self._x[~mask])
+        num_components = len(components)
+        if 'tmp_normalization_offset_' in components:
+            num_components -= 1
+        if num_components > 1:
+            eval_index = 0
             for modelname, y in components.items():
+                if modelname == 'tmp_normalization_offset_':
+                    continue
+                if modelname == '_eval':
+                    modelname = f'eval{eval_index}'
+                if len(modelname) > 20:
+                    modelname = f'{modelname[0:16]} ...'
                 if isinstance(y, (int, float)):
-                    y *= np.ones(len(self._x))
-                plots += ((self._x, y, '--'),)
-#                if modelname[-1] == '_':
-#                    legend.append(modelname[:-1])
-#                else:
-#                    legend.append(modelname)
-        quickPlot(plots, legend=legend, block=True)
+                    y *= np.ones(self._x[~mask].size)
+                plots += [(self._x[~mask], y, '--')]
+                if plot_comp_legends:
+                    if modelname[-1] == '_':
+                        legend.append(modelname[:-1])
+                    else:
+                        legend.append(modelname)
+        title = kwargs.get('title')
+        if title is not None:
+            kwargs.pop('title')
+        quick_plot(tuple(plots), legend=legend, title=title, block=True, **kwargs)
 
     @staticmethod
     def guess_init_peak(x, y, *args, center_guess=None, use_max_for_center=True):
         """ Return a guess for the initial height, center and fwhm for a peak
         """
+#        print(f'\n\nargs = {args}')
+#        print(f'center_guess = {center_guess}')
+#        quick_plot(x, y, vlines=center_guess, block=True)
         center_guesses = None
+        x = np.asarray(x)
+        y = np.asarray(y)
         if len(x) != len(y):
-            logging.error(f'Illegal x and y lengths ({len(x)}, {len(y)}), skip initial guess')
-            return None, None, None
+            logging.error(f'Invalid x and y lengths ({len(x)}, {len(y)}), skip initial guess')
+            return(None, None, None)
         if isinstance(center_guess, (int, float)):
             if len(args):
                 logging.warning('Ignoring additional arguments for single center_guess value')
+            center_guesses = [center_guess]
         elif isinstance(center_guess, (tuple, list, np.ndarray)):
             if len(center_guess) == 1:
                 logging.warning('Ignoring additional arguments for single center_guess value')
                 if not isinstance(center_guess[0], (int, float)):
-                    raise ValueError(f'Illegal center_guess type ({type(center_guess[0])})')
+                    raise ValueError(f'Invalid parameter center_guess ({type(center_guess[0])})')
                 center_guess = center_guess[0]
             else:
                 if len(args) != 1:
-                    raise ValueError(f'Illegal number of arguments ({len(args)})')
+                    raise ValueError(f'Invalid number of arguments ({len(args)})')
                 n = args[0]
                 if not is_index(n, 0, len(center_guess)):
-                    raise ValueError('Illegal argument')
+                    raise ValueError('Invalid argument')
                 center_guesses = center_guess
                 center_guess = center_guesses[n]
         elif center_guess is not None:
-            raise ValueError(f'Illegal center_guess type ({type(center_guess)})')
+            raise ValueError(f'Invalid center_guess type ({type(center_guess)})')
+#        print(f'x = {x}')
+#        print(f'y = {y}')
+#        print(f'center_guess = {center_guess}')
 
         # Sort the inputs
         index = np.argsort(x)
@@ -319,12 +1001,21 @@
 #        print(f'miny = {miny}')
 #        print(f'x_range = {x[0]} {x[-1]} {len(x)}')
 #        print(f'y_range = {y[0]} {y[-1]} {len(y)}')
+#        quick_plot(x, y, vlines=center_guess, block=True)
 
 #        xx = x
 #        yy = y
         # Set range for current peak
+#        print(f'n = {n}')
 #        print(f'center_guesses = {center_guesses}')
         if center_guesses is not None:
+            if len(center_guesses) > 1:
+                index = np.argsort(center_guesses)
+                n = list(index).index(n)
+#                print(f'n = {n}')
+#                print(f'index = {index}')
+                center_guesses = np.asarray(center_guesses)[index]
+#                print(f'center_guesses = {center_guesses}')
             if n == 0:
                low = 0
                upp = index_nearest(x, (center_guesses[0]+center_guesses[1])/2)
@@ -338,7 +1029,7 @@
 #            print(f'upp = {upp}')
             x = x[low:upp]
             y = y[low:upp]
-#            quickPlot(x, y, vlines=(x[0], center_guess, x[-1]), block=True)
+#            quick_plot(x, y, vlines=(x[0], center_guess, x[-1]), block=True)
 
         # Estimate FHHM
         maxy = y.max()
@@ -377,7 +1068,7 @@
                 fwhm_index2 = i
                 break
 #        print(f'fwhm_index2 = {fwhm_index2} {x[fwhm_index2]}')
-#        quickPlot((x,y,'o'), vlines=(x[fwhm_index1], center, x[fwhm_index2]), block=True)
+#        quick_plot((x,y,'o'), vlines=(x[fwhm_index1], center, x[fwhm_index2]), block=True)
         if fwhm_index1 == 0 and fwhm_index2 < len(x)-1:
             fwhm = 2*(x[fwhm_index2]-center)
         elif fwhm_index1 > 0 and fwhm_index2 == len(x)-1:
@@ -389,53 +1080,442 @@
 #        print(f'fwhm = {fwhm}')
 
         # Return height, center and FWHM
-#        quickPlot((x,y,'o'), (xx,yy), vlines=(x[fwhm_index1], center, x[fwhm_index2]), block=True)
-        return height, center, fwhm
+#        quick_plot((x,y,'o'), (xx,yy), vlines=(x[fwhm_index1], center, x[fwhm_index2]), block=True)
+        return(height, center, fwhm)
+
+    def _check_linearity_model(self):
+        """Identify the linearity of all model parameters and check if the model is linear or not
+        """
+        if not self._try_linear_fit:
+            logging.info('Skip linearity check (not yet supported for callable models)')
+            return(False)
+        free_parameters = [name for name, par in self._parameters.items() if par.vary]
+        for component in self._model.components:
+            if 'tmp_normalization_offset_c' in component.param_names:
+                continue
+            if isinstance(component, ExpressionModel):
+                for name in free_parameters:
+                    if diff(component.expr, name, name):
+#                        print(f'\t\t{component.expr} is non-linear in {name}')
+                        self._nonlinear_parameters.append(name)
+                        if name in self._linear_parameters:
+                            self._linear_parameters.remove(name)
+            else:
+                model_parameters = component.param_names.copy()
+                for basename, hint in component.param_hints.items():
+                    name = f'{component.prefix}{basename}'
+                    if hint.get('expr') is not None:
+                        model_parameters.remove(name)
+                for name in model_parameters:
+                    expr = self._parameters[name].expr
+                    if expr is not None:
+                        for nname in free_parameters:
+                            if name in self._nonlinear_parameters:
+                                if diff(expr, nname):
+#                                    print(f'\t\t{component} is non-linear in {nname} (through {name} = "{expr}")')
+                                    self._nonlinear_parameters.append(nname)
+                                    if nname in self._linear_parameters:
+                                        self._linear_parameters.remove(nname)
+                            else:
+                                assert(name in self._linear_parameters)
+#                                print(f'\n\nexpr ({type(expr)}) = {expr}\nnname ({type(nname)}) = {nname}\n\n')
+                                if diff(expr, nname, nname):
+#                                    print(f'\t\t{component} is non-linear in {nname} (through {name} = "{expr}")')
+                                    self._nonlinear_parameters.append(nname)
+                                    if nname in self._linear_parameters:
+                                        self._linear_parameters.remove(nname)
+#        print(f'\nfree parameters:\n\t{free_parameters}')
+#        print(f'linear parameters:\n\t{self._linear_parameters}')
+#        print(f'nonlinear parameters:\n\t{self._nonlinear_parameters}\n')
+        if any(True for name in self._nonlinear_parameters if self._parameters[name].vary):
+            return(False)
+        return(True)
+
+    def _fit_linear_model(self, x, y):
+        """Perform a linear fit by direct matrix solution with numpy
+        """
+        # Construct the matrix and the free parameter vector
+#        print(f'\nparameters:')
+#        self._parameters.pretty_print()
+#        print(f'\nparameter_norms:\n\t{self._parameter_norms}')
+#        print(f'\nlinear_parameters:\n\t{self._linear_parameters}')
+#        print(f'nonlinear_parameters:\n\t{self._nonlinear_parameters}')
+        free_parameters = [name for name, par in self._parameters.items() if par.vary]
+#        print(f'free parameters:\n\t{free_parameters}\n')
+        expr_parameters = {name:par.expr for name, par in self._parameters.items()
+                if par.expr is not None}
+        model_parameters = []
+        for component in self._model.components:
+            if 'tmp_normalization_offset_c' in component.param_names:
+                continue
+            model_parameters += component.param_names
+            for basename, hint in component.param_hints.items():
+                name = f'{component.prefix}{basename}'
+                if hint.get('expr') is not None:
+                    expr_parameters.pop(name)
+                    model_parameters.remove(name)
+#        print(f'expr parameters:\n{expr_parameters}')
+#        print(f'model parameters:\n\t{model_parameters}\n')
+        norm = 1.0
+        if self._normalized:
+            norm = self._norm[1]
+#        print(f'\n\nself._normalized = {self._normalized}\nnorm = {norm}\nself._norm = {self._norm}\n')
+        # Add expression parameters to asteval
+        ast = Interpreter()
+#        print(f'Adding to asteval sym table:')
+        for name, expr in expr_parameters.items():
+#            print(f'\tadding {name} {expr}')
+            ast.symtable[name] = expr
+        # Add constant parameters to asteval
+        # (renormalize to use correctly in evaluation of expression models)
+        for name, par in self._parameters.items():
+            if par.expr is None and not par.vary:
+                if self._parameter_norms[name]:
+#                    print(f'\tadding {name} {par.value*norm}')
+                    ast.symtable[name] = par.value*norm
+                else:
+#                    print(f'\tadding {name} {par.value}')
+                    ast.symtable[name] = par.value
+        A = np.zeros((len(x), len(free_parameters)), dtype='float64')
+        y_const = np.zeros(len(x), dtype='float64')
+        have_expression_model = False
+        for component in self._model.components:
+            if isinstance(component, ConstantModel):
+                name = component.param_names[0]
+#                print(f'\nConstant model: {name} {self._parameters[name]}\n')
+                if name in free_parameters:
+#                    print(f'\t\t{name} is a free constant set matrix column {free_parameters.index(name)} to 1.0')
+                    A[:,free_parameters.index(name)] = 1.0
+                else:
+                    if self._parameter_norms[name]:
+                        delta_y_const = self._parameters[name]*np.ones(len(x))
+                    else:
+                        delta_y_const = (self._parameters[name]*norm)*np.ones(len(x))
+                    y_const += delta_y_const
+#                    print(f'\ndelta_y_const ({type(delta_y_const)}):\n{delta_y_const}\n')
+            elif isinstance(component, ExpressionModel):
+                have_expression_model = True
+                const_expr = component.expr
+#                print(f'\nExpression model:\nconst_expr: {const_expr}\n')
+                for name in free_parameters:
+                    dexpr_dname = diff(component.expr, name)
+                    if dexpr_dname:
+                        const_expr = f'{const_expr}-({str(dexpr_dname)})*{name}'
+#                        print(f'\tconst_expr: {const_expr}')
+                        if not self._parameter_norms[name]:
+                            dexpr_dname = f'({dexpr_dname})/{norm}'
+#                        print(f'\t{component.expr} is linear in {name}\n\t\tadd "{str(dexpr_dname)}" to matrix as column {free_parameters.index(name)}')
+                        fx = [(lambda _: ast.eval(str(dexpr_dname)))(ast(f'x={v}')) for v in x]
+#                        print(f'\tfx:\n{fx}')
+                        if len(ast.error):
+                            raise ValueError(f'Unable to evaluate {dexpr_dname}')
+                        A[:,free_parameters.index(name)] += fx
+#                        if self._parameter_norms[name]:
+#                            print(f'\t\t{component.expr} is linear in {name} add "{str(dexpr_dname)}" to matrix as column {free_parameters.index(name)}')
+#                            A[:,free_parameters.index(name)] += fx
+#                        else:
+#                            print(f'\t\t{component.expr} is linear in {name} add "({str(dexpr_dname)})/{norm}" to matrix as column {free_parameters.index(name)}')
+#                            A[:,free_parameters.index(name)] += np.asarray(fx)/norm
+                # FIX: find another solution if expr not supported by simplify
+                const_expr = str(simplify(f'({const_expr})/{norm}'))
+#                print(f'\nconst_expr: {const_expr}')
+                delta_y_const = [(lambda _: ast.eval(const_expr))(ast(f'x = {v}')) for v in x]
+                y_const += delta_y_const
+#                print(f'\ndelta_y_const ({type(delta_y_const)}):\n{delta_y_const}\n')
+                if len(ast.error):
+                    raise ValueError(f'Unable to evaluate {const_expr}')
+            else:
+                free_model_parameters = [name for name in component.param_names
+                        if name in free_parameters or name in expr_parameters]
+#                print(f'\nBuild-in model ({component}):\nfree_model_parameters: {free_model_parameters}\n')
+                if not len(free_model_parameters):
+                    y_const += component.eval(params=self._parameters, x=x)
+                elif isinstance(component, LinearModel):
+                    if f'{component.prefix}slope' in free_model_parameters:
+                        A[:,free_parameters.index(f'{component.prefix}slope')] = x
+                    else:
+                        y_const += self._parameters[f'{component.prefix}slope'].value*x
+                    if f'{component.prefix}intercept' in free_model_parameters:
+                        A[:,free_parameters.index(f'{component.prefix}intercept')] = 1.0
+                    else:
+                        y_const += self._parameters[f'{component.prefix}intercept'].value* \
+                                np.ones(len(x))
+                elif isinstance(component, QuadraticModel):
+                    if f'{component.prefix}a' in free_model_parameters:
+                        A[:,free_parameters.index(f'{component.prefix}a')] = x**2
+                    else:
+                        y_const += self._parameters[f'{component.prefix}a'].value*x**2
+                    if f'{component.prefix}b' in free_model_parameters:
+                        A[:,free_parameters.index(f'{component.prefix}b')] = x
+                    else:
+                        y_const += self._parameters[f'{component.prefix}b'].value*x
+                    if f'{component.prefix}c' in free_model_parameters:
+                        A[:,free_parameters.index(f'{component.prefix}c')] = 1.0
+                    else:
+                        y_const += self._parameters[f'{component.prefix}c'].value*np.ones(len(x))
+                else:
+                    # At this point each build-in model must be strictly proportional to each linear
+                    #   model parameter. Without this assumption, the model equation is needed
+                    #   For the current build-in lmfit models, this can only ever be the amplitude
+                    assert(len(free_model_parameters) == 1)
+                    name = f'{component.prefix}amplitude'
+                    assert(free_model_parameters[0] == name)
+                    assert(self._parameter_norms[name])
+                    expr = self._parameters[name].expr
+                    if expr is None:
+#                        print(f'\t{component} is linear in {name} add to matrix as column {free_parameters.index(name)}')
+                        parameters = deepcopy(self._parameters)
+                        parameters[name].set(value=1.0)
+                        index = free_parameters.index(name)
+                        A[:,free_parameters.index(name)] += component.eval(params=parameters, x=x)
+                    else:
+                        const_expr = expr
+#                        print(f'\tconst_expr: {const_expr}')
+                        parameters = deepcopy(self._parameters)
+                        parameters[name].set(value=1.0)
+                        dcomp_dname = component.eval(params=parameters, x=x)
+#                        print(f'\tdcomp_dname ({type(dcomp_dname)}):\n{dcomp_dname}')
+                        for nname in free_parameters:
+                            dexpr_dnname =  diff(expr, nname)
+                            if dexpr_dnname:
+                                assert(self._parameter_norms[name])
+#                                print(f'\t\td({expr})/d{nname} = {dexpr_dnname}')
+#                                print(f'\t\t{component} is linear in {nname} (through {name} = "{expr}", add to matrix as column {free_parameters.index(nname)})')
+                                fx = np.asarray(dexpr_dnname*dcomp_dname, dtype='float64')
+#                                print(f'\t\tfx ({type(fx)}): {fx}')
+#                                print(f'free_parameters.index({nname}): {free_parameters.index(nname)}')
+                                if self._parameter_norms[nname]:
+                                    A[:,free_parameters.index(nname)] += fx
+                                else:
+                                    A[:,free_parameters.index(nname)] += fx/norm
+                                const_expr = f'{const_expr}-({dexpr_dnname})*{nname}'
+#                                print(f'\t\tconst_expr: {const_expr}')
+                        const_expr = str(simplify(f'({const_expr})/{norm}'))
+#                        print(f'\tconst_expr: {const_expr}')
+                        fx = [(lambda _: ast.eval(const_expr))(ast(f'x = {v}')) for v in x]
+#                        print(f'\tfx: {fx}')
+                        delta_y_const = np.multiply(fx, dcomp_dname)
+                        y_const += delta_y_const
+#                        print(f'\ndelta_y_const ({type(delta_y_const)}):\n{delta_y_const}\n')
+#            print(A)
+#            print(y_const)
+        solution, residual, rank, s = np.linalg.lstsq(A, y-y_const, rcond=None)
+#        print(f'\nsolution ({type(solution)} {solution.shape}):\n\t{solution}')
+#        print(f'\nresidual ({type(residual)} {residual.shape}):\n\t{residual}')
+#        print(f'\nrank ({type(rank)} {rank.shape}):\n\t{rank}')
+#        print(f'\ns ({type(s)} {s.shape}):\n\t{s}\n')
+
+        # Assemble result (compensate for normalization in expression models)
+        for name, value in zip(free_parameters, solution):
+            self._parameters[name].set(value=value)
+        if self._normalized and (have_expression_model or len(expr_parameters)):
+            for name, norm in self._parameter_norms.items():
+                par = self._parameters[name]
+                if par.expr is None and norm:
+                    self._parameters[name].set(value=par.value*self._norm[1])
+#        self._parameters.pretty_print()
+#        print(f'\nself._parameter_norms:\n\t{self._parameter_norms}')
+        self._result = ModelResult(self._model, deepcopy(self._parameters))
+        self._result.best_fit = self._model.eval(params=self._parameters, x=x)
+        if self._normalized and (have_expression_model or len(expr_parameters)):
+            if 'tmp_normalization_offset_c' in self._parameters:
+                offset = self._parameters['tmp_normalization_offset_c']
+            else:
+                offset = 0.0
+            self._result.best_fit = (self._result.best_fit-offset-self._norm[0])/self._norm[1]
+            if self._normalized:
+                for name, norm in self._parameter_norms.items():
+                    par = self._parameters[name]
+                    if par.expr is None and norm:
+                        value = par.value/self._norm[1]
+                        self._parameters[name].set(value=value)
+                        self._result.params[name].set(value=value)
+#        self._parameters.pretty_print()
+        self._result.residual = self._result.best_fit-y
+        self._result.components = self._model.components
+        self._result.init_params = None
+#        quick_plot((x, y, '.'), (x, y_const, 'g'), (x, self._result.best_fit, 'k'), (x, self._result.residual, 'r'), block=True)
+
+    def _normalize(self):
+        """Normalize the data and initial parameters
+        """
+        if self._normalized:
+            return
+        if self._norm is None:
+            if self._y is not None and self._y_norm is None:
+                self._y_norm = np.asarray(self._y)
+        else:
+            if self._y is not None and self._y_norm is None:
+                self._y_norm = (np.asarray(self._y)-self._norm[0])/self._norm[1]
+            self._y_range = 1.0
+            for name, norm in self._parameter_norms.items():
+                par = self._parameters[name]
+                if par.expr is None and norm:
+                    value = par.value/self._norm[1]
+                    _min = par.min
+                    _max = par.max
+                    if not np.isinf(_min) and abs(_min) != float_min:
+                        _min /= self._norm[1]
+                    if not np.isinf(_max) and abs(_max) != float_min:
+                        _max /= self._norm[1]
+                    par.set(value=value, min=_min, max=_max)
+            self._normalized = True
+
+    def _renormalize(self):
+        """Renormalize the data and results
+        """
+        if self._norm is None or not self._normalized:
+            return
+        self._normalized = False
+        for name, norm in self._parameter_norms.items():
+            par = self._parameters[name]
+            if par.expr is None and norm:
+                value = par.value*self._norm[1]
+                _min = par.min
+                _max = par.max
+                if not np.isinf(_min) and abs(_min) != float_min:
+                    _min *= self._norm[1]
+                if not np.isinf(_max) and abs(_max) != float_min:
+                    _max *= self._norm[1]
+                par.set(value=value, min=_min, max=_max)
+        if self._result is None:
+            return
+        self._result.best_fit = self._result.best_fit*self._norm[1]+self._norm[0]
+        for name, par in self._result.params.items():
+            if self._parameter_norms.get(name, False):
+                if par.stderr is not None:
+                    par.stderr *= self._norm[1]
+                if par.expr is None:
+                    _min = par.min
+                    _max = par.max
+                    value = par.value*self._norm[1]
+                    if par.init_value is not None:
+                        par.init_value *= self._norm[1]
+                    if not np.isinf(_min) and abs(_min) != float_min:
+                        _min *= self._norm[1]
+                    if not np.isinf(_max) and abs(_max) != float_min:
+                        _max *= self._norm[1]
+                    par.set(value=value, min=_min, max=_max)
+        if hasattr(self._result, 'init_fit'):
+            self._result.init_fit = self._result.init_fit*self._norm[1]+self._norm[0]
+        if hasattr(self._result, 'init_values'):
+            init_values = {}
+            for name, value in self._result.init_values.items():
+                if name not in self._parameter_norms or self._parameters[name].expr is not None:
+                    init_values[name] = value
+                elif self._parameter_norms[name]:
+                    init_values[name] = value*self._norm[1]
+            self._result.init_values = init_values
+            for name, par in self._result.init_params.items():
+                if par.expr is None and self._parameter_norms.get(name, False):
+                    value = par.value
+                    _min = par.min
+                    _max = par.max
+                    value *= self._norm[1]
+                    if not np.isinf(_min) and abs(_min) != float_min:
+                        _min *= self._norm[1]
+                    if not np.isinf(_max) and abs(_max) != float_min:
+                        _max *= self._norm[1]
+                    par.set(value=value, min=_min, max=_max)
+                par.init_value = par.value
+        # Don't renormalize chisqr, it has no useful meaning in physical units
+        #self._result.chisqr *= self._norm[1]*self._norm[1]
+        if self._result.covar is not None:
+            for i, name in enumerate(self._result.var_names):
+                if self._parameter_norms.get(name, False):
+                    for j in range(len(self._result.var_names)):
+                        if self._result.covar[i,j] is not None:
+                            self._result.covar[i,j] *= self._norm[1]
+                        if self._result.covar[j,i] is not None:
+                            self._result.covar[j,i] *= self._norm[1]
+        # Don't renormalize redchi, it has no useful meaning in physical units
+        #self._result.redchi *= self._norm[1]*self._norm[1]
+        if self._result.residual is not None:
+            self._result.residual *= self._norm[1]
+
+    def _reset_par_at_boundary(self, par, value):
+        assert(par.vary)
+        name = par.name
+        _min = self._parameter_bounds[name]['min']
+        _max = self._parameter_bounds[name]['max']
+        if np.isinf(_min):
+            if not np.isinf(_max):
+                if self._parameter_norms.get(name, False):
+                    upp = _max-0.1*self._y_range
+                elif _max == 0.0:
+                    upp = _max-0.1
+                else:
+                    upp = _max-0.1*abs(_max)
+                if value >= upp:
+                    return(upp)
+        else:
+            if np.isinf(_max):
+                if self._parameter_norms.get(name, False):
+                    low = _min+0.1*self._y_range
+                elif _min == 0.0:
+                    low = _min+0.1
+                else:
+                    low = _min+0.1*abs(_min)
+                if value <= low:
+                    return(low)
+            else:
+                low = 0.9*_min+0.1*_max
+                upp = 0.1*_min+0.9*_max
+                if value <= low:
+                    return(low)
+                elif value >= upp:
+                    return(upp)
+        return(value)
 
 
 class FitMultipeak(Fit):
     """Fit data with multiple peaks
     """
-    def __init__(self, x, y, normalize=True):
-        super().__init__(x, deepcopy(y))
-        self._norm = None
+    def __init__(self, y, x=None, normalize=True):
+        super().__init__(y, x=x, normalize=normalize)
         self._fwhm_max = None
         self._sigma_max = None
-        if normalize:
-            self._normalize()
-        #quickPlot((self._x,self._y), block=True)
 
     @classmethod
-    def fit_multipeak(cls, x, y, centers, peak_models='gaussian', center_exprs=None, fit_type=None,
-            background_order=None, fwhm_max=None, plot_components=None):
+    def fit_multipeak(cls, y, centers, x=None, normalize=True, peak_models='gaussian',
+            center_exprs=None, fit_type=None, background_order=None, background_exp=False,
+            fwhm_max=None, plot_components=False):
         """Make sure that centers and fwhm_max are in the correct units and consistent with expr
            for a uniform fit (fit_type == 'uniform')
         """
-        fit = cls(x, y)
+        fit = cls(y, x=x, normalize=normalize)
         success = fit.fit(centers, fit_type=fit_type, peak_models=peak_models, fwhm_max=fwhm_max,
                 center_exprs=center_exprs, background_order=background_order,
-                plot_components=plot_components)
+                background_exp=background_exp, plot_components=plot_components)
         if success:
-            return fit.best_fit, fit.residual, fit.best_values, fit.best_errors, fit.redchi, \
-                    fit.success
+            return(fit.best_fit, fit.residual, fit.best_values, fit.best_errors, fit.redchi, \
+                    fit.success)
         else:
-            return np.array([]), np.array([]), {}, {}, sys.float_info.max, False
+            return(np.array([]), np.array([]), {}, {}, float_max, False)
 
     def fit(self, centers, fit_type=None, peak_models=None, center_exprs=None, fwhm_max=None,
-                background_order=None, plot_components=None, param_constraint=False):
+                background_order=None, background_exp=False, plot_components=False,
+                param_constraint=False):
         self._fwhm_max = fwhm_max
         # Create the multipeak model
         self._create_model(centers, fit_type, peak_models, center_exprs, background_order,
-                param_constraint)
+                background_exp, param_constraint)
+
+        # RV: Obsolete Normalize the data and results
+#        print('\nBefore fit before normalization in FitMultipeak:')
+#        self._parameters.pretty_print()
+#        self._normalize()
+#        print('\nBefore fit after normalization in FitMultipeak:')
+#        self._parameters.pretty_print()
 
         # Perform the fit
         try:
             if param_constraint:
-                super().fit(fit_kws={'xtol' : 1.e-5, 'ftol' : 1.e-5, 'gtol' : 1.e-5})
+                super().fit(fit_kws={'xtol': 1.e-5, 'ftol': 1.e-5, 'gtol': 1.e-5})
             else:
                 super().fit()
         except:
-            return False
+            return(False)
 
         # Check for valid fit parameter results
         fit_failure = self._check_validity()
@@ -447,21 +1527,27 @@
             else:
                 logging.info('  -> Retry fitting with constraints')
                 self.fit(centers, fit_type, peak_models, center_exprs, fwhm_max=fwhm_max,
-                        background_order=background_order, plot_components=plot_components,
-                        param_constraint=True)
+                        background_order=background_order, background_exp=background_exp,
+                        plot_components=plot_components, param_constraint=True)
         else:
-            # Renormalize the data and results
-            self._renormalize()
+            # RV: Obsolete Renormalize the data and results
+#            print('\nAfter fit before renormalization in FitMultipeak:')
+#            self._parameters.pretty_print()
+#            self.print_fit_report()
+#            self._renormalize()
+#            print('\nAfter fit after renormalization in FitMultipeak:')
+#            self._parameters.pretty_print()
+#            self.print_fit_report()
 
             # Print report and plot components if requested
-            if plot_components is not None:
+            if plot_components:
                 self.print_fit_report()
                 self.plot()
 
-        return success
+        return(success)
 
     def _create_model(self, centers, fit_type=None, peak_models=None, center_exprs=None,
-                background_order=None, param_constraint=False):
+                background_order=None, background_exp=False, param_constraint=False):
         """Create the multipeak model
         """
         if isinstance(centers, (int, float)):
@@ -476,10 +1562,10 @@
                     f'{num_peaks})')
         if num_peaks == 1:
             if fit_type is not None:
-                logging.warning('Ignoring fit_type input for fitting one peak')
+                logging.debug('Ignoring fit_type input for fitting one peak')
             fit_type = None
             if center_exprs is not None:
-                logging.warning('Ignoring center_exprs input for fitting one peak')
+                logging.debug('Ignoring center_exprs input for fitting one peak')
                 center_exprs = None
         else:
             if fit_type == 'uniform':
@@ -493,10 +1579,10 @@
                     logging.warning('Ignoring center_exprs input for unconstrained fit')
                     center_exprs = None
             else:
-                raise ValueError(f'Illegal fit_type in fit_multigaussian {fit_type}')
+                raise ValueError(f'Invalid fit_type in fit_multigaussian {fit_type}')
         self._sigma_max = None
         if param_constraint:
-            min_value = sys.float_info.min
+            min_value = float_min
             if self._fwhm_max is not None:
                 self._sigma_max = np.zeros(num_peaks)
         else:
@@ -510,13 +1596,18 @@
         # Add background model
         if background_order is not None:
             if background_order == 0:
-                self.add_model('constant', prefix='background', c=0.0)
+                self.add_model('constant', prefix='background', parameters=
+                        {'name': 'c', 'value': float_min, 'min': min_value})
             elif background_order == 1:
                 self.add_model('linear', prefix='background', slope=0.0, intercept=0.0)
             elif background_order == 2:
                 self.add_model('quadratic', prefix='background', a=0.0, b=0.0, c=0.0)
             else:
-                raise ValueError(f'background_order = {background_order}')
+                raise ValueError(f'Invalid parameter background_order ({background_order})')
+        if background_exp:
+            self.add_model('exponential', prefix='background', parameters=(
+                        {'name': 'amplitude', 'value': float_min, 'min': min_value},
+                        {'name': 'decay', 'value': float_min, 'min': min_value}))
 
         # Add peaks and guess initial fit parameters
         ast = Interpreter()
@@ -534,9 +1625,9 @@
                 sig_max = ast(fwhm_factor[peak_models[0]])
                 self._sigma_max[0] = sig_max
             self.add_model(peak_models[0], parameters=(
-                    {'name' : 'amplitude', 'value' : amp_init, 'min' : min_value},
-                    {'name' : 'center', 'value' : cen_init, 'min' : min_value},
-                    {'name' : 'sigma', 'value' : sig_init, 'min' : min_value, 'max' : sig_max}))
+                    {'name': 'amplitude', 'value': amp_init, 'min': min_value},
+                    {'name': 'center', 'value': cen_init, 'min': min_value},
+                    {'name': 'sigma', 'value': sig_init, 'min': min_value, 'max': sig_max}))
         else:
             if fit_type == 'uniform':
                 self.add_parameter(name='scale_factor', value=1.0)
@@ -556,107 +1647,921 @@
                     self._sigma_max[i] = sig_max
                 if fit_type == 'uniform':
                     self.add_model(peak_models[i], prefix=f'peak{i+1}_', parameters=(
-                            {'name' : 'amplitude', 'value' : amp_init, 'min' : min_value},
-                            {'name' : 'center', 'expr' : center_exprs[i], 'min' : min_value},
-                            {'name' : 'sigma', 'value' : sig_init, 'min' : min_value,
-                            'max' : sig_max}))
+                            {'name': 'amplitude', 'value': amp_init, 'min': min_value},
+                            {'name': 'center', 'expr': center_exprs[i]},
+                            {'name': 'sigma', 'value': sig_init, 'min': min_value,
+                            'max': sig_max}))
                 else:
                     self.add_model('gaussian', prefix=f'peak{i+1}_', parameters=(
-                            {'name' : 'amplitude', 'value' : amp_init, 'min' : min_value},
-                            {'name' : 'center', 'value' : cen_init, 'min' : min_value},
-                            {'name' : 'sigma', 'value' : sig_init, 'min' : min_value,
-                            'max' : sig_max}))
+                            {'name': 'amplitude', 'value': amp_init, 'min': min_value},
+                            {'name': 'center', 'value': cen_init, 'min': min_value},
+                            {'name': 'sigma', 'value': sig_init, 'min': min_value,
+                            'max': sig_max}))
 
     def _check_validity(self):
         """Check for valid fit parameter results
         """
         fit_failure = False
-        index = re.compile(r'\d+')
-        for parameter in self.best_parameters:
-            name = parameter['name']
-            if ((('amplitude' in name or 'height' in name) and parameter['value'] <= 0.0) or
-                    (('sigma' in name or 'fwhm' in name) and parameter['value'] <= 0.0) or
-                    ('center' in name and parameter['value'] <= 0.0) or
-                    (name == 'scale_factor' and parameter['value'] <= 0.0)):
-                logging.info(f'Invalid fit result for {name} ({parameter["value"]})')
+        index = compile(r'\d+')
+        for name, par in self.best_parameters.items():
+            if 'background' in name:
+#                if ((name == 'backgroundc' and par['value'] <= 0.0) or
+#                        (name.endswith('amplitude') and par['value'] <= 0.0) or
+                 if ((name.endswith('amplitude') and par['value'] <= 0.0) or
+                        (name.endswith('decay') and par['value'] <= 0.0)):
+                    logging.info(f'Invalid fit result for {name} ({par["value"]})')
+                    fit_failure = True
+            elif (((name.endswith('amplitude') or name.endswith('height')) and
+                    par['value'] <= 0.0) or
+                    ((name.endswith('sigma') or name.endswith('fwhm')) and par['value'] <= 0.0) or
+                    (name.endswith('center') and par['value'] <= 0.0) or
+                    (name == 'scale_factor' and par['value'] <= 0.0)):
+                logging.info(f'Invalid fit result for {name} ({par["value"]})')
                 fit_failure = True
-            if 'sigma' in name and self._sigma_max is not None:
+            if name.endswith('sigma') and self._sigma_max is not None:
                 if name == 'sigma':
                     sigma_max = self._sigma_max[0]
                 else:
                     sigma_max = self._sigma_max[int(index.search(name).group())-1]
-                i = int(index.search(name).group())-1
-                if parameter['value'] > sigma_max:
-                    logging.info(f'Invalid fit result for {name} ({parameter["value"]})')
+                if par['value'] > sigma_max:
+                    logging.info(f'Invalid fit result for {name} ({par["value"]})')
+                    fit_failure = True
+                elif par['value'] == sigma_max:
+                    logging.warning(f'Edge result on for {name} ({par["value"]})')
+            if name.endswith('fwhm') and self._fwhm_max is not None:
+                if par['value'] > self._fwhm_max:
+                    logging.info(f'Invalid fit result for {name} ({par["value"]})')
                     fit_failure = True
-                elif parameter['value'] == sigma_max:
-                    logging.warning(f'Edge result on for {name} ({parameter["value"]})')
-            if 'fwhm' in name and self._fwhm_max is not None:
-                if parameter['value'] > self._fwhm_max:
-                    logging.info(f'Invalid fit result for {name} ({parameter["value"]})')
-                    fit_failure = True
-                elif parameter['value'] == self._fwhm_max:
-                    logging.warning(f'Edge result on for {name} ({parameter["value"]})')
-        return fit_failure
+                elif par['value'] == self._fwhm_max:
+                    logging.warning(f'Edge result on for {name} ({par["value"]})')
+        return(fit_failure)
+
+
+class FitMap(Fit):
+    """Fit a map of data
+    """
+    def __init__(self, ymap, x=None, models=None, normalize=True, transpose=None, **kwargs):
+        super().__init__(None)
+        self._best_errors = None
+        self._best_fit = None
+        self._best_parameters = None
+        self._best_values = None
+        self._inv_transpose = None
+        self._max_nfev = None
+        self._memfolder = None
+        self._new_parameters = None
+        self._out_of_bounds = None
+        self._plot = False
+        self._print_report = False
+        self._redchi = None
+        self._redchi_cutoff = 0.1
+        self._skip_init = True
+        self._success = None
+        self._transpose = None
+        self._try_no_bounds = True
+
+        # At this point the fastest index should always be the signal dimension so that the slowest 
+        #    ndim-1 dimensions are the map dimensions
+        if isinstance(ymap, (tuple, list, np.ndarray)):
+            self._x = np.asarray(x)
+        elif have_xarray and isinstance(ymap, xr.DataArray):
+            if x is not None:
+                logging.warning('Ignoring superfluous input x ({x}) in Fit.__init__')
+            self._x = np.asarray(ymap[ymap.dims[-1]])
+        else:
+            illegal_value(ymap, 'ymap', 'FitMap:__init__', raise_error=True)
+        self._ymap = ymap
+
+        # Verify the input parameters
+        if self._x.ndim != 1:
+            raise ValueError(f'Invalid dimension for input x {self._x.ndim}')
+        if self._ymap.ndim < 2:
+            raise ValueError('Invalid number of dimension of the input dataset '+
+                    f'{self._ymap.ndim}')
+        if self._x.size != self._ymap.shape[-1]:
+            raise ValueError(f'Inconsistent x and y dimensions ({self._x.size} vs '+
+                    f'{self._ymap.shape[-1]})')
+        if not isinstance(normalize, bool):
+            logging.warning(f'Invalid value for normalize ({normalize}) in Fit.__init__: '+
+                    'setting normalize to True')
+            normalize = True
+        if isinstance(transpose, bool) and not transpose:
+            transpose = None
+        if transpose is not None and self._ymap.ndim < 3:
+            logging.warning(f'Transpose meaningless for {self._ymap.ndim-1}D data maps: ignoring '+
+                    'transpose')
+        if transpose is not None:
+            if self._ymap.ndim == 3 and isinstance(transpose, bool) and transpose:
+                self._transpose = (1, 0)
+            elif not isinstance(transpose, (tuple, list)):
+                logging.warning(f'Invalid data type for transpose ({transpose}, '+
+                        f'{type(transpose)}) in Fit.__init__: setting transpose to False')
+            elif len(transpose) != self._ymap.ndim-1:
+                logging.warning(f'Invalid dimension for transpose ({transpose}, must be equal to '+
+                        f'{self._ymap.ndim-1}) in Fit.__init__: setting transpose to False')
+            elif any(i not in transpose for i in range(len(transpose))):
+                logging.warning(f'Invalid index in transpose ({transpose}) '+
+                        f'in Fit.__init__: setting transpose to False')
+            elif not all(i==transpose[i] for i in range(self._ymap.ndim-1)):
+                self._transpose = transpose
+            if self._transpose is not None:
+                self._inv_transpose = tuple(self._transpose.index(i)
+                        for i in range(len(self._transpose)))
+
+        # Flatten the map (transpose if requested)
+        # Store the flattened map in self._ymap_norm, whether normalized or not
+        if self._transpose is not None:
+            self._ymap_norm = np.transpose(np.asarray(self._ymap), list(self._transpose)+
+                    [len(self._transpose)])
+        else:
+            self._ymap_norm = np.asarray(self._ymap)
+        self._map_dim = int(self._ymap_norm.size/self._x.size)
+        self._map_shape = self._ymap_norm.shape[:-1]
+        self._ymap_norm = np.reshape(self._ymap_norm, (self._map_dim, self._x.size))
+
+        # Check if a mask is provided
+        if 'mask' in kwargs:
+            self._mask = kwargs.pop('mask')
+        if self._mask is None:
+            ymap_min = float(self._ymap_norm.min())
+            ymap_max = float(self._ymap_norm.max())
+        else:
+            self._mask = np.asarray(self._mask).astype(bool)
+            if self._x.size != self._mask.size:
+                raise ValueError(f'Inconsistent mask dimension ({self._x.size} vs '+
+                        f'{self._mask.size})')
+            ymap_masked = np.asarray(self._ymap_norm)[:,~self._mask]
+            ymap_min = float(ymap_masked.min())
+            ymap_max = float(ymap_masked.max())
+
+        # Normalize the data
+        self._y_range = ymap_max-ymap_min
+        if normalize and self._y_range > 0.0:
+            self._norm = (ymap_min, self._y_range)
+            self._ymap_norm = (self._ymap_norm-self._norm[0])/self._norm[1]
+        else:
+            self._redchi_cutoff *= self._y_range**2
+        if models is not None:
+            if callable(models) or isinstance(models, str):
+                kwargs = self.add_model(models, **kwargs)
+            elif isinstance(models, (tuple, list)):
+                for model in models:
+                    kwargs = self.add_model(model, **kwargs)
+            self.fit(**kwargs)
+
+    @classmethod
+    def fit_map(cls, x, ymap, models, normalize=True, **kwargs):
+        return(cls(x, ymap, models, normalize=normalize, **kwargs))
+
+    @property
+    def best_errors(self):
+        return(self._best_errors)
+
+    @property
+    def best_fit(self):
+        return(self._best_fit)
 
-    def _normalize(self):
-        """Normalize the data
+    @property
+    def best_results(self):
+        """Convert the input data array to a data set and add the fit results.
         """
-        y_min = self._y.min()
-        self._norm = (y_min, self._y.max()-y_min)
-        if self._norm[1] == 0.0:
-            self._norm = None
+        if self.best_values is None or self.best_errors is None or self.best_fit is None:
+            return(None)
+        if not have_xarray:
+            logging.warning('Unable to load xarray module')
+            return(None)
+        best_values = self.best_values
+        best_errors = self.best_errors
+        if isinstance(self._ymap, xr.DataArray):
+            best_results = self._ymap.to_dataset()
+            dims = self._ymap.dims
+            fit_name = f'{self._ymap.name}_fit'
         else:
-            self._y = (self._y-self._norm[0])/self._norm[1]
+            coords = {f'dim{n}_index':([f'dim{n}_index'], range(self._ymap.shape[n]))
+                    for n in range(self._ymap.ndim-1)}
+            coords['x'] = (['x'], self._x)
+            dims = list(coords.keys())
+            best_results = xr.Dataset(coords=coords)
+            best_results['y'] = (dims, self._ymap)
+            fit_name = 'y_fit'
+        best_results[fit_name] = (dims, self.best_fit)
+        if self._mask is not None:
+            best_results['mask'] = self._mask
+        for n in range(best_values.shape[0]):
+            best_results[f'{self._best_parameters[n]}_values'] = (dims[:-1], best_values[n])
+            best_results[f'{self._best_parameters[n]}_errors'] = (dims[:-1], best_errors[n])
+        best_results.attrs['components'] = self.components
+        return(best_results)
+
+    @property
+    def best_values(self):
+        return(self._best_values)
+
+    @property
+    def chisqr(self):
+        logging.warning('property chisqr not defined for fit.FitMap')
+        return(None)
+
+    @property
+    def components(self):
+        components = {}
+        if self._result is None:
+            logging.warning('Unable to collect components in FitMap.components')
+            return(components)
+        for component in self._result.components:
+            if 'tmp_normalization_offset_c' in component.param_names:
+                continue
+            parameters = {}
+            for name in component.param_names:
+                if self._parameters[name].vary:
+                    parameters[name] = {'free': True}
+                elif self._parameters[name].expr is not None:
+                    parameters[name] = {'free': False, 'expr': self._parameters[name].expr}
+                else:
+                    parameters[name] = {'free': False, 'value': self.init_parameters[name]['value']}
+            expr = None
+            if isinstance(component, ExpressionModel):
+                name = component._name
+                if name[-1] == '_':
+                    name = name[:-1]
+                expr = component.expr
+            else:
+                prefix = component.prefix
+                if len(prefix):
+                    if prefix[-1] == '_':
+                        prefix = prefix[:-1]
+                    name = f'{prefix} ({component._name})'
+                else:
+                    name = f'{component._name}'
+            if expr is None:
+                components[name] = {'parameters': parameters}
+            else:
+                components[name] = {'expr': expr, 'parameters': parameters}
+        return(components)
+
+    @property
+    def covar(self):
+        logging.warning('property covar not defined for fit.FitMap')
+        return(None)
+
+    @property
+    def max_nfev(self):
+        return(self._max_nfev)
+
+    @property
+    def num_func_eval(self):
+        logging.warning('property num_func_eval not defined for fit.FitMap')
+        return(None)
 
-    def _renormalize(self):
-        """Renormalize the data and results
-        """
-        if self._norm is None:
+    @property
+    def out_of_bounds(self):
+        return(self._out_of_bounds)
+
+    @property
+    def redchi(self):
+        return(self._redchi)
+
+    @property
+    def residual(self):
+        if self.best_fit is None:
+            return(None)
+        if self._mask is None:
+            return(np.asarray(self._ymap)-self.best_fit)
+        else:
+            ymap_flat = np.reshape(np.asarray(self._ymap), (self._map_dim, self._x.size))
+            ymap_flat_masked = ymap_flat[:,~self._mask]
+            ymap_masked = np.reshape(ymap_flat_masked,
+                    list(self._map_shape)+[ymap_flat_masked.shape[-1]])
+            return(ymap_masked-self.best_fit)
+
+    @property
+    def success(self):
+        return(self._success)
+
+    @property
+    def var_names(self):
+        logging.warning('property var_names not defined for fit.FitMap')
+        return(None)
+
+    @property
+    def y(self):
+        logging.warning('property y not defined for fit.FitMap')
+        return(None)
+
+    @property
+    def ymap(self):
+        return(self._ymap)
+
+    def best_parameters(self, dims=None):
+        if dims is None:
+            return(self._best_parameters)
+        if not isinstance(dims, (list, tuple)) or len(dims) != len(self._map_shape):
+            illegal_value(dims, 'dims', 'FitMap.best_parameters', raise_error=True)
+        if self.best_values is None or self.best_errors is None:
+            logging.warning(f'Unable to obtain best parameter values for dims = {dims} in '+
+                    'FitMap.best_parameters')
+            return({})
+        # Create current parameters
+        parameters = deepcopy(self._parameters)
+        for n, name in enumerate(self._best_parameters):
+            if self._parameters[name].vary:
+                parameters[name].set(value=self.best_values[n][dims])
+            parameters[name].stderr = self.best_errors[n][dims]
+        parameters_dict = {}
+        for name in sorted(parameters):
+            if name != 'tmp_normalization_offset_c':
+                par = parameters[name]
+                parameters_dict[name] = {'value': par.value, 'error': par.stderr,
+                        'init_value': self.init_parameters[name]['value'], 'min': par.min,
+                        'max': par.max, 'vary': par.vary, 'expr': par.expr}
+        return(parameters_dict)
+
+    def freemem(self):
+        if self._memfolder is None:
+            return
+        try:
+            rmtree(self._memfolder)
+            self._memfolder = None
+        except:
+            logging.warning('Could not clean-up automatically.')
+
+    def plot(self, dims, y_title=None, plot_residual=False, plot_comp_legends=False,
+            plot_masked_data=True):
+        if not isinstance(dims, (list, tuple)) or len(dims) != len(self._map_shape):
+            illegal_value(dims, 'dims', 'FitMap.plot', raise_error=True)
+        if self._result is None or self.best_fit is None or self.best_values is None:
+            logging.warning(f'Unable to plot fit for dims = {dims} in FitMap.plot')
             return
-        self._y = self._norm[0]+self._norm[1]*self._y
-        self._result.best_fit = self._norm[0]+self._norm[1]*self._result.best_fit
-        for name in self._result.params:
-            par = self._result.params[name]
-            if 'amplitude' in name or 'height' in name or 'background' in name:
-                par.value *= self._norm[1]
-                if par.stderr is not None:
-                    par.stderr *= self._norm[1]
-                if par.init_value is not None:
-                    par.init_value *= self._norm[1]
-                if par.min is not None and not np.isinf(par.min):
-                    par.min *= self._norm[1]
-                if par.max is not None and not np.isinf(par.max):
-                    par.max *= self._norm[1]
-            if 'intercept' in name or 'backgroundc' in name:
-                par.value += self._norm[0]
-                if par.init_value is not None:
-                    par.init_value += self._norm[0]
-                if par.min is not None and not np.isinf(par.min):
-                    par.min += self._norm[0]
-                if par.max is not None and not np.isinf(par.max):
-                    par.max += self._norm[0]
-        self._result.init_fit = self._norm[0]+self._norm[1]*self._result.init_fit
-        init_values = {}
-        for name in self._result.init_values:
-            init_values[name] = self._result.init_values[name]
-            if init_values[name] is None:
+        if y_title is None or not isinstance(y_title, str):
+            y_title = 'data'
+        if self._mask is None:
+            mask = np.zeros(self._x.size).astype(bool)
+            plot_masked_data = False
+        else:
+            mask = self._mask
+        plots = [(self._x, np.asarray(self._ymap[dims]), 'b.')]
+        legend = [y_title]
+        if plot_masked_data:
+            plots += [(self._x[mask], np.asarray(self._ymap)[(*dims,mask)], 'bx')]
+            legend += ['masked data']
+        plots += [(self._x[~mask], self.best_fit[dims], 'k-')]
+        legend += ['best fit']
+        if plot_residual:
+            plots += [(self._x[~mask], self.residual[dims], 'k--')]
+            legend += ['residual']
+        # Create current parameters
+        parameters = deepcopy(self._parameters)
+        for name in self._best_parameters:
+            if self._parameters[name].vary:
+                parameters[name].set(value=
+                        self.best_values[self._best_parameters.index(name)][dims])
+        for component in self._result.components:
+            if 'tmp_normalization_offset_c' in component.param_names:
                 continue
-            if 'amplitude' in name or 'height' in name or 'background' in name:
-                init_values[name] *= self._norm[1]
-            if 'intercept' in name or 'backgroundc' in name:
-                init_values[name] += self._norm[0]
-        self._result.init_values = init_values
-        # Don't renormalized chisqr, it has no useful meaning in physical units
-        #self._result.chisqr *= self._norm[1]*self._norm[1]
-        if self._result.covar is not None:
-            for i, name in enumerate(self._result.var_names):
-                if 'amplitude' in name or 'height' in name or 'background' in name:
-                    for j in range(len(self._result.var_names)):
-                        if self._result.covar[i,j] is not None:
-                            self._result.covar[i,j] *= self._norm[1]
-                        if self._result.covar[j,i] is not None:
-                            self._result.covar[j,i] *= self._norm[1]
-        # Don't renormalized redchi, it has no useful meaning in physical units
-        #self._result.redchi *= self._norm[1]*self._norm[1]
-        self._result.residual *= self._norm[1]
+            if isinstance(component, ExpressionModel):
+                prefix = component._name
+                if prefix[-1] == '_':
+                    prefix = prefix[:-1]
+                modelname = f'{prefix}: {component.expr}'
+            else:
+                prefix = component.prefix
+                if len(prefix):
+                    if prefix[-1] == '_':
+                        prefix = prefix[:-1]
+                    modelname = f'{prefix} ({component._name})'
+                else:
+                    modelname = f'{component._name}'
+            if len(modelname) > 20:
+                modelname = f'{modelname[0:16]} ...'
+            y = component.eval(params=parameters, x=self._x[~mask])
+            if isinstance(y, (int, float)):
+                y *= np.ones(self._x[~mask].size)
+            plots += [(self._x[~mask], y, '--')]
+            if plot_comp_legends:
+                legend.append(modelname)
+        quick_plot(tuple(plots), legend=legend, title=str(dims), block=True)
+
+    def fit(self, **kwargs):
+#        t0 = time()
+        # Check input parameters
+        if self._model is None:
+            logging.error('Undefined fit model')
+        if 'num_proc' in kwargs:
+            num_proc = kwargs.pop('num_proc')
+            if not is_int(num_proc, ge=1):
+                illegal_value(num_proc, 'num_proc', 'FitMap.fit', raise_error=True)
+        else:
+            num_proc = cpu_count()
+        if num_proc > 1 and not have_joblib:
+            logging.warning(f'Missing joblib in the conda environment, running FitMap serially')
+            num_proc = 1
+        if num_proc > cpu_count():
+            logging.warning(f'The requested number of processors ({num_proc}) exceeds the maximum '+
+                    f'number of processors, num_proc reduced to ({cpu_count()})')
+            num_proc = cpu_count()
+        if 'try_no_bounds' in kwargs:
+            self._try_no_bounds = kwargs.pop('try_no_bounds')
+            if not isinstance(self._try_no_bounds, bool):
+                illegal_value(self._try_no_bounds, 'try_no_bounds', 'FitMap.fit', raise_error=True)
+        if 'redchi_cutoff' in kwargs:
+            self._redchi_cutoff = kwargs.pop('redchi_cutoff')
+            if not is_num(self._redchi_cutoff, gt=0):
+                illegal_value(self._redchi_cutoff, 'redchi_cutoff', 'FitMap.fit', raise_error=True)
+        if 'print_report' in kwargs:
+            self._print_report = kwargs.pop('print_report')
+            if not isinstance(self._print_report, bool):
+                illegal_value(self._print_report, 'print_report', 'FitMap.fit', raise_error=True)
+        if 'plot' in kwargs:
+            self._plot = kwargs.pop('plot')
+            if not isinstance(self._plot, bool):
+                illegal_value(self._plot, 'plot', 'FitMap.fit', raise_error=True)
+        if 'skip_init' in kwargs:
+            self._skip_init = kwargs.pop('skip_init')
+            if not isinstance(self._skip_init, bool):
+                illegal_value(self._skip_init, 'skip_init', 'FitMap.fit', raise_error=True)
+
+        # Apply mask if supplied:
+        if 'mask' in kwargs:
+            self._mask = kwargs.pop('mask')
+        if self._mask is not None:
+            self._mask = np.asarray(self._mask).astype(bool)
+            if self._x.size != self._mask.size:
+                raise ValueError(f'Inconsistent x and mask dimensions ({self._x.size} vs '+
+                        f'{self._mask.size})')
+
+        # Add constant offset for a normalized single component model
+        if self._result is None and self._norm is not None and self._norm[0]:
+            self.add_model('constant', prefix='tmp_normalization_offset_', parameters={'name': 'c',
+                    'value': -self._norm[0], 'vary': False, 'norm': True})
+                    #'value': -self._norm[0]/self._norm[1], 'vary': False, 'norm': False})
+
+        # Adjust existing parameters for refit:
+        if 'parameters' in kwargs:
+#            print('\nIn FitMap before adjusting existing parameters for refit:')
+#            self._parameters.pretty_print()
+#            if self._result is None:
+#                raise ValueError('Invalid parameter parameters ({parameters})')
+#            if self._best_values is None:
+#                raise ValueError('Valid self._best_values required for refitting in FitMap.fit')
+            parameters = kwargs.pop('parameters')
+#            print(f'\nparameters:\n{parameters}')
+            if isinstance(parameters, dict):
+                parameters = (parameters, )
+            elif not is_dict_series(parameters):
+                illegal_value(parameters, 'parameters', 'Fit.fit', raise_error=True)
+            for par in parameters:
+                name = par['name']
+                if name not in self._parameters:
+                    raise ValueError(f'Unable to match {name} parameter {par} to an existing one')
+                if self._parameters[name].expr is not None:
+                    raise ValueError(f'Unable to modify {name} parameter {par} (currently an '+
+                            'expression)')
+                value =  par.get('value')
+                vary =  par.get('vary')
+                if par.get('expr') is not None:
+                    raise KeyError(f'Illegal "expr" key in {name} parameter {par}')
+                self._parameters[name].set(value=value, vary=vary, min=par.get('min'),
+                        max=par.get('max'))
+                # Overwrite existing best values for fixed parameters when a value is specified
+#                print(f'best values befored resetting:\n{self._best_values}')
+                if isinstance(value, (int, float)) and vary is False:
+                    for i, nname in enumerate(self._best_parameters):
+                        if nname == name:
+                            self._best_values[i] = value
+#                print(f'best values after resetting (value={value}, vary={vary}):\n{self._best_values}')
+#RV            print('\nIn FitMap after adjusting existing parameters for refit:')
+#RV            self._parameters.pretty_print()
+
+        # Check for uninitialized parameters
+        for name, par in self._parameters.items():
+            if par.expr is None:
+                value = par.value
+                if value is None or np.isinf(value) or np.isnan(value):
+                    value = 1.0
+                    if self._norm is None or name not in self._parameter_norms:
+                        self._parameters[name].set(value=value)
+                    elif self._parameter_norms[name]:
+                        self._parameters[name].set(value=value*self._norm[1])
+
+        # Create the best parameter list, consisting of all varying parameters plus the expression
+        #   parameters in order to collect their errors
+        if self._result is None:
+            # Initial fit
+            assert(self._best_parameters is None)
+            self._best_parameters = [name for name, par in self._parameters.items()
+                    if par.vary or par.expr is not None]
+            num_new_parameters = 0
+        else:
+            # Refit
+            assert(len(self._best_parameters))
+            self._new_parameters = [name for name, par in self._parameters.items()
+                if name != 'tmp_normalization_offset_c' and name not in self._best_parameters and
+                    (par.vary or par.expr is not None)]
+            num_new_parameters = len(self._new_parameters)
+        num_best_parameters = len(self._best_parameters)
+
+        # Flatten and normalize the best values of the previous fit, remove the remaining results
+        #   of the previous fit
+        if self._result is not None:
+#            print('\nBefore flatten and normalize:')
+#            print(f'self._best_values:\n{self._best_values}')
+            self._out_of_bounds = None
+            self._max_nfev = None
+            self._redchi = None
+            self._success = None
+            self._best_fit = None
+            self._best_errors = None
+            assert(self._best_values is not None)
+            assert(self._best_values.shape[0] == num_best_parameters)
+            assert(self._best_values.shape[1:] == self._map_shape)
+            if self._transpose is not None:
+                self._best_values = np.transpose(self._best_values,
+                        [0]+[i+1 for i in self._transpose])
+            self._best_values = [np.reshape(self._best_values[i], self._map_dim)
+                for i in range(num_best_parameters)]
+            if self._norm is not None:
+                for i, name in enumerate(self._best_parameters):
+                    if self._parameter_norms.get(name, False):
+                        self._best_values[i] /= self._norm[1]
+#RV            print('\nAfter flatten and normalize:')
+#RV            print(f'self._best_values:\n{self._best_values}')
+
+        # Normalize the initial parameters (and best values for a refit)
+#        print('\nIn FitMap before normalize:')
+#        self._parameters.pretty_print()
+#        print(f'\nparameter_norms:\n{self._parameter_norms}\n')
+        self._normalize()
+#        print('\nIn FitMap after normalize:')
+#        self._parameters.pretty_print()
+#        print(f'\nparameter_norms:\n{self._parameter_norms}\n')
+
+        # Prevent initial values from sitting at boundaries
+        self._parameter_bounds = {name:{'min': par.min, 'max': par.max}
+                for name, par in self._parameters.items() if par.vary}
+        for name, par in self._parameters.items():
+            if par.vary:
+                par.set(value=self._reset_par_at_boundary(par, par.value))
+#        print('\nAfter checking boundaries:')
+#        self._parameters.pretty_print()
+
+        # Set parameter bounds to unbound (only use bounds when fit fails)
+        if self._try_no_bounds:
+            for name in self._parameter_bounds.keys():
+                self._parameters[name].set(min=-np.inf, max=np.inf)
+
+        # Allocate memory to store fit results
+        if self._mask is None:
+            x_size = self._x.size
+        else:
+            x_size = self._x[~self._mask].size
+        if num_proc == 1:
+            self._out_of_bounds_flat = np.zeros(self._map_dim, dtype=bool)
+            self._max_nfev_flat = np.zeros(self._map_dim, dtype=bool)
+            self._redchi_flat = np.zeros(self._map_dim, dtype=np.float64)
+            self._success_flat = np.zeros(self._map_dim, dtype=bool)
+            self._best_fit_flat = np.zeros((self._map_dim, x_size),
+                    dtype=self._ymap_norm.dtype)
+            self._best_errors_flat = [np.zeros(self._map_dim, dtype=np.float64)
+                    for _ in range(num_best_parameters+num_new_parameters)]
+            if self._result is None:
+                self._best_values_flat = [np.zeros(self._map_dim, dtype=np.float64)
+                        for _ in range(num_best_parameters)]
+            else:
+                self._best_values_flat = self._best_values
+                self._best_values_flat += [np.zeros(self._map_dim, dtype=np.float64)
+                        for _ in range(num_new_parameters)]
+        else:
+            self._memfolder = './joblib_memmap'
+            try:
+                mkdir(self._memfolder)
+            except FileExistsError:
+                pass
+            filename_memmap = path.join(self._memfolder, 'out_of_bounds_memmap')
+            self._out_of_bounds_flat = np.memmap(filename_memmap, dtype=bool,
+                    shape=(self._map_dim), mode='w+')
+            filename_memmap = path.join(self._memfolder, 'max_nfev_memmap')
+            self._max_nfev_flat = np.memmap(filename_memmap, dtype=bool,
+                    shape=(self._map_dim), mode='w+')
+            filename_memmap = path.join(self._memfolder, 'redchi_memmap')
+            self._redchi_flat = np.memmap(filename_memmap, dtype=np.float64,
+                    shape=(self._map_dim), mode='w+')
+            filename_memmap = path.join(self._memfolder, 'success_memmap')
+            self._success_flat = np.memmap(filename_memmap, dtype=bool,
+                    shape=(self._map_dim), mode='w+')
+            filename_memmap = path.join(self._memfolder, 'best_fit_memmap')
+            self._best_fit_flat = np.memmap(filename_memmap, dtype=self._ymap_norm.dtype,
+                    shape=(self._map_dim, x_size), mode='w+')
+            self._best_errors_flat = []
+            for i in range(num_best_parameters+num_new_parameters):
+                filename_memmap = path.join(self._memfolder, f'best_errors_memmap_{i}')
+                self._best_errors_flat.append(np.memmap(filename_memmap, dtype=np.float64,
+                        shape=self._map_dim, mode='w+'))
+            self._best_values_flat = []
+            for i in range(num_best_parameters):
+                filename_memmap = path.join(self._memfolder, f'best_values_memmap_{i}')
+                self._best_values_flat.append(np.memmap(filename_memmap, dtype=np.float64,
+                        shape=self._map_dim, mode='w+'))
+                if self._result is not None:
+                    self._best_values_flat[i][:] = self._best_values[i][:]
+            for i in range(num_new_parameters):
+                filename_memmap = path.join(self._memfolder,
+                        f'best_values_memmap_{i+num_best_parameters}')
+                self._best_values_flat.append(np.memmap(filename_memmap, dtype=np.float64,
+                        shape=self._map_dim, mode='w+'))
+
+        # Update the best parameter list
+        if num_new_parameters:
+            self._best_parameters += self._new_parameters
+
+        # Perform the first fit to get model component info and initial parameters
+        current_best_values = {}
+#        print(f'0 before:\n{current_best_values}')
+#        t1 = time()
+        self._result = self._fit(0, current_best_values, return_result=True, **kwargs)
+#        t2 = time()
+#        print(f'0 after:\n{current_best_values}')
+#        print('\nAfter the first fit:')
+#        self._parameters.pretty_print()
+#        print(self._result.fit_report(show_correl=False))
+
+        # Remove all irrelevant content from self._result
+        for attr in ('_abort', 'aborted', 'aic', 'best_fit', 'best_values', 'bic', 'calc_covar',
+                'call_kws', 'chisqr', 'ci_out', 'col_deriv', 'covar', 'data', 'errorbars',
+                'flatchain', 'ier', 'init_vals', 'init_fit', 'iter_cb', 'jacfcn', 'kws',
+                'last_internal_values', 'lmdif_message', 'message', 'method', 'nan_policy',
+                'ndata', 'nfev', 'nfree', 'params', 'redchi', 'reduce_fcn', 'residual', 'result',
+                'scale_covar', 'show_candidates', 'calc_covar', 'success', 'userargs', 'userfcn',
+                'userkws', 'values', 'var_names', 'weights', 'user_options'):
+            try:
+                delattr(self._result, attr)
+            except AttributeError:
+#                logging.warning(f'Unknown attribute {attr} in fit.FtMap._cleanup_result')
+                pass
+
+#        t3 = time()
+        if num_proc == 1:
+            # Perform the remaining fits serially
+            for n in range(1, self._map_dim):
+#                print(f'{n} before:\n{current_best_values}')
+                self._fit(n, current_best_values, **kwargs)
+#                print(f'{n} after:\n{current_best_values}')
+        else:
+            # Perform the remaining fits in parallel
+            num_fit = self._map_dim-1
+#            print(f'num_fit = {num_fit}')
+            if num_proc > num_fit:
+                logging.warning(f'The requested number of processors ({num_proc}) exceeds the '+
+                        f'number of fits, num_proc reduced to ({num_fit})')
+                num_proc = num_fit
+                num_fit_per_proc = 1
+            else:
+                num_fit_per_proc = round((num_fit)/num_proc)
+                if num_proc*num_fit_per_proc < num_fit:
+                    num_fit_per_proc +=1
+#            print(f'num_fit_per_proc = {num_fit_per_proc}')
+            num_fit_batch = min(num_fit_per_proc, 40)
+#            print(f'num_fit_batch = {num_fit_batch}')
+            with Parallel(n_jobs=num_proc) as parallel:
+                parallel(delayed(self._fit_parallel)(current_best_values, num_fit_batch,
+                        n_start, **kwargs) for n_start in range(1, self._map_dim, num_fit_batch))
+#        t4 = time()
+
+        # Renormalize the initial parameters for external use
+        if self._norm is not None and self._normalized:
+            init_values = {}
+            for name, value in self._result.init_values.items():
+                if name not in self._parameter_norms or self._parameters[name].expr is not None:
+                    init_values[name] = value
+                elif self._parameter_norms[name]:
+                    init_values[name] = value*self._norm[1]
+            self._result.init_values = init_values
+            for name, par in self._result.init_params.items():
+                if par.expr is None and self._parameter_norms.get(name, False):
+                    _min = par.min
+                    _max = par.max
+                    value = par.value*self._norm[1]
+                    if not np.isinf(_min) and abs(_min) != float_min:
+                        _min *= self._norm[1]
+                    if not np.isinf(_max) and abs(_max) != float_min:
+                        _max *= self._norm[1]
+                    par.set(value=value, min=_min, max=_max)
+                par.init_value = par.value
+
+        # Remap the best results
+#        t5 = time()
+        self._out_of_bounds = np.copy(np.reshape(self._out_of_bounds_flat, self._map_shape))
+        self._max_nfev = np.copy(np.reshape(self._max_nfev_flat, self._map_shape))
+        self._redchi = np.copy(np.reshape(self._redchi_flat, self._map_shape))
+        self._success = np.copy(np.reshape(self._success_flat, self._map_shape))
+        self._best_fit = np.copy(np.reshape(self._best_fit_flat,
+               list(self._map_shape)+[x_size]))
+        self._best_values = np.asarray([np.reshape(par, list(self._map_shape))
+                for par in self._best_values_flat])
+        self._best_errors = np.asarray([np.reshape(par, list(self._map_shape))
+                for par in self._best_errors_flat])
+        if self._inv_transpose is not None:
+            self._out_of_bounds = np.transpose(self._out_of_bounds, self._inv_transpose)
+            self._max_nfev = np.transpose(self._max_nfev, self._inv_transpose)
+            self._redchi = np.transpose(self._redchi, self._inv_transpose)
+            self._success = np.transpose(self._success, self._inv_transpose)
+            self._best_fit = np.transpose(self._best_fit,
+                    list(self._inv_transpose)+[len(self._inv_transpose)])
+            self._best_values = np.transpose(self._best_values,
+                    [0]+[i+1 for i in self._inv_transpose])
+            self._best_errors = np.transpose(self._best_errors,
+                    [0]+[i+1 for i in self._inv_transpose])
+        del self._out_of_bounds_flat
+        del self._max_nfev_flat
+        del self._redchi_flat
+        del self._success_flat
+        del self._best_fit_flat
+        del self._best_values_flat
+        del self._best_errors_flat
+#        t6 = time()
+
+        # Restore parameter bounds and renormalize the parameters
+        for name, par in self._parameter_bounds.items():
+            self._parameters[name].set(min=par['min'], max=par['max'])
+        self._normalized = False
+        if self._norm is not None:
+            for name, norm in self._parameter_norms.items():
+                par = self._parameters[name]
+                if par.expr is None and norm:
+                    value = par.value*self._norm[1]
+                    _min = par.min
+                    _max = par.max
+                    if not np.isinf(_min) and abs(_min) != float_min:
+                        _min *= self._norm[1]
+                    if not np.isinf(_max) and abs(_max) != float_min:
+                        _max *= self._norm[1]
+                    par.set(value=value, min=_min, max=_max)
+#        t7 = time()
+#        print(f'total run time in fit: {t7-t0:.2f} seconds')
+#        print(f'run time first fit: {t2-t1:.2f} seconds')
+#        print(f'run time remaining fits: {t4-t3:.2f} seconds')
+#        print(f'run time remapping results: {t6-t5:.2f} seconds')
+
+#        print('\n\nAt end fit:')
+#        self._parameters.pretty_print()
+#        print(f'self._best_values:\n{self._best_values}\n\n')
+
+        # Free the shared memory
+        self.freemem()
+
+    def _fit_parallel(self, current_best_values, num, n_start, **kwargs):
+        num = min(num, self._map_dim-n_start)
+        for n in range(num):
+#            print(f'{n_start+n} before:\n{current_best_values}')
+            self._fit(n_start+n, current_best_values, **kwargs)
+#            print(f'{n_start+n} after:\n{current_best_values}')
+
+    def _fit(self, n, current_best_values, return_result=False, **kwargs):
+#RV        print(f'\n\nstart FitMap._fit {n}\n')
+#RV        print(f'current_best_values = {current_best_values}')
+#RV        print(f'self._best_parameters = {self._best_parameters}')
+#RV        print(f'self._new_parameters = {self._new_parameters}\n\n')
+#        self._parameters.pretty_print()
+        # Set parameters to current best values, but prevent them from sitting at boundaries
+        if self._new_parameters is None:
+            # Initial fit
+            for name, value in current_best_values.items():
+                par = self._parameters[name]
+                par.set(value=self._reset_par_at_boundary(par, value))
+        else:
+            # Refit
+            for i, name in enumerate(self._best_parameters):
+                par = self._parameters[name]
+                if name in self._new_parameters:
+                    if name in current_best_values:
+                        par.set(value=self._reset_par_at_boundary(par, current_best_values[name]))
+                elif par.expr is None:
+                    par.set(value=self._best_values[i][n])
+#RV        print(f'\nbefore fit {n}')
+#RV        self._parameters.pretty_print()
+        if self._mask is None:
+            result = self._model.fit(self._ymap_norm[n], self._parameters, x=self._x, **kwargs)
+        else:
+            result = self._model.fit(self._ymap_norm[n][~self._mask], self._parameters,
+                    x=self._x[~self._mask], **kwargs)
+#        print(f'\nafter fit {n}')
+#        self._parameters.pretty_print()
+#        print(result.fit_report(show_correl=False))
+        out_of_bounds = False
+        for name, par in self._parameter_bounds.items():
+            value = result.params[name].value
+            if not np.isinf(par['min']) and value < par['min']:
+                out_of_bounds = True
+                break
+            if not np.isinf(par['max']) and value > par['max']:
+                out_of_bounds = True
+                break
+        self._out_of_bounds_flat[n] = out_of_bounds
+        if self._try_no_bounds and out_of_bounds:
+            # Rerun fit with parameter bounds in place
+            for name, par in self._parameter_bounds.items():
+                self._parameters[name].set(min=par['min'], max=par['max'])
+            # Set parameters to current best values, but prevent them from sitting at boundaries
+            if self._new_parameters is None:
+                # Initial fit
+                for name, value in current_best_values.items():
+                    par = self._parameters[name]
+                    par.set(value=self._reset_par_at_boundary(par, value))
+            else:
+                # Refit
+                for i, name in enumerate(self._best_parameters):
+                    par = self._parameters[name]
+                    if name in self._new_parameters:
+                        if name in current_best_values:
+                            par.set(value=self._reset_par_at_boundary(par,
+                                    current_best_values[name]))
+                    elif par.expr is None:
+                        par.set(value=self._best_values[i][n])
+#            print('\nbefore fit')
+#            self._parameters.pretty_print()
+#            print(result.fit_report(show_correl=False))
+            if self._mask is None:
+                result = self._model.fit(self._ymap_norm[n], self._parameters, x=self._x, **kwargs)
+            else:
+                result = self._model.fit(self._ymap_norm[n][~self._mask], self._parameters,
+                    x=self._x[~self._mask], **kwargs)
+#            print(f'\nafter fit {n}')
+#            self._parameters.pretty_print()
+#            print(result.fit_report(show_correl=False))
+            out_of_bounds = False
+            for name, par in self._parameter_bounds.items():
+                value = result.params[name].value
+                if not np.isinf(par['min']) and value < par['min']:
+                    out_of_bounds = True
+                    break
+                if not np.isinf(par['max']) and value > par['max']:
+                    out_of_bounds = True
+                    break
+#            print(f'{n} redchi < redchi_cutoff = {result.redchi < self._redchi_cutoff} success = {result.success} out_of_bounds = {out_of_bounds}')
+            # Reset parameters back to unbound
+            for name in self._parameter_bounds.keys():
+                self._parameters[name].set(min=-np.inf, max=np.inf)
+        assert(not out_of_bounds)
+        if result.redchi >= self._redchi_cutoff:
+            result.success = False
+        if result.nfev == result.max_nfev:
+#            print(f'Maximum number of function evaluations reached for n = {n}')
+#            logging.warning(f'Maximum number of function evaluations reached for n = {n}')
+            if result.redchi < self._redchi_cutoff:
+                result.success = True
+            self._max_nfev_flat[n] = True
+        if result.success:
+            assert(all(True for par in current_best_values if par in result.params.values()))
+            for par in result.params.values():
+                if par.vary:
+                    current_best_values[par.name] = par.value
+        else:
+            logging.warning(f'Fit for n = {n} failed: {result.lmdif_message}')
+        # Renormalize the data and results
+        self._renormalize(n, result)
+        if self._print_report:
+            print(result.fit_report(show_correl=False))
+        if self._plot:
+            dims = np.unravel_index(n, self._map_shape)
+            if self._inv_transpose is not None:
+                dims= tuple(dims[self._inv_transpose[i]] for i in range(len(dims)))
+            super().plot(result=result, y=np.asarray(self._ymap[dims]), plot_comp_legends=True,
+                skip_init=self._skip_init, title=str(dims))
+#RV        print(f'\n\nend FitMap._fit {n}\n')
+#RV        print(f'current_best_values = {current_best_values}')
+#        self._parameters.pretty_print()
+#        print(result.fit_report(show_correl=False))
+#RV        print(f'\nself._best_values_flat:\n{self._best_values_flat}\n\n')
+        if return_result:
+            return(result)
+
+    def _renormalize(self, n, result):
+        self._redchi_flat[n] = np.float64(result.redchi)
+        self._success_flat[n] = result.success
+        if self._norm is None or not self._normalized:
+            self._best_fit_flat[n] = result.best_fit
+            for i, name in enumerate(self._best_parameters):
+                self._best_values_flat[i][n] = np.float64(result.params[name].value)
+                self._best_errors_flat[i][n] = np.float64(result.params[name].stderr)
+        else:
+            pars = set(self._parameter_norms) & set(self._best_parameters)
+            for name, par in result.params.items():
+                if name in pars and self._parameter_norms[name]:
+                    if par.stderr is not None:
+                        par.stderr *= self._norm[1]
+                    if par.expr is None:
+                        par.value *= self._norm[1]
+                        if self._print_report:
+                            if par.init_value is not None:
+                                par.init_value *= self._norm[1]
+                            if not np.isinf(par.min) and abs(par.min) != float_min:
+                                par.min *= self._norm[1]
+                            if not np.isinf(par.max) and abs(par.max) != float_min:
+                                par.max *= self._norm[1]
+            self._best_fit_flat[n] = result.best_fit*self._norm[1]+self._norm[0]
+            for i, name in enumerate(self._best_parameters):
+                self._best_values_flat[i][n] = np.float64(result.params[name].value)
+                self._best_errors_flat[i][n] = np.float64(result.params[name].stderr)
+            if self._plot:
+                if not self._skip_init:
+                    result.init_fit = result.init_fit*self._norm[1]+self._norm[0]
+                result.best_fit = np.copy(self._best_fit_flat[n])