pymodelfit.builtins.TwoPowerModel¶

class pymodelfit.builtins.TwoPowerModel[source]¶

Bases: pymodelfit.core.FunctionModel1DAuto

A model that smoothly transitions between two power laws. The specific form is:

$A x^a (x+x_s)^{b-a}$

The turnover point is given by the parameter xs. a is the inner slope, b is the outer slope, and A is the overall normalization.

__init__()¶: x.__init__(...) initializes x; see help(type(x)) for signature

Methods

`chi2Data`([x, y, weights, ddof])	Computes the chi-squared statistic for the data assuming this model.
`derivative`(x[, dx])	Compute the derivative.
`f`(x[, A, xs, a, b])
`findroot`(x0[, method])	Finds a root for the model (location where the model is 0).
`findval`(val, x0[, method])	Finds where the model is equal to a specified value.
`fitData`([x, y, fixedpars, weights, ...])	Fit the provided data using algorithms from scipy.optimize, and adjust the model parameters to match.
`getCall`()	Retreives infromation about the calling function.
`getCov`()	Computes the covariance matrix for the last `fitData()` call.
`getMCMC`(x, y[, priors, datamodel])	Generate an Markov Chain Monte Carlo sampler for the data and model.
`integrate`(lower, upper[, method, n, jac])	Compute the definite integral of this model.
`integrateCircular`(lower, upper, args, *kwargs)	Integrate this model on the 2D circle. This calls `integrate()` with
`integrateSpherical`(lower, upper, args, *kwargs)	Integrate this model on the 3D sphere. This calls `integrate()` with
`inv`(yval, args, *kwargs)	Find the x value matching the requested y-value.
`isVarnumModel`()	Determines if the model represented by this class accepts a variable number of parameters (i.e.
`maximize`(x0[, method])	Finds a local maximum for the model.
`minimize`(x0[, method])	Finds a local minimum for the model.
`pixelize`(xorxl[, xu, n, edge, sampling])	Generate a discretized version o the model.
`plot`([lower, upper, n, clf, logplot, data, ...])	Plot the model function and possibly data and error bars with `matplotlib.pyplot`.
`plotResiduals`([x, y, clf, logplot])	Plots the residuals of the provided data ( $y-y_{mod}$ ) against this model.
`resampleFit`([x, y, xerr, yerr, bootstrap, ...])	Estimates errors via resampling.
`residuals`([x, y, retdata])	Compute residuals of the provided data against the model.
`setCall`([calltype, xtrans, ytrans])	Sets the type of function evaluation to occur when the model is called.
`stdData`([x, y])	Determines the standard deviation of the model from data.

Attributes

`A`	int(x=0) -> int or long
`a`	int(x=0) -> int or long
`b`	int(x=0) -> int or long
`data`	The fitting data for this model.
`defaultIntMethod`	str(object=’‘) -> string
`defaultInvMethod`	str(object=’‘) -> string
`defaultparval`	int(x=0) -> int or long
`errors`	Error on the data.
`fittype`	str(object=’‘) -> string
`fittypes`	A Sequence of the available valid values for the `fittype`
`fixedpars`	tuple() -> empty tuple
`fxs`	The function’s value at
`params`	A tuple of the parameter names.
`pardict`	A dictionary mapping parameter names to the associated values.
`parvals`	The values of the parameters in the same order as `params`
`rangehint`
`weightstype`	Determines the statistical interpretation of the weights in data.
`xaxisname`
`xs`	int(x=0) -> int or long
`yaxisname`

chi2Data(x=None, y=None, weights=None, ddof=1)¶

Computes the chi-squared statistic for the data assuming this model.

Parameters:

x (array-like or None) – Input data value or None to use stored data
y (array-like or None) – Output data value or None to use stored data
weights (array-like or None) – Weights to adjust chi-squared, typically for error bars. Statistically interpreted based on the weightstype attribute. If None, any stored data will be used.
ddof (int) – Delta Degrees of Freedom. The divisor used for the reduced chi-squared is n-m-ddof, where N is the number of points and m is the number of parameters in the model.

Returns:

tuple of floats (chi2,reducedchi2,p-value)

data None¶: The fitting data for this model. Should be either None, or a tuple(datain,dataout,weights). Note that the weights are interpreted statistically as errors based on the weightstype attribute.

derivative(x, dx=None)¶

Compute the derivative. This implementation numerically estimates the derivative at x using the following formula:

$\frac{df}{dx} \approx \frac{f(x+\Delta x)-f(x)}{\Delta x}$

Parameters:	x (float or array-like) – the value at which to compute the derivative dx – The spacing to assume for the numerically-computed derivative ( $\Delta x$ above). If None, and an array is provided for x, spacing will be inferred from the spacing between the x values. Otherwise, it defaults to 1.

Note

If overridden in a subclass for efficient analytical computation, the signature should be derivative(self,x,dx=None), such that if dx is None, the analytical computation is used, and otherwise this numerical technique is used. i.e. all subclasses should have this at the beginning:

if dx is not None:
    return FunctionModel1D.integrate(self,x,dx)

errors None¶: Error on the data. Sets the weights on data assuming the interpretation for errors given by weightstype. If data is None/missing, a TypeError will be raised.

findroot(x0, method='fmin', **kwargs)¶

Finds a root for the model (location where the model is 0).

Parameters:	x0 (float) – The location to start the search method (string) – Can be ‘fmin’ or ‘fmin_powell’, to use scipy.optimize.fmin and scipy.optimize.fmin_powell.

kwargs are passed into the method function

Returns:	the x value where the model is 0

findval(val, x0, method='fmin', **kwargs)¶

Finds where the model is equal to a specified value.

x0 is the location to start the search method can be ‘fmin’ or ‘fmin_powell’ (from scipy.optimize) kwargs are passed into the scipy.optimize function

fitData(x=None, y=None, fixedpars='auto', weights=None, savedata=True, updatepars=True, fitf=False, contraction='sumsq', **kwargs)¶

Fit the provided data using algorithms from scipy.optimize, and adjust the model parameters to match.

The fitting technique is sepcified by the fittype attribute of the object, which by default can be any of the optimization types in the scipy.optimize module (except for scalar minimizers)

The full fitting output is available in lastfit attribute after this method completes.

Parameters:

x (array-like) – The input values at which to evaluate the model. Valid shapes are those that this model will accept.
y (array-like) – The expected output values for the model at the given x values. Valid shapes are those that this model will output.
fixedpars (sequence of strings, ‘auto’ or None) – Parameter names to leave fixed. If ‘auto’ the fixed parameters are inferred from self.fixedpars (or all will be free parameters if self.fixedpars is absent). If None, all parameters will be free.
weights –
Weights to use for fitting, statistically interpreted as inverse errors (not inverse variance). May be one of the following forms:
- None for equal weights
- an array of points that must match the output
- a 2-sequence of arrays (xierr,yierr) such that xierr matches the x-data and yierr matches the y-data
- a function called as f(params) that returns an array of weights that match one of the above two conditions
savedata (bool) – If True, x,`y`,and weights will be saved to data. Otherwise, data will be discarded after fitting.
updatepars (bool) – If True, sets the parameters on the object to the best-fit values.
fitf (bool) – If True, the fit is performed directly against the f() method instead of against the model as evaluated if called (as altered using setCall()).
contraction –
Only applies for optimize-based methods and is the technique used to convert vectors to figures of merit. this is composed of multiple string segments:
1. these will be applied to each element of the vector first:
  - ‘sq’: square
  - ‘abs’:absolute value
  - ‘’:raw value
2. the contraction is performed after:
  - ‘sum’: sum of all the elements
  - ‘median’: median of all the elements
  - ‘mean’: mean of all the elements
  - ‘prod’: product of all the elements
- optionally,the string can include ‘frac’ to use the fractional version of the differnce vector instead of raw values. For the leastsq method, this is the only applicable value

kwargs are passed into the fitting function.

Raises ModelTypeError:
Returns:	array of the best fit parameters
	If the output of the model does not match the shape of y.

See also

getMCMC()

fittypes None¶: A Sequence of the available valid values for the fittype attribute. (Read-only)

fxs None¶: The function’s value at xs- this maps one-to-one onto the A parameter, so setting fxs also sets A (and vice versa).

getCall()¶

Retreives infromation about the calling function.

Returns:	The type of evaluation to perform when this model is called - a string like that of the type passed into `setCall()`, or None if the model function itself is to be called.

getCov()¶

Computes the covariance matrix for the last fitData() call.

Returns:	The covariance matrix with variables in the same order as `params`. Diagonal entries give the variance in each parameter.

Warning

This is not guaranteed to work for custom fit-types, but will always work with the default (leastsq) fit.

getMCMC(x, y, priors={}, datamodel=None)¶

Generate an Markov Chain Monte Carlo sampler for the data and model. This function requires the PyMC package for the MCMC internals and sampling.

Raises ValueError:
Parameters:	x (array-like) – Input data value y (array-like) – Output data value priors (dictionary) – Maps parameter names to the priors to assume for that parameter. There must be an entry for every parameter in the model. The prior specification can be in any of the following forms: A `pymc.Stochastric` object A 2-tuple (lower,upper) for a uniform prior A scalar > 0 to use a gaussian prior of the provided width centered at the current value of the parameter 0 for a Poisson prior with k set by the current value of the parameter datamodel – Specifies the model to assume for the fitting data points. May be any of the following: None A normal distribution with sigma given by the data’s standard deviation. A tuple (dist,dataname,kwargs) The first element is the pymc.distribution to be used as the distribution representing the data and the second is the name of the argument to be associated with the FunctionModel1D’s output, and the third is kwargs for the distribution (“observed” and “data” will be ignored, as will the data argument) A sequence A normal distribution is used with sigma for each data point specified by the sequence. The length must match the model. A scalar A normal distribution with the given standard deviation.
	If a prior is not provided for any parameter.
Returns:	A `pymc.MCMC` object ready to sample for this model.

integrate(lower, upper, method=True, n=100, jac=None, **kwargs)¶

Compute the definite integral of this model. This implememntation numerically estimates the integral using scipy.integrate functions. The integral computed is:

$\int_l^u \! f(x) \, j(x) \, dx$

where $j(x)$ is the jacobian set from the jac argument.

Parameters:	lower (float) – the lower limit of the integral upper (float) – the upper limit of the integral method (string or None) – The name of a function from `scipy.integrate`, or if True, the class attribute `defaultIntMethod` will be used. n (int) – Only has an effect for integration techniques that use samples. If an integer, it specifies the number of evenly spaced samples. If it as a sequence, it is an array of samples to use for the integral (this renders lower and upper meaningless and their values have no effect) jac – The jacobian factor to include in the integrand.
Typ jac:	a callable f(x,*params) or None
Returns:	The value of the computed definite integral.

Note

Integration methods will store their full output to the attribute lastintegrate upon completion.

Note

If overridden in a subclass, the signature should be integrate(self,lower,upper,method=None,**kwargs). if method is anything other than None, it should fall back on this version. e.g. the following should be at the top of the overriding method:

if method is not None:
    return FunctionModel1D.integrate(self,lower,upper,method,**kwargs)

integrateCircular(lower, upper, *args, **kwargs)¶

Integrate this model on the 2D circle. This calls integrate() with the jacobian set appropriately assuming the model is the radial profile for an azimuthally symmetric 2D surface density.

$\int_l^u \! f(x) \, 2 \pi x \, dx$

If a jac keyword is provided, it is taken as an additional factor to multiply the circular jacobian.

integrateSpherical(lower, upper, *args, **kwargs)¶

Integrate this model on the 3D sphere. This calls integrate() with the jacobian set appropriately assuming the model is the radial profile for a spherically symmetric 3D density.

$\int_l^u \! f(x) \, 4 \pi x^2 \, dx$

If a jac keyword is provided, it is taken as an additional factor to multiply the circular jacobian.

inv(yval, *args, **kwargs)¶

Find the x value matching the requested y-value. The inverse is computed using root finders from the scipy.optimize module.

Parameters:	yval (float) – the output y-value at which to compute the inverse

Other args and kwargs are those appropriate for the chosen root-finder, except for the keyword method which can be a name of any of the root finders from scipy.optimize. method can also be a function that should take f(g(x),*args,**kwargs) and return the x value at which g(x) is 0.

The default method depends on the input arguments as follows:

inv(yval)

Uses scipy.optimize.newton()
inv(yval,x0)

Uses scipy.optimize.newton(), starting the search at x0
inv(yval,a,b)

Uses scipy.optimize.brentq(), searching the bracketing interval [a,b] for the lower and upper edges of the search range.

Returns:	the x-value at which the model equals the given yval

Examples

This finds the x value of the (very simple) function $y(x) = 4x+2$ at the point y=3

>>> from pymodelfit.builtins import LinearModel
>>> m = LinearModel(m=4,b=2)
>>> '%.2f'%m.inv(3)
'0.25'

These examples use Newton’s, Brent’s, and Ridder’s method, to find the inverse of :math`y(x) = x^2` for 2,9,and 16, respectively (i.e. they should give sqrt(2),3, and 4)

>>> from pymodelfit.builtins import QuadraticModel
>>> m = QuadraticModel()
>>> '%.2f'%m.inv(2,1)
'1.41'
>>> '%.2f'%m.inv(9,2,4)
'3.00'
>>> '%.2f'%m.inv(16,3,5,method='ridder')
'4.00'

All these methods require a guide for the range of x values to search. The first requires a guess, although the default guess of 0 will be assumed if the second argument is not present. (for this example, sqrt(2) and -sqrt(2) are both valid answers, so the default guess of 0 is ambiguous).

classmethod isVarnumModel()¶

Determines if the model represented by this class accepts a variable number of parameters (i.e. number of parameters is set when the object is created).

Returns:	True if this model has a variable number of parameters.

maximize(x0, method='fmin', **kwargs)¶

Finds a local maximum for the model.

Parameters:	x0 (float) – The location to start the search method (string) – Can be ‘fmin’ or ‘fmin_powell’, to use scipy.optimize.fmin and scipy.optimize.fmin_powell.

kwargs are passed into the method function

Returns:	a x value where the model is a local maximum

minimize(x0, method='fmin', **kwargs)¶

Finds a local minimum for the model.

Parameters:	x0 (float) – The location to start the search method (string) – Can be ‘fmin’ or ‘fmin_powell’, to use scipy.optimize.fmin and scipy.optimize.fmin_powell.

kwargs are passed into the method function

Returns:	a x value where the model is a local minimum

params None¶: A tuple of the parameter names. (read-only)

pardict None¶: A dictionary mapping parameter names to the associated values.

parvals None¶: The values of the parameters in the same order as params

pixelize(xorxl, xu=None, n=None, edge=False, sampling=None)¶

Generate a discretized version o the model. This method integrates over the model for a number of ranges to get a 1D “pixelized” version of the model.

Parameters:

xoroxl – If array, specifies the location of each of the pixels. If float, specifies the lower edge of the pixelized section.
xu (float) – Specifies the upper edge of the pixelized section. Ignored if xorxl is array-like.
n (int) – Specifies the number of pixels. Ignored if xorxl is array-like.
edge (bool) – If True, pixel locations are for the edges (xorxl lower edges, xu upper edge), or if False, they are pixel centers.
sampling (int or None) – If None, each pixel will be computed by integrating. Otherwise, the number of samples to use in each pixel.

Returns:

Integrated values for the function as an array of size n or matching xorxl.

plot(lower=None, upper=None, n=100, clf=True, logplot='', data='auto', errorbars=True, *args, **kwargs)¶

Plot the model function and possibly data and error bars with matplotlib.pyplot. The plot will reflect any changes applied with setCall().

Parameters:

lower (scalar or None) – The starting x value for the plot. If None, the bound will be inferred from the data argument to this function or the rangehint attribute of the model.
upper (scalar or None) – The ending x value for the plot. If None, the bound will be inferred from the data argument to this function or the rangehint attribute of the model.
n (int) – The number of samples for the plot
clf (boolean) – If True, the figure will be cleared before the plot is drawn.
logplot (‘’,’x’,’y’, or ‘xy’ string) – Sets which axes are logarithmic
data (‘auto’, None, or array like of the form (x,y) or (x,y,(xerr,yerr))) – Determines what (if any) data to display. If None, no data is displayed. If ‘auto’, data from the data attribute of the FunctionModel1D will be used if present, or nothing if the data attribute is empty or None. Otherwise, the data to be plotted (and possibly errors) can be provided as arrays. Or if a dictionary is given, it will be treated as keyword arguments to the matplotlib.pyplot.scatter() function (possibly with data as ‘xdata’,’ydata’,and ‘weights’ entries).
errorbars (boolean) – If True and weights are present in the data, error bars are shown on the data based on the interpretation given by the weightstype attribute. If False, no error bars are shown.

Additional arguments and keywords are passed into matplotlib.pyplot.plot().

Note

By default, the model is plotted over the data points. If the points should be drawn on top of the model, set the keyword argument zorder to 0.

plotResiduals(x=None, y=None, clf=True, logplot='', **kwargs)¶

Plots the residuals of the provided data ( $y-y_{mod}$ ) against this model.

Parameters:	x (array-like or None) – The x data to plot the residuals for, or None to get it from any fitted data. y (array-like or None) – The x data to plot the residuals for, or None to get it from any fitted data. clf (bool) – If True, the plot will be cleared first. logplot (‘’,’x’,’y’, or ‘xy’ string) – Sets which axes are logarithmic

Additional arguments and keywords are passed into matplotlib.pyplot.scatter().

resampleFit(x=None, y=None, xerr=None, yerr=None, bootstrap=False, modely=False, n=250, prefit=True, medianpars=False, plothist=False, **kwargs)¶

Estimates errors via resampling. Uses the fitData function to fit the function many times while either using the “bootstrap” technique (resampling w/replacement), monte carlo estimates for the error, or both to estimate the error in the fit.

Parameters:

x (array-like or None) – The input data - if None, will be taken from the data attribute.
y (array-like or None) – The output data - if None, will be taken from the data attribute.
xerr (array-like, callable, or None) – Errors for the input data (assumed to be normally distributed), or if None, will be taken from the data. Alternatively, it can be a function that accepts the input data as the first argument and returns corresponding monte carlo sampled values.
yerr (array-like, callable, or None) – Errors for the output data (assumed to be normally distributed), or if None, will be taken from the data. Alternatively, it can be a function that accepts the output data as the first argument and returns corresponding monte carlo sampled values.
bootstrap (bool) – If True, the data is also resampled (with replacement).
modely – If True, the fitting data will be generated by offseting from model values (evaluated at the x-values) instead of y-values.
n (int) – The number of times to draw samples.
prefit (bool) – If True, the data will be fit without resampling once before the samples are recorded.
medianpars (bool) – If True, the median from the histogram will be set as the value for the parameter.
plothist (bool or string) – If True, histograms will be plotted using matplotlib.pyplot.hist() for each of the parameters, or if it is a string, only the histogram for the requested parameter will be shown.

kwargs are passed into fitData

Returns:	(histd,cov) where histd is a dictionary mapping parameters to their histograms and cov is the covariance matrix of the parameters in parameter order.

Note

If x, y, xerr, or yerr are provided, they do not overwrite the stored data, unlike most other methods for this class.

residuals(x=None, y=None, retdata=False)¶

Compute residuals of the provided data against the model. E.g. $y-{\rm model}(x)$ .

Parameters:	x (array-like or None) – Input data value or None to use stored `data` y (array-like or None) – Output data value or None to use stored `data` retdata (bool) – If True, returns the data along with the model.
Returns:	Residuals of model from y or if retdata is True, a tuple (x,y,residuals).
Return type:	array-like

setCall(calltype=None, xtrans=None, ytrans=None, **kwargs)¶

Sets the type of function evaluation to occur when the model is called. Changes the output of a function call to be $t_y(g(f(t_x(x))))$ , where $g(x)$ is set by calltype, $t_x$ is given by the xtrans argument, and $t_y$ is given by ytrans.

Parameters:

calltype –
Specifies what should be output when the model is called. Can be:
- None
  
  basic function evaluation
- ‘derivative’
  
  derivative at the location (see derivative())
- ‘integrate’
  
  integral - specify upper or lower kwargs and the evaluation location will be treated as the other bound. If neither is given, lower = 0 is assumed. (see integrate())
- ‘integrateCircular’
  
  Polar integration – see integrateCircular()
- ‘integrateSpherical’
  
  Spherical integration – see integrateSpherical()
- Any other string that is the name of a method on this object. It will be called with the function value as its first argument.
xtrans (string or None) – Transformations applied to the input value before being passed into the model. See below for valid forms.
ytrans (string or None) – Transformations applied to the output value. See below for valid forms.

Any kwargs are passed into the function specified in calltype.

xtrans and ytrans transformation functions can accept the following values:

None

$t(x) = x$

‘log’

$t(x) = \log_{10}(x)$

‘ln’

$t(x) = \ln(x)$

‘log##.#’

$t(x) = \log_{\#\#.\#}(x)$

‘pow’

$t(x) = 10^x$

‘exp’

$t(x) = e^x$

‘pow##.#’

$t(x) = (\#\#.\#)^x$

Warning

There may be unintended consequences of this method due to methods using the call value instead of the default function evaluation result. You have been warned...

stdData(x=None, y=None)¶

Determines the standard deviation of the model from data. Data can either be provided or (by default) will be taken from the stored data.

Parameters:	x (array-like or None) – Input data value or None to use stored `data`. y (array-like or None) – Output data value or None to use stored `data`.
Returns:	standard deviation of model from y

weightstype None¶

Determines the statistical interpretation of the weights in data. Can be:

‘ierror’

Weights act as inverse errors (default)
‘ivar’

Weights act as inverse variance
‘error’

Weights act as errors (non-standard - this makes points with larger error bars count more towards the fit).
‘var’

Weights act as variance (non-standard - this makes points with larger error bars count more towards the fit).

pymodelfit.builtins.TwoPowerModel¶

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