# Welcome to squmfit’s documentation¶

This is the documentation for squmfit (pronounced “squim-fit”), a Python library for convenient non-linear least-squares fitting of multiple analytical models to data. This library is similar to the excellent lmfit package but is a fresh implementation designed with support for global fitting of multiple curves with parameter tying.

You may want to start at Getting started with squmfit or go directly to the squmfit module documentation.

Contents:

## Overview¶

squmfit is a general-purpose library for non-linear least-squared curve fitting. The library is designed to enable modular models which can be easily composed to describe complex data sets. Moreover, the library supports the ability to simultaneously fit multiple data sets, each with its own model, over a common set of parameters.

In contrast to lmfit, squmfit treats the free parameters of the the fit as first-class objects. This allows models to be built up, added, multiplied, and generally treated as standard Python expressions. Moreover, no assumptions are imposed regarding how parameters interact, allowing fitting of a common set of parameters over several data sets.

### A simple example¶

Let’s say that we have a exponential decay with Poissonian noise,

>>> import numpy as np
>>> noise_std = 0.1
>>> xs = np.arange(4000)
>>> ys = np.random.poisson(400 * np.exp(-xs / 400.))


We first define the functional form which we believe describes our data,

>>> import squmfit
>>> def exponential_model(t, amplitude, rate):
>>>     return amplitude * np.exp(-t * rate)


We then create a Fit object which we will use to define the parameters and objective of our fit,

>>> fit = squmfit.Fit()


Say we want to fit this model to our generated ys, allowing both the amplitude and rate parameters to vary. We first need to tell squmfit about these parameters,

>>> amp = fit.param('amp', initial=100)
>>> tau = fit.param('tau', initial=100)


Now we can use amp and tau as normal Python variables,

>>> model = exponential_model(xs, amp, 1. / tau)


Alternatively we could simply do away with the function altogether,

>>> model = amp * np.exp(-t / tau)


Note how we can write expressions involving parameters with the usual Python arithmetic operations, such as 1. / tau, greatly simplifying model composition.

Next we add our curve to our Fit, specifying the model to which we wish to fit along with some weights (taking care to avoid division-by-zero, of course),

>>> weights = np.zeros_like(ys, dtype='f')
>>> weights[ys > 0] = 1 / np.sqrt(ys[ys > 0])
>>> fit.add_curve('curve1', model, ys, weights=weights)


Finally we can run our fit and poke around at the results,

>>> res = fit.fit()
>>> print res.params
{'amp': 403.01725751512635, 'tau': 393.19866908823133}
>>> print res.curves['curve1'].reduced_chi_sqr
0.949579885697


res is a FitResult object, which contains a variety of information about the fit, including the residuals, covariances, and fit values.

squmfit has a variety of options for presenting the results of a fit. squmfit.pretty has several utilities for producing a quantitative summary of the fit. For instance, squmfit.pretty.markdown_fit_result() will produce a Markdown document describing the fit parameters and various goodness-of-fit metrics. If you use IPython Notebook, squmfit.pretty.ipynb_fit_result() can be used to generate a presentation that can be rendered in rich HTML within the notebook.

Finally, squmfit.plot.plot_fit() can be used to plot fits and residuals.

### How does it work?¶

Expressions in squmfit are represented by the Expr class. An Expr captures an abstract syntax tree that describes how a result can be computed. The elementary expressions could represent a fitted parameter (e.g. amp in the example above), or an expression depending upon a fitted parameter. The Expr class implements basic arithmetic operations (e.g. __add__) and numpy’s ufuncs (e.g. sqrt), allowing it to be treated as a scalar or an array.

Of course, some functions require more structure beyond the operations supported by Expr evaluate their result. In this case, you can tell squmfit that you want the function arguments to be evaluated before they are provided to your function with the model() decorator,

>>> @squmfit.model
>>> def sum_odds(vec):
>>>     return vec[1::2].sum()


In this case, we could invoke sum_odds with an Expr, which squmfit would automatically evaluate. It would then evaluate sum_odds with the value of the expression, and pack the result back into an Expr.

squmfit.model(func)[source]

Transforms a function to ensure that all of its parameters are evaluated. Calls to the transformed function will result in an Expr when any of its parameters are Exprs.

## Fitting¶

Fitting begins with the Fit class which is used to define the data sets and models to be fit and ultimately

class squmfit.Fit[source]

This represents a fit configuration.

## Inspecting fit results¶

class squmfit.FitResult(fit, params, covar_p=None, initial_result=None)[source]

This embodies a set of parameter values, possibly originating from a fit.

## Functional interpretation¶

The design of squmfit is largely inspired by patterns introduced to the author by the functional programming community. If one is so inclined it can be useful to think of Expr as an reader applicative having access to an environment of packed parameter values. Composition is standard function composition. Pure values can be lifted with the squmfit.expr.Constant class although this is rarely explicitly needed as pure values are automatically lifted by the operations of Expr. The Expr.map() function provides functorial map. Functions can also be lifted with the model() decorator.