# 3. numkit.fitting — Fitting data¶

The module contains functions to do least square fits of functions of one variable f(x) to data points (x,y).

## 3.1. Example¶

For example, to fit a un-normalized Gaussian with FitGauss to data distributed with mean 5.0 and standard deviation 3.0:

from numkit.fitting import FitGauss
import numpy, numpy.random

# generate suitably noisy data
mu, sigma = 5.0, 3.0
Y,edges = numpy.histogram(sigma*numpy.random.randn(10000), bins=100, density=True)
X = 0.5*(edges[1:]+edges[:-1]) + mu

g = FitGauss(X, Y)

print(g.parameters)
# [ 4.98084541  3.00044102  1.00069061]
print(numpy.array([mu, sigma, 1]) - g.parameters)
# [ 0.01915459 -0.00044102 -0.00069061]

import matplotlib.pyplot as plt
plt.plot(X, Y, 'ko', label="data")
plt.plot(X, g.fit(X), 'r-', label="fit")


If the initial parameters for the least square optimization do not lead to a solution then one can provide customized starting values in the parameters keyword argument:

g = FitGauss(X, Y, parameters=[10, 1, 1])


The parameters have different meaning for the different fit functions; the documentation for each function shows them in the context of the fit function.

## 3.2. Creating new fit functions¶

New fit function classes can be derived from FitFunc. The documentation and the methods FitFunc.f_factory() and FitFunc.initial_values() must be overriden. For example, the class FitGauss is implemented as

class FitGauss(FitFunc):
'''y = f(x) = p[2] * 1/sqrt(2*pi*p[1]**2) * exp(-(x-p[0])**2/(2*p[1]**2))'''
def f_factory(self):
def fitfunc(p,x):
return p[2] * 1.0/(p[1]*numpy.sqrt(2*numpy.pi)) * numpy.exp(-(x-p[0])**2/(2*p[1]**2))
return fitfunc
def initial_values(self):
return [0.0,1.0,0.0]


The function to be fitted is defined in fitfunc(). The parameters are accessed as p[0], p[1], … For each parameter, a suitable initial value must be provided.

## 3.3. Functions and classes¶

numkit.fitting.Pearson_r(x, y)[source]

Pearson’s r (correlation coefficient).

Pearson(x,y) –> correlation coefficient

x and y are arrays of same length.

Historical note – Naive implementation of Pearson’s r ::
Ex = scipy.stats.mean(x) Ey = scipy.stats.mean(y) covxy = numpy.sum((x-Ex)*(y-Ey)) r = covxy/math.sqrt(numpy.sum((x-Ex)**2)*numpy.sum((y-Ey)**2))
numkit.fitting.linfit(x, y, dy=None)[source]

Fit a straight line y = a + bx to the data in x and y.

Errors on y should be provided in dy in order to assess the goodness of the fit and derive errors on the parameters.

linfit(x,y[,dy]) –> result_dict

Fit y = a + bx to the data in x and y by analytically minimizing chi^2. dy holds the standard deviations of the individual y_i. If dy is not given, they are assumed to be constant (note that in this case Q is set to 1 and it is meaningless and chi2 is normalised to unit standard deviation on all points!).

Returns the parameters a and b, their uncertainties sigma_a and sigma_b, and their correlation coefficient r_ab; it also returns the chi-squared statistic and the goodness-of-fit probability Q (that the fit would have chi^2 this large or larger; Q < 10^-2 indicates that the model is bad — Q is the probability that a value of chi-square as _poor_ as the calculated statistic chi2 should occur by chance.)

Returns: result_dict with components intercept, sigma_intercept a +/- sigma_a slope, sigma_slope b +/- sigma_b parameter_correlation correlation coefficient r_ab between a and b chi_square chi^2 test statistic Q_fit goodness-of-fit probability

Based on ‘Numerical Recipes in C’, Ch 15.2.

class numkit.fitting.FitFunc(x, y, parameters=None)[source]

Fit a function f to data (x,y) using the method of least squares.

The function is fitted when the object is created, using scipy.optimize.leastsq(). One must derive from the base class FitFunc and override the FitFunc.f_factory() (including the definition of an appropriate local fitfunc() function) and FitFunc.initial_values() appropriately. See the examples for a linear fit FitLin, a 1-parameter exponential fit FitExp, or a 3-parameter double exponential fit FitExp2.

The object provides two attributes
FitFunc.parameters
list of parameters of the fit
FitFunc.message
message from scipy.optimize.leastsq()

After a successful fit, the fitted function can be applied to any data (a 1D-numpy array) with FitFunc.fit().

f_factory()[source]

Stub for fit function factory, which returns the fit function. Override for derived classes.

fit(x)[source]

Applies the fit to all x values

initial_values()[source]

List of initital guesses for all parameters p[]

class numkit.fitting.FitLin(x, y, parameters=None)[source]

y = f(x) = p[0]*x + p[1]

class numkit.fitting.FitExp(x, y, parameters=None)[source]

y = f(x) = exp(-p[0]*x)

class numkit.fitting.FitExp2(x, y, parameters=None)[source]

y = f(x) = p[0]*exp(-p[1]*x) + (1-p[0])*exp(-p[2]*x)

class numkit.fitting.FitGauss(x, y, parameters=None)[source]

y = f(x) = p[2] * 1/sqrt(2*pi*p[1]**2) * exp(-(x-p[0])**2/(2*p[1]**2))

Fits an un-normalized gaussian (height scaled with parameter p[2]).

• p[0] == mean $mu$
• p[1] == standard deviation $sigma$
• p[2] == scale $a$