Would somebody be able to explain to me how to use the location parameter with the gamma.fit function in Scipy?
It seems to me that a location parameter (μ) changes the support of the distribution from x ≥ 0 to y = ( x - μ ) ≥ 0. If μ is positive then aren't we losing all the data which doesn't satisfy x - μ ≥ 0?
Thanks!
The fit function takes all of the data into consideration when finding a fit. Adding noise to your data will alter the fit parameters and can give a distribution that does not represent the data very well. So we have to be a bit clever when we are using fit.
Below is some code that generates data, y1, with loc=2 and scale=1 using numpy. It also adds noise to the data over the range 0 to 10 to create y2. Fitting y1 yield excellent results, but attempting to fit the noisy y2 is problematic. The noise we added smears out the distribution. However, we can also hold 1 or more parameters constant when fitting the data. In this case we pass floc=2 to the fit, which forces the location to be held at 2 when performing the fit, returning much better results.
from scipy.stats import gamma
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(0,10,.1)
y1 = np.random.gamma(shape=1, scale=1, size=1000) + 2 # sets loc = 2
y2 = np.hstack((y1, 10*np.random.rand(100))) # add noise from 0 to 10
# fit the distributions, get the PDF distribution using the parameters
shape1, loc1, scale1 = gamma.fit(y1)
g1 = gamma.pdf(x=x, a=shape1, loc=loc1, scale=scale1)
shape2, loc2, scale2 = gamma.fit(y2)
g2 = gamma.pdf(x=x, a=shape2, loc=loc2, scale=scale2)
# again fit the distribution, but force loc=2
shape3, loc3, scale3 = gamma.fit(y2, floc=2)
g3 = gamma.pdf(x=x, a=shape3, loc=loc3, scale=scale3)
And make some plots...
# plot the distributions and fits. to lazy to do iteration today
fig, axes = plt.subplots(1, 3, figsize=(13,4))
ax = axes[0]
ax.hist(y1, bins=40, normed=True);
ax.plot(x, g1, 'r-', linewidth=6, alpha=.6)
ax.annotate(s='shape = %.3f\nloc = %.3f\nscale = %.3f' %(shape1, loc1, scale1), xy=(6,.2))
ax.set_title('gamma fit')
ax = axes[1]
ax.hist(y2, bins=40, normed=True);
ax.plot(x, g2, 'r-', linewidth=6, alpha=.6)
ax.annotate(s='shape = %.3f\nloc = %.3f\nscale = %.3f' %(shape2, loc2, scale2), xy=(6,.2))
ax.set_title('gamma fit with noise')
ax = axes[2]
ax.hist(y2, bins=40, normed=True);
ax.plot(x, g3, 'r-', linewidth=6, alpha=.6)
ax.annotate(s='shape = %.3f\nloc = %.3f\nscale = %.3f' %(shape3, loc3, scale3), xy=(6,.2))
ax.set_title('gamma fit w/ noise, location forced')
