Statistics Tools¶
NOTE: In this notebook I use thestats submodule of scipy for all statistics functions, including generation of random numbers. There are other modules with some overlapping functionality, e.g., the regular python random module, and the np.random module, but I do not use them here. The stats submodule includes tools for a large number of distributions, it includes a large and growing set of statistical functions, and there is a unified class structure. In addition, potential namespace issues are minimized. See https://docs.scipy.org/doc/scipy/reference/stats.html.
Marty Ligare, August 2020
import numpy as np
from scipy import stats
import matplotlib as mpl
import matplotlib.pyplot as plt
# Following is an Ipython magic command that puts figures in the notebook.
%matplotlib notebook
# Following set up LateX fonts
#mpl.rc('font', **{'family': 'serif', 'serif': ['Computer Modern']})
#mpl.rc('text', usetex=True)
# M.L. modification of matplotlib defaults
# Changes can also be put in matplotlibrc file,
# or effected using mpl.rcParams[]
plt.style.use('classic')
plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures
plt.rc('axes', labelsize=16, titlesize=14)
plt.rc('xtick', labelsize=12)
plt.rc('ytick', labelsize=12)
plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size
Generating random integers¶
# Generate n integers between low and high:
low = -3
high = 6
n = 100
# or equivalently:
# loq, high, n = (-3, 6, 100)
stats.randint.rvs(low, high+1, size=n)
Sampling random numbers from a uniform p.d.f.¶
# Sample n random numbers in interval [0.0,1.0):
n = 10
stats.uniform.rvs(size=10)
Sampling random numbers from a normal distribution¶
Sample $n$ random numbers from the normal distribution with mean $\mu$, standard deviation $\sigma$, and pdf of Eq.(2.4) of Hughes & Hase: \begin{equation} p(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(x-\mu)^2}{\sigma^2}\right] \end{equation}
# Sampling from normal distribution
n = 10
mean = 10.
sigma = 2.
stats.norm.rvs(mean, sigma, size=n)
Graph the pdf of the normal distribution.¶
plt.figure()
x = np.linspace(mean-3.*sigma, mean+3.*sigma,200)
y = stats.norm.pdf(x, mean, sigma)
plt.title("pdf of normal distribution")
plt.xlabel("$x$")
plt.ylabel("$p(x)$")
plt.grid()
plt.plot(x, y);
Graph of the cdf of normal distribution¶
plt.figure()
x = np.linspace(mean-3.*sigma, mean+3.*sigma, 200)
y = stats.norm.cdf(x, mean, sigma)
plt.title("cdf of normal distribution")
plt.xlabel("$x$")
plt.ylabel("$\int_{-\infty}^x p(x^\prime)\, dx^\prime$")
plt.grid()
plt.plot(x, y);
Sampling random numbers from a Poisson distribution¶
Sample $n$ random numbers from the Poisson distribution with average count $\overline{N}$, and probability distibution given by Eq.(3.1) of Hughes & Hase: \begin{equation} p(N;\overline{N}) = \frac{\exp\left(-\overline{N}\right)\overline{N}^N}{N!} \end{equation} The standard deviation of the Poisson distribution is given by $$ \sigma = \sqrt{\overline{N}}. $$
# Sampling from a Poisson distribution
n = 100
mean = 5
stats.poisson.rvs(mean, size=n)
np.mean(_) # The underscore "_" is similar to Mathematicas "%"
# It refers to the output of the previously executed cell
np.std(__) # Notice the double underscore "__"
np.sqrt(mean)
Graph of pmf ($\sim$pdf) of Poisson distribution¶
plt.figure()
x = np.linspace(0, 12, 13)
y = stats.poisson.pmf(x, mean)
plt.title("pmf of Poisson distribution")
plt.xlabel("$n$")
plt.ylabel("$p(n)$")
plt.grid()
plt.axhline(0)
plt.scatter(x, y);
Graph of the cdf of Poisson distribution¶
plt.figure()
x = np.linspace(0, 12, 13)
y = stats.poisson.cdf(x, mean)
plt.title("cdf of Poisson distribution")
plt.xlabel("$x$")
plt.ylabel("$C_{DF}$")
plt.xlim(-1, 13)
plt.grid()
plt.axhline(0)
plt.scatter(x, y);
Sampling random numbers from a binomial distribution¶
Consider $n$ trials, with probability of success $p$ in each trial. The array below is the number successes in each of $size$ trials.
# Sampling from a binomial distribution
n = 2
p = 0.4
stats.binom.rvs(n, p, size=100)
np.mean(_)
The probablity of getting $x$ successes is given by the probability mass function (pmf). This is analogous to the continous pdf (and it's called the PDF in Mathematica). As an example, the probability of 2 successes in 3 trials with a probability of success in each trial of 0.4 is 29%:
n, s, p = (3, 2, 0.4)
stats.binom.pmf(s, n, p)
Graph of pmf (~pdf) of binomial distribution¶
plt.figure()
n = 5
x = np.linspace(0, n, n+1)
y = stats.binom.pmf(x, n, p)
plt.title("pmf ($\sim$pdf) of binom. dist.; $n=5$, $p = 0.4$")
plt.xlabel("$n$")
plt.ylabel("probability of $n$ successes")
plt.grid()
plt.axhline(0)
plt.scatter(x, y);
Graph of cdf of binomial distribution¶
plt.figure()
n = 5
x = np.linspace(0, n, n+1)
y = stats.binom.cdf(x, n, p)
plt.title("cdf of binomial dist.; $n=5$, $p = 0.4$")
plt.xlabel("$n$")
plt.ylabel("cdf")
plt.grid()
plt.axhline(0)
plt.scatter(x, y);
Histograms¶
Generate some random data from a normal distribution.¶
n = 100
mean = 10.
sigma = 2.
data = stats.norm.rvs(mean, sigma, size=n)
Select number of bins between low and high values.¶
NOTE: plt.hist plots the histogram, and ouputs the binned data
plt.figure()
nbins = 6
low = mean - 3*sigma
high = mean + 3*sigma
plt.xlabel("value")
plt.ylabel("occurences")
plt.title("Histogram; equal sized bins")
out = plt.hist(data, nbins, [low,high])
out[0],out[1] # occurrences and bin boundaries
OR ... Select specific bin boundaries¶
Again, plt.hist outputs the binned data and plots the histogram.
plt.figure()
bins = [4, 7, 8, 10, 13, 16]
plt.xlabel("value")
plt.ylabel("occurences")
plt.title("Histogram; specified (nonuniform) bins")
out = plt.hist(data, bins)
out[0],out[1] # occurrences and bin boundaries
Version Information¶
version_information is from J.R. Johansson (jrjohansson at gmail.com); see Introduction to scientific computing with Python for more information and instructions for package installation.
version_information is installed on the linux network at Bucknell
%load_ext version_information
version_information numpy, scipy, matplotlib