Python, MATLAB, Julia, R code: Chapter 1

Intro to Probability for Data Science

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Acknowledgement: The Julia code is written by the contributors listed here.

Acknowledgement: The R code is written by contributors listed here.

Chapter 1.1

Visualizing a geometric series

p = 1/2;
n = 1:10;
X = p.^n;
bar(n,X,'FaceColor',[0.8, 0.2,0.2]);

import numpy as np
import matplotlib.pyplot as plt
p = 1/2
n = np.arange(1,10)
X = np.power(p,n)
plt.bar(n,X)

using StatsPlots:bar
p = 0.5
n = 1:10
X = p .^ n
bar(n, X, legend=false)

p <- 1/2
n <- seq(1, 10)
X <- p^n
barplot(X, names.arg=1:10)

Compute N choose K

{N choose K}

n = 10;
k = 2;
nchoosek(n,k)
factorial(k)

from scipy.special import comb, factorial
n = 10
k = 2
comb(n, k)
factorial(k)

n = 10
k = 2
binomial(n, k)
factorial(k)

n <- 10
k <- 2
choose(n,k)
factorial(k)

Chapter 1.4

Inner product of two vectors

$mathbf{x}^Tmathbf{y}$

x = [1 0 -1]';
y = [3 2 0]';
z = x'*y;

import numpy as np
x = np.array([1,0,-1])
y = np.array([3,2,0])
z = np.dot(x,y)
print(z)

x = [1, 0, -1]
y = [3, 2, 0]
x'y

x <- c(1,0,-1)
y <- c(3,2,0)
z <- t(x) %*% y
print(z)

Norm of a vector

|mathbf{x}|

x = [1 0 -1];
x_norm = norm(x);

import numpy as np
x = np.array([1,0,-1])
x_norm = np.linalg.norm(x)

using LinearAlgebra:norm
x = [1, 0, -1]
norm(x)

x <- c(1,0,-1)
x_norm <- norm(x, type = "2")

print(x_norm)

Weighted norm of a vector

$|mathbf{x}|_{mathbf{W}}^2$

W = [1 2 3; 4 5 6; 7 8 9];
x = [2; -1; 1];
z = x'*W*x

import numpy as np
W = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
x = np.array([[2],[-1],[1]])
z = np.dot(x.T, np.dot(W,x))
print(z)

W = [1. 2. 3.; 4. 5. 6.; 7. 8. 9.]
x = [2, -1, 1]
z = x'W*x

W <- matrix(1:9, ncol = 3, nrow = 3)
x <- matrix(c(2,-1,1))
z <- t(x) %*% (W %*% x)
print(z)

Matrix inverse

$(mathbf{X}^Tmathbf{X})^{-1}$

X = [1 3; -2 7; 0 1]
XtXinv = inv(X'X)

import numpy as np
X      = np.array([[1, 3], [-2, 7], [0, 1]])
XtX    = np.dot(X.T, X)
XtXinv = np.linalg.inv(XtX)
print(XtXinv)

X = [1 3; -2 7; 0 1]
XtXinv = inv(X'X)

X <- matrix(c(1,-2,0,3,7,1), nrow = 3, ncol = 2)
XtX <- t(X) %*% X
Xtxinv <- solve(XtX)
print(Xtxinv)

System of linear equation

$beta = (mathbf{X}^Tmathbf{X})^{-1}mathbf{X}^Tmathbf{y}$

X      = [1 3; -2 7; 0 1];
y      = [2; 1; 0];
beta   = X\y;

import numpy as np
X      = np.array([[1, 3], [-2, 7], [0, 1]])
y      = np.array([[2],[1],[0]])
beta   = np.linalg.lstsq(X, y, rcond=None)[0]
print(beta)

X = [1. 3.; -2. 7.; 0. 1.]
y = [2., 1., 0.]
beta = X\y

X <- matrix(c(1,-2,0,3,7,1), nrow = 3, ncol = 2)
y <- matrix(c(2,1,0))
beta <- solve(qr(X), y)
print(beta)