Appendix - Introduction

x = 10;             % assign x the value 10, silencing output print with ;
disp(x)             % display x

y = [1,3,5,7,9]            % lists are denoted by square brackets
y(3)                       % calling 3rd element (MATLAB indices start at 1)
size(y)                    % shows that y is 1 x 5 (a row vector, by default)
length(y)                  % as expected, y has length 5
v = nan(2,3)               % fill a 2 x 3 matrix with NaN values
size(v)                    % as expected, v is size (2,3)
w = repmat([1,2,3]', 2, 3) % repeats matrix twice by rows, thrice by columns
s = 1:10                   % s is a list of integers from 1 to 10 inclusive

y = [1,3,5,7,9]       # lists in square brackets are stored as arrays
print(y)
print(y[2])           # 3rd element (Python indices start at 0)
print(len(y))         # as expected, y has length 5
import numpy as np    # NumPy: Numeric Python package
v = np.full([2,3], np.nan) # create a 2x3 matrix with NaN values
print(v)           
print(v.shape)        # as expected, v is size (2,3)
w = np.tile([1,2,3], (3,2)) # repeats thrice by rows, twice by columns
print(w)
s = range(10)         # an iterator from 0 to 9
print([x for x in s]) # return  elements using list comprehension

y = [3.14; 15; 9.26; 5];    % List with semicolons is a column vector
sum(y)         % sum of all elements of y
prod(y)        % product of all elements of y
max(y)         % maximum value of y
min(y)         % minimum value of y
range(y)       % difference between max and min value of y
mean(y)        % arithmetic mean
median(y)      % median
var(y)         % variance
cov(y)         % covar matrix = variance for single vector
corrcoef(y)    % corr matrix = [1] for single vector
sort(y)        % sorting in ascending order
log(y)         % natural log

import numpy as np
y = [3.14, 15, 9.26, 5]
print(sum(y))          # sum of all elements of y
print(max(y))          # maximum value of y
print(min(y))          # minimum value of y
print(np.mean(y))      # arithmetic mean
print(np.median(y))    # median
print(np.var(y))       # population variance
print(np.cov(y))       # covar matrix = sample variance for single vector
print(np.corrcoef(y))  # corr matrix = [1] for single vector
print(np.sort(y))      # sort in ascending order
print(np.log(y))       # natural log

mean(y)      % mean
var(y)       % variance
std(y)       % unbiased standard deviation, by default
skewness(y)  % skewness
kurtosis(y)  % kurtosis

from scipy import stats
print(np.mean(y))                        # mean
print(np.var(y))                         # variance
print(np.std(y, ddof = 1))               # ddof = 1 for unbiased standard deviation
print(stats.skew(y))                     # skewness
print(stats.kurtosis(y, fisher = False)) # fisher = False gives Pearson definition

z = [1, 2; 3, 4]  % z is a 2 x 2 matrix (Note the use of ; as row separator)
x = [1, 2]        % x is a 1 x 2 matrix
z * x'            % this evaluates to a 2 x 1 matrix
vertcat(z,x)      % "stacking" z and x vertically
horzcat(z,x')     % "stacking z and x' horizontally

z = np.matrix([[1, 2], [3, 4]])                   # z is a 2 x 2 matrix
x = np.matrix([1, 2])                             # x is a 1 x 2 matrix
print(z * np.transpose(x))                        # this evaluates to a 2 x 1 matrix          
b = np.concatenate((z,x), axis = 0)               # "stacking" z and x vertically
print(b)
c = np.concatenate((z,np.transpose(x)), axis = 1) # "stacking" z and x horizontally
print(c)

q = -3:1:3                       % specify a set of values
p = 0.1:0.1:0.9                  % specify a set of probabilities
norminv(p, 0, 1)                 % element-wise inverse Normal quantile
tcdf(q, 4)                       % element-wise cdf under Student-t(4)
chi2pdf(q, 2)                    % element-wise pdf under Chisq(2)
x = trnd(5, 100, 1);             % Sampling 100 times from t dist with 5 df
y = normrnd(0, 1, 100, 1);       % Sampling 50 times from a standard normal 
res = fitdist(x, "Normal")       % Fitting x to normal dist

q = np.arange(-3,4,1)        # specify a set of values, syntax arange(min, exclusive-max, step)
p = np.arange(0.1,1.0,0.1)   # specify a set of probabilities
print(stats.norm.ppf(p))     # element-wise inverse Normal quantile
print(stats.t.cdf(q,4))      # element-wise cdf under Student-t(4)
print(stats.chi2.pdf(q,2))   # element-wise pdf under Chisq(2)
x = np.random.standard_t(df=5, size=100)   # Sampling 100 times from TDist with 5 df
y = np.random.normal(size=50)              # Sampling 50 times from a standard normal 
res = stats.norm.fit(x)                    # Fitting x to normal dist
print(res)                                 # First element is mean, second sd

x = trnd(5, 500, 1);                    % Create hypothetical dataset x
[h1, p1, jbstat] = jbtest(x)            % Jarque-Bera test for normality
[h2, p2, lbstat] = lbqtest(x,'lags',20) % Ljung-Box test for serial correlation - Needs Econometrics Toolbox

from statsmodels.stats.diagnostic import acorr_ljungbox
x = np.random.standard_t(df=5, size=500)  # Create dataset x
print(stats.jarque_bera(x))               # Jarque-Bera test - prints statistic and p-value
print(acorr_ljungbox(x, lags=20))         # Ljung-Box test - prints array of statistics and p-values

x = trnd(5, 60, 1); % Create hypothetical dataset x
subplot(1,2,1)
autocorr(x, 20)     % autocorrelation for lags 1:20
subplot(1,2,2)
parcorr(x,20)       % partial autocorrelation for lags 1:20

import statsmodels.api as sm
import matplotlib.pyplot as plt
y = np.random.standard_t(df = 5, size = 60)   # Create hypothetical dataset y
q1 = sm.tsa.stattools.acf(y, nlags=20)        # autocorrelation for lags 1:20
plt.bar(x = np.arange(1,len(q1)), height = q1[1:])
plt.show()
plt.close()
q2 = sm.tsa.stattools.pacf(y, nlags=20)       # partial autocorr for lags 1:20
plt.bar(x = np.arange(1,len(q2)), height = q2[1:])
plt.show()
plt.close()

for i = 3:7        % iterates through [3,4,5,6,7]
    i^2  
end
X = 10;
if (rem(X,3) == 0)
    disp("X is a multiple of 3")
else 
    disp("X is not a multiple of 3")
end

for i in range(3,8):      # NOTE: range(start, end), end excluded
    print(i**2)           # range(3,8) iterates through [3,4,5,6,7)
X = 10
if X % 3 == 0:
    print("X is a multiple of 3")
else:
    print("X is not a multiple of 3")
def excess_kurtosis(x, excess = 3):        # note: excess optional, default = 3
    m4=np.mean((x-np.mean(x))**4)          # note: exponentiation in Python uses **
    excess_kurt=m4/(np.std(x)**4)-excess
    return excess_kurt
x = np.random.standard_t(df=5,size=60)     # Create hypothetical dataset x
print(excess_kurtosis(x))

y = normrnd(0, 1, 50, 1);
z = trnd(4, 50, 1);
subplot(2,2,1)
bar(y)            % bar plot
title("Bar plot")
subplot(2,2,2)
plot(y)           % line plot
title("Line plot")
subplot(2,2,3)
histogram(y)      % histogram
title("Histogram")
subplot(2,2,4)
scatter(y,z)      % scatter plot
title("Scatter plot")

y = np.random.normal(size = 50)
z = np.random.standard_t(df = 4, size = 50)
plt.subplot(2,2,1)
plt.bar(range(len(y)), y);
plt.subplot(2,2,2)
plt.plot(y);
plt.subplot(2,2,3)
plt.hist(y);
plt.subplot(2,2,4)
plt.scatter(y,z);

x = 8.0;
isfloat(x)
x = int8(x);
isinteger(x)

x = 8.0
print(type(x))
x = int(x)
print(type(x))