 # Appendix - Introduction (in Python/Julia)

Copyright 2011 - 2022 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at: https://www.gnu.org/licenses/.

##### Listing P.1: Entering and Printing Data in Python Last updated June 2018

x = 10   # assign x the value 10
print(x) # print the value of x

##### Listing J.1: Entering and Printing Data in Julia Last updated June 2018

x = 10     # assign x the value 10
println(x) # print x
## println() puts next output on new line, while print() doesn't


##### Listing P.2: Vectors, Matrices and Sequences in Python Last updated July 2020

y = [1,3,5,7,9]      # lists in square brackets are stored as arrays
print(y)
print(y)          # 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

##### Listing J.2: Vectors, Matrices and Sequences in Julia Last updated July 2020

y = [1,3,5,7,9]      # lists in square brackets are stored as arrays
println(y)
println(y)        # calling 3rd element (Julia indices start at 1)
println(size(y))     # size of y
println(length(y))   # as expected, y has length 5
v = fill!(Matrix{Float64}(undef, 2,3),NaN) # 2x3 Float64 matrix of NaNs - computationally better
v = fill(NaN, (2,3))           # 2x3 Float64 matrix of NaNs - direct
println(v)           # Julia prints matrices in a single line
println(size(v))     # as expected, v is size (2,3)
w = repeat([1,2,3]', outer = [3,2])      # repeats matrix thrice by rows, twice by columns
println(w)
s = 1:10   # s is an sequence which one can loop across
println(collect(s))  # return sequence elements as an array


##### Listing P.3: Basic Summary Statistics in Python Last updated July 2020

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 =  for single vector
print(np.sort(y))     # sort in ascending order
print(np.log(y))     # natural log

##### Listing J.3: Basic Summary Statistics in Julia Last updated July 2020

y = [3.14,15,9.26,5]
using Statistics;    # load package needed
println("sum: ", sum(y))       # return sum of all elements of y
println("product: ", prod(y))   # return product of all elements of y
println("max: ", maximum(y))    # return maximum value of y
println("min: ", minimum(y))    # return minimum value of y
println("mean: ", mean(y))     # arithmetic mean
println("median: ", median(y))  # median
println("variance: ", var(y))   # variance
println("cov_matrix: ", cov(y)) # covar matrix = variance for single vector
println("cor_matrix: ", cor(y)) # corr matrix =  for single vector
println(sort(y))     # sorts y in ascending order
println(log.(y))     # natural log, note . denotes elementwise operation


##### Listing P.4: Calculating Moments in Python Last updated June 2018

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

##### Listing J.4: Calculating Moments in Julia Last updated June 2018

using StatsBase;
println("mean: ", mean(y))     # mean
println("variance: ", var(y))  # variance
println("std dev: ", std(y))   # unbiased standard deviation
println("skewness: ", skewness(y)) # skewness
println("kurtosis: ", kurtosis(y)) # EXCESS kurtosis (note the different default)


##### Listing P.5: Basic Matrix Operations in Python Last updated June 2018

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
## Note: z * x is undefined since the two matrices are not conformable
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)
## note: dimensions must match along the combining axis

##### Listing J.5: Basic Matrix Operations in Julia Last updated June 2018

z = Matrix([[1 2];[3 4]]) # z is a 2 x 2 matrix
x = Matrix([1 2])    # x is a 1 x 2 matrix
## Note: z * x is undefined since the two matrices are not conformable
println(z * x')      # this evaluates to a 2 x 1 matrix
b = vcat(z,x)        # "stacking" z and x vertically
c = hcat(z,x')       # "stacking" z and x' horizontally
## Note: dimensions must match along the combining axis


##### Listing P.6: Statistical Distributions in Python Last updated July 2020

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)
## One can also obtain pseudorandom samples from distributions using numpy.random
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
## Given data, we obtain MLE estimates of parameters with stats:
res = stats.norm.fit(x)        # Fitting x to normal dist
print(res)           # First element is mean, second sd

##### Listing J.6: Statistical Distributions in Julia Last updated July 2020

## Julia has a wide range of functions contained in the package Distributions.jl
## Vectorized versions of the functions are used here as they are relevant for FRF
using Distributions;
q = collect((-3:1:3))          # specify a set of values
p = collect((0.1:0.1:0.9))     # specify a set of probabilities
println(quantile.(Normal(0,1),p)) # element-wise inverse Normal quantile
println(cdf.(TDist(4), q))     # element-wise cdf calculation under Student-t(4)
println(pdf.(Chisq(2), q))     # element-wise pdf calculation under Chisq(2)
## Similar syntax for other dists, e.g. Bernoulli(p), Binomial(n,p), Poisson(λ)
## For full list of supported distributions, see Distributions.jl documentation
## One can also obtain pseudorandom samples from distributions using rand()
x = rand(TDist(5), 100)        # Sampling 100 times from TDist with 5 df
y = rand(Normal(0,1), 50)      # Sampling 50 times from a standard normal
## Given data, we obtain MLE estimates of parameters with fit_mle():
fit_mle(Normal, x)   # Fitting x to normal dist
## Some distributions like the Student-t cannot be fitted yet (as of July 2020)
## Supported dists: https://juliastats.org/Distributions.jl/stable/fit/


##### Listing P.7: Statistical Tests in Python Last updated July 2020

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

##### Listing J.7: Statistical Tests in Julia Last updated July 2020

Random.seed!(100)    # set random seed
x = rand(TDist(5), 500)        # create hypothetical dataset x
println(JarqueBeraTest(x))     # Jarque-Bera test for normality
println(LjungBoxTest(x,20))    # Ljung-Box test for serial correlation


##### Listing P.8: Time Series in Python Last updated June 2018

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()

##### Listing J.8: Time Series in Julia Last updated July 2020

using Plots;
Random.seed!(100)
x = rand(TDist(5), 60)  # Create hypothetical dataset x
acf = autocor(x, 1:20)  # autocorrelation for lags 1:20
pacf = autocor(x, 1:20) # partial autocorrelation for lags 1:20
## Plotting using Plots.jl
plot(bar(acf, title = "Autocorrelation", legend = false), bar(pacf, title = "Partial autocorrelation", legend = false))


##### Listing P.9: Loops and Functions in Python Last updated June 2018

## For loops
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)
## If-else loops
X = 10
if X % 3 == 0:
print("X is a multiple of 3")
else:
print("X is not a multiple of 3")
## Functions (example: a simple excess kurtosis function)
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))

##### Listing J.9: Loops and Functions in Julia Last updated July 2020

## For loops
for i in range(3,length = 5)   # using range with the "length" option
println(i^2)     # where n = number of terms
end    # this iterates over [3,4,5,6,7]
## If-else loops
X = 10
if X % 3 == 0
println("X is a multiple of 3")
else
println("X is not a multiple of 3")
end
## Functions (example: a simple excess kurtosis function)
using Statistics;
function excess_kurtosis(x, excess = 3)::Float64 # excess optional, default = 3
m4 = mean((x .- mean(x)).^4)         # element-wise exponentiation .^
excess_kurt = m4/(std(x)^4) - excess
return excess_kurt
end
using Random, Distributions;
Random.seed!(100)
x = rand(TDist(5), 60)         # Create hypothetical dataset x
excess_kurtosis(x)
## Note: we have forced output to be of type Float64 by the type declaration above


##### Listing P.10: Basic Graphs in Python Last updated July 2020

y = np.random.normal(size = 50)
z = np.random.standard_t(df = 4, size = 50)
## using Matplotlib to plot bar, line, histogram and scatter plots
## subplot(a,b,c) creates a axb grid and plots the next plot in position c
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);

##### Listing J.10: Basic Graphs in Julia Last updated July 2020

## For the simple plots in FRF we use Plots.jl
## Full documentation at http://docs.juliaplots.org/latest/
## By default, Plots.jl uses the GR backend, sufficient for plots done in FRF
## Alternative backends are also available, e.g. Plotly, PlotlyJS
y = rand(Normal(0,1), 50)
## Plotting barplot, lineplot, histogram, scatterplot of y
return plot(bar(y, title = "Bar plot"), plot(y, title = "Line plot"),
histogram(y, title = "Histogram"), scatter(y, title = "Scatter plot"), legend = false)
## Wrapping plot(...) around multiple plots allows for automatic subplotting
## Options in wrapped plot(...) apply to all subplots
## Plot individual graphs using histogram(y), bar(y), etc. directly
## More examples using GR (plus syntax for customizations) can be found online:
## https://docs.juliaplots.org/latest/generated/gr/


##### Listing P.11: Miscellaneous Useful Functions in Python Last updated June 2018

## Convert objects from one type to another with int(), float() etc
## To check type, use type(object)
x = 8.0
print(type(x))
x = int(x)
print(type(x))

##### Listing J.11: Miscellaneous Useful Functions in Julia Last updated July 2020

## 1) To convert objects from one type to another, use convert(Type, object)
##    To check type, use typeof(object)
x = 8.0
println(typeof(x))
x = convert(Int, 8.0)
println(typeof(x))
## 2) To type Greek letters, type \ + name + Tab in succession
##    e.g. \gammaTab gives you γ and \GammaTab gives you Γ
##
##    Greek letters are sometimes essential in retrieving parameters from functions
##    e.g. res = fit_mle(Normal, x) will return an object res of type Distribution
##    with fitted parameters res.μ and res.σ
y = rand(Normal(0,1), 100)
res = fit_mle(Normal, y)
println("Fitted mean: ", res.μ)
println("Fitted sd: ", res.σ)