 Appendix - Introduction (in R/Julia)

Copyright 2011 - 2020 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 R.1: Entering and Printing Data in R Last updated June 2018

x = 10   # assign x the value 10
print(x) # print 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 R.2: Vectors, Matrices and Sequences in R Last updated June 2018

y = c(1,3,5,7,9)     # create vector using c()
print(y)
print(y)          # calling 3rd element (R indices start at 1)
print(dim(y))        # gives NULL since y is a vector, not a matrix
print(length(y))     # as expected, y has length 5
v = matrix(nrow=2,ncol=3)      # fill a 2 x 3 matrix with NaN values (default)
print(dim(v))        # as expected, v is size (2,3)
w = matrix(c(1,2,3),nrow=6,ncol=3) # repeats matrix twice by rows, thrice by columns
print(w)
s = 1:10   # s is a list of integers from 1 to 10 inclusive
print(s)

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 R.3: Basic Summary Statistics in R Last updated June 2018

y=matrix(c(3.1,4.15,9))
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)  # min, max value of y
mean(y)   # arithmetic mean
median(y) # median
var(y)    # variance
cov(y)    # covar matrix = variance for single vector
cor(y)    # corr matrix =  for single vector
sort(y)   # sorting in ascending order
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 R.4: Calculating Moments in R Last updated June 2018

library(moments)
mean(y)     # mean
var(y)     # variance
sd(y)      # unbiased standard deviation, by default
skewness(y) # skewness
kurtosis(y) # kurtosis

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 R.5: Basic Matrix Operations in R Last updated June 2018

z = matrix(c(1,2,3,4),2,2) # z is a 2 x 2 matrix
x = matrix(c(1,2),1,2)     # x is a 1 x 2 matrix
## Note: z * x is undefined since the two matrices are not conformable
z %*% t(x)           # this evaluates to a 2 x 1 matrix
rbind(z,x)           # "stacking" z and x vertically
cbind(z,t(x))        # "stacking z and x' horizontally
## 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 R.6: Statistical Distributions in R Last updated June 2018

q = seq(from = -3, to = 3, length = 7)    # specify a set of values
p = seq(from = 0.1, to = 0.9, length = 9) # specify a set of probabilities
qnorm(p, mean = 0, sd = 1)     # element-wise inverse Normal quantile
pt(q, df = 4)        # element-wise cdf under Student-t(4)
dchisq(q, df = 2)    # element-wise pdf under Chisq(2)
## Similar syntax for other distributions
## q for quantile, p for cdf, d for pdf
## followed by the abbreviation of the distribution
## One can also obtain pseudorandom samples from distributions
x = rt(100, df = 5)  # Sampling 100 times from TDist with 5 df
y = rnorm(50, mean = 0, sd = 1)          # Sampling 50 times from a standard normal
## Given data, we obtain MLE estimates of distribution parameters with package MASS:
library(MASS)
res = fitdistr(x, densfun = "normal")     # Fitting x to normal dist
print(res)

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 R.7: Statistical Tests in R Last updated June 2018

library(tseries)
x = rt(500, df = 5)  # Create hypothetical dataset x
jarque.bera.test(x)  # Jarque-Bera test for normality
Box.test(x, lag = 20, type = c("Ljung-Box")) # Ljung-Box test for serial correlation

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 R.8: Time Series in R Last updated June 2018

x = rt(60, df = 5) # Create hypothetical dataset x
par(mfrow=c(1,2), pty='s')
acf(x,20)  # autocorrelation for lags 1:20
pacf(x,20)           # partial autocorrelation for lags 1:20

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 R.9: Loops and Functions in R Last updated June 2018

## For loops
for (i in 3:7)       # iterates through [3,4,5,6,7]
print(i^2)
## 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)
excess_kurtosis = function(x, excess = 3){ # note: excess optional, default=3
m4 = mean((x-mean(x))^4)
excess_kurt = m4/(sd(x)^4) - excess
excess_kurt
}
x = rt(60, df = 5)   # Create hypothetical dataset x
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 R.10: Basic Graphs in R Last updated June 2018

y = rnorm(50, mean = 0, sd = 1)
par(mfrow=c(2,2)) # sets up space for subplots
barplot(y)           # bar plot
plot(y,type='l')  # line plot
hist(y)    # histogram
plot(y)    # scatter plot

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 R.11: Miscellaneous Useful Functions in R Last updated June 2018

## Convert objects from one type to another with as.integer() etc
## To check type, use typeof(object)
x = 8.0
print(typeof(x))
x = as.integer(x)
print(typeof(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)