# Entering and Printing Data in Julia
# Listing J.1
# Last updated June 2018
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x = 10             # assign x the value 10
println(x)         # print x

## println() puts next output on new line, while print() doesn't

# Vectors, Matrices and Sequences in Julia
# Listing J.2
# Last updated July 2020
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y = [1,3,5,7,9]     # lists in square brackets are stored as arrays

println(y)

println(y[3])       # 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

# Basic Summary Statistics in Julia
# Listing J.3
# Last updated July 2020
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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 = [1] for single vector
println(sort(y))                 # sorts y in ascending order
println(log.(y))                 # natural log, note . denotes elementwise operation

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

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

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

# Statistical Tests in Julia
# Listing J.7
# Last updated July 2020
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using Random, HypothesisTests; # loading required packages

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

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

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

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

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

# Download S&P 500 data in Julia
# Listing 1.1/1.2
# Last updated June 2022
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using CSV, DataFrames;

price = CSV.read("index.csv",DataFrame)
y = diff(log.(price[:,1]))

using Plots;
plot(y, title = "S&P500 returns", legend = false)

# Sample statistics in Julia
# Listing 1.3/1.4
# Last updated July 2020
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using Statistics, StatsBase;

println("Standard deviation: ", std(y), "\n")
println("Minimum value: ", minimum(y), "\n")
println("Maximum value: ", maximum(y), "\n")
println("Skewness: ", skewness(y), "\n")
println("Kurtosis: ", kurtosis(y), "\n")
println("Autocorrelation of returns:", "\n", autocor(y, 1:20), "\n")
println("Autocorrelation of returns squared:", "\n", autocor(y.^2, 1:20), "\n")

using HypothesisTests;

println(JarqueBeraTest(y))
println(LjungBoxTest(y,20))
println(LjungBoxTest(y.^2, 20))

# ACF plots and the Ljung-Box test in Julia
# Listing 1.5/1.6
# Last updated July 2020
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q1 = autocor(y, 1:20)
q2 = autocor(y.^2, 1:20)

plot(bar(q1, title = "ACF of returns"), 
    bar(q2, title = "ACF of returns squared"), legend = false)

# QQ plots in Julia
# Listing 1.7/1.8
# Last updated July 2020
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using StatsPlots, Distributions;

plot(qqplot(Normal,float(y),qqline=:quantile, title = "QQPlot vs Normal"), 
    qqplot(TDist(5),float(y),qqline=:quantile, title = "QQPlot vs Student-t(5)"),
    qqplot(TDist(4),float(y),qqline=:quantile, title = "QQPlot vs Student-t(4)"),    
    qqplot(TDist(3),float(y),qqline=:quantile, title = "QQPlot vs Student-t(3)"))