Rename year directories to allow natural ordering

year2/semester2/CT248/Assignments/Assignment-02/Lab1.m

@@ -0,0 +1,8 @@
+clear;
+d = roll_2_dice(10000, 100);
+[freq, prop] = tabulate_2_dice(d);
+
+disp(freq);
+disp(prop);
+
+

year2/semester2/CT248/Assignments/Assignment-02/roll_2_dice.m

@@ -0,0 +1,7 @@
+function d = roll_2_dice(N, seed)
+  % function to roll 2 dice and return the combination of each device row in a vector
+  rng(seed);
+
+  % generating two 1 * N vectors of 6 simulated dice rolls and adding them
+  d = randi([1 6], 1, N) + randi([1 6], 1, N);
+end

year2/semester2/CT248/Assignments/Assignment-02/tabulate_2_dice.m

@@ -0,0 +1,14 @@
+function [freq, prop] = tabulate_2_dice(d)
+  % function to calculate the frequency and proportion of each outcome based on a set of dice throws
+  freq = zeros(1,12);
+  % looping through all the values in d and incrementing the corresponding counter in freq
+  for i = d
+    freq(i) = freq(i) + 1;
+  end
+
+  % looping through each index in prop and calculating the proportion pertaining to that index
+  prop = zeros(1,12);
+  for i = [1:12]
+    prop(i) = freq(i) / sum(freq); % proportion of i is the frequency of i divided by the sum of all freqs
+  end
+end

year2/semester2/CT248/Assignments/Assignment-03/mystack.m

@@ -0,0 +1,28 @@
+function[push, pop, peek] = mystack()
+% function to return handles to the subfunction push, pop, & peek
+    push = @push;
+    pop = @pop;
+    peek = @peek;
+end
+
+function [stack] = push(stack, value)
+% function to push a value onto the stack at location 1 and return the
+% stack
+    stack = [value; stack];
+end
+
+function [stack] = pop(stack)
+% function to pop the value at location 1 off the stack and return the
+% stack
+    stack(1) = [];
+end
+
+function [value] = peek(stack)
+% function to return the top value from the stack (arrau location 1)
+% returns NaN if there is no value at location 1
+    if isempty(stack)
+        value = NaN;
+    else
+        value = stack(1);
+    end
+end

year2/semester2/CT248/Assignments/Assignment-03/test.m

@@ -0,0 +1,13 @@
+% test script as specified in assignment spec
+[push, pop, peek] = mystack(); 
+
+stack = []
+stack = push(stack, 100)
+
+stack = push(stack, 200) 
+
+peek(stack) 
+
+stack = pop(stack) 
+
+peek(stack)

year2/semester2/CT248/Assignments/Assignment-04/imageprocessing.m

@@ -0,0 +1,15 @@
+clear;
+eng1 = imread("Engineering-Building.jpg");
+eng1_gs = pic2grayscale(eng1);
+eng1_gs_inv = transform_pic(eng1_gs);
+eng1_gs_bin_50 = transform_threshold(eng1_gs,50);
+eng1_gs_bin_75 = transform_threshold(eng1_gs,75);
+eng1_gs_bin_100 = transform_threshold(eng1_gs,100);
+
+% plotting images
+subplot(3,2,1),imshow(eng1),title("Original Picture"); 
+subplot(3,2,2),imshow(eng1_gs),title("Greyscale"); 
+subplot(3,2,3),imshow(eng1_gs_inv),title("Inverted Greyscale"); 
+subplot(3,2,4),imshow(eng1_gs_bin_50),title("Binary Threshold = 50"); 
+subplot(3,2,5),imshow(eng1_gs_bin_75),title("Binary Threshold = 75"); 
+subplot(3,2,6),imshow(eng1_gs_bin_100),title("Binary Threshold = 100"); 

year2/semester2/CT248/Assignments/Assignment-04/pic2grayscale.m

@@ -0,0 +1,17 @@
+function [returnImg] = pic2grayscale(img) 
+% which uses the NTSC Standard transformation to convert RGB to grayscale. 
+    %0.2989 * R + 0.5870 * G + 0.1140 * B
+
+    % img to be returned
+    returnImg = zeros(size(img,1), size(img, 2)); 
+
+    % looping through the RGB image and calculating the grayscale value for
+    % each pixel in the corresponding returnImg
+    for i = 1:size(img,1) 
+        for j = 1:size(img,2)
+            returnImg(i,j) = (0.2989 * img(i,j,1)) + (0.5870 * img(i,j,2)) + (0.1140 * img(i,j,3));
+        end 
+    end
+    
+    returnImg = uint8(returnImg);
+end

year2/semester2/CT248/Assignments/Assignment-04/transform_pic.m

@@ -0,0 +1,5 @@
+function [img] = transform_pic(img)
+% function which converts a 255 colour code to 0, 254 to 1, etc, and 0 to
+% 255. 
+    img = 255 - img;
+end

year2/semester2/CT248/Assignments/Assignment-04/transform_threshold.m

@@ -0,0 +1,18 @@
+function [img] = transform_threshold(img, threshold) 
+% function which converts the picture to binary format where any value 
+% above the threshold is white (1), and all values equal to or below are
+% black (0). 
+
+    % looping through each element in the matrix, and setting it to 1 if
+    % above the threshold, 0 otherwise
+    for i = 1:numel(img) 
+        if img(i) > threshold 
+            img(i) = 1; 
+        else 
+            img(i) = 0; 
+        end
+    end
+    
+    % casting the matrix to type logical once each element is either 1 or 0
+    img = logical(img);
+end

year2/semester2/CT248/Assignments/Assignment-06/shark_tuna_model.m

@@ -0,0 +1,14 @@
+function dydt = shark_tuna_model(t,x)
+% function to model the prey-predator population relations over time of
+% sharks & tuna 
+
+    global k;   % mentioning global variable x so that it can be used here 
+
+    dydt = [0; 0]; 
+    
+    % dS/dt = k1 S T - k2 S
+    dydt(1) = k(1)*x(1)*x(2)-k(2)*x(1); 
+    
+    % dT/dt = k3 T -k4 S T
+    dydt(2) = k(3)*x(2) - k(4) * x(1) * x(2); 
+end

year2/semester2/CT248/Assignments/Assignment-06/test.m

@@ -0,0 +1,12 @@
+clear; 
+
+global k; 
+
+k = [0.015 0.7 0.5 0.01].'; 
+
+[t,y] = ode45(@shark_tuna_model, [0 50], [100 100]);
+plot(t,y); 
+title("SHARK-TUNA POPULATION DYNAMICS LIMIT CYCLING");
+xlabel("TIME"); 
+ylabel("POPULATION NUMBERS");
+legend("SHARKS", "TUNA", "Location", "northwest");

year2/semester2/CT248/Assignments/Assignment-07/SIR.m

@@ -0,0 +1,15 @@
+function dydt = SIR(t, x, c, i, alpha, beta, gamma)
+    dydt = [0; 0; 0; 0; 0];
+    S = x(1);
+    I = x(2);
+    R = x(3);
+    H = x(4);
+    RH = x(5);
+    N = S + I + R + H + RH;
+
+    dydt(1) = (-c*S) * (I/N) * i;
+    dydt(2) = (c*S) * (I/N) * i - (alpha*I);
+    dydt(3) = (alpha*I) - (beta*R);
+    dydt(4) = (beta*R) - (gamma*H);
+    dydt(5) = gamma*H;
+end

year2/semester2/CT248/Assignments/Assignment-07/test.m

@@ -0,0 +1,40 @@
+clear; 
+
+i = 0.125; 
+alpha = 0.25; 
+beta = 0.02; 
+gamma = 0.10; 
+
+time_vec = 0:.25:100;
+init_vec = [9999 1 0 0 0];
+c = linspace(3, 8, 20); 
+
+infected_stock = zeros(length(time_vec), 20); 
+in_hospital = zeros(length(time_vec), 20); 
+
+for loopcounter = 1:20
+    [t,y] = ode45(@SIR, ...
+                   time_vec, ...
+                   init_vec, ...
+                   odeset, ...
+                   c(loopcounter), ...
+                   i, ...
+                   alpha, ...
+                   beta, ...
+                   gamma); 
+    
+    infected_stock(:, loopcounter) = y(:,2); 
+    in_hospital(:,loopcounter) = y(:,4); 
+end
+
+subplot(3, 1, 1);
+plot(time_vec, infected_stock); 
+title("Infected Stock"); 
+
+subplot(3, 1, 2);
+plot(time_vec, infected_stock); 
+title("People in Hospital"); 
+
+subplot(3,1,3); 
+scatter(c, max(in_hospital)); 
+title("Contacts v Peak in Hospital");

year2/semester2/CT248/Assignments/Assignment-08/six.m

@@ -0,0 +1,76 @@
+clear;
+
+% 1. read in the file and confirm the number of records (336,766)
+flights = readtable("Flights.csv"); 
+numRecords = height(flights)
+
+% 2. convert "origin" & "dest" to strings (from cell type) 
+flights.origin = string(flights.origin); 
+flights.dest = string(flights.dest); 
+
+
+% 3. check the number of missing values for the departure time 
+numMissing = sum(ismissing(flights.dep_delay))
+
+% 4. filter all the missing values from the departure delay and check the
+% difference in the number of records
+flights_clean = flights(~isnan(flights.dep_delay), :);
+numRecordsClean = height(flights_clean);
+disp("Number of records in flights = " + numRecords); 
+disp("Number of records in flights_clean = " + numRecordsClean);
+
+% 5. confirm the difference in records between the two tables 
+diff = numRecords - numRecordsClean
+
+% 6. Remove any departure delay greater than 2 hours (120 minutes). This 
+% leaves 318,798 observations. 
+flights_final = flights_clean(flights_clean.dep_delay <= 120, :); 
+height(flights_final)
+
+% 7. Generate the following table and graph, showing the average delay per 
+% month. 
+months = unique(flights_final.Month);
+
+res1 = table(months, zeros(size(months)), 'VariableNames', {'Month', 'AvgDelayMonth'});
+for i = months(1):length(months)
+    month_delays = flights_final.dep_delay(flights_final.Month == i);
+    avg_delay_month = mean(month_delays); 
+    res1.AvgDelayMonth(i) = avg_delay_month;
+end
+res1
+plot(res1.Month, res1.AvgDelayMonth, '-o');
+title('Average Delay by Month');
+
+% 8. Generate the following table and graph, showing the average delay per
+% hour.
+hours = transpose(1:24);
+
+res2 = table(hours, zeros(size(hours)), 'VariableNames', {'Hour', 'AvgDelayHour'});
+for i = hours(1):length(hours)
+    hour_delays = flights_final.dep_delay(flights_final.hour == i);
+    avg_delay_hour = mean(hour_delays); 
+    res2.AvgDelayHour(i) = avg_delay_hour;
+end
+res2 = res2(~isnan(res2.AvgDelayHour),:);
+res2
+plot(res2.Hour, res2.AvgDelayHour, '-o');
+title('Average Delay by Hour of the Day');
+
+% 9. Generate the following table and graph, showing the average delay by
+% month and by origin 
+res3 = renamevars(removevars(groupsummary(flights_final,["Month","origin"],"mean","dep_delay"),'GroupCount'), 'mean_dep_delay', 'AvrDelayMonthOrigin')
+
+jfk = res3(res3.origin == 'JFK', {'Month', 'AvrDelayMonthOrigin'});
+subplot(3,1,1);
+plot(jfk.Month, jfk.AvrDelayMonthOrigin, '-o');
+title("JFK");
+
+ewr = res3(res3.origin == 'EWR', {'Month', 'AvrDelayMonthOrigin'});
+subplot(3,1,2);
+plot(ewr.Month, ewr.AvrDelayMonthOrigin, '-o');
+title("EWR");
+
+lga = res3(res3.origin == 'LGA', {'Month', 'AvrDelayMonthOrigin'});
+subplot(3,1,3);
+plot(lga.Month, lga.AvrDelayMonthOrigin, '-o');
+title("LGA");

year2/semester2/CT248/Assignments/LabTest-1/clean_grades.m

@@ -0,0 +1,9 @@
+function [grades] = clean_grades(grades) 
+   for i = 1:(size(grades,1)*size(grades,2))
+       if grades(i) < 0 
+           grades(i)  = 0; 
+       elseif grades(i) > 100 
+           grades(i) = 0; 
+       end 
+   end
+end

year2/semester2/CT248/Assignments/LabTest-1/create_grades.m

@@ -0,0 +1,4 @@
+function [grades] = create_grades(rows, columns, min, max, seed) 
+    rng(seed);
+    grades = randi([min max], rows, columns);
+end

year2/semester2/CT248/Assignments/LabTest-1/fill_estimates.m

@@ -0,0 +1,25 @@
+function [grades] = fill_estimates(grades)
+    for i = 1:size(grades, 2) 
+        % making copy
+        copy = grades(:, i);
+        nonzero = 0; % count of nonzero elements to calculate avg
+        sum = 0;    % sum of nonzero elements
+
+        % looping through copy and counting number of nonzero elements
+        for j = 1:size(copy,1) 
+            if copy(j) ~= 0
+                nonzero = nonzero + 1;
+                sum = sum + copy(j);
+            end 
+        end
+        
+        avg = round(sum / nonzero);
+
+        % looping through original and replacing any nonzero with average 
+        for j = 1:size(copy,1)
+            if grades(j,i) == 0
+                grades(j,i) = avg; 
+            end
+        end
+    end
+end

year2/semester2/CT248/Assignments/LabTest-1/test.m

@@ -0,0 +1,4 @@
+clear;
+gr1 = create_grades(10,5,-10,110,100);
+gr2 = clean_grades(gr1);
+gr3 = fill_estimates(gr2);

year2/semester2/CT248/Assignments/LabTest-2/test.m

@@ -0,0 +1,22 @@
+clear;
+
+r = 0.2; % random val for r
+K = linspace(1000, 1000000, 50);
+time_vec = linspace(0, 100, 50);
+P = 100; % random val for P
+population = zeros(length(time_vec), 50);
+
+% implementing the model as an anonymous function 
+dpdt = @(t,P,r,K) r*P * (1 - (P/K)); 
+
+for loopcounter = 1:50
+    [t,y] = ode45(dpdt, ...
+                  time_vec, ...
+                  P, ...
+                  odeset, ...
+                  r, ...
+                  K(loopcounter));
+    population(:, loopcounter) = y(:,1); 
+end
+
+plot(time_vec, population);

year2/semester2/CT248/Lecture Files/Lecture04.m

@@ -0,0 +1,26 @@
+clear;
+
+A = [1 2 ; 3 4 ];
+B = eye(2);
+
+% MATRIX multiplication
+C = A*B;
+
+% element-wise multiplication
+D = A.* B;
+
+rng(100);
+rolls = randi([1 6], 1, 10);
+
+% checking each element of rolls one by one to see which ones are greater than 3
+gt3 = rolls > 3;
+rolls(gt3); % filtering rolls to find the values > 3
+
+A ~= 3;
+
+% remove duplicates from rolls
+unique(rolls)
+
+for i = rolls
+  disp(i)
+ end

year2/semester2/CT248/Notes/CT248-Notes.tex

@@ -0,0 +1,598 @@
+\documentclass[11pt]{article}
+\usepackage{report}
+
+\usepackage[utf8]{inputenc} % allow utf-8 input
+\usepackage[T1]{fontenc}    % use 8-bit T1 fonts
+\usepackage[colorlinks=true, linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref}       % hyperlinks
+\usepackage{url}            % simple URL typesetting
+\usepackage{booktabs}       % professional-quality tables
+\usepackage{amsfonts}       % blackboard math symbols
+\usepackage{nicefrac}       % compact symbols for 1/2, etc.
+\usepackage{microtype}      % microtypography
+\usepackage{lipsum}		% Can be removed after putting your text content
+\usepackage{graphicx}
+\graphicspath{ {./images/} }
+\usepackage{natbib}
+\usepackage{doi}
+\setcitestyle{aysep={,}}
+\usepackage{array}
+\usepackage{listings}
+
+\usepackage{xcolor}
+\definecolor{codegreen}{rgb}{0,0.6,0}
+\definecolor{codegray}{rgb}{0.5,0.5,0.5}
+\definecolor{codeorange}{rgb}{1,0.49,0}
+\definecolor{backcolour}{rgb}{0.95,0.95,0.96}
+
+\lstdefinestyle{mystyle}{
+    backgroundcolor=\color{backcolour},   
+    commentstyle=\color{codegray},
+    keywordstyle=\color{codeorange},
+    numberstyle=\tiny\color{codegray},
+    stringstyle=\color{codegreen},
+    basicstyle=\ttfamily\footnotesize,
+    breakatwhitespace=false,         
+    breaklines=true,                 
+    captionpos=b,                    
+    keepspaces=true,                 
+    numbers=left,                    
+    numbersep=5pt,                  
+    showspaces=false,                
+    showstringspaces=false,
+    showtabs=false,                  
+    tabsize=2,
+    xleftmargin=10pt,
+}
+
+\lstset{style=mystyle}
+\title{CT248 - Introduction to Modelling}
+
+\author{Andrew Hayes\\
+\AND
+Student ID: 21321503\\
+\AND
+\AND
+\AND
+\AND
+	1BCT1\\
+\AND
+	NUI Galway\\
+}
+
+% Uncomment to remove the date
+\date{February 2022}
+
+% Uncomment to override  the `A preprint' in the header
+\renewcommand{\headeright}{Introduction to Modelling}
+\renewcommand{\undertitle}{Introduction to Modelling}
+\renewcommand{\shorttitle}{}
+
+%%% Add PDF metadata to help others organize their library
+%%% Once the PDF is generated, you can check the metadata with
+%%% $ pdfinfo template.pdf
+% \hypersetup{
+% pdftitle={A template for the arxiv style},
+% pdfsubject={q-bio.NC, q-bio.QM},
+% pdfauthor={David S.~Hippocampus, Elias D.~Striatum},
+% pdfkeywords={First keyword, Second keyword, More},
+% }
+
+\begin{document}
+\maketitle
+
+\newpage
+\tableofcontents
+\thispagestyle{empty}
+
+
+\newpage
+\setcounter{page}{1}
+\section{Lecture 01 - MATLAB Overview}
+\textbf{MATLAB} (MAtrix LABoratory) is a powerful technical computing system for handling scientific \& engineering 
+calculations, designed to make \textbf{matrix computations} particularly easy. 
+It is a \textbf{high-level language} with an interactive development environment.
+MATLAB is \textbf{loosely typed}.
+
+MATLAB is used for:
+\begin{itemize}
+    \item Numerical computation. 
+    \item Data analysis \& visualisation. 
+    \item Algorithm development \& programming. 
+    \item Application development \& deployment.
+\end{itemize}
+
+\subsection{Variables}
+A variable name in MATLAB may consist only of letters a-z, digits 0-9, and the underscore (\_) and they must start
+with a letter. 
+A variable name may not exceed 63 characters. 
+MATLAB is case-sensitive, including command \& function names. 
+
+The use of a semi-colon \verb|;| at the end of a variable initialisation prevents the value from being displayed. 
+
+\verb|carSpeed = 20;|
+
+\subsubsubsection{The MATLAB Workspace}
+All variables created during a session remain in the workspace until they are \textbf{cleared}. 
+\begin{lstlisting}[language=MATLAB]
+clear;
+mph = input("Please enter the speed in mph: ");
+kph = mph * 1.6;
+fprintf("%d mph = %d kph\n",mph,kph);
+\end{lstlisting}
+
+The command \verb|who| lists all the names in the current workspace. 
+The command \verb|whos| lists the size of each variable. 
+
+Comments can be made in MATLAB using the \verb|%| sign. 
+
+\subsection{Expressions}
+An \textbf{expression} is a formula consisting of variables, numbers, operators, \& function names. 
+Expressions are evaluated when you enter them into the MATLAB prompt. 
+Note that MATLAB uses the function \verb|ans| to return the last expression that was evaluated but not assigned
+to a variable. 
+
+\subsubsection{Statements}
+MATLAB \textbf{statements} are frequently of the form \verb|variable = expression|. \\
+E.g.: \verb|s=u*t-g/2*t.^2|. 
+\\
+This is an example of an \textbf{assignment statement}, with the value of the expression assigned to the 
+variable on the left-hand side.
+A semicolon is at the end to suppress the output. 
+
+Statements that are too long for a line can be extended to several lines with an ellipsis of at least 3 dots \verb|...|
+\\
+Statements on the same line can be separated by commas or semicolons.
+
+\newpage
+
+\subsection{Input \& Output}
+The \verb|input()| statement is used to read in input data. 
+\begin{lstlisting}[language=MATLAB]
+name = input ("Enter your name: ");
+\end{lstlisting}
+
+The \verb|disp()| statement is used to \textbf{display} data. 
+The general form of the \verb|disp| statement is \verb|disp(variable)|. 
+
+\begin{lstlisting}[language=MATLAB]
+>> disp("Hello World!")
+>> x = 2;
+>> disp(["The answer is ", num2str(x)])
+The answer is 2. 
+>> disp([2 3 x])
+2.00 3.00 2.00
+\end{lstlisting}
+
+To display more than one variable, embed the variables into an \textbf{array}. 
+All components of a MATLAB array must be the same type.
+
+\begin{lstlisting}[language=MATLAB]
+% Script file for converting temperatures from F to C. 
+% Step 1: Get the input
+F = input ("Enter the temperature in degrees F: ");
+
+% Step 2: Convert to C 
+ C = (F -32) * 5/9;
+
+% Step 3: Display the result
+fprintf("The temperature of %f (F) is %f (C)\n",F,C);
+\end{lstlisting}
+
+\subsection{Arithmetic Operators}
+The evaluation of an expression is achieved by means of arithmetic operators. 
+The arithmetic operations on two scalar constants / variables is shown. 
+Left division may seem curious, however \textit{matrix} left division has an entirely different meaning. 
+
+\begin{center}
+\begin{tabular}{ |c|c|c| } 
+    \hline
+    \textbf{Operation} & \textbf{Algebraic Form} & \textbf{MATLAB} \\ 
+    \hline
+    Addition        & $ a + b $                 & \verb|a + b|  \\ 
+    \hline
+    Subtraction     & $ a - b $                 & \verb|a - b|  \\ 
+    \hline
+    Multiplication  & $ a \times b  $           & \verb|a*b|    \\
+    \hline
+    Right Division  & $ a/b $                   & \verb|a/b|    \\
+    \hline
+    Left Division   & $ b \textbackslash a $    & \verb|b\a|    \\
+    \hline
+    Exponent        & $ a^b $                   & \verb|a^b|    \\
+    \hline
+\end{tabular}
+\end{center}
+
+MATLAB uses PEMDAS as to determine \textbf{precedence}. 
+
+\subsection{Repetition \& Loops - The For Loop}
+The \verb|for| statement repeats some statement a specific number of times. 
+The two basic \verb|for| loop structures are as follows:
+\begin{lstlisting}[language=MATLAB]
+for index = j:k 
+    statements;
+end 
+
+for index = j:m:k 
+    statements;
+end
+\end{lstlisting}
+
+\verb|j:k| is a \textit{vector} with elements $j, j+1, j+2, \cdots, k$. 
+\verb|j:m:k| is a vector with elements $j, j+m, j+2m, \cdots$ such that the final element does not exceed $k$. 
+\verb|index| must be a variable. Loop statements should be indented.
+
+
+\begin{lstlisting}[language=MATLAB]
+for i = 1:5
+    disp(i);
+end
+\end{lstlisting}
+
+This for loop repeats the \verb|disp()| statement 5 times, starting with \verb|i = 1| and ending with \verb| i = 5|, incrementing by 1 each iteration of the loop. 
+
+For a for loop of the form \verb|for i = a:b|, the number of iterations will be $b - a +1$.
+
+The following example program takes in $n$ numbers and displays their average. 
+\begin{lstlisting}
+clear; 
+sum = 0;
+
+n = input("How many numbers: ");
+
+for i=1:n 
+    num = input("Please enter a number: ");
+    sum += num;
+end 
+
+avg = sum/n; 
+
+fprintf("The average is %f\n.", avg); 
+\end{lstlisting}
+
+
+\subsection{Relational Operators} 
+\textbf{Relational Operators} form a logical expression that is either \textit{true} or \textit{false}.
+Relational operators form the basis for decision logic. 
+
+In MATLAB, "true" \& "false" can be represented both lexically (\verb|true|, \verb|false|) and logically (\verb|1|, \verb|0|). 
+When returning the value of a Boolean expression or variable, MATLAB will use logical true (\verb|1|) \& false (\verb|0|). 
+
+\begin{center}
+    \includegraphics[width=0.25\textwidth]{truefalse.png}
+\end{center}
+
+\begin{center}
+\begin{tabular}{ |c|c| } 
+    \hline
+    \textbf{Relational Operator} & \textbf{Meaning} \\ 
+    \hline
+    \verb|<|    & Less than                 \\ 
+    \hline 
+    \verb|<=|   & Less than or equal to     \\ 
+    \hline 
+    \verb|==|   & Equal to                  \\
+    \hline 
+    \verb|~=|   & Not equal to              \\ 
+    \hline 
+    \verb|>|    & Greater than              \\
+    \hline 
+    \verb|>=|   & Greater than or equal to  \\
+    \hline
+\end{tabular}
+\end{center}
+
+\subsubsection{The "If" Statement} 
+In MATLAB, \verb|if| statements take the general form(s):
+\begin{lstlisting}[language=MATLAB]
+if condition; statements; end 
+
+if condition
+    statements
+else
+    statements
+end
+
+if condition
+    statements
+elseif condition
+    statements 
+else 
+    statements
+end
+    
+\end{lstlisting}
+
+\begin{lstlisting}[language=MATLAB]
+num = input("Enter a number: ");
+if num < 0; disp("Negative number"); end
+\end{lstlisting}
+
+\section{Matrices}
+In linear algebra, a \textbf{matrix} is a rectangular grid of numbers arranged into \textit{rows} \& \textit{columns}. 
+An $r \times c$ matrix has $r$ rows \& $c$ columns. 
+Uppercase letters (e.g., $A$) are usually used to denote variables that are matrices. 
+$a_{ij}$ denotes the element in $A$ at row $i$ \& column $j$. 
+
+\begin{center}
+    \includegraphics[width=0.25\textwidth]{matrix.png}
+\end{center}
+
+\subsection{Multiplying a Matrix by a Scalar}
+A matrix $A$ can be multiplied by a scalar $k$, resulting in a matric of the same dimensions as $A$. 
+The multiplication takes place in a straightforward fashion, with each element in the new matrix being the product of $k$ times the corresponding element in $A$. 
+
+\begin{center}
+    \includegraphics[width=0.5\textwidth]{matrixscalar.png}
+\end{center}
+
+Let matrix $C$ be the $r \times c$ product $AB$ of the $r \times n$ matrix $A$ with the $n \times c$ matrix B. 
+Each element of $C$, $C_{ij}$, is equal to the vector dot product of row $i$ of $A$ with column $j$ of $B$. 
+
+\begin{center}
+    \includegraphics[width=0.5\textwidth]{2x2example.png}
+\end{center}
+
+In MATLAB, a matrix is a rectangular object consisting of rows \& columns. 
+Matrices have comprehensive support with functions \& operators.
+Rows can be separated when initialising a matrix using the semicolon \verb|;|. 
+
+\begin{lstlisting}
+>>a = [1 2 3; 4 5 6]
+a = 
+    1   2   3 
+    4   5   6
+
+>>a'
+ans= 
+        1   4 
+        2   5 
+        3   6
+\end{lstlisting}
+
+
+Individual elements are usually referenced with two subscripts in the form \verb|matrixname(row, column)|. 
+However, elements can also be referenced using \textit{one} subscript. 
+In this case, MATLAB will treat the entire vector as a single column.
+
+\begin{lstlisting}[language=MATLAB]
+A=[1 2 3;4 5 6;7 8 9] 
+A(1,1)
+A(1)
+A(9)
+\end{lstlisting}
+
+Using the colon \verb|:| operator in place of a subscript denotes all the elements in the corresponding row or column. 
+\verb|A(3,:)| means ``all the elements in the third row''. 
+\verb|A(:,3)| means ``all the elements in the third column''. 
+You can extract subsets of a matrix using something like \verb|A(1:2, 2:3)|.
+
+\subsubsection{Matrix Functions}
+\begin{center} \begin{tabular}{ |c|c| } 
+    \hline
+    \textbf{Function}   & \textbf{Description}                  \\ 
+    \hline
+    \verb|eye()|        & Identity Matrix                       \\
+    \hline 
+    \verb|linspace()|   & Vector with linearly spaced elements  \\
+    \hline 
+    \verb|ones()|       & Matrix of 1s                                      \\ 
+    \hline 
+    \verb|rand()|       & Uniformly distributed random numbers \& arrays    \\ 
+    \hline 
+    \verb|randn()|      & Normally distributed random numbers \& arrays     \\ 
+    \hline 
+    \verb|zeros()|      & Matrix of 0s.     \\ 
+    \hline 
+    \verb|det()|        & Determinant       \\ 
+    \hline 
+    \verb|eig()|        & Eigenvalues \& eigenvectors. \\
+    \hline 
+    \verb|expm()|       & Matrix exponential \\
+    \hline 
+    \verb|inv()|        & Matrix inverse    \\ 
+    \hline 
+    \verb|trace()|      & Sum of diagonal elements \\ 
+    \hline 
+    \verb|{}\| and \verb|/| & Linear equation solutions. \\
+    \hline
+\end{tabular}
+\end{center}
+
+
+\subsection{Vectors} 
+A \textbf{vector} is a special type of matrix, having only one row or one column. 
+MATLAB handles matrices \& vectors in the same way. 
+
+Rules:
+\begin{itemize}
+    \item Elements in the list must be enclose in square brackets. 
+    \item Elements in the list must be separated by either spaces or by commas. 
+\end{itemize}
+
+\begin{lstlisting}[language=MATLAB]
+x = [1 2 3 4 5 6]
+y = [1,2,3,4,5,6]
+\end{lstlisting}
+
+Vectors can also be initialised using the colon \verb|:| operator. 
+\begin{lstlisting}[language=MATLAB]
+x = 1:10 
+y = 1:5:10
+\end{lstlisting}
+
+Operations can be performed on vectors, e.g \verb|sum(x)|, \verb|mean(x)|, etc. 
+Elements in a vector can be accessed via \verb|x(1)| or \verb|x(1:3)| to get a range of elements. 
+Note: vectors in MATLAB are indexed from 1, not 0. 
+
+\subsubsection{Transposing Vectors} 
+The vectors generated thus far have been \textit{row vectors}. 
+\textit{Column vectors} may be needed for matrix operations. 
+The single quote \verb|'| can be used to transpose a vector. 
+
+\begin{center}
+    \includegraphics[width=0.25\textwidth]{singlequote.png}
+\end{center}
+
+\section{Lecture 02 - Logical Vectors \& Arrays}
+\subsection{Element-Wise Operations}
+
+\section{MATLAB Functions} 
+MATLAB allows you to create your own function \textbf{M-files}. 
+A function file differs from a script file in that the function M-file communicates with the MATLAB workspace through 
+specially designated input \& output arguments. 
+
+A function M-file \verb|name.m| has the following general form:
+\begin{lstlisting}[language=MATLAB]
+function [outarg1, outarg2, ...] = name(inarg1, inarg2, ...)
+% help text
+outarg1 = ...;
+outarg2 = ...; 
+end 
+\end{lstlisting}
+
+\begin{itemize}
+    \item   \textbf{Function Keyword -} The function file must start with the keyword \verb|function|. 
+    \item   \textbf{Input \& Output Arguments -} The input \& output arguments define the function's means of communication with the workspace. 
+        \begin{itemize}
+            \item   If there is more than one output argument, the output arguments must be separated by commas \& enclosed in square brackets. 
+            \item   Input \& output variables can be vectors.
+        \end{itemize}
+    \item   \textbf{Help Text -} When you type \verb|help function_name|, MATLAB displays the comment lines that appear between the function definition \& the first non-comment line.
+    \item   Function names should follow the MATLAB rules for variable names. 
+\end{itemize}
+    
+If a function changes the value of any of its input arguments, the change is \textbf{not} reflected in the actual argument. 
+An input argument is only passed by value if a function modifies it. 
+If a function does not modify an input argument, it is passed by reference. 
+
+A variable number of arguments can be passed to a function. 
+A function may be called with all, some, or none of its arguments. 
+The same applies to output arguments. 
+\verb|nargin| displays the number of arguments passed. 
+
+
+\subsection{Scope}
+Any \textbf{local variables} defined inside a function are not accessible outside the function. 
+These local variables exist only inside the function, which has its own workspace, separate from the base workspace.
+
+Variables which are defined in the base workspace and are not normally accessible inside functions. 
+If functions (\& possibly the base workspace) declare variables as \textbf{global}. then they will all share a single copy of those variables. 
+
+\textbf{Persistent variables}, unlike local variables, remain persistent between function calls. 
+A persistent variable is initially an empty array.
+Persistent variables can be declared with the keyword \verb|persistent <variablename>|.
+
+\subsection{Subfunctions}
+The function M-file may contain code for more than one function. 
+The first function in the M-file is called the \textbf{primary function}. 
+Additional functions are known as \textbf{subfunctions}, and are visible only to the primary functions \& to the other subfunctions. 
+Subfunctions follow each other ina ny order \textit{after} the primary function.
+
+\section{Processing Images in MATLAB}
+\subsection{Examples of Image Types}
+\begin{itemize}
+    \item   \textbf{Binary Images:} 2D arrays that assign one numerical value from the set (0,1) to each pixel in the image.
+            Often called \textbf{logical images}. 
+            \verb|0| corresponds to black, while \verb|1| corresponds to white. 
+            Binary images can be represented as a simple bit stream.
+
+    \item   \textbf{Intensity/Grayscale Images:} 2D arrays that assign one numerical value to each pixel, representing the intensity 
+            at said pixel.
+            The range is bounded by the bit resolution of the image.
+
+    \item   \textbf{RGB Images:} 3D arrays that assign three numerical values to each pixel, with each value corresponding to the red, 
+            green, \& blue image channel. 
+            Pixels are accessed by \verb|I(Row,Column,Channel)| in MATLAB.
+\end{itemize}
+
+\subsection{Key MATLAB Functions}
+MATLAB's image processing toolbox contains an extensive image processing library, including:
+\begin{itemize}
+    \item   \verb|imread(filename)|     - Reads an image from a graphical file and converts it to a MATLAB array object.
+    \item   \verb|imshow(object)|       - Displays an image.
+    \item   \verb|linspace(X1,X2,N)|    - Generates a row vector of \verb|N| linearly spaced points between \verb|X1| \& \verb|X2|.
+                                          If \verb|N| is excluded, 100 points will be generated. 
+    \item   \verb|subplot(m,n,p)|       - Creates axes in tiled positions by dividing the current figure into an \verb|m|-by-\verb|n| 
+                                          grid and creating axes in the position specified by \verb|p|.
+                                          MATLAB numbers subplot positions by row. 
+                                          The first subplot is the first column of the first row, the second subplot is the second column 
+                                          of the second row, etc.
+\end{itemize}
+
+\subsection{Image Colour}
+An image contains one or more colour channels that define the intensity or colour at a particular pixel location \verb|I(m.n)|. 
+In the simplest case, each pixel location contains only a single numerical value representing the signal level at that point in the image. 
+The most common colour map is \textbf{greyscale}.
+The maximum numerical value representing the signal level is $2^8-2=255$.
+
+\subsection{Operations on Pixels}
+The most basic type of image processing is a \textbf{point transform} which maps the value at individual points (pixels) in the input 
+image to corresponding points in an output image. 
+In a mathematical sense, it's a one-to=one functional mapping from input to output.
+
+Types of operations include:
+\begin{itemize}
+    \item   Pixel addition \& subtraction. 
+    \item   Thresholding. 
+    \item   RGB to Greyscale. 
+    \item   Rotation (90º).
+    \item   Simple cropping.
+\end{itemize}
+
+\subsubsection{Arithmetic Operations on Images}
+Arithmetic operations can be performed quickly and easily on images. 
+The example below shows contrast adjustment.
+\begin{lstlisting}[language=MATLAB]
+I = imread("cameraman.tif"); 
+
+O = I + 50; 
+O1 = I + 100; 
+O2 = I - 100; 
+
+subplot(2,2,1),imshow(I),title("Original Image"); 
+subplot(2,2,2),imshow(O),title("+50"); 
+subplot(2,2,3),imshow(O1),title("+100");
+subplot(2,2,4),imshow(O2),title("-100"); 
+\end{lstlisting}
+
+\begin{center}
+    \includegraphics[width=0.7\textwidth]{arithmeticoperationsonimages.png}
+\end{center}
+
+\subsubsection{Thresholding}
+\textbf{Thresholding} produces a binary image from a greyscale or colour image by setting pixels to \verb|1| or \verb|0| depending on 
+whether they are above or below the threshold value.
+This is useful to help separate the image foreground from the background.
+Logical operators are very useful for this.
+
+\begin{lstlisting}[language=MATLAB]
+I = imread('rice.png');
+T1 = I > 100;
+T2 = I > 105;
+T3 = I > 110;
+T4 = I > 115;
+T5 = I > 120;
+subplot(2,3,1),imshow(I),title('Original Image');
+subplot(2,3,2),imshow(T1),title('Threshold @100');
+subplot(2,3,3),imshow(T2),title('Threshold @105');
+subplot(2,3,4),imshow(T3),title('Threshold @110');
+subplot(2,3,5),imshow(T4),title('Threshold @115');
+subplot(2,3,6),imshow(T5),title('Threshold @120');
+\end{lstlisting}
+
+\section{Data Science}
+\subsection{Introducing the Table in MATLAB} 
+\textbf{Tables} are used to collect heterogeneous data \& metadata in a single container. 
+Tables are suitable for storing column-oriented or tabular data that is often stored as columns in a text file or in a spreadsheet.
+Tables can accomodate variables of different types, sizes, units, etc. 
+Tables are often used to store experimental data, with rows representing the different observations and columns representing different measured variables.
+
+Tables can be subscripted using parentheses much like ordinary numeric arrays, but in addition to numeric and logical indices, one can use a table's variable and row names (if defined) as \textbf{indices}.  
+One can access individual variables in a table much like fields in a structure, using \textbf{dot subscripting}.  
+One can access the contents of one or more variables using \textbf{brace subscripting}.
+
+\bibliographystyle{unsrtnat}
+\bibliography{references}
+\end{document}
+

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