Mathematics

Capstone: Mathematics

Abstract

Convolutional neural networks are the Machine Learning industry standard for computer vision, and they generally excel at image recognition. The operations performed in a convolutional neural network are mathematic in nature, yet the inner layers of the neural networks are often referred to as a ‘black box’ due to their complex inner workings. Melanoma, a rare but deadly form of skin cancer, can be seen in images, implying the possibility of image recognition using a convolutional neural network. This project aims to develop said network and use its mathematical properties to optimize its performance.

Paper Slides Poster

Real Analysis: The Inverse Function Theorem

My partner and I researched and presented information about the Inverse Function Theorem and its proof. This is the theorem that dictates that the derivative of ln(x) = 1/x.

Slides

Mathematical Modeling: The Stock Market

See the Original Assignment

The objective of the project was to model the stock market using random variables in a stochastic model.

As a part of this project, we examined a few random tickers using

this tool.

The random tickers selected were:

• AMTB: Amerant Bancorp, Inc.
• FRAK: VanEck Vectors Unconventional Oil & Gas ET
• REED: Reed's, Inc.
• SQQQ: ProShares UltraPro Short QQQ

We used these stocks to calculate the average p-value to use in our simulations. Then, we wrote code to display these simulations on a line graph. For each ticker, we plotted fifteen simulations using our parameters for the model, and we also plotted the average of those simulations against real-world data. We also examined Apple's stock, first running our simulations with Apple's true p-value, then by running the simulations with our derived p-value.

The code below is written in Octave, similar to Matlab.

                                
pkg load statistics;

data = xlsread("AAPL.xlsx");
%data = xlsread("AMTB.xlsx");
%data = xlsread("ANH.xlsx");
%data = xlsread("FRAK.xlsx");
%data = xlsread("REED.xlsx");
%data = xlsread("SQQQ.xlsx");


t = data(:,1); %time is first column of data
price = data(:,5); % closing price is fifth column


% Define parameters

x0 = data(2,5); % closing price on day 1
N = max(t)-1; % max time
%p = .477; %AVERAGE OF RANDOM STOCK TICKER P'S
p = .52988; %APPLE'S P-value
% Make time vector
n = 0:N;

% Allocate space for x
x = zeros(size(n));

% Assign initial condition
x(1)=x0;

% Create for loop
for j = 1:10 %how many repetitions?
    for i = 1:N %simulate through entire time period
        u = rand; %compared to p - between 0 and 1
        a=(normrnd(x(i)*.02393,x(i)*.0308)); %price change is scaled by the current price
        if (u < p)
            x(i+1) = x(i)+a; %price increase
        elseif (u >= p)
            x(i+1) = x(i)-a; %price decrease
        endif
    end
    alldata(j,:) = x;
    % Plot x vs. n using a bold line 
    figure(1)
    plot(n,x,'linewidth',3)
    hold on;
    xlabel('n')
    ylabel('x','rotation',0)
end

% Change the axis font size
set(gca,'fontsize',18)
plot(t,price,'x')
hold off;

figure(2)
plot(t,price,'x')
hold on;
plot(n, mean(alldata), 'linewidth', 5)
set(gca, 'fontsize', 18)
xlabel('day')
ylabel('x', 'rotation', 0)

resid = price - x';
sum(resid.^2)

hold off;
                                
                            

See the Outputs

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Linear Algebra: The Wassily-Leontief Model

An application of linear algebra in economics

My Work (PDF)

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Theory of Proof: Inductive Algebraic Proof

See Proof (JPG)

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Theory of Proof: Distributive Law (Set Theory)

See Proof (JPG)

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