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.

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.

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

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;