Understanding Data Plotting and Dimensionality in Regression

Introduction

In the world of data analysis and machine learning, understanding how to visualize and manipulate data is essential. In this article, we will work with a dataset containing 25 one-dimensional points. The dataset comprises two columns: the first column represents the x-values, and the second column corresponds to the y-values. We will explore the process of plotting this data and discuss the significance of adding a new dimension to the x-values.

1. Plotting Data with y as a Function of x

The first step in our analysis is to visualize the relationship between the x and y values. A simple scatter plot is an effective way to represent this data. Below is the MATLAB code to achieve this:

code% Load the dataset
data = load('Data.mat'); % Make sure to adjust the filename
x = data(:, 1); % First column is x
y = data(:, 2); % Second column is y
% Plot the data
figure;
plot(x, y, 'b*'); % Blue stars for the data points
xlabel('X');
ylabel('Y');
title('Scatter Plot of Y vs X');
grid on;
% Save the plot
saveas(gcf, 'scatter_plot.png');

This code loads the data, extracts the x and y values, and plots them. The resulting scatter plot provides a clear visual representation of the data points.

2. Adding a New Dimension to the X Data

Next, we will enhance our x-values by adding a new dimension, where all values are equal to 1. This step can be implemented in MATLAB as follows:

% Add a new dimension to x
x_new = [ones(size(x)), x]; % New x matrix with a column of ones

Explanation of the Effect

Adding a column of ones to the x matrix introduces a bias term, which is crucial in regression models. Here’s why this transformation is important:

• Bias Term: The column of ones allows the model to learn an intercept (or bias) in addition to the slope when fitting a line to the data. Without this term, the model will always assume that the line passes through the origin, which may not represent the data accurately.
• Effect on Plot: Although this transformation does not change the scatter plot visually, it alters the model’s capacity to capture the underlying relationship more flexibly by accounting for a possible intercept.

In summary, adding a column of ones enables the model to fit the data better by considering a constant bias (intercept) alongside the input values. This adjustment is a fundamental aspect of linear regression and helps improve the accuracy of predictions.

For full code visit this page (https://github.com/raghdafaris/AUT-Machine-Learning/tree/main/1-%20SimplePlot-Overfitting)