ML Regression in MATLAB^®

Simple Linear Regression

This example shows how to perform simple linear regression using the accidents dataset. The example also shows you how to calculate the coefficient of determination R² to evaluate the regressions. The accidents dataset contains data for fatal traffic accidents in U.S. states.

Linear regression models the relation between a dependent, or response, variable y and one or more independent, or predictor, variables x₁,...,x_n. Simple linear regression considers only one independent variable using the relation

• y=β0+β1x+ϵ,

where β₀ is the y-intercept, β₁ is the slope (or regression coefficient), and ϵ is the error term. This can be simplified to Y=XB

From the dataset accidents, load accident data in y and state population data in x. Find the linear regression relation y=β₁x between the accidents in a state and the population of a state using the \ operator. The \ operator performs a least-squares regression.

load accidents
x = hwydata(:,14); %Population of states
y = hwydata(:,4); %Accidents per state
format long
b1 = x\y

b1 =

     1.372716735564871e-04

b1 is the slope or regression coefficient. The linear relation is y=β₁x=0.0001372x.

Calculate the accidents per state yCalc from x using the relation. Visualize the regression by plotting the actual values y and the calculated values yCalc.

load accidents
x = hwydata(:,14); %Population of states
y = hwydata(:,4); %Accidents per state
format long
b1 = x\y;

yCalc1 = b1*x;
scatter(x,y)
hold on
plot(x,yCalc1)
xlabel('Population of state')
ylabel('Fatal traffic accidents per state')
title('Linear Regression Relation Between Accidents & Population')
grid on

fig2plotly(gcf);

Improve the fit by including a y-intercept β₀ in your model as y=β₀+β₁x. Calculate β₀ by padding x with a column of ones and using the \ operator.

load accidents
x = hwydata(:,14); %Population of states
y = hwydata(:,4); %Accidents per state

X = [ones(length(x),1) x];
b = X\y

b =

   1.0e+02 *

   1.427120171726538
   0.000001256394274

This result represents the relation y=β₀+β₁x=142.7120+0.0001256x.

Visualize the relation by plotting it on the same figure.

load accidents;
x = hwydata(:,14); %Population of states
y = hwydata(:,4); %Accidents per state

X = [ones(length(x),1) x];
b = X\y;

yCalc2 = X*b;
plot(x,yCalc2,'--');
legend({'Data'},'Location','best');

fig2plotly(gcf);

If with to plot the data alongside the slope, you can do it in the following way.

load accidents
x = hwydata(:,14); %Population of states
y = hwydata(:,4); %Accidents per state

X = [ones(length(x),1) x];
b = X\y;

yCalc2 = X*b;
plot(x,yCalc2,'--');
legend({'Data'},'Location','best');

hold on
plot(x,y,'o');

fig2plotly(gcf);