Linear Regression

clear all
clc

X=[1 2 3 4 5 6 7 8 9 10] %array X
Y=[12 14 24 28 19 28 30 37 41 45] %array Y

scatter(X,Y) %to obtain scatter plot

%for axis labels
xlabel('X')
ylabel('Y')

%to obtain line of best fit
f=polyfit(X,Y,1);
v=polyval(f,X);
hold on
grid on;
plot(X,v)

%calculation of estimates b0 and b1
AvgX=mean(X); %mean value of array X
AvgY=mean(Y); %mean value of array Y

n=length(X); %setting n to 10(number of entries in arrays)
sum1=sum(X.*Y); %summation of Xi*Yi
sum2=sum(X); %summation of Xi
sum3=sum(Y); %summation of Yi
sum4=sum(X.^2); %summation of Xi^2
sum5=sum2^2; %squaring the summation of Xi

b1=(((n*sum1)-(sum2*sum3))/((n*sum4)-sum5)); %to obtain value of b1
b0=AvgY-(b1*AvgX); %to obtain value of b0

disp("Equation of line of best fit is: Y = "+b0+" + "+b1+"X") %to display equation
disp(" ")%empty line in command window

%computing R^2 coefficient
sum6=sum((X-AvgX).*(Y-AvgY)); %summation of (Xi-AvgX)(Yi-AvgY)
sum7=sum((X-AvgX).^2); %summation of (Xi-Avg(X))^2
sum8=sum((Y-AvgY).^2); %summation of (Yi-Avg(Y))^2

rsqrdcoeff = ((sum6)/(sqrt(sum7*sum8)))^2; %to obtain value of R^2 coefficient
disp("R^2 Coefficient = "+rsqrdcoeff) %to display r^2 coefficient
disp(" ") %empty line in command window

%to calculate length of unstandardised residuals
for k=1:length(X)
    residualss(k)=(((-b1)*X(k))+(Y(k))-b0)/(sqrt((b1^2)+(1^2)));
end

LengthOfResiduals = abs(residualss') %to display absolute value of lengths in a vector

%to plot unstandardised residuals
GradientOfResiduals=-1/(b1) %to obtain gradient of residuals

%loop to obtain y-intercepts of residuals
c=zeros(1,10);
for i=1:10
    c(i)=Y(i)-(GradientOfResiduals*X(i));
    xInt=(c(i)- b0)/(b1-GradientOfResiduals);
    yInt=(GradientOfResiduals*xInt)+c(i);

    line([X(i) xInt],[Y(i) yInt], 'Color', 'b');
end