% Data from presentation "Statistical View of Regression"

% Hint: Highlight code and press F9 to run in workspace


clear all   % Clears all data

% The (x,y) data
x =[
    0.0532
    0.5323
   -0.8114
    0.6316
   -0.6111
   -0.1549
   -0.5356
    0.1772
   -0.9918
    0.2546
    0.4079
    0.7411
    0.9125
   -0.2603
   -0.4453]
Y =[
    7.8644
    5.4082
    1.1687
    6.7322
    4.8942
    8.6170
    6.4206
    6.8425
    0.2626
    6.6100
    7.1609
    4.5558
    3.1863
    6.1780
    5.6934]

% Plot the data with a scatterplot
figure,
scatter(x,Y)
title('Scatter Plot')
xlabel('x')
ylabel('y(x)')

%% Estimate Parameters

%% First order model (line fit)

% Get data into matrix form
    X = [ones(n,1),x]

% Estimate parameters b0,b1
    b = inv(X'*X)*X'*Y

% Built in Matlab commands
    % Backslash
        b = X\Y
        
    % regress function
        b = regress(Y,X)
    
        % to see more options  with regress function  
        help regress  
        % or search in help window

% Calculate Residuals
    Yhat = X*b
    e = Y - Yhat
    
    % Plot residuals
        figure, scatter(x,e)
        hold on, plot([-1,1],[0,0],'k')    % Line at 0
        title('Residual Scatter Plot - Linear')
        xlabel('x')
        ylabel('e(x)')
    
%% Second order model (quadratic fit)
X2 = [ones(n,1),x,x.^2]
b2 = X2\Y
Yhat2 = X2*b2
e2 = Y-Yhat2

        figure, scatter(x,e2)
        hold on, plot([-1,1],[0,0],'k')    % Line at 0
        title('Residual Scatter Plot - Quadratic')
        xlabel('x')
        ylabel('e(x)')
        
%% Plotting fits
fit1=@(x) b(1) + b(2)*x
fit2=@(x) b2(1) + b2(2)*x + b2(3)*x.^2
t=-1:.01:1;     % Range for plotting
figure, scatter(x,Y)
hold on, plot(t,fit1(t))
plot(t,fit2(t),'r')
title('Fitted functions')
legend('Obs','Linear','Quadratic')

%% Introduce curve fitting tool
cftool