clc
format shortG

%% Part 1: first data set  
% data is given by an mx2 matrix with x-values in column 1 
% and y-values in column 2
data1=[ 1 8 ; 2 6 ; 4 5 ; 5 7 ; 8 16 ];
% extracts the x-values in column 1
x=data1(:,1);
% extracts the y-values in column 2
y=data1(:,2);


%% first model: linear y=a0+a1*x
% constructs the design matrix which is the coefficient matrix of the system
% Ax=b obtained from plugging in data points into the model
A=[1 x(1) ; 
   1 x(2) ; 
   1 x(3) ; 
   1 x(4) ; 
   1 x(5) ]
% b vector is just the y-values
b=y;
% gets xhat least squares solution using fact that A has linearly
% independent columns so A^T*A is invertible - note A' is MATLAB transpose
xhat=inv(A'*A)*A'*b;
% extracts alpha and beta values from the xhat vector
a0=xhat(1);
a1=xhat(2);
% defines the function for plotting
% WARNING: you must use .* for 'times' and .^ for 'exponent' or else 
% MATLAB will give an error code
f1=@(x) a0+a1.*x;
% compute the least squares error ||A*xhat-b||
errorLine=sqrt((A*xhat-b)'*(A*xhat-b))
% draws figure
figure(1)
fplot(f1,[0 8])
hold on
plot(x,y,'.','markersize',28)
hold off


%% second model: full quadratic y=a0+a1*x+a2*x^2
% [STUDENT FILL IN:] student should construct the design matrix which 
% is the coefficient matrix of the system Ax=b obtained from plugging 
% in data points into the model y=a0+a1*x+a2*x^2
A=[ '_fill_in_matrix_here_'
    



        ]
% copy-pasted/modified code from part 1
b=y;
xhat=inv(A'*A)*A'*b;
a0=xhat(1);
a1=xhat(2);
a2=xhat(3);
% defines the function for plotting
% WARNING: you must use .* for 'times' and .^ for 'exponent' or else 
% MATLAB will give an error code
f2=@(x) a0+a1.*x+a2.*x.^2;
% least squares error ||A*xhat-b||
errorQuad=sqrt((A*xhat-b)'*(A*xhat-b))
% draws figure
figure(2)
fplot(f2,[0 8])
hold on
plot(x,y,'.','markersize',28)
hold off

%% third model: [STUDENT FILL IN:] _model_name_here_ _model_here_
% [STUDENT FILL IN:] student shout construct the design matrix which 
% is the coefficient matrix of the system Ax=b obtained from plugging 
% in data points into the model
A=[ '_fill_in_matrix_here_'
    


        ]
b=y;
xhat=inv(A'*A)*A'*b;
% [STUDENT FILL IN:] you may need to add/delete variables depending on how
% many terms your model has, e.g. three terms needs up to a2=xhat(3)
a0=xhat(1);
a1=xhat(2);
a2=xhat(3);
% [STUDENT FILL IN:] define the function for plotting
% WARNING: you must use .* for 'times' and .^ for 'exponent' or else 
% MATLAB will give an error code
f3=@(x) '_fill_in_model_here_' ;
% least squares error is ||A*xhat-b||
errorStudentModelforData1=sqrt((A*xhat-b)'*(A*xhat-b))
% draws figure
figure(3)
fplot(f3,[0 8])
hold on
plot(x,y,'.','markersize',28)
hold off

%% answer questions: [STUDENT FILL IN:] answer questions as comments
% Q1: which model has the best error (smallest error value)?
% Ans1:
% Q2: what does the value of the error ||A*hat-b|| represent geometrically?
% Ans2:

%% Part 2: student created data set must have at least 6 data points
% [STUDENT FILL IN:] create a data2 matrix where the data is 
% given by an mx2 matrix with m>5 (at least 6 data points)
% with x-values in column 1 
% and y-values in column 2
data2=[ '_fill_in_data_here' ];
% extracts the x-values in column 1
x=data2(:,1);
% extracts the y-values in column 2
y=data2(:,2);

%% fourth model: [STUDENT FILL IN:]  _model_name_here_ _model_here_
% [STUDENT FILL IN:] student should construct the design matrix which 
% is the coefficient matrix of the system Ax=b obtained from plugging 
% in data points into the model
A=[ '_fill_in_matrix_here_'
    



        ]
b=y;
xhat=inv(A'*A)*A'*b;
% [STUDENT FILL IN:] you may need to add/delete variables depending on how
% many terms your model has
a0=xhat(1);
a1=xhat(2);
a2=xhat(3);
% [STUDENT FILL IN:] define the function for plotting
% WARNING: you must use .* for 'times' and .^ for 'exponent' or else 
% MATLAB will give an error code
f4=@(x) '_fill_in_model_here_' ;
% least squares error is ||A*xhat-b||
errorStudentModelforData2=sqrt((A*xhat-b)'*(A*xhat-b))
% draws figure
% [STUDENT FILL IN:] You may need to modify the range of the fplot second
% argument [0 8] if your x-values go outside the range 0 <= x <= 8
figure(4)
fplot(f4,[0 8])
hold on
plot(x,y,'.','markersize',28)
hold off