linear regression with multiple variables in matlab, formula and code do not match

Question

I have the following datasets:

X

X =

1.0000    0.1300   -0.2237
1.0000   -0.5042   -0.2237
1.0000    0.5025   -0.2237
1.0000   -0.7357   -1.5378
1.0000    1.2575    1.0904
1.0000   -0.0197    1.0904
1.0000   -0.5872   -0.2237
1.0000   -0.7219   -0.2237
1.0000   -0.7810   -0.2237
1.0000   -0.6376   -0.2237
1.0000   -0.0764    1.0904
1.0000   -0.0009   -0.2237
1.0000   -0.1393   -0.2237
1.0000    3.1173    2.4045
1.0000   -0.9220   -0.2237
1.0000    0.3766    1.0904
1.0000   -0.8565   -1.5378
1.0000   -0.9622   -0.2237
1.0000    0.7655    1.0904
1.0000    1.2965    1.0904
1.0000   -0.2940   -0.2237
1.0000   -0.1418   -1.5378
1.0000   -0.4992   -0.2237
1.0000   -0.0487    1.0904
1.0000    2.3774   -0.2237
1.0000   -1.1334   -0.2237
1.0000   -0.6829   -0.2237
1.0000    0.6610   -0.2237
1.0000    0.2508   -0.2237
1.0000    0.8007   -0.2237
1.0000   -0.2034   -1.5378
1.0000   -1.2592   -2.8519
1.0000    0.0495    1.0904
1.0000    1.4299   -0.2237
1.0000   -0.2387    1.0904
1.0000   -0.7093   -0.2237
1.0000   -0.9584   -0.2237
1.0000    0.1652    1.0904
1.0000    2.7864    1.0904
1.0000    0.2030    1.0904
1.0000   -0.4237   -1.5378
1.0000    0.2986   -0.2237
1.0000    0.7126    1.0904
1.0000   -1.0075   -0.2237
1.0000   -1.4454   -1.5378
1.0000   -0.1871    1.0904
1.0000   -1.0037   -0.2237

theta

0
0
0

y

y =

  399900
  329900
  369000
  232000
  539900
  299900
  314900
  198999
  212000
  242500
  239999
  347000
  329999
  699900
  259900
  449900
  299900
  199900
  499998
  599000
  252900
  255000
  242900
  259900
  573900
  249900
  464500
  469000
  475000
  299900
  349900
  169900
  314900
  579900
  285900
  249900
  229900
  345000
  549000
  287000
  368500
  329900
  314000
  299000
  179900
  299900
  239500

The X set represents values for multiple variable regression, the first colum stands for X0, second X1; and so on.

The implementation formula is something like:

I have implemented a matlab code which is:

for i=1:size(theta,1)
    h=X*theta;
    sumE=sum((h-y).*X(:,i));
    theta(i)=theta(i)-alpha*(1/m)*sumE;
end

which is inside a for loop going from 1 to a n number of iterations (the value of m is not relevant, it can be set up to 40 for example). The problem is that even though the code works and the result is the one expected, when I submit it to a online checking program it appears that my results are wrong. The reason is that I should update theta simultaneously.

I have gotten the following Matlab code from Internet:

h = X*theta;
theta = theta - alpha / m * (X'*(h - y));

when I run the Internet solution it gives me almost the same answer as mine, with only a subtle difference in the 6th decimal position. When I submit that answer to the online program it is fully accepted, but I was wondering where the summation goes? in the formula explicitly indicates a summation which is no longer in the Internet solution. Maybe both codes are fine, but I do not know if the Internet author has made some linear algebra trick. Any help?

Thanks

Answer 1

Their code simultaneously updates theta. Your code iterates through the rows of theta using newer values of theta to regenerate the h which is used in updating the later rows of theta. I'd bet heavily that this is the difference.

---

For clarity, let's keep track of each iteration of theta in a matrix. Their code for iteration j is:

h = X*theta(:,j);
theta(:,j+1) = theta(:,j) - alpha / m * (X'*(h - y));

On the other hand, your code would be:

for i=1:size(theta,1)
    my_mismatched_theta = [theta(1:i-1, j+1); theta(i:end, j)];
    h=X * my_mismatched_theta;
    sumE=sum((h-y).*X(:,i));
    theta(i,j)=theta(i,j)-alpha*(1/m)*sumE;
end

It doesn't simultaneously update theta in one step. You're using newer versions of theta (i.e. theta(:,j+1) ) to generate h when updating the later rows of theta.

something you should try

Change your code to what I have below and see if you then get the same answer:

h=X*theta;   %placed outside of loop so it doesn't get updated by new theta values
for i=1:size(theta,1)      
  sumE=sum((h-y).*X(:,i));
  theta(i)=theta(i)-alpha*(1/m)*sumE;
end

Your algorithm may converge to the same point as theirs in this case, but there's a chance that the sort of casecade updating you're doing creates weirdness in other cases. Who knows.

Answer 2

I am not sure if I understood your question, but the formula you copied from internet is X'(h-y). Note that there is a tranposition signal after X! So, this is a matrices product. Your sum (your loop) is replaced by this matrices product.