matlab matrix declaration

Question

I am struggling with declaring some more complex matrices in matlab, perhaps you could help me out, I have an array of $T$ values / lets call it $y = [y_0, \hdots, y_T]$ (its a digital signal representing a sound).

I am using formula:

\begin{equation}
    y_t= a_0 + \sum_{i=1}^p  (a_i y_{t-i} + \epsilon_t), t \geq p,
\end{equation}

in order to create a synthetic signal based on the one give using only previous $p$ values of $y_t$, where $p$ is significantly smaller than y. What I have to do, is find those $a_0, \vdots, a_p$ parametres in order to use LSE method.

Now here is what I need you guys to help me with: How do I create a matrix that looks like this:

\begin{equation}
    M =
    \begin{bmatrix}
        1 & 0 & 0 & 0 & 0 & \hdots & 0 \\
        1 & y_0 & 0 & 0 & 0 & \hdots & 0 \\
        1 & y_1 & y_0 & 0 & 0 & \hdots & 0 \\
        1 & y_2 & y_1 & y_0 & 0 & \hdots & 0 \\ 
        \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
        1 & y_{T-1} & y_{T-2} & \hdots & \hdots & \hdots & y_{T-p} \\
    \end{bmatrix}
    \in R^{T+1xp+1}
\end{equation}

Thanks for any help

edit: how to format LaTeX here?

Answer 1

You seem to need a matrix that represents a sort of convolution. In Matlab the toeplitz function is relevant here.

See the following example

>> y=[1 2 3 4 5 6 7];
>> toeplitz(y,[y(1) zeros(1,length(y)-1)])

ans =

 1     0     0     0     0     0     0
 2     1     0     0     0     0     0
 3     2     1     0     0     0     0
 4     3     2     1     0     0     0
 5     4     3     2     1     0     0
 6     5     4     3     2     1     0
 7     6     5     4     3     2     1

So your code should look like the follwing

 M = [ones(length(y),1)  toeplitz(y,[y(1) zeros(1,length(y)-1)]) ];
 M = M(:,1:p+1);  

Answer 2

It's really hard to see what's going on because of the formatting of the quesiton but how about this:

M = zeros(length(y) + 1);

M(1) = 1;

for row = 2:length(y)+1
    M(1:row) = [1 y(row-1:-1:1)];
end