# Vectorize linear trend estimates

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David Wang on 24 May 2013
I have a few thousand time series of the same length, and for each I'd like to obtain its linear trend and the corresponding p-value. I use for loops and estimate the trend/p-value for each time series sequentially using the robustfit function.
The whole thing doesn't take much time. But still I wonder whether it is possible to vectorize the trend estimates. I understand I can get all linear trends at once using the backslash operator. But that doesn't give p-values.

Matt J on 24 May 2013
Edited: Matt J on 24 May 2013
I understand I can get all linear trends at once using the backslash operator. But that doesn't give p-values.
First, it might not be a good idea to use backslash straightforwardly. Polynomial fitting has certain numerically sensitive properties. POLYFIT uses a QR decomposition:
[Q,R] = qr(V,0);
ws = warning('off','all');
p = R\(Q'*y); % Same as p = V\y;
warning(ws);
if size(R,2) > size(R,1)
warning(message('MATLAB:polyfit:PolyNotUnique'))
elseif warnIfLargeConditionNumber(R)
if nargout > 2
warning(message('MATLAB:polyfit:RepeatedPoints'));
else
warning(message('MATLAB:polyfit:RepeatedPointsOrRescale'));
end
end
But anyway, once you've found a solution to the system V*x=Y, it seems to me that you could vectorize the p-values by analyzing the residuals,
R=V*x-Y; %residuals
Rmean=mean(R,2);
Rstd=std(R,0,2);
R=bsxfun(@minus,R,Rmean); %Normalize the residuals
R=bsxfun(@rdivide,R,Rstd);
pvalues=sum(R>=quantile,2);