# Fitting time series when errors are correlated

I am working with software fro online analysis of PHOS/ALICE detector data
for the high level trigger (HLT).

Our detector response is shaped with a second order semi Gaussian
shaper with a rise time of ~4us. The signal is over sampled (That is errors are correlated). Im trying to find the ROOT packages to anylse my data, but the TMinuit
package, and the other packages in ROOT doesnt seems to taker into account correlated errors.

Is there any ROOT packages to handle the general case of the optimum fit
problem. That is, when errors might be correlated and the model for the
detetor response is nonlinear.

Fortunatly, one can most likely assume that the measurement correlation matrix
is Hermitan symetric.

I know one is doing something similar for track reconstruction where the scattering angle is is correalted with the spatial dispacement.

Regards
Per Thomas

Hi Per Thomas,

This statement is not correct ! Minuit is a general function minimizer . It
is up to the user to specify the correct objective function which can
among others contain terms for correlation between data points .

Eddy

Ok, I realize that have to be alittle bit more specific abouth my question.
May quastion is.

Is there any rootpackages available to handle the genral
Leas Mean Square (LMS) problem. Where the word "Mean"
implies that one is working with the least square fit based on the
knowledge of the statistical properties of the process.
(Same as Leas square fit in the special case when error are uncorrelated).

Per Thomas

Hi,

Let me try to write down what I think you are talking about :

You have a data set (x_ij, y_i, e_i) for i=1,N points
where there are j=1,M control variables and a covariance
structure so that

``````      E(e_l e_k) = Cov_lk  where E() means expectation value
``````

You want to determine a function yFit(par,x) , assuming Gaussian
statistics … , this will lead to the following objective function:

``````     (y-yFit)^T Cov (y-yFit)
``````

Plug this expression in a function for Minuit and minimize wrt the
parameters par . The matrix /vector manipulations can be done
through TVectorD/TMatrixD …

Eddy

ps Googling “fitting auto-correlated time series”, you will find the
following interesting write-up:

`` cran.r-project.org/doc/contrib/ Fox-Companion/appendix-timeseries-regression.pdf``