So I am computing the LogLikelihood of a FractionalGaussianNoiseProcess, say :
data=RandomFunction[FractionalGaussianNoiseProcess[0.,0.15,0.9],{1,1000,1}][[2,1,1]];
Then,
LogLikelihood[FractionalGaussianNoiseProcess[0.,0.15,0.9],data]
gives something around 900. What if I want to know the individual contributions of each point? One way to do it would be to simply compute the expression:
Differences@Prepend[Table[LogLikelihood[FractionalGaussianNoiseProcess[0.,0.15,0.9],Take[data,i]],{i,1,Length[data]}],0]
This works if I have a data series of only a couple tens of points, but when I reach series which are 100s or 1000s of point it obviously becomes painfully slow since it is recalculating the whole Loglikelihood function each time. If I am not mistaken, the LogLikelihood function for such a process with increments which are not independent, is the linear sum over all previous points, so there must be a way to obtain directly the terms of this sum and thus find the information my previous code does without recomputing everything at for each data point that is added to the series.