[R] comparing predicted sequence A'(t) to observed sequence A(t)

[Spencer Graves]

      What do you mean by the following: 

      A(t) = B(t) + C(t) - D(t)? 

      Since you speak of regressing predicted against actual A(t), I 
gather this is not what you mean. 

      Another question:  Do you have numbers <-0 for either predicted or 
actual A(t)?  If yes but only a very few, I might replace the 0's by 0.5 
and any negatives by 0.25, take their logarithms, then try acf, pacf, 
ar, arima(..., xreg=A.pred), etc. 

      There are doubtless better methods.  However, if I had to have an 
answer today, I think I'd try this, then discuss implications and 
limitations.  If I needed a more sophisticated answer and I had a few 
weeks or months to work on it, I might develop some way to simulate a 
process that seemed to describe what I thought generated these numbers 
and compare simulated results with actual, under a variety of 
hypotheses, obtaining various kinds of p-values, etc. 

      hope this helps. 
      spencer graves

Suresh Krishna wrote:

>
> Hi,
>
> I have a question that I have not been succesful in finding a 
> definitive answer to; and I was hoping someone here could give me some 
> pointers to the right place in the literature.
>
> A. We have 4 sets of data, A(t), B(t), C(t), and D(t). Each of these 
> consists of a series of counts obtained in sequential time-intervals: 
> so  for example, A(t) would be something like:
>
> Count A(t):  25,    28,    26,   34   ......
> Time (ms):  0-10, 10-20, 20-30, 30-40 .......
>
> Each count in the series A(t) is obtained by summing the total number 
> of observed counts over multiple (say 50), independent repetitions of 
> that time-series. These counts are generally known to be Poisson 
> distributed, and the 4 processes A(t), B(t), C(t) and D(t) are 
> independent of each other.
>
> B. It appears on visual observation that the following relationship 
> holds; and such a relationship would also be expected on mechanistic 
> considerations.
>
> A(t) = B(t) + C(t) - D(t)
>
> We now want to test this hypothesis statistically.
>
> Because successive counts in the sequence are likely to be correlated, 
> isnt it true that none of these methods are valid ? Perhaps for other 
> reasons as well ?
>
> a)Doing a chi-squared test to see if the predicted curve for A(t) 
> deviates significantly from the observed A(t); this also seems to not 
> take the variability of the predicted curve into account.
>
> b)Doing a regression of the predicted values of A(t) against the 
> actual values of A(t) and checking for deviations of slope from 1 and 
> intercept from 0 ? Here, in addition to lack of independence, the fact 
> that X-values are not fixed (i.e. are variable) and the fact that X 
> and Y are Poisson distributed counts should also be taken into 
> account, right ?
>
> I would be very grateful if someone could point me to methods to 
> handle this kind of situation, or where to look for them. Is there 
> something in the time-series literature, for instance ?
>
> Thanks !!
>
> Suresh
>
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