# Prediction variance The prediction variance of the forecast for lead times l=1,...,maxLead is computed given theoretical autocovariances. ```r PredictionVariance(r, maxLead = 1, DLQ = TRUE) ``` ## Arguments - `r`: the autocovariances at lags 0, 1, 2, ... - `maxLead`: maximum lead time of forecast - `DLQ`: Using Durbin-Levinson if TRUE. Otherwise Trench algorithm used. ## Details Two algorithms are available which are described in detail in McLeod, Yu and Krougly (2007). The default method, DLQ=TRUE, uses the autocovariances provided in r to determine the optimal linear mean-square error predictor of order length(r)-1. The mean-square error of this predictor is the lead-one error variance. The moving-average expansion of this model is used to compute any remaining variances (McLeod, Yu and Krougly, 2007). With the other Trench algorithm, when DLQ=FALSE, a direct matrix representation of the forecast variances is used (McLeod, Yu and Krougly, 2007). The Trench method is exact. Provided the length of r is large enough, the two methods will agree. ## Returns vector of length maxLead containing the variances ## References McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software. ## Author(s) A.I. McLeod ## See Also `predict.Arima`, `TrenchForecast`, `exactLoglikelihood` ## Examples ```r #Example 1. Compare using DL method or Trench method va<-PredictionVariance(0.9^(0:10), maxLead=10) vb<-PredictionVariance(0.9^(0:10), maxLead=10, DLQ=FALSE) cbind(va,vb) # #Example 2. Compare with predict.Arima #general script, just change z, p, q, ML z<-sqrt(sunspot.year) n<-length(z) p<-9 q<-0 ML<-10 #for different data/model just reset above out<-arima(z, order=c(p,0,q)) sda<-as.vector(predict(out, n.ahead=ML)$se) # phi<-theta<-numeric(0) if (p>0) phi<-coef(out)[1:p] if (q>0) theta<-coef(out)[(p+1):(p+q)] zm<-coef(out)[p+q+1] sigma2<-out$sigma2 r<-sigma2*tacvfARMA(phi, theta, maxLag=n+ML-1) sdb<-sqrt(PredictionVariance(r, maxLead=ML)) cbind(sda,sdb) # # #Example 3. DL and Trench method can give different results # when the acvf is slowly decaying. Trench is always # exact based on a finite-sample. L<-5 r<-1/sqrt(1:(L+1)) va<-PredictionVariance(r, maxLead=L) vb<-PredictionVariance(r, maxLead=L, DLQ=FALSE) cbind(va,vb) #results are slightly different r<-1/sqrt(1:(1000)) #larger number of autocovariances va<-PredictionVariance(r, maxLead=L) vb<-PredictionVariance(r, maxLead=L, DLQ=FALSE) cbind(va,vb) #results now agree ```

PredictionVariance function