# Statistics for Environmental Engineers

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1.7

For lead time € = 5:

Zt(5) = 10.0 + 0.725( 10.6 — 10.0) = 10.10 As € increases, the forecasts will exponentially converge to:

Z (€ )»n = 10

A statement of the forecast precision is needed. The one-step-ahead forecast error is: e,(1) = z,+1z,(1) = fiz, + a,+1 — фг, = at+1

The a,+i is white noise. The expected value of the forecast error is zero so the forecasts are unbiased. Because a, is white noise, the forecast error et(1) is independent of the history of the process. (If this were not so, the dependence could be used to improve the forecast.) This result also implies that the one-step-ahead forecast error variance is given by:

Var[ e, (1)] = o;

The AR(1) model can be written as the infinite MA model:

z, = a, + фа,-1 + ф2 a,-2 + • • • which leads to the general forecast error:

e,(€) = a,+( + фС11+(-1 + фai+(-2 +■■■ + \$ la,+ 1

This shows that the forecast error increases as the lead € increases. The forecast error variance also increases:

Var[ e,( €)] = о* (1 + ф+ф2 + • + ф^1)2

which simplifies to:

Var[ e, (€)] = О*--1-;:

1 — ф

For long lead times, the numerator becomes 1.0 and:

Var[e,(€)] = о* -i—1 — ф

Because the lag 1 autocorrelation coefficient p = ф for an AR(1) process, this is equivalent to О; = o„212, as used in Chapter 41.

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