Statistics for Environmental Engineers

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The White Noise Model

When the data vary about a fixed level and contain nonnegligible random measurement error, sampling error, etc., the statistical analysis often relies on the model:

yt = П + et


yt = observation at t = 1, 2,…, n П = true (and unobservable) mean value of yi

et = random error, assumed to be independently distributed according to normal distribution with mean 0 and variance a]; that is, et « N (0, a]).

This is called the white noise model because the random errors (et) are white noise.

Figure 54.2a illustrates nj observations made at one level, followed by n2 observations at a new level, and represents a case where the white noise model would be appropriate. A deliberate intervention has caused the change in conditions at time T The intervention model is:

yt = П + 8I + et


yt = value observed at time t T = time the intervention takes place 8 = effect of the intervention, 8 = ц2 — ц1

I = an indicator function: I = 0 before the intervention; I = 1 after the intervention П1 = mean value of yt for t < T


—    I 8

У2 ■■■    .•

(a) White noise model — rat    andom walk model — no

independent variation about fixed fixed mean levels; white noise mean levels.    is assumed negligible.

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