# Statistics for Environmental Engineers

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### The Autocorrelation Function

The autocorrelation function is the fundamental tool for diagnosing the structure of a time series. The correlation of two variables (x and y) is:

r (x, y) =

X(Xi — x)2X(yt — y )2

The denominator scales the correlation coefficient so -1 < r (x, y) < 1.

In a time series, adjacent and nearby observations are correlated, so we want a correlation of zt and zt-k, where k is the lag distance, which is measured as the number of sampling intervals between the observations. For lag = 2, we correlate z1 and z3, z2 and z4, etc. The general formula for the sample autocorrelation at lag k is:

n

X (zt — ‘z)(zt-k — z)

r = >=*+1_

rk =    n

X(zt — z)2

t=1

where n is the total number of observations in the time series. The sample autocorrelation (rk) estimates the population autocorrelation (pk). The numerator is calculated with a few less terms than n; the denominator is calculated with n terms. Again, the denominator scales the correlation coefficient so it falls in the range -1 < rx < 1.

The autocorrelation function is the collection of rk’s for k = 0, 1, 2,…, m, where m is not larger than about n/4. In practice, at least 50 observations are needed to estimate the autocorrelation function (ACF).

First-Order Autotregressive Time Series

(b) z, = -0.7zM+a1

(a) z,= 0.7zM+ai

 +1-| +1l Скачать в pdf «Statistics for Environmental Engineers» P. M. Berthouex, L. C. BrownISBN 1-56670-592-4Метки учебные пособия