Statistics for Environmental Engineers

Скачать в pdf «Statistics for Environmental Engineers»

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»