Statistics for Environmental Engineers

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The test examines whether the first-order autocorrelation parameter p is zero. In the case where p = 0, the errors are independent. The test statistic is:


n


et e,-1 )2 D = -2-


n


!(ei )2


i=1


where the et are the residuals determined by fitting a model using least squares.


Durbin and Watson (1971) obtained approximate upper and lower bounds (dL and dU) on the statistic D. If dL < D < dU, the test is inconclusive. However, if D > dU, conclude p = 0; and if D < dL, conclude p > 0. A few Durbin-Watson test bounds for the 0.05 level of significance are given in Table 41.3. Note that this test is for positive p. If p < 0, a test for negative correlation is required; the test statistic to be used is 4 — D, where D is calculated as before.


Results of Trend Analysis of Data in Figure 41.3


Result


Time Series A


Time Series B


Generating model


yt = 10 + at


yt = 10 + 0.8et_1 + at


Fitted model


y t = 9.98 + 0.0051


y t = 9.71 + 0.033Г


Confidence interval of Д


[ -0.012 to 0.023]


[0.005 to 0.061]


Standard error of Д


0.009


0.014


Conclusion regarding в


в = 0


в > 0


Durbin-Watson statistic


2.17


0.44

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