G.E.P. Box’s scientific contributions

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Publications (1)


Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models
  • Article

January 1970

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137 Reads

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2,202 Citations

G.E.P. Box

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D.A. Pierce

Many statistical models, and in particular autoregressive-moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrelated sequence of errors. If the parameters are known exactly, this random sequence can be computed directly from the observations; when this calculation is made with estimates substituted for the true parameter values, the resulting sequence is referred to as the "residuals," which can be regarded as estimates of the errors. If the appropriate model has been chosen, there will be zero autocorrelation in the errors. In checking adequacy of fit it is therefore logical to study the sample autocorrelation function of the residuals. For large samples the residuals from a correctly fitted model resemble very closely the true errors of the process; however, care is needed in interpreting the serial correlations of the residuals. It is shown here that the residual autocorrelations are to a close approximation representable as a singular linear transformation of the autocorrelations of the errors so that they possess a singular normal distribution. Failing to allow for this results in a tendency to overlook evidence of lack of fit. Tests of fit and diagnostic checks are devised which take these facts into account.

Citations (1)


... Time series anomaly detection has been extensively studied, with massive of statistical, machine learning, and deep learning methods being proposed. The classical statistical methods learn the statistical characteristics of time series data, such as the autoregressive integrated moving average (ARIMA) approach (Box & Pierce, 1970). These methods are computation-lightweight but non-effective for complex multivariate time series anomaly detection. ...

Reference:

GDformer: Going Beyond Subsequence Isolation for Multivariate Time Series Anomaly Detection
Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models
  • Citing Article
  • January 1970