Statistics for Environmental Engineers

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A time series of a finite number of successive observations consists of the data Zi, z2, ■■, zt_i, zt, zt+i,…, zn. Our discussion will be limited to discrete (sampled-data) systems where observations occur at equally spaced intervals. The zt may be known precisely, as the price of IBM stock at the day’s closing of the stock market, or they may be measured imperfectly, as the biochemical oxygen demand (BOD) of a treatment plant effluent. The BOD data will contain a component of measurement error; the IBM stock prices will not. In both cases there are forces, some unknown, that nudge the series this way and that. The effect of these forces on the system can be “remembered” to some extent by the process. This memory makes adjacent observations dependent on the recent past. Time series analysis provides tools for analyzing and describing this dependence. The goal usually is to obtain a model that can be used to forecast future values of the same series, or to obtain a transfer function to predict the value of an output from knowledge of a related input.


Time series data are common in environmental work. Data may be monitored frequently (pH every second) or at long intervals and at regular or irregular intervals. The records may be complete, or have missing data. They may be homogeneous over time, or measurement methods may have changed, or some intervention has shifted the system to a new level (new treatment plant or new flood control dam). The data may show a trend or cycle, or they may vary about a fixed mean value. All of these are possible complications in times series data. The common features are that time runs with the data and we do not expect neighboring observations to be independent. Otherwise, each time series is unique and its interpretation and modeling are not straightforward. It is a specialty. Most of us will want a specialist’s help even for simple time series analysis. The authors have always teamed with an experienced statistician on these jobs.

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