# Statistics for Environmental Engineers

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An expanded mechanistic model may still have a simple discrete time-series-like form. The mechanistic model for a CSTR with a first-order chemical reaction, with inputs and outputs that are functions of time, is:

v ^dT = Qx{ r) — Qy{ r) — kyy{ t)

which can be written as :

yt =

1.

1 + т + кт X

т

yt = 01 Xt + 02 yt_i + a,

For к = 1 and т = 2, this becomes yt = 0.111 xt + 0.4yt_1 + a,.

Imagine now a process that has some characteristics of the CSTR reactor but which is more complicated. Perhaps the mixing is not efficient, or there is some time delay in the system, or the input contains interfering impurities in random amounts, or that the temperature changes so the reaction rate coefficient (k) is not constant. This makes it more difficult to write an adequate differential equation. Still, it should be possible to derive a dynamic model in terms of the inputs and outputs. Instead of simply yt= 01xt+ 02yt_1 + at, we might add more terms, perhaps a time delay (xt_d) or more memory of past conditions (zt_2 or at_2), to get yt = 0jxt + 02xt_d + 03yt_j + 04yt_2 + at — фat_1. With some gentle persuasion the time series data will reveal the necessary terms. A tentative model is fitted to the data. The residuals of the model are examined. If they are not random and independent, they contain additional information and the model is modified and refitted. This iterative process continues until an adequate model of the process emerges.

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