Time Series Analysis
A time series consists of a series of values of a variable at successive times in regular intervals. Time series are the bases for the treatment of many engineering questions. In the analysis of time series we try to uncover regularities in the succession
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Time Series Analysis
A time series consists of a series of values of a variable at successive times in regular intervals. Time series are the bases for the treatment of many engineering questions. In the analysis of time series we try to uncover regularities in the succession of the individual measurements and to derive information from these regularities. In addition, we will use time series analysis to develop stochastic models for the simulation of variables.
14.1 Time Series In urban water management, we use many time series which we obtain over long time periods with entirely different frequencies. Examples of time series are: • The development of the population in the distribution area of a waterworks, which is frequently compiled annually • the daily production of water in a water supply enterprise • the measurement of a flow rate in a sewer in 1-min intervals • the measurement of a pollutant concentration on line in 10-min intervals or daily in a mixed sample • the measurement of the rain intensity in 1-, 5- or 10-min intervals • etc. All these time series have in common that they express the condition of a (measured) variable in regular intervals; they are the basis for many statistical evaluations and allow us to derive important information about these variables which serve us in designing and analyzing systems and plants.
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14 Time Series Analysis
14.2 Stationary Time Series A time series is stationary if its expected value is not subject to a trend in the course of time and the dispersion of the individual values remains constant over time. Nonstationary time series are subject to a trend and/or the dispersion of the individual values is not independent of time. We test for stationarity of a time series by obtaining the average value and the variance of the first half of the time series and comparing it with similar results of the second half. Example 14.1: Stationary and nonstationary time series The development of the population of a city over the years is an nonstationary time series, the expected value typically increases (or decreases) continuously. The treatment performance of a wastewater treatment plant is frequently a stationary time series. Its value changes from day to day; its expected value and its dispersion remain, however, constant over longer time periods. Example 14.2: Test for stationarity of a time series Figure 14.1 shows the elements of a (simulated) time series over 360 days. Is this series stationary? For the test, the time series is split in the middle and average and standard deviation of these partial time series are obtained with the following code in BM: {The time series is read in as #x(i)} STARTTIME = 0 STOPTIME = 0 ; Initialization is sufficient DT = 1 ; Irrelevant, STARTTIME = STOPTIME low[1..180] = #x(i) ; Elements of first half of time series high[181..360] = #x(i) ; Elements of second half of time series m_low = arraymean(low[*]) ; Average of low Result: 3.05 ± 0.09
Fig. 14.1 Example of an instationary time series
14.3 Case study: Yearly Variation of
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