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2.1.2 Trends A trend, or long-term change in the data, is much more clear at this stage. The analysis procedure quantifies the severity of the trend and indicates whether or not it is statistically significant. This to is done by finding a straight line which slopes over time and fits ~ through the data better in a least squares sense than any other ~J possible line. A test is made which shows if the slope of this best- ~ fit line is significantly different from zero. If the slope is statistically insignificant, it can be inferred that the data are not changing in a consistent up or down direction over time. When a trend is found, it must be ~emoved from the data set. 2.1.3 Serial Correlation Another important structural property of time series which the analysis must inspect is the correlation between consecutive data values. Occasionally in annual and especially in monthly or daily streamflow data there is a strong dependence on past values. This serial dependence as it is called must be investigated and removed if necessary. It is quantified by a procedure very similar to regres- sion analysis. Each value in the data set is considered dependent upon the values preceding it one at a time until the maximum likely I~emory" of the process has been studied, When only one past value is considered, the model structure is called first-order linear autoregressive. This is also one form of a Markov model since the process is essentially a Markov chain. Higher order dependence can also be found and sometimes second-or third-order models are indi- cated. In these cases each value in the series is a function of two or three past values. Least squares estimates are used to evaluate the relationship between the variables. This shows not only the degree of correlation between past values but also the coefficients of the equation relating them. By testing the strength of the dependence on the past values, the analysis indicates which order model is most applicable. The actual calculations in this analysis proceed from the residual series of equation (5). The. stationary y series may exhibit serial correlation if adjacent values show a significant degree of correla- tion. In order words, the values of Yi, at time i gives some information about the value of Yi+l in time i+l. Note that since the y series is now stationary in the mean and vari- ance, the subscript for the year is not retained. The correlation among subsequent values can be investigated by computing autocorrela- tion coefficients at lags k and incorporating the values into the first-order Markov model, 8