Introduction to Time Series Modeling 2.1 INTRODUCTION Time series modeling is the analysis of a temporally distributed sequence of data or the synthesis of a model for prediction in which time is an independent variable. In many cases, time is not actually used to predict the magnitude of a random variable such as peak discharge, but the data are ordered by time. Time series are analyzed for a number of reasons. One might be to detect a trend due to another random variable. For example, an annual maximum flood series may be analyzed to detect an increasing trend due to urban development over all or part of the period of record. Second, time series may be analyzed to formulate and calibrate a model that would describe the time-dependent characteristics of a hydrologic variable. For example, time series of low-flow discharges might be analyzed in order to develop a model of the annual variation of base flow from agricultural watersheds. Third, time series models may be used to predict future values of a time-dependent variable. A con- tinuous simulation model might be used to estimate total maximum daily loads from watersheds undergoing deforestation. Methods used to analyze time series can also be used to analyze spatial data of hydrologic systems, such as the variation of soil moisture throughout a watershed or the spatial transport of pollutants in a groundwater aquifer. Instead of having measurements spaced in time, data can be location dependent, possibly at some equal interval along a river or down a hill slope. Just as time-dependent data may be temporally correlated, spatial data may be spatially correlated. The extent of the correlation or independence is an important factor in time- and space-series model- ing. While the term time series modeling suggests that the methods apply to time series, most such modeling techniques can also be applied to space series. Time and space are not causal variables; they are convenient parameters by which we bring true cause and effect into proper relationships. As an example, evapotranspiration is normally highest in June. This maximum is not caused by the month, but because insolation is highest in June. The seasonal time of June can be used as a model parameter only because it connects evapotranspiration and insolation. In its most basic form, time series analysis is a bivariate analysis in which time is used as the independent or predictor variable. For example, the annual variation of air temperature can be modeled by a sinusoidal function in which time determines the point on the sinusoid. However, many methods used in time series analysis differ from the bivariate form of regression in that regression assumes independence among the individual measurements. In bivariate regression, the order of the x-y data pairs is not important. Conversely, time series analysis recognizes a time dependence and 2 L1600_Frame_C02 Page 9 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC attempts to use this dependence to improve either the understanding of the underlying physical processes or the accuracy of prediction. More specifically, time series are analyzed to separate the systematic variation from the nonsystematic variation in order to explain the time-dependence characteristics of the data where some of the variation is time dependent. Regression analysis is usually applied to unordered data, while the order in a time series is an important characteristic that must be considered. Actually, it may not be fair to compare regression with time series analysis because regression is a method of calibrating the coefficients of an explicit function, while time series analysis is much broader and refers to an array of data analysis techniques that handle data in which the independent variable is time (or space). The principle of least squares is often used in time series analysis to calibrate the coefficients of explicit time-dependent models. A time series consists of two general types of variation, systematic and nonsys- tematic. For example, an upward-sloping trend due to urbanization or the annual variation of air temperature could be modeled as systematic variation. Both types of variation must be analyzed and characterized in order to formulate a model that can be used to predict or synthesize expected values and future events. The objective of the analysis phase of time series modeling is to decompose the data so that the types of variation that make up the time series can be characterized. The objective of the synthesis phase is to formulate a model that reflects the characteristics of the systematic and nonsystematic variations. Time series modeling that relies on the analysis of data involves four general phases: detection, analysis, synthesis, and verification. For the detection phase, effort is made to identify systematic components, such as secular trends or periodic effects. In this phase, it is also necessary to decide whether the systematic effects are significant, physically and possibly statistically. In the analysis phase, the systematic components are analyzed to identify their characteristics, including magnitudes, form, and duration over which the effect exists. In the synthesis phase, the informa- tion from the analysis phase is used to assemble a model of the time series and evaluate its goodness of fit. In the final phase, verification, the model is evaluated using independent data, assessed for rationality, and subjected to a complete sensi- tivity analysis. Poor judgment in any of the four phases will result in a less-than- optimum model. 2.2 COMPONENTS OF A TIME SERIES In the decomposition of a time series, five general components may be present, all of which may or may not be present in any single time series. Three components can be characterized as systematic: secular, periodic, and cyclical trends. Episodic events and random variation are components that reflect sources of nonsystematic variation. The process of time series analysis must be viewed as a process of identifying and separating the total variation in measured data into these five com- ponents. When a time series has been analyzed and the components accurately characterized, each component present can then be modeled. L1600_Frame_C02 Page 10 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC 2.2.1 S ECULAR T RENDS A secular trend is a tendency to increase or decrease continuously for an extended period of time in a systematic manner. The trend can be linear or nonlinear. If urbanization of a watershed occurs over an extended period, the progressive increase in peak discharge characteristics may be viewed as a secular trend. The trend can begin slowly and accelerate upward as urban land development increases with time. The secular trend can occur throughout or only over part of the period of record. If the secular trend occurs over a short period relative to the length of the time series, it is considered an abrupt change. It may appear almost like an episodic event, with the distinction that a physical cause is associated with the change and the cause is used in the modeling of the change. If the secular trend occurs over a major portion or all of the duration of the time series, it is generally referred to as a gradual change. Secular trends are usually detected by graphical analyses. Filtering tech- niques can be used to help smooth out random functions. External information, such as news reports or building construction records, can assist in identifying potential periods of secular trends. Gradual secular trends can be modeled using typical linear and nonlinear func- tional forms, such as the following: linear: y = a + bt (2.1a) polynomial: y = a + bt + ct 2 (2.1b) power: y = at b (2.1c) reciprocal: (2.1d) exponential: y = ae − bt (2.1e) logistic: (2.1f) in which y is the time series variable; a , b , and c are empirical constants; and t is time scaled to some zero point. In addition to the forms of Equations 2.1, composite or multifunction forms can be used (McCuen, 1993). Example 2.1 Figure 2.1 shows the annual peak discharges for the northwest branch of the Ana- costia River at Hyattsville, Maryland (USGS gaging station 01651000) for water years 1939 to 1988. While some development occurred during the early years of the record, the effect of that development is not evident from the plot. The systematic variation associated with the development is masked by the larger random variation that is inherent to flood peaks that occur under different storm events and when the antecedent soil moisture of the watershed is highly variable. During the early 1960s, y abt = + 1 y a e bt = + − 1 L1600_Frame_C02 Page 11 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC development increased significantly with the hydrologic effects apparent in Figure 2.1. The peak flows show a marked increase in both the average and variation of the peaks. To model the secular variation evident in Figure 2.1, a composite model (McCuen, 1993) would need to be fit with a “no-effect” constant used before the mid-1950s and a gradual, nonlinear secular trend for the 1970s and 1980s. After 1980, another “no-effect” flat line may be appropriate. The specific dates of the starts and ends of these three sections of the secular trend should be based on records indicating when the levels of significant development started and ended. The logistic model of Equation 2.1f may be a reasonable model to represent the middle portion of the secular trend. 2.2.2 P ERIODIC AND C YCLICAL V ARIATIONS Periodic trends are common in hydrologic time series. Rainfall, runoff, and evapo- ration rates often show periodic trends over an annual period. Air temperature shows distinct periodic behavior. Seasonal trends may also be apparent in hydrologic data and may be detected using graphical analyses. Filtering methods may be helpful to reduce the visual effects of random variations. Appropriate statistical tests can be used to test the significance of the periodicity. The association of an apparent periodic or cyclical trend with a physical cause is generally more important than the results of a statistical test. Once a periodic trend has been shown, a functional form can be used to represent the trend. Quite frequently, one or more sine functions are used to represent the trend: (2.2) FIGURE 2.1 Annual maximum peak discharge for the Northwest Branch of the Anacostia River near Hyattsville, Maryland. ft A ft Y( ) sin( )=++2 0 πθ L1600_Frame_C02 Page 12 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC in which is s the mean magnitude of the variable, A is the amplitude of the trend, f 0 the frequency, θ the phase angle, and t the time measured from some zero point. The phase angle will vary with the time selected as the zero point. The frequency is the reciprocal of the period of the trend, with the units depending on the dimensions of the time-varying variable. The phase angle is necessary to adjust the trend so that the sine function crosses the mean of the trend at the appropriate time. The values of A , f 0 , and θ can be optimized using a numerical optimization method. In some cases, f 0 may be set by the nature of the variable, such as the reciprocal of 1 year, 12 months, or 365 days for an annual cycle. Unlike periodic trends, cyclical trends occur irregularly. Business cycles are classic examples. Cyclical trends are less common in hydrology, but cyclical behav- ior of some climatic factors has been proposed. Sunspot activity is cyclical. Example 2.2 Figure 2.2 shows elevation of the Great Salt Lake surface for the water years 1989 to 1994. The plot reveals a secular trend, probably due to decreased precipitation in the region, periodic or cyclical variation in each year, and a small degree of random variation. While the secular decline is fairly constant for the first 4 years, the slope of the trend appears to decline during the last 2 years. Therefore, a decreasing nonlinear function, such as an exponential, may be appropriate as a model representation of the secular trend. The cyclical component of the time series is not an exact periodic function. The peaks occur in different months, likely linked to the timing of the spring snowmelt. The peak occurs as early as April (1989 and 1992) and as late as July (1991). Peaks also occur in May (1994) and June (1990, 1993). While the timing of the maximum amplitude of the cyclical waves is likely related to the temperature cycle, it may be appropriate to model the cyclical variation evident in Figure 2.2 using a periodic function (Equation 2.2). This would introduce some error, but since the actual month FIGURE 2.2 Variation of water surface elevation of Great Salt Lake (October 1988 to August 1994). Y Record High 4,211.85 feet June 3–8, 1986, and April 1–15, 1987 Record Low 4,191.35 feet October-November 1963 1989 1990 1991 1992 1993 1994 Oct. Oct. Oct. Oct. Oct. Oct. 4215 4210 4205 4200 4195 Elevation (feet) L1600_Frame_C02 Page 13 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC of the peak cannot be precisely predicted in advance, the simplification of a constant 12-month period may be necessary for a simple model. If a more complex model is needed, the time of the peak may depend on another variable. If the dependency can be modeled, better accuracy may be achieved. 2.2.3 E PISODIC V ARIATION Episodic variation results from “one-shot” events. Over a long period, only one or two such events may occur. Extreme meteorological events, such as monsoons or hurricanes, may cause episodic variation in hydrological data. The change in the location of a recording gage may also act as an episodic event. A cause of the variation may or may not be known. If the cause can be quantified and used to estimate the magnitude and timing of the variation, then it is treated as an abrupt secular effect. Urbanization of a small watershed may appear as an episodic event if the time to urbanize is very small relative to the period of record. The failure of an upstream dam may produce an unusually large peak discharge that may need to be modeled as an episodic event. If knowledge of the cause cannot help predict the magnitude, then it is necessary to treat it as random variation. The identification of an episodic event often is made with graphical analyses and usually requires supplementary information. Although extreme changes may appear in a time series, one should be cautious about labeling a variation as an episodic event without supporting data. It must be remembered that extreme events can be observed in any set of measurements on a random variable. If the supporting data do not provide the basis for evaluating the characteristics of the episodic event, one must characterize the remaining components of the time series and use the residual to define the characteristics of the episodic event. It is also necessary to distinguish between an episodic event and a large random variation. Example 2.3 Figure 2.3 shows the time series of the annual maximum discharges for the Saddle River at Lodi, New Jersey (USGS gaging station 01391500), from 1924 to 1988. The watershed was channelized in 1968, which is evident from the episodic change. The characteristics of the entire series and the parts of the series before and after the channelization are summarized below. The discharge series after completion of the project has a higher average discharge than prior to the channelization. The project reduced the roughness of the channel and increased the slope, both of which contributed to the higher average flow rate. Discharge (cfs) Logarithms Series n Mean Standard Deviation Mean Standard Deviation Skew Total 65 1660 923 3.155 0.2432 0.0 Pre 44 1202 587 3.037 0.1928 0.2 Post 21 2620 746 3.402 0.1212 0.1 L1600_Frame_C02 Page 14 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC The reduction in variance of the logarithms is due to the removal of pockets of natural storage that would affect small flow rates more than the larger flow rates. The skew is essentially unchanged after channelization. The flood frequency characteristics (i.e., moments) before channelization are much different than those after channelization, and different modeling would be nec- essary. For a flood frequency analysis, separate analyses would need to be made for the two periods of record. The log-Pearson type-III models for pre- and post- channelization are: x = 3.037 + 0.1928 K and x = 3.402 + 0.1212 K , respectively, in which K is the log-Pearson deviate for the skew and exceedance probability. For developing a simulation model of the annual maximum discharges, defining the stochastic properties for each section of the record would be necessary. 2.2.4 R ANDOM V ARIATION Random fluctuations within a time series are often a significant source of variation. This source of variation results from physical occurrences that are not measurable; these are sometimes called environmental factors since they are considered to be uncontrolled or unmeasured characteristics of the physical processes that drive the system. Examples of such physical processes are antecedent moisture levels, small amounts of snowmelt runoff that contribute to the overall flow, and the amount of vegetal cover in the watershed at the times of the events. FIGURE 2.3 Annual maximum peak discharges for Saddle River at Lodi, New Jersey. L1600_Frame_C02 Page 15 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC The objective of the analysis phase is to characterize the random variation. Generally, the characteristics of random variation require the modeling of the secular, periodic, cyclical, and episodic variations, subtracting these effects from the mea- sured time series, and then fitting a known probability function and the values of its parameters to the residuals. The normal distribution is often used to represent the random fluctuations, with a zero mean and a scale parameter equal to the standard error of the residuals. The distribution selected for modeling random variation can be identified using a frequency analysis. Statistical hypothesis tests can be used to verify the assumed population. For example, the chi-square goodness-of-fit test is useful for large samples, while the Kolmogorov–Smirnov one-sample test can be used for small samples. These methods are discussed in Chapter 9. 2.3 MOVING-AVERAGE FILTERING Moving-average filtering is a computational technique for reducing the effects of nonsystematic variations. It is based on the premise that the systematic components of a time series exhibit autocorrelation (i.e., correlation between adjacent and nearby measurements) while the random fluctuations are not autocorrelated. Therefore, the averaging of adjacent measurements will eliminate the random fluctuations, with the remaining variation converging to a description of the systematic trend. The moving-average computation uses a weighted average of adjacent observa- tions. The averaging of adjacent measurements eliminates some of the total variation in the measured data. Hopefully, the variation smoothed out or lost is random rather than a portion of the systematic variation. Moving-average filtering produces a new time series that should reflect the systematic trend. Given a time series Y , the filtered series is derived by: (2.3) in which m is the number of observations used to compute the filtered value (i.e., the smoothing interval), and w j is the weight applied to value j of the series Y . The smoothing interval is generally an odd integer, with 0.5 ( m − 1) values of Y before observation i and 0.5 ( m − 1) values of Y after observation i used to estimate the smoothed value . A total of ( m − 1) observations is lost; that is, while the length of the measured time series equals n , the smoothed series, , only has n − m + 1 values. The simplest weighting scheme would be the arithmetic mean (i.e., w j = 1/ m ): (2.4) Other weighting schemes often give the greatest weight to the central point in the interval, with successively smaller weights given to points farther removed from the central point. For example, if weights of 0.25 were applied to the two adjacent ˆ Y ˆ . ( ), , ( ) .( ) YwY im nm ij ij m j m ==+−− +− + = ∑ 05 1 1 05 1 1 2 1 for K ˆ Y ˆ Y ˆ .( ) Y m Y t ij m j m = +− − = ∑ 1 05 1 1 L1600_Frame_C02 Page 16 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC time periods and a weight of 0.5 to the value at the time of interest, then the moving- average filter would have the form: (2.5) Moving-average filtering has several disadvantages. First, m − 1 observations are lost, which may be a serious limitation for short record lengths. Second, a moving- average filter is not itself a mathematical representation, and thus forecasting with the filter is not possible; a functional form must still be calibrated to forecast any systematic trend identified by the filtering. Third, the choice of the smoothing interval is not always obvious, and it is often necessary to try several intervals to identify the best separation of systematic and nonsystematic variation. Fourth, if the smooth- ing interval is not properly selected, it is possible to eliminate both systematic and nonsystematic variation. Filter characteristics are important in properly identifying systematic variation. As the length of the filter is increased, an increasingly larger portion of the systematic variation will be eliminated along with the nonsystematic variation. For example, if a moving-average filter is applied to a sine curve that does not include any random variation, the smoothed series will also be a sine curve with an amplitude that is smaller than that of the time series. When the smoothing interval equals the period of the sine curve, the entire systematic variation will be eliminated, with the smoothed series equal to the mean of the series (i.e., of Equation 2.2). Generally, the moving-average filter is applied to a time series using progressively longer intervals. Each smoothed series is interpreted, and decisions are made based on the knowledge gained from all analyses. A moving-average filter can be used to identify a trend or a cycle. A smoothed series may make it easier to identify the form of the trend or the period of the cycle to be fitted. A model can then be developed to represent the systematic component and the model coefficients evaluated with an analytical or numerical optimization method. The mean square variation of a time series is a measure of the information content of the data. The mean square variation is usually standardized to a variance by dividing by the number of degrees of freedom, which equals n − 1, where n is the number of observations in the time series. As a series is smoothed, the variance will decrease. Generally speaking, if the nonsystematic variation is small relative to the signal, smoothing will only reduce the variation by a small amount. When the length of the smoothing interval is increased and smoothing begins to remove variation associated with the signal, then the variance of the smoothed series begins to decrease at a faster rate relative to the variance of the raw data. Thus, a precipitous drop in the variance with increasing smoothing intervals may be an indication that the smoothing process is eliminating some of the systematic variation. When com- puting the raw-data variance to compare with the variance of the smoothed series, it is common to only use the observations of the measured data that correspond to points on the smoothed series, rather than the variance of the entire time series. This series is called the truncated series. The ratio of the variances of the smoothed series to the truncated series is a useful indicator of the amount of variance reduction associated with smoothing. ˆ YYYY tttt =++ −+ 025 05 025 11 Y L1600_Frame_C02 Page 17 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC Example 2.4 Consider the following time series with a record length of 8: Y = {13, 13, 22, 22, 22, 31, 31, 34} (2.6) While a general upward trend is apparent, the data appear to resemble a series of step functions rather than a predominantly linear trend. Applying a moving-average filter with equal weights of one-third for a smoothing interval of three yields the following smoothed series: = {16, 19, 22, 25, 28, 32} (2.7) While two observations are lost, one at each end, the smoothed series still shows a distinctly linear trend. Of course, if the physical processes would suggest a step function, then the smoothed series would not be rational. However, if a linear trend were plausible, then the smoothed series suggests a rational model structure for the data of Equation 2.6. The model should be calibrated from the data of Equation 2.6, not the smoothed series of Equation 2.7. The nonsystematic variation can be assessed by computing the differences between the smoothed and measured series, that is, e i = − Y i : e 3 = {3, − 3, 0, 3, − 3, 1} The differences suggest a pattern; however, it is not strong enough, given the small record length, to conclude that the data of Equation 2.6 includes a second systematic component. The variance of the series of Equation 2.6 is 64.29, and the variance of the smoothed series of Equation 2.7 is 34.67. The truncated portion of Equation 2.6 has a variance of 45.90. Therefore, the ratio of the smoothed series to the truncated series is 0.76. The residuals have a variance of 7.37, which is 16% of the variance of the truncated series. Therefore, the variation in the residuals relative to the variation of the smoothed series is small, and thus the filtering probably eliminated random variation. A moving-average filtering with a smoothing interval of five produces the fol- lowing series and residual series: = {18.4, 22.0, 25.6, 28.0} (2.8a) and e 5 = {−3.6, 0.0, 3.6, −3.0}. (2.8b) The variance of the truncated series is 20.25, while the variances of and e 5 are 17.64 and 10.89, respectively. Thus, the variance ratios are 0.87 and 0.53. While the ˆ Y ˆ Y i ˆ Y 5 ˆ Y 5 L1600_Frame_C02 Page 18 Friday, September 20, 2002 10:05 AM © 2003 by CRC Press LLC [...]... 60 90 120 150 180 21 0 24 0 27 0 300 330 360 30 60 90 120 150 180 21 0 Y(t) 3 5 7 9 20 25 28 .66 30 28 .66 25 20 15 11.34 10 11.34 15 20 25 28 .66 30 28 .66 25 20 15 — 24 .55 27 .89 29 .11 27 .89 24 .55 20 .00 15.45 12. 11 10.89 12. 11 15.45 20 .00 24 .55 27 .89 29 .11 27 .89 24 .55 20 .00 — — — 26 .46 27 .46 26 .46 23 .73 20 .00 16 .27 13.54 12. 54 13.54 16 .27 20 .00 23 .73 26 .46 27 .46 26 .46 23 .73 — — — — — 25 .33 24 . 62 22. 67 20 .00... process X = {20 .55, 22 .30, 20 .86, 20 .24 , 22 .11, 22 .36, 21 .67, 19.11, 17. 42, 17.74, 17.55, 17.11, 16.87, 15. 42, 14.75, 12. 92, 16. 12, 14.43, 13. 12, 11.49, 12. 36, 18. 12, 19.55, 19.93, 17. 12, 16.93, 12. 86, 17.61, 16.11, 15 .24 , 19. 12} Y = { 324 , 28 8, 27 8, 28 8, 27 1, 25 7, 22 9, 21 8, 25 7, 368, 336, 328 , 28 8, 26 8, 29 9, 320 , 344, 29 2, 21 2, 177, 177, 158, 151, 1 42, 131, 122 , 151, 125 , 109, 109, 109} 2- 2 6 How might... degree-day factor (July and August, 1979) for the Conejos River near Magote, Colorado Characterize the systematic and nonsystematic variation D = {25 .45, 24 .57, 24 .95, 23 .57, 24 .49, 24 .99, 24 .87, 27 .34, 27 .57, 29 .57, 29 .40, 30.69, 29 .84, 29 .25 , 28 .13, 25 .83, 27 .95, 25 .14, 24 .93, 25 .19, 27 .07, 26 .16, 26 .78, 29 .57, 27 . 52, 27 .61, 28 .23 , 27 .81, 27 .93, 27 .99, 24 .28 , 24 .49, 27 .16, 25 .95, 26 . 72, 28 .13, 29 .30, 27 . 92, ... 24 .28 , 24 .49, 27 .16, 25 .95, 26 . 72, 28 .13, 29 .30, 27 . 92, 26 .98, 24 .60, 24 .22 , 23 .34, 24 .98, 21 .95, 22 .60, 19.37, 20 .10, 20 .28 , 17.90, 17 .28 , 17.04, 17.45, 22 .28 , 23 .49, 23 .40, 21 .28 , 22 .25 , 19.80, 23 .63, 22 .13, 21 . 72, 23 .28 } © 20 03 by CRC Press LLC L1600_Frame_C 02 Page 38 Friday, September 20 , 20 02 10:05 AM 2- 1 8 Use three-point and five-point moving-average filters to smooth the daily discharge (July 1979)... 20 03 by CRC Press LLC L1600_Frame_C 02 Page 27 Friday, September 20 , 20 02 10:05 AM TABLE 2. 4 Analysis of Time Series (1) (2) t Xt 1 2 3 4 5 6 7 Total 4 3 5 4 6 5 8 35 (3) Offset Xt+1 + — 4 3 5 4 6 5 27 (4) (5) (6) (7) (8) (9) (10) XtXt+1 + Xt2 2 X t+1 Xt +2 + XtXt +2 + Xt2 Xt2 +2 — — 4 3 5 4 6 22 — — 20 12 30 20 48 130 — 12 15 20 24 30 40 141 — 9 25 16 36 25 64 175 — 16 9 25 16 36 25 127 — — 25 16 36 25 ... Total Xt Yt Yt+1 + Yt +2 + XtYt Xt2 Yt2 5.0 4.8 3.7 2. 8 3.6 3.3 2. 9 26 .1 2. 5 2. 1 2. 0 1.3 1.7 2. 0 1.8 13.4 2. 1 2. 0 1.3 1.7 2. 0 1.8 — 10.9 2. 0 1.3 1.7 2. 0 1.8 — — 9.8 12. 50 10.08 7.40 3.64 6. 12 6.60 5 .22 51.56 25 .00 23 .04 13.69 7.84 12. 96 10.89 8.41 101.83 6 .25 4.41 4.00 1.69 2. 89 4.00 3 .24 26 .48 © 20 03 by CRC Press LLC L1600_Frame_C 02 Page 32 Friday, September 20 , 20 02 10:05 AM Since May rainfall cannot... with Xt−1 Thus, the lag-1 cross-correlation coefficient is: Rc (1) = 42. 81 − (23 .2) (10.9)/ 6 = 0. 526 0 [93. 42 − (23 .2) 2 / 6]0.5 [20 .23 − (10.9) 2 / 6]0.5 (2. 22) The lag -2 cross-correlation coefficient relates April rainfall and June runoff and other values of Xt 2 and Yt: Rc (2) = 34.61 − (19.9)(8.8)/ 5 = −0.3939 [ 82. 53 − (19.9) 2 / 5]0.5 [15. 82 − (8.8) 2 / 5]0.5 (2. 23) The cross-correlogram shows a trend... 0.18 −0.03 −0 .26 −0.35 −0.41 −0.37 −0 .24 −0.03 0 .27 0.39 0.43 0.41 0 .20 −0.01 28 29 30 31 32 33 34 35 36 37 38 39 40 41 −0 .24 −0.38 −0.39 −0.38 −0 .23 −0.03 0 .21 0. 42 0.47 0.47 0 .20 −0.06 −0 .22 −0.36 42 43 44 45 46 47 48 49 50 51 52 53 54 −0.41 −0.36 −0 .27 −0. 12 0.13 0.33 0.49 0.49 0 .22 −0.04 −0 .23 −0.37 −0.39 FIGURE 2. 6 Correlogram for Chestuee Creek monthly streamflow watershed Figure 2. 6 suggests that... the © 20 03 by CRC Press LLC L1600_Frame_C 02 Page 29 Friday, September 20 , 20 02 10:05 AM TABLE 2. 6 Serial Correlation Coefficients (R) as a Function of Time Lag (τ) in Months for Chestuee Creek Monthly Water-Yield Data τ R(τ ) τ R(τ ) τ R(τ ) τ R(τ ) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1.00 0.44 0.19 −0.08 −0 .24 −0.34 −0.37 −0.34 −0.17 0.01 0 .23 0.39 0.47 0. 32 14 15 16 17 18 19 20 21 22 23 24 25 26 27 0.18... cross-regression model is not good © 20 03 by CRC Press LLC L1600_Frame_C 02 Page 34 Friday, September 20 , 20 02 10:05 AM TABLE 2. 9 Cross-Correlation and Cross-Regression Analyses between SCA and Degree-Day Factor FT Cross-Regression Coefficients Lag Rc Se (Standard Error) Se/Sy b0 b1 0 1 2 3 4 5 6 7 8 9 −0. 721 −0.709 −0.698 −0.691 −0.685 −0.678 −0.671 −0.669 −0.686 −0.731 0 .25 2 0 .25 3 0 .25 4 0 .25 4 0 .25 2 0 .25 1 . of 3579 020 ———— 30 25 24 .55 — — — 60 28 .66 27 .89 26 .46 — — 90 30 29 .11 27 .46 25 .33 — 120 28 .66 27 .89 26 .46 24 . 62 22. 63 150 25 24 .55 23 .73 22 .67 21 . 52 180 20 20 .00 20 .00 20 .00 20 .00 21 0 15 15.45 16 .27 . 21 . 52 60 28 .66 27 .89 26 .46 24 . 62 22. 63 90 30 29 .11 27 .46 25 .33 23 .03 120 28 .66 27 .89 26 .46 24 . 62 — 150 25 24 .55 23 .73 — — 180 20 20 .00 — — — 21 015 ———— L1600_Frame_C 02 Page 21 Friday, September 20 , 20 02. —— 2 3 4 12 916—— —— 3 5 3 15 25 9 4 20 25 16 4 4 5 20 16 25 3 12 16 9 5 6 4 24 36 16 5 30 36 25 6 5 6 30 25 36 4 20 25 16 7 8 5 40 64 25 6 48 64 36 Total 35 27 141 175 127 22 130 166 1 02 X t 2