Modeling Hydrologic Change: Statistical Methods - Chapter 7 pot

Statistical Detection of Nonhomogeneity 7.1 INTRODUCTION Data independent of the flood record may suggest that a flood record may not be stationary. Knowing that changes in land cover occurred during the period of record will necessitate assessing the effect of the land cover change on the peaks in the record. Statistical hypothesis testing is the fundamental approach for analyzing a flood record for nonhomogeneity. Statistical testing only suggests whether a flood record has been affected; it does not quantify the effect. Statistical tests have been used in flood frequency and hydrologic analyses for the detection of nonhomogeneity (Natural Environment Research Council, 1975; Hirsch, Slack, and Smith, 1982; Pilon and Harvey, 1992; Helsel and Hirsch, 1992). The runs test can be used to test for nonhomogeneity due to a trend or an episodic event. The Kendall test tests for nonhomogeneity associated with a trend. Correlation analyses can also be applied to a flood series to test for serial independence, with significance tests applied to assess whether an observed dependency is significant; the Pearson test and the Spearman test are commonly used to test for serial corre- lation. If a nonhomogeneity is thought to be episodic, separate flood frequency analyses can be done to detect differences in characteristics, with standard techniques used to assess the significance of the differences. The Mann–Whitney test is useful for detecting nonhomogeneity associated with an episodic event. Four of these tests (all but the Pearson test) are classified as nonparametric. They tests can be applied directly to the discharges in the annual maximum series without making a logarithmic transform. The exact same solution results when the test is applied to the logarithms and to the untransformed data with all four tests. This is not true for the Pearson test, which is parametric. Because a logarithmic transform is cited in Bulletin 17B (Interagency Advisory Committee on Water Data, 1982), the transform should also be applied when making the statistical test for the Pearson correlation coefficient. The tests presented for detecting nonhomogeneity follow the six steps of hypoth- esis testing: (1) formulate hypotheses; (2) identify theory that specifies the test statistic and its distribution; (3) specify the level of significance; (4) collect the data and compute the sample value of the test statistic; (5) obtain the critical value of the test statistic and define the region of rejection; and (6) make a decision to reject the null hypothesis if the computed value of the test statistic lies in the region of rejection. 7 L1600_Frame_C07 Page 135 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC 7.2 RUNS TEST Statistical methods generally assume that hydrologic data measure random variables, with independence among measured values. The runs (or run) test is based on the mathematical theory of runs and can test a data sample for lack of randomness or independence (or conversely, serial correlation) (Siegel, 1956; Miller and Freund, 1965). The hypotheses follow: H 0 : The data represent a sample of a single independently distributed random variable. H A : The sample elements are not independent values. If one rejects the null hypothesis, the acceptance of nonrandomness does not indicate the type of nonhomogeneity; it only indicates that the record is not homogeneous. In this sense, the runs test may detect a systematic trend or an episodic change. The test can be applied as a two-tailed or one-tailed test. It can be applied to the lower or upper tail of a one-tailed test. The runs test is based on a sample of data for which two outcomes are possible, x 1 or x 2 . These outcomes can be membership in two groups, such as exceedances or nonexceedances of a user-specified criterion such as the median. In the context of flood-record analysis, these two outcomes could be that the annual peak discharges exceed or do not exceed the median value for the flood record. A run is defined as a sequence of one or more of outcome x 1 or outcome x 2 . In a sequence of n values, n 1 and n 2 indicate the number of outcomes x 1 and x 2 , respectively, where n 1 + n 2 = n . The outcomes are determined by comparing each value in the data series with a user-specified criterion, such as the median, and indicating whether the data value exceeds ( + ) or does not exceed ( − ) the criterion. Values in the sequence that equal the median should be omitted from the sequences of + and − values. The solution procedure depends on sample size. If the values of n 1 and n 2 are both less than 20, the critical number of runs, n α , can be obtained from a table. If n 1 or n 2 is greater than 20, a normal approximation is made. The theorem that specifies the test statistic for large samples is as follows: If the ordered (in time or space) sample data, contains n 1 and n 2 values for the two possible outcomes, x 1 and x 2 , respectively, in n trials, where both n 1 and n 2 are not small, the sampling distribution of the number of runs is approximately normal with mean, , and variance, , which are approximated by: (7.1a) and (7.1b) U S u 2 U nn nn =+ 2 1 12 12 () () ()( ) S nn nn n n nn nn u 2 12 12 1 2 12 2 12 22 1 = −− ++− L1600_Frame_C07 Page 136 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC in which n 1 + n 2 = n . For a sample with U runs, the test statistic is (Draper and Smith, 1966): (7.2) where z is the value of a random variable that has a standard normal distribution. The 0.5 in Equation 7.2 is a continuity correction applied to help compensate for the use of a continuous (normal) distribution to approximate the discrete distribution of U . This theorem is valid for samples in which n 1 or n 2 exceeds 20. If both n 1 and n 2 are less than 20, it is only necessary to compute the number of runs U and obtain critical values of U from appropriate tables (see Appendix Table A.5). A value of U less than or equal to the lower limit or greater than or equal to the upper limit is considered significant. The appropriate section of the table is used for a one-tailed test. The critical value depends on the number of values, n 1 and n 2 . The typically available table of critical values is for a 5% level of significance when applied as a two-tailed test. When it is applied as a one-tailed test, the critical values are for a 2.5% level of significance. The level of significance should be selected prior to analysis. For consistency and uniformity, the 5% level of significance is commonly used. Other significance levels can be justified on a case-by-case basis. Since the basis for using a 5% level of significance with hydrologic data is not documented, it is important to assess the effect of using the 5% level on the decision. The runs test can be applied as a one-tailed or two-tailed test. If a direction is specified, that is, the test is one-tailed, then the critical value should be selected accordingly to the specification of the alternative hypothesis. After selecting the characteristic that determines whether an outcome should belong to group 1 ( + ) or group 2 ( − ), the runs should be identified and n 1 , n 2 , and U computed. Equations 7.1a and 7.1b should be used to compute the mean and variance of U . The computed value of the test statistic z can then be determined with Equation 7.2. For a two-tailed test, if the absolute value of z is greater than the critical value of z , the null hypothesis of randomness should be rejected; this implies that the values of the random variable are probably not randomly distributed. For a one- tailed test where a small number of runs would be expected, the null hypothesis is rejected if the computed value of z is less (i.e., more negative) than the critical value of z . For a one-tailed test where a large number of runs would be expected, the null hypothesis is rejected if the computed value of z is greater than the critical value of z . For the case where either n l or n 2 is greater than 20, the critical value of z is − z α or + z α depending on whether the test is for the lower or upper tail, respectively. When applying the runs test to annual maximum flood data for which watershed changes may have introduced a systematic effect into the data, a one-sided test is typically used. Urbanization of a watershed may cause an increase in the central tendency of the peaks and a decrease in the coefficient of variation. Channelization may increase both the central tendency and the coefficient of variation. Where the primary effect of watershed change is to increase the central tendency of the annual z UU S u = −−05. L1600_Frame_C07 Page 137 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC maximum floods, it is appropriate to apply the runs test as a one-tailed test with a small number of runs. Thus, the critical z value would be a negative number, and the null hypothesis would be rejected when the computed z is more negative than the critical z α , which would be a negative value. For a small sample test, the null hypothesis would be rejected if the computed number of runs was smaller than the critical number of runs. Example 7.1 The runs test can be used to determine whether urban development caused an increase in annual peak discharges. It was applied to the annual flood series of the rural Nolin River and the urbanized Pond Creek watersheds to test the following null ( H 0 ) and alternative ( H A ) hypotheses: H 0 : The annual peak discharges are randomly distributed from 1945 to 1968, and thus a significant trend is not present. H A : A significant trend in the annual peak discharges exists since the annual peaks are not randomly distributed. The flood series is represented in Table 7.1 by a series of + and − symbols. The criterion that designates a + or − event is the median flow (i.e., the flow exceeded or not exceeded as an annual maximum in 50% of the years). For the Pond Creek and North Fork of the Nolin River watersheds, the median values are 2175 ft 3 /sec and 4845 ft 3 /sec, respectively (see Table 7.1). If urbanization caused an increase in discharge rates, then the series should have significantly more + symbols in the part of the series corresponding to greater urbanization and significantly more − symbols before urbanization. The computed number of runs would be small so a one-tailed TABLE 7.1 Annual Flood Series for Pond Creek (q p , median == == 2175 ft 3 /s) and the Nolin River (Q p , median == == 4845 ft 3 /s) Year q p Sign Q p Sign Year q p Sign Q p Sign 1945 2000 − 4390 − 1957 2290 + 6510 + 1946 1740 − 3550 − 1958 2590 + 8300 + 1947 1460 − 2470 − 1959 3260 + 7310 + 1948 2060 − 6560 + 1960 + 1640 − 1949 1530 − 5170 + 1961 + 4970 + 1950 1590 − 4720 − 1962 + 2220 − 1951 1690 − 2720 − 1963 + 2100 − 1952 1420 − 5290 + 1964 + 8860 + 1953 1330 − 6580 + 1965 + 2300 − 1954 607 − 548 − 1966 4380 + 4280 − 1955 1380 − 6840 + 1967 3220 + 7900 + 1956 1660 − 3810 − 1968 4320 + 5500 + L1600_Frame_C07 Page 138 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC test should be applied. While rejection of the null hypothesis does not necessarily prove that urbanization caused a trend in the annual flood series, the investigator may infer such a cause. The Pond Creek series has only two runs (see Table 7.1). All values before 1956 are less than the median and all values after 1956 are greater than the median. Thus, n l = n 2 = 12. The critical value of 7 was obtained from Table A.5. The null hypothesis should be rejected if the number of runs in the sample is less than or equal to 7. Since a one-tailed test was used, the level of significance is 0.025. Because the sequence includes only two runs for Pond Creek, the null hypothesis should be rejected. The rejection indicates that the data are nonrandom. The increase in urban- ization after 1956 may be a causal factor for this nonrandomness. For the North Fork of the Nolin River, the flood series represents 14 runs (see Table 7.1). Because n 1 and n 2 are the same as for the Pond Creek analysis, the critical value of 7 applies here also. Since the number of runs is greater than 7, the null hypothesis of randomness cannot be rejected. Since the two watersheds are located near each other, the trend in the flood series for Pond Creek is probably not due to an increase in rainfall. (In a real-world application, rainfall data should be examined for trends as well.) Thus, it is probably safe to conclude that the flooding trend for Pond Creek is due to urban development in the mid-1950s. 7.2.1 RATIONAL ANALYSIS OF RUNS TEST Like every statistical test, the runs test is limited in its ability to detect the influence of a systematic factor such as urbanization. If the variation of the systematic effect is small relative to the variation introduced by the random processes, then the runs test may suggest randomness. In such a case, all of the variation may be attributed to the effects of the random processes. In addition to the relative magnitudes of the variations due to random processes and the effects of watershed change, the ability of the runs test to detect the effects of watershed change will depend on its temporal variation. Two factors are important. First, change can occur abruptly over a short time or gradually over the duration of a flood record. Second, an abrupt change may occur near the center, beginning, or end of the period of record. These factors must be understood when assessing the results of a runs test of an annual maximum flood series. Before rationally analyzing the applicability of the runs test for detecting hydro- logic change, summarizing the three important factors is worthwhile. 1. Is the variation introduced by watershed change small relative to the variation due to the randomness of rainfall and watershed processes? 2. Has the watershed change occurred abruptly over a short part of the length of record or gradually over most of the record length? 3. If the watershed change occurred over a short period, was it near the center of the record or at one of the ends? Answers to these questions will help explain the rationality of the results of a runs test and other tests discussed in this chapter. L1600_Frame_C07 Page 139 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC Responses to the above three questions will include examples to demonstrate the general concepts. Studies of the effects of urbanization have shown that the more frequent events of a flood series may increase by a factor of two for large increases in imperviousness. For example, the peaks in the later part of the flood record for Pond Creek are approximately double those from the preurbanization portion of the flood record. Furthermore, variation due to the random processes of rainfall and watershed conditions appears relatively minimal, so the effects of urbanization are apparent (see Figure 2.4). The annual maximum flood record for the Ramapo River at Pompton Lakes, New Jersey (1922 through 1991) is shown in Figure 7.1. The scatter is very significant, and an urbanization trend is not immediately evident. Most urban development occurred before 1968, and the floods of record then appear smaller than floods that occurred in the late 1960s. However, the random scatter largely prevents the identification of effects of urbanization from the graph. When the runs test is applied to the series, the computed test statistic of Equation 7.2 equals zero, so the null hypothesis of randomness cannot be rejected. In contrast to the series for Pond Creek, the large random scatter in the Ramapo River series masks the variation due to urbanization. The nature of a trend is also an important consideration in assessing the effect of urbanization on the flows of an annual maximum series. Urbanization of the Pond Creek watershed occurred over a short period of total record length; this is evident in Figure 2.4. In contrast, Figure 7.2 shows the annual flood series for the Elizabeth River, at Elizabeth, New Jersey, for a 65-year period. While the effects of the random processes are evident, the flood magnitudes show a noticeable increase. Many floods at the start of the record are below the median, while the opposite is true for later years. This causes a small number of runs, with the shorter runs near the center of record. The computed z statistic for the run test is −3.37, which is significant at the FIGURE 7.1 Annual maximum peak discharges for Ramapo River at Pompton Lakes, New Jersey. 20,000 18,000 16,000 14,000 12,000 10,000 8000 6000 4000 2000 0 1887 1904 1921 1938 1955 1972 1989 Peak Discharge in Cubic Feet per Second L1600_Frame_C07 Page 140 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC 0.0005 level. Thus, a gradual trend, especially with minimal variation due to random processes, produces a significant value for the runs test. More significant random effects may mask the hydrologic effects of gradual urban development. Watershed change that occurs over a short period, such as that in Pond Creek, can lead to acceptance or rejection of the null hypothesis for the runs test. When the abrupt change is near the middle of the series, the two sections of the record will have similar lengths; thus, the median of the series will fall in the center of the two sections, with a characteristic appearance of two runs, but it quite possibly will be less than the critical number of runs. Thus, the null hypothesis will be rejected. Conversely, if the change due to urbanization occurs near either end of the record length, the record will have short and long sequences. The median of the flows will fall in the longer sequence; thus, if the random effects are even moderate, the flood series will have a moderate number of runs, and the results of a runs test will suggest randomness. It is important to assess the type (gradual or abrupt) of trend and the location (middle or end) of an abrupt trend. This is evident from a comparison of the series for Pond Creek, Kentucky, and Rahway River in New Jersey. Figure 7.3 shows the annual flood series for the Rahway River. The effect of urbanization appears in the later part of the record. The computed z statistic for the runs test is −1.71, which is not significant at the 5% level, thus suggesting that randomness can be assumed. 7.3 KENDALL TEST FOR TREND Hirsch, Slack, and Smith (1982) and Taylor and Loftis (1989) provide assessments of the Kendall nonparametric test. The test is intended to assess the randomness of a data sequence X i ; specifically, the hypotheses (Hirsch, Slack, and Smith, 1982) are: FIGURE 7.2 Annual maximum peak discharges for Elizabeth River, New Jersey. 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 1925 1936 1947 1958 1969 1980 1991 Peak Discharge in Cubic Feet per Second L1600_Frame_C07 Page 141 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC H 0 : The annual maximum peak discharges (x i ) are a sample of n independent and identically distributed random variables. H A : The distributions of x j and x k are not identical for all k, j ≤ n with k ≤ j. The test is designed to detect a monotonically increasing or decreasing trend in the data rather than an episodic or abrupt event. The above H A alternative is two-sided, which is appropriate if a trend can be direct or inverse. If a direction is specified, then a one-tailed alternative must be specified. Gradual urbanization would cause a direct trend in the annual flood series. Conversely, afforestation can cause an inverse trend in an annual flood series. For the direct (inverse) trend in a series, the one- sided alternative hypothesis would be: H A : A direct (inverse) trend exists in the distribution of x j and x k . The theorem defining the test statistic is as follows. If x j and x k are independent and identically distributed random values, the statistic S is defined as: (7.3) where (7.4) FIGURE 7.3 Annual maximum peak discharges for Rahway River, New Jersey. 6000 5400 4800 4200 3600 3000 2400 1800 1200 600 0 1925 1936 1947 1958 1969 1980 1991 Peak Discharge in Cubic Feet per Second Sxx j k jk n k n =− =+= − ∑∑ sgn( ) 11 1 z = > = −<        10 00 10 if if if Θ Θ Θ L1600_Frame_C07 Page 142 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC For sample sizes of 30 or larger, tests of the hypothesis can be made using the following test statistic: (7.5a) in which z is the value of a standard normal deviate, n is the sample size, and V is the variance of S, given by: (7.5b) in which g is the number of groups of measurements that have equal value (i.e., ties) and t i is the number of ties in group i. Mann (1945) provided the variance for series that did not include ties, and Kendall (1975) provided the adjustment shown as the second term of Equation 7.5b. Kendall points out that the normal approxima- tion of Equation 7.5a should provide accurate decisions for samples as small as 10, but it is usually applied when N ≥ 30. For sample sizes below 30, the following τ statistic can be used when the series does not include ties: (7.6) Equation 7.6 should not be used when the series includes discharges of the same magnitude; in such cases, a correction for ties can be applied (Gibbons, 1976). After the sample value of the test statistic z is computed with Equation 7.5 and a level of significance α selected, the null hypothesis can be tested. Critical values of Kendall’s τ are given in Table A.6 for small samples. For large samples with a two-tailed test, the null hypothesis H 0 is rejected if z is greater than the standard normal deviate z α /2 or less than −z α /2 . For a one-sided test, the critical values are z α for a direct trend and −z α for an inverse trend. If the computed value is greater than z α for the direct trend, then the null hypothesis can be rejected; similarly, for an inverse trend, the null hypothesis is rejected when the computer z is less (i.e., more negative) than −z α . Example 7.2 A simple hypothetical set of data is used to illustrate the computation of τ and decision making. The sample consists of 10 integer values, as shown, with the values of sgn(Θ) shown immediately below the data for each sample value. z SV S S SV S = −> = +<        ()/ ()/ . . 10 00 10 05 05 for for for V nn n t t t ii i i g = −+− −+ = ∑ ()( ) ()( )12 5 12 5 18 1 τ =−21Snn/[ ( )] L1600_Frame_C07 Page 143 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC Since there are 33 + and 12 − values, S of Equation 7.3 is 21. Equation 7.6 yields the following sample value of τ : Since the sample size is ten, critical values are obtained from tables, with the following tabular summary of the decision for a one-tailed test: Thus, for a 5% level the null hypothesis is rejected, which suggests that the data contain a trend. At smaller levels of significance, the test would not suggest a trend in the sequence. Example 7.3 The 50-year annual maximum flood record for the Northwest Branch of the Anacostia River watershed (Figure 2.1) was analyzed for trend. Since the record length is greater than 30, the normal approximation of Equation 7.5 is used: (7.7) Because the Northwest Branch of the Anacostia River has undergone urbanization, the one-sided alternative hypothesis for a direct trend is studied. Critical values of z for 5% and 0.1% levels of significance are 1.645 and 3.09, respectively. Thus, the computed value of 3.83 is significant, and the null hypothesis is rejected. The test suggests that the flood series reflects an increasing trend that we may infer resulted from urban development within the watershed. 2503714968 +−++−++++ −−+−−+++ +++++++ +−++++ −−+−+ ++++ +++ −− + Level of Significance Critical ττ ττ Decision 0.05 0.422 Reject H 0 0.025 0.511 Accept H 0 0.01 0.600 Accept H 0 0.005 0.644 Accept H 0 τ == 221 10 9 0 467 () () . z == 459 119 54 383 . . L1600_Frame_C07 Page 144 Friday, September 20, 2002 10:16 AM © 2003 by CRC Press LLC [...]... X = {680, 940, 870 , 1230, 1050, 1160, © 2003 by CRC Press LLC L1600_Frame_C 07 Page 169 Friday, September 20, 2002 10:16 AM 7- 8 7- 9 7- 1 0 7- 1 1 7- 1 2 7- 1 3 7- 1 4 7- 1 5 7- 1 6 7- 1 7 7- 1 8 7- 1 9 800, 77 0, 890, 540, 950, 1230, 1460, 1080, 1 970 , 1550, 940, 880, 1000, 1350, 1610, 1880, 1540} The following sequence is the daily degree-day factor for the Conejos River near Magote, Colorado, for June 1 979 Using the runs... 18.61 16.63 13. 37 6.22 5.95 101 91 82 88 109 174 250 360 402 378 21 22 23 24 25 26 27 28 29 30 10.10 11 .77 15 .75 12.92 13.01 10.30 10 .72 11.86 8 .72 6 .77 378 418 486 506 534 530 488 583 619 588 7- 2 0 Use the Pearson correlation coefficient to assess the correlation between Manning’s n and the tree density d for the data of Problem 7- 3 Test the significance of the computed correlation 7- 2 1 It is acceptable... 1.03 0.30 2.18 2 .78 1.14 1.31 1.51 2.48 0.88 0.50 0 .74 3.52 0.50 0.32 3.96 2.03 0.99 0.82 1.69 4.95 2.54 a 3. 47 0.56 0 .71 1. 47 1.24 0.55 0.48 6.94 1 .73 0.91 0.83 8.96 1.55 0. 67 0.39 1.05 2.24 0.58 0.32 6.34 2.65 2 .70 0.99 4 .73 1 .79 1.33 1 .70 0.51 0. 47 3.39 1.98 0.49 0.90 4.86 2.65 0.86 1.35 2.82 1.03 0.51 0.54 5 .72 0. 87 0.51 3.65 3.32 1 .70 0 .78 1.59 3.99 1.86 0.95 Ya Y Y Y Y Y Y Y Y Y 2-TIE Y Y Y indicates... L1600_Frame_C 07 Page 160 Friday, September 20, 2002 10:16 AM ns Equation 7. 15 z Equation 7. 19 14 0.00088 3.241 0.0006 13 0.00 372 2 .76 7 0.0028 12 0.012 97 2.293 0.0083 11 0.0 376 4 1.818 0.00346 10 0.09190 1.344 0.0895 If n is equal to 30, then the agreement is better ns Equation 7. 15 z Equation 7. 19 19 0.00 07 3.292 0.0005 18 0.0025 2.905 0.0018 17 0.0 072 2.5 17 0.0059 16 0.0188 2.130 0.0166 15 0.0435 1 .74 3 0.04 07. .. Residual (cfs) 243 392 658 559 644 2034 623 1143 71 8 1511 613 583 − 172 168 −6 97 12 27 733 −13 −964 −5221 −1664 −1998 122 −1242 Intensity (in./hr) 0. 37 0.29 0.28 0.64 0.32 1. 37 0.42 0.60 0.23 0.28 0.19 0. 37 0.28 0 .74 0.60 1.43 0 .78 0.56 0.48 0 .77 0.66 0. 47 1.16 0. 97 L1600_Frame_C 07 Page 165 Friday, September 20, 2002 10:16 AM is below the lower value, the null hypothesis should be rejected The assumption of... Time period I Time period I Time period I 1 2 17 5 33 8 2 3 18 6 34 6 3 0 19 7 35 7 4 1 20 4 36 11 5 2 21 8 37 9 6 2 22 7 38 8 7 4 23 8 39 7 8 1 24 5 40 9 9 3 25 9 41 6 10 0 26 7 42 10 11 5 27 9 43 7 12 4 28 8 44 12 13 3 29 6 45 13 14 4 30 8 46 8 15 3 31 10 47 10 16 6 32 11 48 10 7- 7 The following sequence of annual maximum discharges occurred over a 23-year period on a watershed undergoing urbanization... TABLE 7. 8 Application of Noether’s Test to Monthly Rainfall Depth (in.) for Chestuee Creek Watershed, March 1944 to August 1950 6.91 2.15 6.62 4.29 3.05 3.85 1.56 5.84 4.91 3. 87 5 .75 4.20 3.28 5.04 2.92 2.69 6.58 1.45 2.50 5 .76 4.63 5.06 3.96 4.19 7. 78 6.01 3.84 a 4.64 2.30 1.15 2.64 4.86 3.84 4 .73 10.42 4.09 4.29 3 .79 12.63 2 .76 3. 47 2.10 4.12 2. 57 5.04 2.88 8 .72 4.44 7. 60 6. 87 8.16 1.43 6.39 3 .70 3.44... L1600_Frame_C 07 Page 148 Friday, September 20, 2002 10:16 AM TABLE 7. 2 Computation of Pearson R for Increasing Trend (A) and Decreasing Trend (B) Year of Record Flow Ai Offset Ai+1 + Product Ai Ai+1 + Ai2 1 2 3 4 5 6 7 12 14 17 22 25 27 31 14 17 22 25 27 31 — 168 238 374 550 675 8 37 — 144 196 289 484 625 72 9 — 1 17 136 2842 24 67 Totals Flow Bi Offset Bi+1 + Product Bi Bi+1 + 196 289 484 625 72 9 961 — 17 14 10... Problem 7- 9 , test the significance of the Spearman correlation coefficient between A and Q 7- 2 9 Test the roughness coefficient (n) data of Problem 7- 3 with the Spearman– Conley statistic for an increasing trend with increases in river mile 7- 3 0 Test the discharge data of Problem 7- 9 with the Spearman–Conley statistic for an increasing trend 7- 3 1 Test the Fishkill Creek discharge data of Problem 7- 1 7 for... 614 172 0 1060 1680 76 0 1380 1030 820 1020 998 3500 1100 1010 830 1030 452 2530 174 0 1860 1 270 2200 79 5 176 0 806 1190 952 1 670 824 70 2 1490 1600 800 3330 1540 2130 377 0 2240 3210 2940 272 0 2440 3130 4500 2890 2 470 1900 1980 2550 3350 2120 1850 2320 1630 − + + − + + + + + + − + − + + + + + + + + + + + + + + + + − + − No of + Symbols Cumulative Probability 20 21 22 23 24 25 26 27 28 29 30 31 32 0.9449 079 . 144 196 17 14 238 289 196 2 14 17 238 196 289 14 10 140 196 100 3 17 22 374 289 484 10 13 130 100 169 4 22 25 550 484 625 13 11 143 169 121 5 25 27 675 625 72 9 11 8 88 121 64 6 27 31 8 37 729 961. 64 73 1—— ——8 — — — — Totals 1 17 136 2842 24 67 3284 73 64 803 939 71 4 A i 2 A i+1 2 B i 2 B i1+ 2 R A = − −− = 2842 1 17 136 6 24 67 1 17 6 3284 136 6 0 983 205 205 ()/ (/)(/) . R B = − −− = 803 73 . 19 57 2290 + 6510 + 1946 174 0 − 3550 − 1958 2590 + 8300 + 19 47 1460 − 2 470 − 1959 3260 + 73 10 + 1948 2060 − 6560 + 1960 + 1640 − 1949 1530 − 5 170 + 1961 + 4 970 + 1950 1590 − 472 0