Graphical Detection of Nonhomogeneity 6.1 INTRODUCTION The preparation phase of data analysis involves compilation, preliminary organiza- tion, and hypothesis formulation. All available physiographic, climatic, and hydro- logic data should be compiled. While a criterion variable, such as annual maximum discharge, is often of primary interest, other hydrologic data can be studied to decide whether a change in the criterion variable occurred. The analysis of daily flows, flow volumes, and low flow magnitudes may be useful for detecting watershed change. Physiographic data, such as land use or channel changes, are useful for assigning responsibility to changes and developing a method that can be used to adjust the flood record. Climatic data, such as rainfall volumes, reflect the extent to which the change in the annual flood series is climate related. If physiographic or climatic data do not suggest a significant watershed change, it may not be necessary to apply trend tests to the flood data. The variation in the annual flood series may simply be a function of random climatic variability, and this hypothesis can be evaluated by applying univariate trend tests to the sequence of data. Graphical analyses are often the first step in data analyses. They are preludes to quantitative analyses on which decisions can be based. Graphical analyses should always be considered initial steps, not conclusive steps. They can be misleading when not accompanied by quantitative analyses. However, failure to graph data may prevent the detection of a trend or the nature of the trend. Graphical analyses should be used in conjunction with other quantitative methods. They will be discussed in this chapter and quantitative analyses will be discussed in other chapters. 6.2 GRAPHICAL ANALYSES After compilation of data, several preliminary analyses can be made in preparing to test for and, if necessary, adjust for the effects of watershed changes. Three general types of analyses can be made. First, one or more graphical analyses of the series can be made, including the standard frequency analysis (e.g., plotting several Pearson Type III frequency curves for different time periods). The purpose of graphical analysis is to study the data to identify the ways watershed changes affected the flood series. For example, does the central tendency change with time? Did the variance of the data change? Did the watershed changes affect only part of the temporal series, thus producing a mixed population series? Graphical analyses can provide some insight into characteristics of the changes and suggest the best path for detect- ing the effects and adjusting the series. 6 L1600_Frame_C06 Page 113 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC Graphical methods that can be initially used to understand the data include plots of data versus time, ranking the annual event versus the water year, the number of occurrences above a threshold versus water year, and histograms or empirical cumu- lative probability plots of the data for two or more periods of the record. Where untransformed data are characterized by considerable random variation, the loga- rithms of the data can be plotted to assist in detecting the effects of watershed change. In most cases, several plots should be made, as different types of plots will identify different characteristics of the data. 6.2.1 U NIVARIATE H ISTOGRAMS Graphical analyses are often the first steps in analyzing data. Univariate graphical analyses in the form of histograms help identify the distribution of the random variable being analyzed. A frequency histogram is a tabulation or plot of the fre- quency of occurrence versus selected intervals of the continuous random variable. It is the equivalent of a bar graph used for graphing discrete random variables. The effectiveness of a graphical analysis in identifying characteristics of a random vari- able or its probability density function depends on the sample size and interval selected to plot the abscissa. For small samples, it is difficult to separate the data into a sufficient number of groups to provide a meaningful indication of data characteristics. With small samples, the impressions of the data will be very sensitive to the cell boundaries and widths selected for the histogram. It is generally wise to try several sets of cell boundaries and widths to ensure accurate assessments of the data. The following are general guidelines for constructing frequency histograms: 1. Set the minimum value ( X m ) as (a) the smallest sample value or (b) a physically limiting value, such as zero. 2. Set the maximum value ( X x ) as (a) the largest sample value or (b) an upper limit considered the largest value expected. 3. Select the number of intervals ( k ), which is usually about 5 for small samples and a maximum of about 20 for large samples. For moderate size samples, the following empirical equation can be used to estimate the number of cells: k = 1 + 3.3 log 10 ( n ) (6.1) 4. Compute the approximate cell width ( w ) where w = ( X x − X m )/ k . 5. Round the computed value of w to a convenient value w 0 . 6. Set the upper bound ( B i ) for cell i using the minimum value X m , and the cell width w 0 : B i = X m + i w 0 for i = 1, 2, … , k (6.2) 7. Using the sample data, compute the sample frequencies for each cell. In addition to the cell width w 0 , assessments of the data characteristics can be influenced by the scale used as the ordinate. For example, a histogram where all L1600_Frame_C06 Page 114 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC frequencies are 10 to 15 per cell will appear quite different when the ordinate is scaled from 10 to 15 and from 0 to 15. The former scale suggests the cell frequencies are varied. The latter suggests a relatively uniform set of frequencies. This can skew the viewer’s impression of the data characteristics. Histograms provide a pictorial representation of the data. They provide for assessing the central tendency of the data; the range and spread of the data; the symmetry (skewness) of the data; the existence of extreme events, which can then be checked for being outliers; and approximate sample probabilities. Frequency histograms can be transformed to relative frequency or probability histograms by dividing the frequencies of every cell by the sample size. Example 6.1 Consider the 38-year discharge record of Table 6.1. To achieve an average frequency of five per cell will require a histogram with no more than seven cells. The use of more cells would produce cells with low frequencies and invite problems in char- acterizing the data. Figure 6.1(a) shows a nine-cell histogram based on a cell width of 50 cfs. With an average of 4.2 floods per cell, only four of the nine cells have frequencies of five or more. The histogram is multimodal and does not suggest an underlying distribu- tion. The cell with a one-count in the middle would discount the use of a normal or lognormal distribution. Figure 6.1(b) shows a histogram of the same data but with a cell width of 100 cfs. With only five cells, the average frequency is 7.6. Except for the 500–600 cfs cell, the data appear to follow a uniform distribution. However, with only five cells, it is difficult to have confidence in the shape of the distribution. Figure 6.1(c) also shows the frequency histogram with a cell width of 100 cfs, but the lowest cell bound is 550 cfs rather than the 500 cfs used in Figure 6.1(b). The histogram of Figure 6.1(c) is characterized by one high-count cell, with the other cells having nearly the same count. The histogram might suggest a lognormal distribution. The important observation about the histograms of Figure 6.1 is that, even with a sample size of 38, it is difficult to characterize the data. When graphing such data, several cell widths and cell bound delineations should be tried. The three histograms of Figure 6.1 could lead to different interpretations. While the data could be trans- formed using logarithms, the same problems would exist. The frequencies in each cell would be limited because of the sample size. TABLE 6.1 Annual Maximum Discharge Record 654 967 583 690 957 814 871 859 843 837 714 725 917 708 618 685 941 822 883 766 827 693 660 902 672 612 742 703 731 637 810 981 646 992 734 565 678 962 L1600_Frame_C06 Page 115 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC Example 6.2 Table 6.2 includes the measured annual maximum flood series for the Rubio Wash Watershed for the 1929–1976 period ( n = 48). During the period of record, the percentage of imperviousness increased from 18% to 40%; thus, the record is nonhomogeneous. Using the method of Section 5.4.2, the measured series was adjusted to a homogeneous record based on 40% imperviousness (Table 6.2). The histograms for the two series are shown in Figure 6.2. Since the homogeneous series has large discharge rates, it appears shifted to the right. The adjusted series shows a more bell-shaped profile than the measured series. However, the small sample size allows the one high-frequency cell for the adjusted series (Figure 6.2b) to dominate the profile of the histogram. In summary, even though the flood record increases to 48 annual maximum discharges, it is difficult to use the graphical analysis alone to identify the underlying population. The analyses suggest that the adjusted series is different from the mea- sured but nonhomogeneous series. 6.2.2 B IVARIATE G RAPHICAL A NALYSIS In addition to univariate graphing with histograms and frequency plots, graphs of related variables can be helpful in understanding data, such as flood peaks versus the level of urbanization or percent forest cover. The first step in examining a FIGURE 6.1 Frequency histograms for annual maximum discharge record of Table 6.1: effects of cell width. L1600_Frame_C06 Page 116 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC TABLE 6.2 Measured ( Y ) and Adjusted ( X ) Annual Maximum Flood Series for the Rubio Wash Watershed, 1929–1976 Year YX Year YX Year YX 1929 661 879 1945 1630 1734 1961 1200 1213 1930 1690 2153 1946 2650 2795 1962 1180 1193 1931 798 1032 1947 2090 2192 1963 1570 1586 1932 1510 1885 1948 530 555 1964 2040 2040 1933 2071 2532 1949 1060 1095 1965 2300 2300 1934 1680 2057 1950 2290 2332 1966 2041 2041 1935 1370 1693 1951 3020 3069 1967 2460 2460 1936 1181 1445 1952 2200 2221 1968 2890 2890 1937 2400 2814 1953 2310 2331 1969 2540 2540 1938 1720 2015 1954 1290 1303 1970 3700 3700 1939 1000 1172 1955 1970 1990 1971 1240 1240 1940 1940 2186 1956 2980 3005 1972 3166 3166 1941 1201 1353 1957 2740 2764 1973 1985 1985 1942 2780 3026 1958 2781 2805 1974 3180 3180 1943 1930 2110 1959 985 996 1975 2070 2070 1944 1780 1912 1960 902 912 1976 2610 2610 FIGURE 6.2 Frequency histograms for annual maximum discharges for Rubio Wash, Cali- fornia: (a) nonhomogeneous series; and (b) homogeneous series. L1600_Frame_C06 Page 117 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC relationship of two variables is to perform a graphical analysis. Visual inspection of the graphed data can identify: 1. The degree of common variation, which is an indication of the degree to which the two variables are related 2. The range and distribution of the sample data points 3. The presence of extreme events 4. The form of the relationship between the two variables (linear, power, exponential) 5. The type of relationship (direct or indirect) All these factors are of importance in the statistical analysis of sample data and decision making. When variables show a high degree of association, one assumes that a causal relationship exists. If a physical reason suggests that a causal relationship exists, the association demonstrated by the sample data provides empirical support for the assumed relationship. Systematic variation implies that when the value of one of the random variables changes, the value of the other variable will change predictably, that is, an increase in the value of one variable occurs when the value of another variable increases. For example, a graph of the mean annual discharge against the percentage of imperviousness may show an increasing trend. If the change in the one variable is highly predictable from a given change in the other variable, a high degree of common variation exists. Figure 6.3 shows graphs of different samples of data for two variables having different degrees of common variation. In Figures 6.3(a) and (e), the degrees of common variation are very high; thus the variables are said to be correlated. In Figure 6.3(c), the two variables are not correlated because, as the value of X is increased, it is not certain whether Y will increase or decrease. In Figures 6.3(b) and (d), the degree of correlation is moderate; in Figure 6.3(b), it is evident that Y will increase as X is increased, but the exact change in Y for a change in X is difficult to estimate. A more quantitative discussion of the concept of common variation appears later in this chapter. It is important to use a graphical analysis to identify the range and distribution of the sample data points so that the stability of the relationship can be assessed and so that one can assess the ability of the data sample to represent the distribution of the population. If the range of the data is limited, a fitted relationship may not be stable; that is, it may not apply to the distribution of the population. Figure 6.4 shows a case where the range of the sample is much smaller than the expected range of the population. If an attempt is made to use the sample to project the relationship between the two random variables, a small change in the slope of the relationship will cause a large change in the predicted estimate of Y for values of X at the extremes of the range of the population. A graph of two random variables might alert an investigator to a sample in which the range of the sample data may cause stability problems in a derived relationship between two random variables, especially when the relationship will be extrapolated beyond the range of the sample data. It is important to identify extreme events in a sample of data for several reasons. First, extreme events can dominate a computed relationship between two variables. L1600_Frame_C06 Page 118 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC For example, in Figure 6.5(a), the extreme point suggests a high correlation between X and Y and the cluster of points acts like a single observation. In Figure 6.5(b), the extreme point causes a poor correlation between the two random variables. Since the cluster of points has the same mean value of Y as the value of Y of the extreme point, the data of Figure 6.5(b) suggest that a change in X is not associated with a change in Y . A correlation coefficient is more sensitive to an extreme point when sample size is small. An extreme event may be due to errors in recording or plotting the data or a legitimate observation in the tail of the distribution. Therefore, an extreme event must be identified and its cause determined. Otherwise, it will not be possible to properly interpret the results of correlation analysis. FIGURE 6.3 Different degrees of correlation between two random variables ( X and Y ): (a) R = 1.0; (b) R = 0.5; (c) R = 0.0; (d) R = − 0.5; (e) R = − 1.0; (f) R = 0.3. L1600_Frame_C06 Page 119 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC Relationships can be linear or nonlinear. Since the statistical methods to be used for the two forms of a relationship differ, it is important to identify the form. In addition, the most frequently used correlation coefficient depends on a linear rela- tionship between the two random variables; thus low correlation may result for a nonlinear relationship even when a strong relationship is obvious. For example, the bivariate relationship of Figure 6.3(f) suggests a predictable trend in the relationship between Y and X ; however, the correlation coefficient will be low, and is certainly not as high as that in Figure 6.3(a). Graphs relating pairs of variables can be used to identify the type of the rela- tionship. Linear trends can be either direct or indirect, with an indirect relationship indicating a decrease in Y as X increases. This information is useful for checking the rationality of the relationship, especially when dealing with data sets that include more than two variables. A variable that is not dominant in the physical relationship may demonstrate a physically irrational relationship with another variable because of the values of the other variables affecting the physical relationship. FIGURE 6.4 Instability in the relationship between two random variables. FIGURE 6.5 Effect of an extreme event in a data sample on correlation: (a) high correlation; and (b) low correlation. Range of population Range of sample y x L1600_Frame_C06 Page 120 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC Consider the X - Y graphs of Figure 6.6. If the Pearson correlation coefficient did not assume linearity, a very high correlation (near 1) could be expected for Figure 6.6(a). The data suggest a high degree of systematic variation, but because the relationship is nonlinear, the correlation coefficient is only 0.7 or 49% explained variance. Figures 6.6(b) and 6.6(c) show far more nonsystematic variation than Figure 6.6(a), but have the same correlation (0.7) because the trend is more linear. The graphs of Figures 6.6(d) and 6.6(e) show less nonsystemic variation than seen in Figures 6.6(b) and 6.6(c), but they have the same correlation of 0.7 because the total variation is less. Thus, the ratio of the variation explained by the linear trend to the total variation is the same in all four graphs. Figures 6.6(f), (g), and (h) show single events that deviate from the remainder of the sample points. In Figure 6.6(f), the deviant point lies at about the mean of the X values but is outside the range of the Y values. The deviant point in Figure 6.6(g) lies at the upper end of the Y values and at the lower end of the X values. Both points are located away from the general linear trend shown by the other sample points. In Figure 6.6(h), the deviant point falls beyond the ranges of the sample values of both X and Y but the one deviant point creates a linear trend. The correlations for the three graphs are the same, 0.7, in spite of the positioning of the deviant points. Figures 6.6(i) and 6.6(j) show two clusters of points. The two clusters in Figure 6.6(i) show greater internal variation than those in Figure 6.6(j) but they are more dispersed along the y-axis. Thus, the two graphs have the same correlation of 0.7 and show that the correlation depends on both the slope of the relationship and the amount of nonsystematic variation or scatter. All the graphs in Figures 6.6(a) through 6.6(j) have the same correlation coef- ficient of 0.7 despite the dissimilar patterns of points. This leads to several important FIGURE 6.6 Graphical assessment of bivariate plots. (a) (b) (c) (d) (e) (f) (k) (l) (m) (n) (g) (h) (i) (j) L1600_Frame_C06 Page 121 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC observations about bivariate graphs. Both graphs and computed correlation coeffi- cients can be very misleading. Either one alone can lead to poor modeling. It is necessary to graph the data and, if the trend is somewhat linear, compute the correlation coefficient. Second, the correlation coefficient is a single-valued index that cannot reflect all circumstances such as clustering of points, extreme deviant points, nonlinearity, and random versus systematic scatter. Third, the correlation coefficient may not be adequate to suggest a model form, as the data of Figure 6.6(a) obviously need a different model form than needed by the data of Figure 6.6(i). Bivariate graphs and correlation coefficients also suffer from the effects of other variables. Specifically, the apparent random scatter in an X - Y graph may be due to a third variable, suggesting that X and Y are not related. Consider Figures 6.6(k) and 6.6(l). Both show considerable random scatter, but if they are viewed as data for different levels of a third variable, the degree of linear association between Y and X is considerably better. Figure 6.6(k) with a correlation of 0.7 shows a smaller effect of a second variable than does Figure 6.6(l), which has a correlation of 0.5. However, if the data of Figure 6.6(l) is separated for the two levels of the second predictor variable, the correlation between Y and X is much better. Figures 6.6(m) and 6.6(n) show the data of Figure 6.6(l) separated into values for the two levels of the second predictor variable. The correlations for Figures 6.6(m) and 6.6(n) are 0.98 and 0.9, respectively. Figures 6.6(k) through 6.6(n) show the importance of considering the effects of other predictor variables when evaluating bivariate plots. Graphing is an important modeling tool, but it cannot be used alone. Numerical indicators such as correlation coefficients must supplement the information extracted from graphical analyses. Example 6.3 Table 6.3 contains data for 22 watersheds in the Western Coastal Plain of Maryland and Virginia. The data includes the drainage area (A, mi 2 ), the percentage of forest cover (F), and the 10-year log-Pearson type III discharge (Q, cfs). The correlation matrix for the three variables follows. The area and discharge are highly correlated, while the correlation between forest cover and discharge is only moderate. Both correlations are rational, as peak dis- charge should increase with area and decrease with forest cover. The relatively high correlation between peak discharge and area suggests a strong linear relationship. The moderate correlation between peak discharge and forest cover may be the result of a nonlinear relationship or lack of common variation. This cannot be known without plotting the data. AFQ 1.000 −0.362 0.933 A 1.000 −0.407 F 1.000 Q L1600_Frame_C06 Page 122 Friday, September 20, 2002 10:14 AM © 2003 by CRC Press LLC [...]... 813 1590 2470 4200 7730 n (record length) SN (USGS station number) 11 10 10 25 14 9 19 11 24 35 18 42 25 21 44 36 10 39 25 25 37 42 66 1430 594445 4 960 80 66 8300 66 0900 66 8200 59 460 0 594800 66 1800 590500 66 160 0 590000 66 1000 66 1050 66 1500 4 960 00 495500 66 9000 594500 65 360 0 65 8000 594000 Figure 6. 7(a) shows the plot of drainage area versus peak discharge The graph shows a cluster of points near the origin... LLC L 160 0_Frame_C 06 Page 129 Friday, September 20, 2002 10:14 AM TABLE 6. 4 Annual Flood Series for Pond Creek (Qp1) and North Fork of Nolin River (Qp2) Watersheds Qp1 (ft3/s) Qp2 (ft3/s) 1945 19 46 1947 1948 1949 1950 1951 1952 1953 1954 1955 19 56 1957 1958 1959 1 960 1 961 1 962 1 963 1 964 1 965 1 966 1 967 1 968 2000 1740 1 460 2 060 1530 1590 169 0 1420 1330 60 7 1380 166 0 2290 2590 3 260 2490 3080 2520 3 360 8020... 23 42 06 57 78 71 67 83 23 16 15 98 03 06 06 08 41 84 © 2003 by CRC Press LLC 26 57 82 67 32 23 12 64 44 62 08 46 87 28 83 66 22 29 71 27 16 90 01 45 48 11 97 20 08 83 75 78 46 19 09 28 67 72 47 19 81 39 05 76 84 L 160 0_Frame_C 06 Page 132 Friday, September 20, 2002 10:14 AM FIGURE 6. 13 An example of a moving-average filtered series for annual-maximum peak discharges and associated event rainfall 6- 3 Create...L 160 0_Frame_C 06 Page 123 Friday, September 20, 2002 10:14 AM TABLE 6. 3 Data Matrix A (mi2) 0.30 1.19 1.70 2.18 2.30 2.82 3.85 6. 73 6. 82 6. 92 6. 98 8.50 10.4 18.5 24.0 24.3 26. 8 28.0 30.2 39.5 54.8 98.4 F (%) Q (10-year discharge, cfs) 25 19 96 68 82 69 46 83 85 70 66 70 66 60 82 22 23 69 47 42 69 22 54 334 67 9 193 325 418 440 192 400 548 1100 350 974 3010... 2003 by CRC Press LLC L 160 0_Frame_C 06 Page 124 Friday, September 20, 2002 10:14 AM 8 22 7 % 20 F= Discharge (x103 ft3/s) 6 5 0% F=8 69 4 23 22 60 3 82 42 2 66 1 0 47 66 90 1 969 46 70 70 62 2 568 83 0 10 69 20 30 40 50 60 70 80 90 100 Drainage area (sq miles) (a) 8 98 7 Discharge (x103 ft3/s) 6 5 55 A=5 4 0m 2 i 27 18 3 24 24 40 2 30 7 1 A=5 mi 2 10 4 1 0 3 2 0.3 0 10 20 30 40 50 60 28 7 8 70 2 2 7 7 80... 4.90 3. 86 2.91 2.13 3.89 3.88 4.95 3.31 4.95 2.42 4 .68 3.88 3.48 3.93 4.97 5.50 3.88 4.54 5.40 3.48 2.53 3.99 3 .65 5.34 5.47 3 .61 1.81 3. 76 4.52 4.31 4.17 4.57 3.78 6. 26 3.35 4.81 4.58 4.55 3.23 4.42 4. 46 5. 56 6 .65 2.12 3.25 4 .66 4.23 2.47 5.28 3.35 3.51 3.41 3.42 6- 4 Discuss the uses of bivariate graphical analyses and indicate the characteristics of graphs that will illustrate each use 6- 5 Provide... 3550 2470 65 60 5170 4720 2720 5290 65 80 548 68 40 3810 65 10 8300 7310 164 0 4970 2220 2100 8 860 2300 4280 7900 5000 Median 2175 4845 Water Year channelization increased from 18 .6 to 56. 7% Most changes occurred between 1954 and 1 965 The Nolin River watershed served as a control since change was minimal for 1945 through 1 968 The annual flood series for Pond Creek and the North Fork for the 24-year period... time series of T (d) Compute the lag-0 cross-correlation coefficient between T and P, and interpret the result 6- 1 3 Discuss the use of moving-average filtering on the criterion variable of a bivariate relationship © 2003 by CRC Press LLC L 160 0_Frame_C 06 Page 134 Friday, September 20, 2002 10:14 AM 6- 1 4 The following is a 13-year record of the annual load of nitrogen-based fertilizer (Xt) applied to a... 1.57 2.94 0. 86 5.17 2.20 4.33 2.01 4 .60 1.72 5.25 2.51 6. 41 2 .62 3.37 1.52 7.29 3 .65 6- 1 1 The following time series represents the annual summer baseflow over 21 years for a moderate-sized watershed that has undergone a land use change Characterize the effect of that land use change Q = {8, 5, 9, 7, 12, 8, 7, 18, 14, 20, 22, 20, 23, 28, 31, 26, 34, 25, 30, 32, 27} 6- 1 2 The following is a time series of... 6. 6 The use of moving-average smoothing for detecting a secular trend is illustrated using data for 1945 through 1 968 for two adjacent watersheds in north-central Kentucky, about 50 miles south of Louisville The data include the annual flood series for Pond Creek, a 64 -square-mile watershed, and the north fork of the Nolin River at Hodgenville, which has an area of 36. 4 square miles From 1945 to 1 968 , . 14 66 0900 2.82 69 418 9 66 8200 3.85 46 440 19 59 460 0 6. 73 83 192 11 594800 6. 82 85 400 24 66 1800 6. 92 70 548 35 590500 6. 98 66 1100 18 66 160 0 8.50 70 350 42 590000 10.4 66 974 25 66 1000 18.5 60 . TABLE 6. 1 Annual Maximum Discharge Record 65 4 967 583 69 0 957 814 871 859 843 837 714 725 917 708 61 8 68 5 941 822 883 766 827 69 3 66 0 902 67 2 61 2 742 703 731 63 7 810 981 64 6 992 734 565 67 8 962 . 2 060 65 60 1949 1530 5170 1950 1590 4720 1951 169 0 2720 1952 1420 5290 1953 1330 65 80 1954 60 7 548 1955 1380 68 40 19 56 166 0 3810 1957 2290 65 10 1958 2590 8300 1959 3 260 7310 1 960 2490 164 0 1 961