»• . < f" > RAINFALL PREDICTION FOR AGRICULTURE AND &TER RESOURCE MANAGEMENT IN THE UNITED STATES VIRG3T ISLANDS Frank L. Mills and Ededet A. Mama i" Technical Report No. 33 March 1990 v •. •v v.* Water Resources Research Center University of the Virgin Islanls . St. Thomas VI 00802 ABSTRACT This paper finds that the gamma model provides a better fit to the rainfall distribution in the United States Virgin Islands, than does the more popular normal distribution model, especially for the dry months. It also shows that in areas of limited longitudinal rainfall data, prediction can be done using shorter rainfall periodsthan usually the case. Finaliyi the paper applies the modeling effort to agriculture, crop insurance, and water resource management. Introduction This paper is an attempt to use statistical models to predict rainfall patterns in the United States Virgin Islands, a microclimatic region of the Caribbean, as an input into urban water management, agriculture, and crop insurance. The paper describes rainfall variability in the United States Virgin Islands. Statistical models of rainfall prediction are examined, especially with regard to their applicability to data that reflect the pattern of rainfall in the territory. Two statistical models are chosen and used to predict monthly rainfall variability for selected stations in the United States Virgin Islands. For each station, prediction is made of the representative wet and dry months. The models are then evaluated to determine which predictive model can be applied to each of these representative rainfall periods. The applications of this modeling effort to urban water resource management and agriculture are discussed. Rainfall modeling and prediction are an essential input into agriculture/especially in regions of light, variable, and unreliable rainfall. In such regions, adequate knowledge is needed of the probability of wet and dry periods, because bumper harvests and lean years may depend on rainfall variability. So far, however, there is very little reliable knowledge in this area of study, especially on small subtropical islands, to be used as inputs into effective agricultural planning (Mills, 1984). In the Caribbean, despite the relationship between rainfall variability and production levels in the plantation economies, very little effort has been devoted to the role of rainfall periodicity on food crops. These crops have a growing season generally shorter than those for plantation crops. For extension officers serving small farmers, especially those growing food crops, understanding rainfall variability will help in making decisions on planting, processing, marketing, and storage activities so as to maximize the farmer's profits. An effective method of rainfall prediction may also be attractive to insurance companies, governments, and other agencies involved in insurance planning for food crops farmers and their properties. Data Limitations Unlike the other English-speaking islands of the Caribbean, the U.S. Virgin Islands lacks time series climatic data for periods before 1950. The main sources of rainfall data for the territory are the Annual reports, Climatological Data for Puerto Rico and the U.S. Virgin Islands, available at the U.S. Weather Bureau in San Juan and the National Climatic Data Center, Asheville, North Carolina. The available data were found to have a few problems. One was that measurements were made in inches, so we were forced to use inches in our data analysis. A second problem was that no data were collected for the territory in the years 1953 and 1954. We used moving averages to interpolate the missing data. This procedure may underestimate the data for these years. A third was that oniy Dorothea on St. Thomas, Cruz Bay on St. John, and Anna's Hope on St. Croix have time series rainfall data extending for over 40 years. Therefore, the averages shown in table 1 were computed using 47 years each for Dorothea and Cruz Bay, 40 years each for Cyril King and Hamilton airports, but only 30 years each for Fort Mylner, Annaly, East Hill, Catherineburg, and Lameshur Bay. 3 A study by Court (1969) in Israel, a semi-arid country with marked rainfall periodicity, showed that shorter periods as little as 15 years can be used successfully in rainfall prediction. However, no such study is known to have been done in the Caribbean. Some stations which were potential sites for selection on the basis of hydrologicai and spatial variability were eliminated because the time series data were limited, or the stations had periods of missing data that were too long to be useful for analysis. However, the selected stations showed as much climatic and hydrologicai variability as data availability would permit. Rainfall Regime of the U.S. Virgin Islands The U.S. Virgin Islands, located about 18° N and 65° W with a total area of about 343.2 km2, is composed primarily of three islands: St. Thomas (72.8 km2), St. Croix (218.4 km2), and St. John (52 km2). Tropical easterlies from the predominant trade winds bring maritime tropical air masses, from moist high-pressure cells to produce orographic rainfall along the northern coasts. Rainfall shows a marked annual cycle with a peak in the summer when the intertropical convergence zone is dominant in the area. A distinct season of low rainfall distinguishes the winter months. The small size of these islands limits the formation of large cloud masses from cdnvectional or frontal processes. Consequently, the annual rainfall on these relatively low tropical islands hardly exceeds 1500mm. (Smith,1986). Each island,however, has highland ranges between 300 and 600m in height running from east to west. These ranges induce orographic rainfall from the prevailing northeast trade winds, so that the islands display a rainfall variability beyond what their small size would suggest. For example, on the northern windward slopes, annual rainfall may exceed 60 inches, while on the southern leeward slopes, it may be less than 30 inches (Smith,1986). This remarkable variability in rainfall is depicted in Table 1, which shows the average monthly and annual rainfalls for nine stations in the U.S. Virgin Islands. These are Dorothea, Fort Mylner, and Cyril King Airport on St. Thomas; Annaly, Hamilton Airport and East Hill on St. Croix; Cruz Bay, Catherineburg, and Lameshur Bay on St. John. These stations were selected to show as much geographic and hydrologicai variations as data availability would permit. Dorothea on the north side of St. Thomas is on the windward side of Crown Mountain (467 m), Fort Mylner is on . Tablel. Mean Monthly Rainfall for selected Stations (inches). Compiled by the authors. Station Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total St. Thomas 1 Dorothea 2.68 1,74 1.97 2.93 4.83 2.90 2.98 3.68 5.51 5.71,5.12 3.04 43.09 2 C. Airport 2.00 1.47 1.76 2.76 4.28 2.88 3.11 4.18 5.53.5.52 4.70 3.08 . 41.27 3 Ft. Mylner .2.56 1.73 2.29 3.40 5.64 3.00 3.26 4.35 5.53 6.38 6.41 4.94. 49.49 St. Croix 4 Annaly .. 3.10 2.15 2.27 3.43 5.70 3.38 3.67. 4.78 6.64 6.17 7.45 4.33 53.07 5 C. Airport 2.21 1.95 1.75 2.62 4.06 2.84 3.15 4.06 4.98 5.39 5.17 3.64 41.83 6 East Hill 1.94 1.45 1.44 2.22 3.87 2.47 2.77 4.05 4.30 4.59 5.16 3.60 37.86 St. John 7 Cruz Bay 2.40 1.89 1.78 2.82 4.93 2.94 3.51; 4.13 5.21 5.31 4.90 3.37 43.19 8 Cath'burg 3.13 2.09 2.49 3.62 5.69 2.99. 3.81 4.81 5.67 5.93 6.64 4.72 51.59 9 Lameshur 2.60 1.79 2.02 3.10 5.50 2.81 3.20 4.21 5.40 5.21 6.39 4.33 46.56 the east side of the island tangential to the prevailing north-east trade winds, while Cyril King Airport, formerly Truman Field, is on the south leeward side of the island! On the island of St. Croix, Annaly is on the windward side of Mt. Stewart (245 m), Hamilton Airport is on the southern leeward side of the island, while East Hill is on the coast of the dry east end of the island. On the island of St. John, Catherineburg is on the more humid windward side of the island, Cruz Bay is on the leeward west coast, while Lameshur Bay is on the leeward south coast. Therefore, on each island, we have selected stations on the windward and leeward sides, and where data availability permits, stations on the east and west of the island. There is marked variability in annual rainfall from island to island (Table 1): St. John, which is the most easterly of the three islands, averages 47.11 inches annually for the three selected stations. St. Thomas and St. Croix, which lie. further west, average 2.5 to 3 inches less, with St. Thomas to the north of St. Croix averaging 44.62 inches, and St. Croix to the south of St. Thomas averaging 44.25 inches. There is greater annual variability in rainfall, within islands. On St.. Thomas, Fort Mylner to the east receives the heaviest fall of 49.49 inches, Dorothea on the northern windward slope receiving 43.09 inches, while Cyril King on the southern leeward coast receives 41.27 inches, giving a range of 6.4 inches for these stations. On the island Of St. Croix, the station with the heaviest rainfall is Annaly with 53.07 inches, while the lowest rainfall is •;6 at East Hill with 37.86 inches, giving a range of 15.21 inches. On St. John, the heaviest rainfall is 51.59 at Catherineburg, while the lowest is at Cruz Bay with 43.19 inches giving a range of 8.4 inches. Thus, St. John, which is smaller in size than St. Thomas, has more annual rainfall variability as indicated by the range for selected stations. St. Croix, which is about three times the size of St. Thomas, experiences almost three times as much annual rainfall variability as the latter station as indicated by the range. Monthly rainfall also displays marked variability, though a very consistent seasonal pattern is displayed by all selected stations. The heaviest rainfall months are September, October and November, each of which receives five inches or more of rain. After November, monthly totals tend to decrease gradually until February or March which are the two driest months. Rainfall totals are about 1.79 inches or below for each of these months. The low rainfall of these months reflect the near- northerly presence of the high pressure cells just north of the Equator where surface winds are hot and diverging, and contribute little moisture. After March, the heat equator moves north and the moist northeastern trades begin their slow but increasing dominance of the area. At all stations the month of May is usually wet compared to April or June. It is also markedby extreme variability. In almost all. cases, the coefficient of variation exceeds 70 percent. ' •'•'•' ' "' 7 • •• '" •''. -; • •• Rainfall periodicity is evident also from a longitudinal profile (fig.i) of monthly rainfalls for Dorothea on St. Thomas (Mills and lniama,1990). For the representative dry month of February, only 4 years out of the 47 between 1941 and 1987 have rainfall in excess of 3 inches. Of these, two years (1949 and 1961) precede 1964, while the other two are after 1964 Years having February rainfall of one inch and below are equally spaced also between the two periods. On the other hand, for the wet month of October, all the years with rain in excess of 9 inches occur after 1964. Therefore, October rainfall at Dorothea generally was heavier after 1964 than in the study period preceding 1964. This periodicity of rainfall raises the question as to whether we could regard the entire rainfall period of the wet months as a single period for statistical analysis, or whether each of these periods is part of a different rainfall cycle. A cursory survey, of longitudinal rainfall indicates that, generally, the period after 1964 had heavier rainfalls than the preceding period, especially for the wet months. The apparent cyclical nature of this rainfall regime will have further implications for our modeling procedure. Modeling Rainfall Variability The use of stochastic modes to predict seasonal and annual rainfall in tropical areas has engaged the attention of researchers since the pioneering work in East Africa in the 1950's when Manning (1950) fitted normal density functions to monthly rainfall data. As a follow up, o _: •Li . ' io.— X - — • i940 February •• A A I • / ~ / # / I ' »"T- 1960 '970 1980 Fig. Jia) .r.ainfa-ii Disiribuiicn far I-orc-inec . .St. :horrcs. .-ebrugry, '941 - "357 95C ^O'J ;?g. I io) ^z:r\fz:\ St. 7hor~Gs. ~ ^.:-sinbui-. ziooer. &~% • > * >SQ en r.or Uorc.ne. :990 .'.;" ' , . 8 •••; ';._;•. V Friedman (1957) as well as Hartley and Lewis (1959) used the incomplete gamma function to describe rainfall in North America. For the Caribbean, Mills (1976) used the normal and the incomplete gamma density functions to predict rainfall in St.-. Kitts, and showed later (Mills, 1984) that the incomplete gamma provides a better prediction of the monthly rainfall pattern in that Caribbean island than does the normal distribution. The methodological approach for this study is as follows: nine rainfall stations, three on each island, are selected for analysis, and the monthly rainfalls for selected stations are tested for normality so as to determine whether the normal density function or some other model should be used to fit these data. The data for the chosen stations are tested for normality using a procedure developed by Snedecor and Cochran (1980). This procedure is used to test the degree of skewness and kurtosis of the distributions to determine whether they are significantly different from those expected in data that are normally distributed. According to Snedecor and Cochran (1980), the degree of skewness, denoted by the third moment about the mean, is dependent on the scale on which the data, are recorded. . In order to generate a coefficient of skewness independent of this scale, the.authors generated a function denoted by g-j. The sample estimate is given by: g1 = {m3 / (m2>/rrr2)} where m2 is obtained from: m2 =£(X-X)2 and m3 from: m3 =^(X-X)37 n If the sample comes from a normally distributed population, than g1 is normally distributed with mean equal to zero and standard deviation (^6/n) provided n > 150. For sample sizes between 25 and 200, the authors provide tabled values of the one-tailed distribution at the p = 0.05 and p = 0.01 levels as accurate measures of g1v Therefore, if the coefficient of skewness exceeds the critical levels shown in this table, the distribution is considered significantly different from normal for given values of n. at the one-tailed p = 0.01 and 0.05 . We shall use this procedure in our test of normality. We shall test the data also to determine whether the degree of kurtosis in the distribution departs significantly from what would have been expected from a normal distribution. Two procedures are recommended. The first is by Geary (1936) who developed a value a which equals the mean deviation divided by the standard deviation. If the population is normally distributed, then a has the value 0.7979. Leptokurtic distributions have higher values while platykurtic ones have lower values than 0.7979. to .. Snedecor and Cochran (1980) have developed another test of kurtosis denoted by g2 such that: g2 - • (m4/ m22 - 3) where m4 is the fourth moment about the mean, and m2 is defined as above. For large samples exceeding 1000, g2 is normally distributed with mean 0 and standard deviationV(24/n) = 0.219. For smaller sample size (of 50 to 1000), the authors developed tabled values for testing at p-values of 0.05 and 0.01. The authors state that both g| and g2 are similar to that for Geary's measure as described above. Our final step involves fitting observed rainfall frequencies to expected frequencies generated by a selected theoretical model. The Kolmogorov-Smirnov goodness-of-fit test is applied to determine the degree of agreement between the distribution of observed rainfall values for eabh stationand those generated by the specified theoretical distribution. Mills (1984) applied this method to determine the degree of agreement between the distribution of observed rainfall values for each station in St. Kitts and those generated by the specified theoretical distribution. Next, each rainfall data series is divided into two periods. The expected frequencies of the earlier period are used to predict the observed rainfall in the latter period. Finally, the implications of this study for agriculture and urban water resource management are discussed. 11 From time to time, various theoretical models have been applied to the prediction of rainfall patterns. These include the normal, binomial, truncated normal and incomplete gamma probability distributions, as well as the regression model. Since the pioneering work of Glover and Robinson (1953), the normal distribution model has been the model most commonly used in rainfall prediction, probably because the normal distribution function table is readily available. Another reason may be that the normal distribution tends to model rainfall data in humid climates fairly well. However, its suitability to data in drier climates has not been tested fully. In this study, we compare the normal distribution and the incomplete gamma functions. Two fundamental assumptions that are made in fitting an observed rainfall frequency distribution to a theoretical one. First, individual annual or monthly totals are considered to be random selections from a set of possible values. A random selection does not necessarily give a good estimate of the population frequencies, but a large number of such selections provides a better fit. However, since data of this kind are limited,; a method which provides the most efficient estimate of the population probabilities from a sample must be identified. Second, since available data limit us to one sample, our task is to use the sample data to estimate the true frequency distribution from which the sample was derived. This approach is based on the assumption •'12 . . "••'. that the frequency distribution of the population from which the sample was drawn possesses the characteristics of a defined mathematical model. Even though mathematical models may not exactly replicate the behavior of natural phenomena, the correspondence is often close enough for practical applications. a\ The Normal Distribution Model One mathematical model which is often used in scientific applications is the normal distribution. Studies of monthly rainfall variability in humid environments tend to show that the normal curve approximates the rainfall distribution pattern. Thus the mean (u) and standard deviation (^) of the data present a good fit to the rainfall distribution pattern and confidence limits can be calculated (Manning,1950). In dry areas, however, the normal distribution has been found inadequate for fitting highly skewed data representative of such areas (Barger and Thorn, 1949; Thorn, 1958, Friedman, 1957). Mills (1984) showed that this is also the case for monthly rainfalls in St. Kitts. In. . these cases, average annual monthly rainfall'tend to be low, and the natural lower bound of zero inches becomes important. Further, the largest frequencies of rainfall tend to be close to the mean, with a trailing off of frequencies toward the higher values. This characteristic of the data generally suggests a positively skewed distribution. We do not present a formula for the normal distribution since these are commonly found in many fundamental statistical texts. '•"•• 13 '/'.••' h) Thp Incomplete Gamma Model The incomplete gamma distribution has been applied to precipitation studies under a wide range of conditions because of the relatively good fit that it provides. This mathematical model had been used as a better approximation to the cumulative probability of a variable X, where X represents weekly, monthly, or annual amount of random rainfall (Barger and Thorn, 19.49; Thorn, 1958). Apart from the fact that the incomplete gamma function is suitable for positively skewed distributions, it has the added power of the exponential distribution which has been used for data which show high periodicity. The exponential function distribution has been shown to be a special case of the gamma function with 2 degrees of freedom (Friedman and Jan es, 1957; Clark and Hosking, 1986). The frequency distribution of the gamma function is given by: / (x) 1 ' •7-1 ~*/fi 'fi r (7)' . • A C [0 > 0, 7 > 0) given that x denotes the random variable which is the. amount of rainfall. 13 is the scale parameter, while Y is the shape parameter. The function is positively skewed with the degree of skewness being an inverse function of 6.. Since / (0) • =0 for X < 0, then the distribution has a lower limit of zero but is unlimited to the right. The cumulative probability function is called the incomplete gamma function and is denoted by: x •...'•.• G (x) =. p •? (1 -P) 7-1 -x/0 X .. e y • • \ .0 r (7) i ax. '•" 14 " •' '••••.' • the parameters of which must be estimated from sample data. Friedman and Janes (1957) stated that efficient estimates of these parameters cannot be made using the method of moments which is used to estimate u and s. Thorn (1958) used the maximum likelihood procedure to estimate these parameters. Following this procedure, the estimate of g is given by: 1 + 1 + |[lnx - i|mXi ] 0-5 [lnr-iflnXi] g > 0 while the estimate of 13 is given by: XV g b >.. 0 Since./ (X=0) V 0, the model can be estimated only for months with rainfall greater than zero. However, the model has the added power that for Y= V, the distribution is exponential with ordinate 1/8 at X = 0. It is also a variation of.X2 distribution with 2 degrees of freedom when Y = 1/2n and B = 1. It is also different from the normal distribution in that it has a zero lower bound and is highly skewed for low values of Y. It approaches the normal curve as T becomes large. Confidence estimates of these parameters based on their variance estimates can be made on the assumption that the estimators b and g are normally distributed (Friedman and Jan es, 1957; Thorn 1958). However, 15 Mills (1984) found that the method used by Hartley and Lewis (1959, pp. 2- 3) is easier to calculate. By this procedure, var(g) = g2{(1.79629 - 1.21163u +.. 0.77255u2)/n}, u < 0.5772 and var(g) = g2{(1.00513 + 0.89538u - 0.3558u2)/n}, u > 0.5772 where u = In X- (£ln Xj)/n Our study uses the normal and incomplete gamma functions to fit the monthly rainfall data for the selected stations, and the Kolmogoroy- Smirnov (KS) procedure is used to test the goodness of fit. Data Analysis Monthly rainfall data and some parameters for Dorothea, Cruz Bay, and Hamilton Airport underscore the rainfall variability at these representative stations on each island. For Dorothea (Table 2), the month with the greatest coefficient of variability is April, a relatively dry .month, while August, a relatively wet month has the lowest. April also has the largest coefficients of skewness and kurtosis. October is the wettest month, with the highest positive kurtosis,. while February has the lowest rainfall. . Therefore for Dorothea, October is a very wet and highly variable month, while February is the driest month. Following the same procedure, the same months have been selected.for Cruz Bay, except that the driest month for Cruz Bay is March, not February (Table 3). For Hamilton airport (Tab)e 4), the representative months are different from those for the two former stations, probably because it is on the leeward southern shore of 16 St. Croix(Table 4). February is the most variable dry month with a coefficient of variability of 0.701, and September is the most variable month with a coefficient of variability of 0.663. November is its wettest month, while March is its driest. Thus, spatial and topographical differences produce rainfall variability on these islands far in excess of what would have been expected, given their relatively small sizes. This characteristic should alert planners who use meteorological data to the dangers inherent in extrapolating parameters from one station in an area to other regions, even on relatively small islands. In order to decide whether the distribution of monthly rainfalls for these stations is Significantly different from normal, the procedure developed by Snedecor and Cochran (1980) is used and the skewness and kurtosis are shown in the last two columns of Tables 2, 3, and 4. Table 2: Rainfall Statistics for Dorothea, St. Thomas, 1941 to 1987 (ins). Coeff.of Month Mean 2T677~ Std.dev. 1.574 Variation Skewness Kurtosis January 0.588 1.603** 2.959** February 1.743 1.632 0.667 1.537** 3.478** March 1.965 . 1,7.41 0.886 1.602** .2.565** April 2.930 2.641 0.901 3.598** 16.021** May 4.831 3.962 0.820 1.725** 3.741** June 2.896 2.166 0.748 1,001** 0.073 July 2.966 1.464 0.493 0.773* 1.880** August 3.680 1.585 0.431 1.170** 1.661 September 5.484 2.880 0.525 0.722* 0.935 October 5.669 2.989 0.527 1.477** . 2.859* November 5.438 4.071 0.749 1.347** 1.794 December .3.020 2.044 0.677 1.272** 1.741. **p-value < 0.01 p-value < 0.05 .:••:•'••'• ...-. . • •17.:':' .' v. v.;:- -••', •/•;• /; •••; /• Using the procedure, a coefficient of skewness of 0.810 or more is neejded to reject the assumption of normality at p> 0.01 while a coefficient of 0.548 is needed to reject normality at p = 0.05. The information in these tables shows that the monthly rainfall distributions for the selected stations are positively skewed, and that the skewness is significantly different from that which would have been expected from a normal distribution at p » 0.01 except July and September at Dorothea, and June to August at Cruz Bay where the p - value is 0.05. Using Snedecor and Cochran's procedure, a value of zero indicates a normal distribution, a negative value suggests a platykurtic curve, while a positive value indicates a leptokurtic curve. Our parameter eistimates indicate that the monthly rainfall distributions are significantly different from normal because these estimates are significantly different from zero. Table 3: Rainfall Statistics for Cruz Bay, St. John 1941 to 1987 (ins). Month Mean Coeff. of Std.dev. Variation January February 2.400 1.885 1.438 1.301 0.599 0.690 March 076 .1.507 0.848 April 2.824 2.482 0.879 May 5.013 3.912 0.780 June 2,937 1.655 0.563 July 3.514 1.587 0.451 August September October 4.147 . 5.209 5.310 . 1.747 2.953 3.29b 0.421 0.567 0.621 November 4.903 3.150 0.642 December 3.371 1.842 0.546 **p-value < 0.01 *p-value < 0.05 Skewness Kurtosis 1.735** 4.737** 1.695** 4.065** 1.870** 3.319** 2.458** 7.105** 1,358** 0,941 0.767* 0.225 0.786* 0.543 0.557* -0.702 1.509** 3.586** 1.896** 5.063** 1.214** 1.684 1.146** ,1.594 —• ' 18:- "'.•'•'. '.•• •, -:.V-;.;';- A kurtosis coefficient of 1.88 is heeded to reject the assumption of normality at p ~ 0.01, while a coefficient of 1.99 or more is needed to reject this assumption at p = 0.05. Zero implies a normal distribution. Using our results, the coefficient of kurtosis is significantly different from what would have been expected from a normal distribution, except for five months at Dorothea, six at Cruz Bay, and two at Hamilton Airport. This suggests that the wet months tend toward normality while the dry months do not. Our findings are that the skewness is significantly different from what would have been expected from a normal population. Table 4: Rainfall Statistics for Hamilton Airport, St. Croix, 1948 to 1987 (ins). Month Mean Std. dev, Variation Skewness Kurtosis January 2.209 1.250 0.566 1.714** . 5.235** February i:953 1.369 0.701 2.587** 9.773** March 1.749 1.218 : . 0.696 1.785**; . 4.883** April 2.624 1,847 0.704 1.562** 2.539** May 4!061 3,647 .0.898 1.868** 4.272** June 2.845 1.952 0.686 1.449** 2.710** July 3.171 1.729 0.545 0.878** -0.084 August 4.091 2.037 0.498 1.621** 4.085** September 4.977 .3.300 ,0.663 3.224** 13:495** October 5.135 2.941 0.573 1.434** 3.287** November 5.173 3.482 0.673 1.494** 1.730 . December 3.641 2.559 0.702 1.926** 3.556** **p-value < 0.01 *p-value <: 0.05 19 Even the kurtosis is significantly different from what would have been expected from a normal distribution, except for some of the wetter months, especially on St. Thomas and St. John, which are more humid than St. Croix. This is the. first substantive indication that a different theoretical model may be required to describe adequately the distribution of monthly rainfall at these stations especially for drier months and drier stations. Since one of our purposes is to examine which theoretical model is better suited to rainfall data for selected months for the given stations, a more detailed analysis of these stations follows. In order to test the strength of the models under varying conditions, we selected, in genera), the two driest and the two wettest months for each of the three stations, one on each island. The four selected months are February, March, September, and November, the first two months fall in the dry period, and the other two in the wet one. Table 2 indicates that for Dorothea, February and March are the two driest months of the year. Their distributions are significantly skewed with tails to the right, but they are not mesokurtic. September and November are also similarly skewed but appear mesokurtic. Similar characteristics of skewness and kurtosis are evident in the data for Cruz Bay and Hamilton Airport in Tables 3 and 4 respectively. 20 For Hamilton airport, Table 4 shows that the skewness for the four months selected and the kurtosis for three out of the four are significantly different from what would be expected from a normal distribution. Only the wettest month, November, exhibits a level of kurtosis of 1.730 which is just below the 1.88 which is the critical value for rejecting the assumption of normality. This shows that all months, except the wettest, have degrees of skewness and kurtosis which are significantly different from what would be expected under the assumption of a normal distribution. The wettest month displays a degree of kurtosis significantly different from normal. In order to determine whether these monthly rainfall data depart significantly from what would have been expected from a normal distribution, the KoimogorovrSmirnov (K-S) test, a goodness-of-fit test is used, it measures the degree of agreement between the distribution of a set of observed values and those of a specified theoretical distribution. It determines whether the observations in the sample can reasonably be thought to have been derived from a population of values that has the same theoretical distribution (Siegel, 1956). 21 Table 5: Normal and incomplete Gamma Distribution Statistics for Three Selected Rain Stations. Normal Distribution Incomplete Gamma Std. Shape Scale Maximum Difference* Month Mean Dev. g b Normal Gamma (2) (3) (4) (5) (6) (7) (a) Dorothea: February 1.743 1.632 2.274 0.783 0.095 0.088 March 1.965 1.741 1.379 1.456 0.131 0.047 September 5.484 2.880 3.252 1.686 0.085 0.086 November 5.438 4.071 1.829 2.972 0.148 0.054 (b) Cruz Bay February 1.885 1.301 2.706 0.712 0.090 0.059 March 1.776 1.507 1.845 0.962 0.144 0.081 September 5.209 2.953 3.373 1.544 0.097 0.070 November 4.903 3.150 2.606 1.881 0.124 0.068 (c) Hamilton Airport February 1.953 1.369 2.616 0.746 0.138 , 0.073 March 1.749 1.218 2.521 0.712 0.145 0.067 September. 4.977 3.300 3.668 1.357 0.154. 0.052 November 5.173 3.482 2.820 i.835 0.171 0.111 •Critical |D| for Dorothea and Cruz Bay at p = 0.05 is 0.198. •Critical.|D| for Hamilton at p = 0.05. is 6.215. . The parameter estimates for the three stations under study are shown in table 5. The normal distribution is completely described by the mean and standard deviation, while the gamma distribution is described by g (the shape parameter) and b (the scale parameter). February and March have been selected as representative dry months each with under two inches of rainfall, while September and November are used as .' 22 ••' '. representative wet months, each with over 5 inches of rainfall. The standard deviations indicate that the wet months have the greatest variability of rainfall, with November having the largest. For the gamma distribution, as g (shown in column 4) increases, the distribution tends to normality. An inspection of the shape parameters measured by g and the rainfall amount measured by the mean suggest a relationship between them. A small value of g indicates a symmetrical distribution. In general there appears to be an increase in the shape parameter as the average rainfall increases. Column 6 shows the maximum absolute difference |D| that results from comparing the observed frequencies of each station with expected frequencies generated by the normal distribution function in the Kolmpgorov-Smirnov test. All of the maxima in this column are smaller than the critical values of 0.198 for Dorothea and Cruz Bay, and 0.215 for Hamilton Airport. None of the maxima, is significant at the 0.05 level. This suggests that the expected frequencies predicted by this model for these months do not depart significantly from the actual rain frequencies. However, the values are not very distant from the critical value. If, shorter.time;periods are used for prediction, this model may not be as reliable. .Critical. |D| values for the differences between the observed values and those predicted by the incomplete gamma distribution demonstrate that this distribution predicts frequencies that are not significantly different from actual frequencies for all months for all the stations 23 (Table 5, column 7). In addition, the relative smallness of the |D| values in column 7, when compared with those values in column 6, suggests further that the incomplete gamma function predicts more closely than does the normal distribution in months with skewed and/or non- mesokurtic distributions. Graphical representations of these rainfall distributions confirm this to be the case (Mills and Iniama, 1990). Examples are figs. 2 (a) and 2 (b) for Dorothea where the gamma distribution model predicts the most frequent rainfall of 5 inches, while the normal distribution model predicts this as 6 inches. The observed number of years with 5 inches of rainfall was 14 years. The gamma model predicted this as 7 1/2 while the normal distribution predicted it as 6, years. The tail of the gamma model includes more data than does the tail, of the normal distribution. Predicting Future Rainfall In order to determine how well knowledge of the rainfall pattern in. an earMer short period can be used to predict the rainfall pattern in a future period, the rainfall of representative wet and dry months for each station was divided into two periods, each of equal duration. The periods varied from 15 years for Fort Mylner, Annaly, East Hill, Catherineburg and Lameshur Bay to 23 years for Dorothea and Cruz Bay, We decided to use nine rainfall stations, three on each island so as to give more validity to 14--i 4-i 0- Wim Observed -— Expected . -or OornTftoii ,^t. Thomas'October ! 941 - '..£>;"?; f I •////i •/i :"vj ',7/'lv , '////l ^ £% £2£E] c—^ — <~t '///.;^ Fin / i h i wsinTftil fi.rrftn Tn 5 O^rnP™01?T POUT! OP CUP."? IP-1 HOP?? P*r ^OOTP for Doroihes.it.Tnoma3 vOctGDer 194!.-• 9^7). 3 24 our analysis. The expected frequencies within class limits of one-inch within the earlier period were used to predict the observed frequencies in the latter period, and the Kolmogorov-Smirnov test was used to determine how closely the expected frequencies compared with the observed frequencies in the latter period. Both the normal and incomplete gamma models predict future rainfall well at the 0.05 level of significance, for ail months (Table 6). The critical |D| values for Fort Mylner, Annaly, East Hill, Catherineburg, and Lameshur Bay where the period of prediction was 15 years was 0.338. For Cyril King and Hamilton Airports where the predicted period was 20 years, |D| was .0.294, and for Cruz Bay and Dorothea where the period was 23 years, |D| was 0.275. A major assumption of our prediction process is that the data from the predictor and predicted periods came from the same population. In other words, there should be no. trend or periodicity in the longitudinal data. Where a trend exists in the data, the central tendency and variability parameters are likely to be significantly different, and. the predictive ability of the model is diminished. In order to examine the influence of trend on the predictive ability of our model, the month of November for those stations whose predictor period is only 15 years was. chosen for further analysis. November was selected because this is a month with notable storm activity in the U.S. Virgin Islands, and our preliminary analysis indicated that both the incomplete gamma and norma! models were poor predictors for the month of November at p '= 0,05. • ' •••••• 25;- Table 6: Kolmpgoroy-Smirnov Test of Significance of the Normal and incomplete Gamma Rainfall Prediction for Wettest and Driest Months Station Months |D| Value Normal Confidence Limits |D| Value Gamma Confidence Limits Dorothea n = 23 February October 0.8110 0.1878 p > .20 p > .20 0.1061 0.1582 P>20 p>.20 C.K. Airport n = 20 February October 0.1610 0.1880 p > .20 p>.20 0.1730 9.1675 p>:20 p > .20 Ft. Mylner n = 15 February October 0.0547 0.2918 p > .20 20>p > .10 0.1120 0-3156 p>.20 .10 > p > .05 Annaly n = 15 February October 0.1673 0.2453 p > .20 p>.20 0.1760 0.1866 p > .20 p>.20 H. Airport n = 20 March October 0.1197 0.1224 • p>.20 p > .20 0.1442 0.1219 P > -20 . P"'>.'20 East Hill n = 15 March October 0.0984 0.1653 p > .20 p> .20 0.0751 0.2122 P > .20 p>.20 Cruz Bay n = 23 March October 0.2170 0.1402 p>,20 p> .20 0.2195 0,1321 p>.20 p>.20 Cath'burg n = 15 February October 0.0660 0.1847 p> .20 p> .20 0.0708 0.0893 . P> 20 p>.20 Lameshur Br n - 15 February October 0.1476 0,1376 P > -20 p > .20 0.0931 0.1880 p>.20 p>.2d 26 Conover (1980, pp. 250-259) provided a test for trend in longitudinal data using Spearman's rho, (p), or T as the test statistic, given that 1 - 6T rho = where and n(n2 -.1) T =£[R(X| j -R(Yj)]2 R(Xj) =the ranking of longitudinal data Xj R (Y:) = the ranking of the time period Y; The test procedure using rho is theoretically as robust as that using T, but we present the procedure using T. because this requires fewer calculations. By this procedure, a trend exists in the longitudinal data at the stated level if the calculated value for T is less than the critical value for a/2, or greater than the critical value fori - a/2, where a is the alpha level. The null hypothesis of no trend is.supported if the calculated value of T lies between the values for a/2 and 1 - a/2 (Conover, 1980). Table 7 shows a T test for trend in the longitudinal rainfall data for those stations whose predictor and predicted period is only 15 years. These stations are used because our model predicted stations with n > 20 quite well. This may well be due to the fact that longer periods of prediction tend to smooth out some of the periodicity in the rainfall data, thereby satisfying the assumption that all the data are from the same population. Periodicity may tend to have a greater effect on prediction using shorter rainfall periods. The longitudinal data for Fort Mylner, Gatherineburg, and Lameshur Bay show a significant trend at p =0.05 (Table 7). Consequently, the poor fit of the model for these stations may be explained by the presence of a trend in the longitudinal data which indicated that the data for the two periods do not come from the same population. No trend was found in the data for Annaly and East Hill on St. Croix. Annaly, therefore, represents a station where the model was a good predictor because there was no trend table 7: Test of Trend in November Rainfall for Selected Stations T-Value1 Confidence Limits Fort Mylner2 .2758 .025