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MLE OF A 3-PARAMETER GAMMA DISTRIBUTION ANALYSIS OF RAINFALL INTENSITY DATA SETS
1David.I.J., 2Adubisi.D.O., 3Ogbaji.O.E., 4Adehi.U.M., and 5Ikwuoche.O.P.
2, 3, 5Department of Mathematics and Statistics, Federal University Wukari, Nigeria "Department of Statistics, Nasarawa State University, Keffi, Nigeria davidij@fuwukari.edu.ng1, adubisiobinna@fuwukari.edu.ng2, ogbajieka@yahoo.com3, maryadehi@yahoo.com4, philonye10@gmail.com5
Abstract
This research presents the maximum likelihood estimation of a three-parameter Gamma distribution with application to four types of average rainfall intensities in Nigeria. These data sets are average half-yearly, yearly, quarterly and monthly rainfall intensities. The fitted three-parameter Gamma is compared to a two-parameter Gamma distribution using empirical distribution function (EDF) tests. The tests used are Cramer-von Mises, Anderson-Darling and Kolmogorov-Smirnov statistics. Based on the results obtained at 10% significance level both the two-parameter and three-parameter Gamma distributions are of good fit to only the average yearly rainfall intensity data. A kernel density plot revealed that the average half-yearly, quarterly and monthly rainfall intensity data sets are multi-modal in nature hence a reason for both Gamma distributions poor fit to the data sets. Also, the PDF, CDF and Q-Q plots are presented which supported the outcome of the analysis.
Keywords: Gamma distribution, Anderson-Darling, Cramer-von Mises, Kernel density, Kolmogorov-Smirnov, Maximum likelihood estimation
1. Introduction
Classical analysis of statistical data in most fields including meteorology and hydrology has assumed that the data being analyzed may be reasonably modeled by distribution with somewhat light tailed where the tail of the density function approaches zero like some kind of exponential function (Arshad, Rasool & Ahmad, [1]). One of the most difficult problems in rainfall modeling is often the fitting of theoretical models to rainfall data (Richard, [10]). According to Hughes [8], the primary objective of modeling is frequently to generate a long representative time series of stream flow volumes from which water supply schemes can be designed. Wolfram [14] stated that Gamma distribution is a general type of statistical distribution that is related to the Beta distribution and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. According to Alghazali & Alawadi [2], the two-parameter Gamma distribution is widely known and used in hydrological analysis. However, Chow et al., [4] stated that the two-parameter Gamma distribution has a lower bound at zero, this condition handicaps its application to hydrological variables with lower bound larger than zero.
In the theory of probability and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. It has a shape and scale parameters, say a and p respectively. If p is an integer, then the distribution represents the sum of p independent exponentially distributed random variables, each of which has a mean of a [which is equivalent to a rate
David,I.J., Adubisi,D.O., Ogbaii,O.E., Adehi,U.M., Ikwuoche,O.P. RT&A, No 4 (71) MLE OF A 3-PARAMETER GAMMA DISTRIBUTION_Volume 17, December 2022
parameter of a'1] (Wackerly et al., [12]). It often appear as solution to problems in Statistical
Physics, for example, the energy density of classical ideal gas or the Wien (Vienna) distribution is
an approximation to the relative intensity of black radiation as a function of the frequency (Crooks,
[5]). The disadvantage of Gamma distribution is that the cumulative distribution function cannot
be plotted. The 1-parameter gamma distribution is very limited in hydrological analysis due to its
relative inflexibility in fitting to frequency distributions of hydrologic variation (Aksoy, [5]).
Gamma distribution is widely used in many fields like reliability, survival analysis, hydrology,
ecology, etc. (Dikko, et al., [7]) Many variant of the gamma distribution exist and different
estimation techniques have been used for estimating the gamma distribution parameters. These
estimation techniques include methods of moment (MOM), percentile method, graphical
estimation technique, maximum likelihood estimation (MLE), etc, with different modifications of
the estimating techniques. The objective of this research is to present the estimation of the three
parameters Gamma distribution using MLE and its application to four average rainfall intensity
data sets for Nigeria.
2. Methods
In this section, the Gamma distribution assumptions for its applicability are presented. The probability density function (PDF) for the Gamma distribution is presented and its parameter estimation is presented using the maximum likelihood estimation technique. Four average rainfall intensity data sets which span for 115 years (1901 - 2015) are fitted for this research. The first data set is a quarterly data while the second data set used was obtained by collapsing the quarterly data to first half (FH) of the year and second half (SH) of the year, that is, average of first and second quarters to produce FH and average of third and fourth quarters to produce SH. The yearly rainfall intensity was used as the third data set and the monthly rainfall intensity data was used as the fourth. Data used was obtained from climate knowledge portal, https://climateknowledgeportal.worldbank.org.
2.1. PDF for A 3-Parameter Gamma Distribution
According to Aksoy [5], the Gamma distribution function is of three different types, 1-parameter, 2-parameters and 3-parameters Gamma distributions. If the continuous random variable x fits to the probability density function of:
f (x) = —^ xk-le~x; x > 0 (1)
w T(k)
it is said that the variable x is 1-parameter Gamma distributed, with the shape parameter k. The Gamma function T(k) in equation (1) is generally expressed as:
T(k) = J
i = J xk-1e~xdx (2)
0
when k = 1, equation (1) becomes a simple exponential distribution function. If x is replaced by x/p in equation (1) the 2-parameter Gamma distribution (2-PGD) with k being the shape parameter and p being the scale parameter is obtained as:
1
Pk~T(k )
which can easily return to 1-parameter Gamma distribution for p =1. Gamma distribution with two parameters k and p denoting the shape and the scale parameters respectively are commonly used in hydrological studies (Alghazali & Alawadi, [2]). The shape of the rainfall distribution is
g(x; k, ß) = —-xk-le /P ; x > 0 (3)
oo
David,I.J., Adubisi,D.O., Ogbaji,O.E., Adehi,U.M., Ikwuoche,O.P. RT&A, No 4 (71) MLE OF A 3-PARAMETER GAMMA DISTRIBUTION_Volume 17 December 2022
regulated by the shape parameter and the scale parameter controls the variation of rainfall
intensity series which is specified in the same unit as the random variable x (Suhaila, & Jemain,
[11]). If x is replaced by (x-X)/P in equation (1) the 3-parameter Gamma distribution (3-PGD)
with k, P and X being the shape, scale and location parameters respectively is obtained as:
1 , _(x-X)/
g(x,0) — —-(x -X)k e /p ; 0 — (P,X,k) > 0 (4)
V 7 pkT(k) J
2.2. Parameter Estimation with Maximum Likelihood Estimation Technique
The likelihood function:
n
Lg (x,0) = n g (x,0) (5)
i=i
Applying the likelihood function of equation (5) to equation (4) we have
n /
l (x_X) /
n 1 n _i=l /
ng(x0) --p^l«-^'' p (6)
Taking the logarithm (ln) of equation (6) we get
In n g(x, 0) ' ln [pkT(k)] n + n (k _ l) ln l (x _ X) _ np-11 (x _ X)
i-1 i=\ i=\
n n
— _nk ln P_ nln T(k) + n (k _ l) ln l (x _ X)_ n P_ l (x _ X)
i=i i=i
n
— _nklnP _nlnT(k) + n(k _ l)lnlxi _n2 (k _ l)ln X
i—1
(7)
_ np~11 xi + n2p~ X
i—l
Differentiating equation (7) with respect to P and setting the derivative to zero, we have
dlnLg(x,0) , i n\_2 m-1
- — _nkPl + nl xi (nP) _ XP
i—l
2
dP
_nkP_l + nl x, (nP)_ _XP~2 — 0 i—l
_nkP~l + x P~2 _XP~2 — 0
x P_ -P2 — nkP_l (8)
Multiply both sides of equation (8) by (nk) l P2 we have P — x (nk) 1 — X (nk) 1
P — (nk)-1 [ x- X] (9)
Differentiating equation (7) with respect to X and setting the derivative to zero, we have
dlnLg(x0 — -n2 (k-l)X-1 + n2P-1 dX
-n2 (k -l)X"1 + n2P~x — 0
n2
P~1 — n2 (1 - k)X-1
n
Multiply both sides of equation (10) by (n~2 ) ip we have
i = ¡3 (1 - k) (11)
Differentiating equation (7) with respect to k and setting the derivative to zero, we have
dlnLg(x,0) =-nln p - nD ]jnr (k)] + nln]T x - n2 lni
dk j—^
n
-nlnp -nD\jnr (k)] + nln^xj -n2 lni = 0
i=1
n
-nlnp + nln^ xj -n2 lni = nD \jnr (k)]
¿=1
n
D\jnr(k)] = ln^x, -nlni-lnp
i=1
D [ln r( k )]= ln
^ x, - ni - P
i=1
where D in equation (12) is the derivative, this implies
d [in r( k )]=dk'n r(. )=rk) k )
1
= -r+Bi--
1
r(k) ' ^ i i + k -1
where y is the Euler-Mascheroni constant and it is given as
' n 1 \
y = lim
- ln (n )+Z1
v
<0.58
«=1 i J
Substituting the value of y in equation (14) into equation (13) we have
-0.58 + fi1--—
y i i + k -1,
-0.58 + jrfi ~ 1
i=1
i i + k -1
= ln
^ xt - npk + np -(nk) 1 xt +(nk ) 1 i
\-1
i=1
(12)
(13)
(14)
(15)
Substituting equation (15) into equation (12) and inserting the estimates of P and i we have
(16)
Equation (16) does not exist in a closed form hence the estimation of k can only be obtained through numerical solution. This can be accomplished using any statistical software. In this research, Statistical Analytical System (SAS) version 9.4 is used to fit both the 2-PGD and 3-PGD.
2.3. Goodness of Fit Test
The goodness-of-fit tests based on empirical distribution function (EDF) are used in this research work. The EDF tests offer advantages over traditional chi-square goodness-of-fit test, including improved power and invariance with respect to the histogram midpoints (D'Agostino and Stephens, [6]). The empirical distribution function is defined for a set of n independent observations Xi, ... ,Xn with a common distribution function F(x). If we Denote the observations ordered from smallest to largest as X(i), ... ,X(n). The empirical distribution function, Fn(x), is defined
as:
F (x) =
0; x < X,
i
(i)
X(i) < x < X(i+i);I = 1,2,...,n
n
1; x > X,
(17)
(n)
Note that Fn(x) is a step function that jump [1/n] in height at each observation, but in the case where two observations or more are equal, that is, when there are nj observations at xj, then Fn(x) becomes a step function that jump [nj/n] in height at each observation xj. This function estimates the distribution function F(x). At any value x, Fn(x) is the proportion or fraction of observations less than or equal to x, while F(x) is the probability of an observation less than or equal to x. EDF statistics measure the discrepancy between Fn(x) and F(x) which are used to conclude whether the empirical distribution Fn(x) fit the hypothesize distribution F(x). In this research, three EDF tests are used in testing the goodness of fit of each distribution fitted to the average monthly, quarterly, half-yearly and yearly rainfall intensity data. The EDF are Kolmogorov-Smirnov, AndersonDarling and Cramer-von Mises. These GOF tests are presented below as follows.
2.3.1. Kolmogrov-Smirnov (D) Statistic
According to Wilks [13], the Kolmogorov-Smirnov (D) Statistic is defined as
D — Supx\Fn (x) - F (x)| (18)
The Kolmogorov-Smirnov statistic belongs to the supremum class of empirical distribution function (EDF) statistics. This class of statistics is based on the largest vertical difference between F(x) and Fn(x). The Kolmogorov-Smirnov statistic is computed as the maximum of D+ and D where D+ is the largest vertical distance between the EDF and the distribution function when the EDF is greater than the distribution function, and D- is the largest vertical distance when the EDF is less than the distribution function.
D — max( D +, D -) (19)
D represents the maximum difference between the empirical and theoretical distributions over all real numbers x, and is referred to as the Kolmogorov-Smirnov value. Fn(x) is the empirical cumulative probability of observing a value less than or equal to y and 1/np is added for each observation (xi) that is greater than zero and less than or equal to y. F(x) is the theoretical cumulative probability at x described by the estimated gamma distribution parameters (P, X, k) . Fn(x) and F(x) are given as (Husak et al., [9])
^ (x)—(i- 2.....niy S >}) (20)
n
.a-1
x) = I f (y)dy = I /P dy (21)
x 1 x
F (x)if (y )dy -ßää I
A smaller value of D implies a better fit between the observed and theoretical distributions for a fixed number of observations, n.
2.3.2. Anderson-Darling Statistic
The Anderson-Darling statistic and the Cramer-von Mises statistic belong to the quadratic class of EDF statistics. This class of statistics is based on the squared difference (Fn (x) - F(x))2. Quadratic statistics have the following general form:
Q = n[ (Fn (x) - F(x))2 W(x)dF(x) (22)
J— w
where, x) is the weight function for the squared differences (Fn (x) — F(x))2.
When the weight function y/(x) = [F (x)(1 — F (x)]—1, then the Anderson-Darling Statistic denoted by A2 is defined as:
A2 = nf (Fn(x) — F(x))2 [F(x)(1 — F(x)]—1 dF(x) (23)
The Anderson-Darling statistic (A2) is computed as follows.
A2 =— n —1 £ [(2i — 1)logU(0 + (2n +1 — 2i) log(1 — U(0)] (24)
< (i) I y^,,. I ± ^v^Sl1 ^(i)„
n — -
where, U is the ith order Statistic.
2.3.3. Cramer-von Mises Statistic
Explained the Cramer-von Mises statistic as similar to Anderson-Darling Statistic, but in the case of Cramer-von Mises statistic, the weights function iy(x) = 1. The Cramer-von Mises statistic denoted by (W2) is defined by:
W2 = n r (Fn (x) — F(x))2 dF(x) (25)
The Cramer-von Mises Statistic (W2) is computed as:
w 2 =£ fu(i) — —]2 +— (26)
£1 { ( ) 2n J 12n
where, U is the ith order Statistic.
3. Results
Results from the fitted distributions are presented below. Table 1 presents the empirical 2-PGD and 3-PGD mean and standard deviation (Std. Dev) values for the average half-yearly rainfall intensity, average yearly rainfall intensity, average quarterly rainfall intensity, and average monthly rainfall intensity data sets, that is, AHYRI, AYRI, AQRI, and AMRI respectively. It is observed that for all the data sets, the 2-PGD and 3-PGD estimates for the mean is the same as the empirical mean estimate. However, both fitted distributions estimates for the standard deviation are different from the empirical standard deviation for each data set except for the AYRI data set. Therefore, both the 2-PGD and 3-PGD estimated equivalent mean and standard deviation values to the that of the empirical mean and standard deviation values of 96.4014 and 7.7945 respectively.
Table 1: Summary Statistics for the Rainfall Data
Data Statistic Observed 2-Gamma 3-Gamma
Type Estimate Estimate
AHYRI Mean 96.401398 96.4014 96.4014
Std. Dev 32.162844 32.74474 35.61137
AYRI Mean 96.401398 96.4014 96.4014
Std. Dev 7.7944973 7.87561 7.860996
AQRI Mean 96.401398 96.4014 96.4014
Std. Dev 78.321223 86.39379 90.5222
AMRI Mean 96.401398 96.4014 96.4014
Std. Dev 85.00559 110.8792 113.5311
The results from the summary statistics clearly give a clue that both the 2-PGD and 3-PGD will fit the average yearly rainfall intensity data better. However, such conclusion cannot be for certain
until the fitted distributions are subjected to goodness of fit tests described earlier in section 2.3. The results for the parameter estimates from the 2-PGD and 3-PGD are presented in Table 2 and Figure 1, 2, 3, and 4 shows the histogram plots, the 2-PGD, and 3-PGD curves with the kernel density curve as well for the AHRI, AYRI, AQRI, and AMRI data sets.
Table 2: Maximum Likelihood Parameter Estimates Results
Data Type Parameter 2-PGD Estimate 3-PGD Estimate
Location **** 41.0887
AHYRI Scale 11.12243 22.92728
Shape 8.667296 2.412528
Location **** -10.213
AYRI Scale 0.643406 0.579615
Shape 149.8298 183.9402
Location **** 4.125437
AQRI Scale 77.42508 88.80176
Shape 1.245093 1.039123
Location **** 0.4245
AMRI Scale 127.5312 134.296
Shape 0.755904 0.714667
Nigeria Average Half-Yearly Rainfall Intensity in mm
ri, y V 1/ 7< v; \
25 40 55
70 85 100 115 100 145 160 175 100 Nigeria Average Half-Yearly Rainfall intensity in mm
Cwves
- GammafTlieta-0 Alpha-8.67 Sigma-11.1)
- GammafTlieta-41.1 Alpha-2.41 Sigma-22.9)
- Kernel(c-0.79)
Figure 1: Fitted Curve for AHYRI Data set
Figure 2: Fitted Curve for AYRI Data set
Figure 3: Fitted Curve for AQRI Data set
Figure 4: Fitted Curve for AMRI Data set
From the figures displayed, it can be seen that the two and three parameter Gamma distributions fits the AYRI data set (Figure 2) better compared to the AHYRI, AQRI, and AMRI data sets. Figure 2 shows a peaked shape with one mode compared to Figure 1, 2, and 3 with two modes, three modes and two modes respectively as depicted by the kernel density curve. To ascertain the 2-PGD and 3-PGD goodness of fit for all data sets, Table 3 presents Cramer-von Mises (W2), Anderson-Darling (A2), and Kolmogorov-Smirnov (D) statistics results for assessing the fitted distributions.
Table 3: Criterion for Assessing Goodness of Fit
Data Type and Goodness of Fit Estimate (P-Values)
GOF Methods 2-PGD 3-PGD
D 0.1809663(<0.001) 0.1972426(<0.001)
AHYRI W2 2.3407958(<0.001) 2.0138826(<0.001)
A2 12.7906188(<0.001) 11.0297705(<0.001)
D 0.06071233(>0.250) 0.05959971(>0.250)
AYRI W2 0.08224762(0.194) 0.07871282(0.217)
A2 0.54769583(0.161) 0.52514845(0.184)
D 0.1095454(<0.001) 0.1179290(<0.001)
AQRI W2 1.9423611(<0.001) 1.6980158(<0.001)
A2 12.1566899(<0.001) 10.5830877(<0.001)
D 4.18050(<0.001) 4.78650(<0.001)
AMRI W2 6.45624(<0.001) 6.00624(<0.001)
A2 38.67804(<0.001) 37.57614(<0.001)
Bold p-values imply good fit
From Table 3 above, it is clearly seen that the 2-PGD and 3-PGD are poor fit to Nigeria average half-yearly, quarterly, and monthly rainfall intensity data sets. The reason is that D, W2 and A2 statistic values produced p-values less than 0.01 but they produced p-values greater than 10% significance level for average yearly rainfall intensity. Therefore, it is clear from the goodness of fit statistics p-values that both the 2-PGD and 3-PGD are good fit to only the average yearly rainfall intensity data. To buttress the results discussed thus far, the cumulative density function (CDF), quantile estimates, and quantile plots (Q-Q plots) are presented. The CDF plots presented in Figure 5, 6, 7, 8, 9, 10, 11 and 12 clearly shows that only the 2-PGD and 3-PGD CDF plots for the AYRI data has a well fitted S-shape as seen in figure 7 and 8 respectively.
Figure 5: 2-Parameter Gamma CDF Curve
Figure 6: 3-Parameter Gamma CDF Curve
Figure 7: 2-Parameter Gamma CDF Curve
Figure 8: 3-Parameter Gamma CDF Curve
Figure 9: 2-Parameter Gamma CDF Curve
Figure 10: 3-Parameter Gamma CDF Curve
Figure 11: 2-Parameter Gamma CDF Curve
Figure 12: 3-Parameter Gamma CDF Curve
The estimated quantile presented in Table 4 shows that the 2-PGD and 3-PGD estimated quantiles are similar to the empirical quantiles for AYRI compared to AHRI, AQRI and AMRI data sets.
Table 4: Quantile Estimates from the three Distributions for the four Quarters
PERCENTAGE OBSERVED 2-Gamma 3-Gamma
1.0 51.2246 36.7578 46.8728
5.0 54.7701 49.5577 53.2972
10.0 59.2066 57.5328 58.4083
25.0 65.4879 72.7802 70.2148
AHYRI 50.0 91.8013 92.7203 88.9753
75.0 126.9564 116.0211 114.5791
90.0 135.8106 140.0208 144.0947
95.0 140.9320 155.8089 164.8805
99.0 147.9636 188.3987 210.4818
1.0 76.5994 79.0310 78.9703
5.0 81.2038 83.8231 83.8090
10.0 86.9932 86.4562 86.4594
25.0 91.0947 90.9790 90.9992
AYRI 50.0 96.7386 96.1870 96.2083
75.0 101.4315 101.5901 101.5931
90.0 106.8712 106.6221 106.5916
95.0 109.5241 109.7110 109.6526
99.0 111.5782 115.6631 115.5364
1.0 7.98870 2.14038 5.20545
5.0 11.63405 8.06373 9.32423
10.0 13.95009 14.59845 14.54737
25.0 24.26035 33.88568 31.70857
AQRI 50.0 67.92154 72.16883 69.04598
75.0 156.24420 133.05217 132.00585
90.0 220.33728 210.27723 214.59713
95.0 228.02246 267.51252 276.84191
99.0 245.61700 398.36593 420.96745
1.0 0.97407 0.25860 0.61226
5.0 2.58519 2.19311 2.22199
10.0 3.91610 5.56936 5.22762
25.0 12.62992 19.93095 18.76462
AMRI 50.0 80.51468 58.62833 56.97441
75.0 171.20598 132.96517 132.42207
90.0 223.01040 237.61879 240.17013
95.0 240.39757 319.16261 324.67099
99.0 266.14557 512.60240 526.04011
The quantile plots for the 2-PGD and 3-PGD are presented in Figure 13, 14, 15, 16, 17, 18, 19 and 20. The 2-PGD and 3-PGD Q-Q plots for the AYRI data set showed almost all points fall on the reference straight line. This implies that the quantiles of the theoretical and data distribution agree for AYRI data set only.
Figure 13: AHYRI 2-P-Gamma Q-Q Plot
Figure 14: AHYRI 3-P-Gamma Q-Q Plot
Figure 15: AYRI 2-P-Gamma Q-Q Plot
Figure 16: AYRI 3-P-Gamma Q-Q Plot
Figure 17: AQRI 2-P-Gamma Q-Q Plot
Figure 18: AQRI 3-P-Gamma Q-Q Plot
Q-Q Plot for AMRI Q-Q Plot for AMRI
0.1 50 75 00 95 99 99.9 0.1 50 75 90 95 99 99.9
0 1 2 3 4 5 6 7 01234567
Gamma Quantiles (Alpha=0 755904) Gamma Quantiles (Alpha=0 714567)
Gamma Line Thrashold-0, Scale-1 27 53 Gamma Line Threshold-0.4245, Scale-134.3
Figure 19: AMRI 2-P-Gamma Q-Q Plot Figure 20: AMRI 3-P-Gamma Q-Q Plot
4. Conclusion
In this research, the maximum likelihood parameter estimation of a 3-PGD is presented. Also, its application to four different average rainfall intensity data sets was performed and compared to a 2-PGD. A goodness of fit test was performed using three criterions, that is, Cramer-von Mises (W2), Anderson-Darling (A2) and Kolmogorov-Smirnov (D) statistics. Based on the results obtained it is concluded that among the four data sets fitted, the 2-PGD and 3-PGD are good fit to Nigeria yearly rainfall intensity data set only. The PDF curves with kernel density curves, CDF curves and Q-Q plots showed supporting evidence as the goodness of fit statistics (W2, A2 and D) results. The kernel density curves showed that AHYRI, AQRI and AMRI data sets are multi-modal data sets and it is a major reason both the 2-PGD and 3-PGD fitted the data sets poorly. Hence, distributions that handle multi-modal data will be more suitable for fitting the AHYRI, AQRI and AMRI data sets.
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