A novel generalized combinative procedure for Multi-Scalar standardized drought Indices-The long average weighted joint aggregative criterion

Table 1.

Range of Index ValuesClass$\mathit{SDI}\ge 2$Extreme Wet (EW)$1.5\le \mathit{SDI}<2$Severe Wet (SW)$1\le \mathit{SDI}<1.5$Moderate Wet (MW)$-1\le \mathit{SDI}<1$Normal Dry (ND) (Near Normal)$-1>\mathit{SDI}\ge -1.5$Moderate Dry (MD)$-1.5>\mathit{SDI}\ge -2$Severe Dry (SD)$\mathit{SDI}<-2$Extreme Dry (ED)

Table 2.

StationMonthsPrecipitationTemperature (max)Temperature (min)MeanKurtosisMeanKurtosisMeanKurtosisAstore140.0790.9943.033−0.616−7.157−0.165245.5450.7794.4180.805−5.1850.610373.8621.5589.097−0.344−0.8120.106479.4230.64115.4720.2024.083−0.191565.6360.66720.207−0.1727.7840.089624.8−0.77224.647−0.27411.4191.348725.2662.24327.123−0.62514.5050.287825.8380.54426.769−0.44914.4970.204925.55115.81023.251−0.95110.216−0.2711020.9232.65517.8340.7764.6320.8091117.1922.15311.6310.103−0.283−0.5591227.6818.7095.734−0.111−4.397−0.559Bunji16.2334.48110.0410.3000.136−0.60229.5928.26012.776−0.0572.8920.548314.1027.40918.474−0.5457.786−0.674424.9562.34224.467−0.19912.269−0.380527.6791.93628.6970.00615.316−0.152610.60.92633.32−0.31319.21.050717.8753.60035.3860.36622.603−0.210822.4343.23334.91−0.09121.7330.429914.5755.69531.455−0.24516.6741.986107.43624.93525.678−0.09210.592−0.094112.9419.02618.751−0.7114.6962.095124.94512.46712.1790.4271.1560.007Gupis17.4032.3444.681−0.426−5.639−0.561212.13621.0597.0210.577−3.2810.463315.861.75912.625−0.3461.776−0.712440.60611.64119.044−0.5217.104−0.675526.9610.92123.787−0.00810.939−0.180617.8593.84528.8560.06414.9720.024716.0914.37031.705−0.34817.5870.491824.7679.03530.546−0.48016.7630.099913.69813.38126.521−0.11612.4580.040108.0512.39220.555−0.1796.545−0.840112.4831.16614.230.1531.046−0.911124.47510.1816.9320.500−3.454−0.202

Table 3.

AstoreBunjiGupisDistributionSPISPEISPTISPISPEISPTISPISPEISPTI2P Beta−735.45−565.00−287.79−722.44−1019.68−125.12−399.44−775.08−254.433P Weibull−1036.51−700.54−483.52−1030.98−1178.06−188.38−777.57−910.74−370.554P Beta−1031.38−700.28−473.37−1020.69−1210.97−181.74−788.69−823.15−374.24Arcsine−853.92−643.79−274.88−791.05−1089.247.71−577.40−817.37−151.87Burr−777.82−656.97−332.22−753.84−1152.29158.50−537.17−905.52−21.11Cauchy−906.22−673.83−357.62−790.41−1147.13−2.03−640.15−884.69−235.77Chi−772.18−571.38−253.75−747.42−1024.70−89.65−535.49−779.77−188.78Chi-Square−778.82−570.07−450.78−771.56−1024.59−49.17−549.29−779.77−146.95Cosine−827.71−706.01−182.80−582.27−1175.57165.44−409.73−880.8013.08Curvilinear Trapezoidal−770.35−590.81−369.28−561.78−1128.83−40.20−399.24−809.67−158.55Exponential−992.52−569.99−398.97−727.50−1024.7030.67−596.82−779.77−206.77F-−735.45−565.75−459.87−863.89−1019.682.54−623.87−775.23−124.81Gamma−1024.97−573.48−482.01−879.99−1019.80−116.91−667.73−775.23−294.28Generalized Extreme Value−959.45−700.89−450.09−886.80−1216.78−88.95−744.99−943.97−338.99Generalized normal−974.59−699.88−472.53−929.73−1208.10−124.81−759.83−938.39−355.54Gumbel−942.14−692.28−382.34−754.06−1159.4529.41−607.56−881.41−205.28Inverse Chi-Square−847.29−573.96−260.44−849.03−1024.7074.59−615.75−779.84−126.69Inverse Gamma−894.39−575.70−393.13−864.79−1019.80−75.36−737.28−775.30−333.93Inverse Gaussian−848.87−565.00−355.22−790.50−1019.68−12.65−740.25−775.08−334.88Johnson SB−971.05−640.08−475.88−925.95−1248.44−122.60−457.52−977.62−351.09Johnson SU−969.88−695.65−467.89−924.84−1203.21−120.21−755.59−936.90−351.09Laplace−914.20−683.10−377.36−756.94−1150.6210.91−630.07−886.00−238.05Logistic−920.09−697.50−356.29−752.10−1166.0937.57−596.22−895.15−187.98Log-normal−970.31−574.42−474.91−882.30−1019.80−105.32−729.92−775.30−338.98Normal−917.65−702.40−348.30−747.38−1170.7946.87−585.58−898.11−172.80Rayleigh−934.02−700.69−365.81−748.60−1166.6042.39−591.38−888.30−182.47Scaled/shifted t-−915.53−698.17−359.42−814.16−1165.90−11.77−642.21−893.57−237.55Skewed-normal−951.45−700.79−378.12−744.10−1218.2744.30−590.96−937.90−185.69Trapezoidal−940.42−710.05−354.12−712.78−1224.0080.19−570.94−946.94−151.71Triangular−945.13−685.96−358.76−664.59−1228.2376.39−557.57−947.69−148.56Uniform−741.92−664.32−220.71−622.63−1084.39121.33−478.66−884.52−37.54von Mises−752.43−568.01−355.83−752.74−1019.8046.82−531.25−775.31−173.35

Table 4.

Sr. NoDistributionsProbability functionCDFRange13P Weibull$P(x)=\frac{\alpha }{\beta }{\left.\frac{x-\gamma }{\beta }\right)}^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-{\left.\frac{x-\gamma }{\beta }\right)}^{\alpha }\right]$F$(x)=1-\mathrm{}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-{\left.\frac{x-\gamma }{\beta }\right)}^{\alpha }\right]$$\gamma \le x<+\text{∞}$2Gamma$f(x)=\frac{{\left.x\right)}^{\alpha -1}}{{\beta }^{\alpha }\Gamma (\alpha )}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.\frac{-\left.x\right)}{\beta }\right)$$F(x)=\frac{\Gamma (\alpha ,\frac{x}{\beta })}{\Gamma (\alpha )}$$x>0$3Normal$f(x)=\frac{1}{\sigma \sqrt{2\pi }}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.\frac{-1}{2}{\left.\frac{x-\mu }{\sigma }\right)}^2\right]$$F(x)=\frac{1}{2}\left.1+\text{erf}\hspace{0.33em}\left.\frac{x-\mu }{\sigma \sqrt{2}}\right)\right]$$-\mathrm{\infty }<x<+\mathrm{\infty }$4Cosine$f(x)=\frac{1}{2s}1+\left.\text{cos}\left.\frac{x-\mu }{s}\pi \right)\right]$$F(x)=\frac{1}{2}1+\left.\frac{x-\mu }{s}+\frac{1}{\pi }\text{sin}\left.\frac{x-\mu }{s}\pi \right)\right]$$\mathrm{\mu }-s\le x\le \mathrm{\mu }+s$5Skewed normal$f(x)=\frac{2}{\omega \sqrt{2\pi }}\text{exp}\hspace{0.17em}\left.-\left.\frac{x-\mathrm{\xi }}{2{\omega }^2}\right)\right){\int }_{-\mathrm{\infty }}^{\alpha \left.\frac{x-\mathrm{\xi }}{\omega }\right)}\frac{1}{\sqrt{2\pi }}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-\left.\frac{t^2}{2}\right)\right)dt$F$(x)=\varphi \left.\frac{x-\mathrm{\xi }}{2\omega }\right)-2T\left.\frac{x-\mathrm{\xi }}{2\omega },\alpha \right)$
$T\left.h,\alpha \right)\mathrm{}is\mathrm{}\mathit{Owe}{n}^{\mathrm{\prime }}s\mathrm{}\mathit{Function}$$-\mathrm{\infty }<x<+\mathrm{\infty }$6Trapezoidal$f(x)=\left.\begin{array}{l}\frac{2}{d+c-a-b}\frac{x-a}{b-a}\mathit{for}a\le x<b\\ \frac{2}{d+c-a-b}\mathit{for}b\le x<c\\ \frac{2}{d+c-a-b}\frac{d-x}{d-c}\mathit{for}c\le x\le d\end{array}\right\}$$F\left.x\right)=\left.\begin{array}{l}\frac{1}{d+c-a-b}\frac{1}{b-a}{\left.x-a\right)}^2\mathit{for}a\le x<b\\ \frac{1}{d+c-a-b}\left.2x-a-b\right)\mathit{for}b\le x<c\\ 1-\frac{1}{d+c-a-b}\frac{1}{d-c}{\left.d-x\right)}^2\mathit{for}c\le x\le d\end{array}\right\}$$a\le x\le d$7Triangular$f(x)=\left.\begin{array}{l}\frac{2\left.x-a\right)}{\left.b-a\right)\left.c-a\right)}\mathit{for}a\le x<c\\ \frac{2}{b-a}\mathit{for}x=c\\ \frac{2\left.b-x\right)}{\left.b-a\right)\left.b-c\right)}\mathit{for}c<x\le b\end{array}\right\}$F$(x)=\left.\begin{array}{l}\frac{(x-a)²}{\left.b-a\right)\left.c-a\right)}\mathit{for}a\le x<c\\ 1-\frac{(b-x)²}{\left.b-a\right)\left.b-c\right)}\mathit{for}x=c\\ 1\mathit{for}c<x\le b\end{array}\right)$$a\le x\le b$8Johnson SU$f\left.x\right)=\frac{\delta }{\lambda \sqrt{2\pi }\mathrm{}\sqrt{z^2+1}}\mathrm{e}\mathrm{xp}\left.-\frac{1}{2}{\left.\mathrm{\gamma }+\mathrm{\delta }\ln \left.z+\sqrt{z^2+1}\right)\right)}^2\right)$$z=\frac{x-\mathrm{\xi }}{\lambda }$$F\left.x\right)=\varphi \left.\mathrm{\gamma }+\mathrm{\delta }\ln \left.z+\sqrt{z^2+1}\right)\right)$$\varphi \mathrm{}is\mathrm{}\mathit{Laplace}\mathrm{}\mathit{integral}$$-\mathrm{\infty }<x<+\mathrm{\infty }$9Johnson SB$f\left.x\right)=\frac{\delta }{\lambda \sqrt{2\pi }\mathrm{}z\left.1-z\right)}\mathrm{e}\mathrm{xp}\left.-\frac{1}{2}{\left.\mathrm{\gamma }+\mathrm{\delta }\ln \left.\frac{z}{1-z}\right)\right)}^2\right)$$F\left.x\right)=\varphi \left.\mathrm{\gamma }+\mathrm{\delta }\ln \left.\frac{z}{1-z}\right)\right)$$\varphi \mathrm{}is\mathrm{}\mathit{Laplace}\mathrm{}\mathit{integral}$$\mathrm{\xi }\le x\le \mathrm{\xi }+\mathrm{\lambda }$10Log normal$f(x)=\frac{\hspace{0.17em}\text{exp}\hspace{0.17em}\left.\frac{-1}{2}{\left.\frac{In(x)-\mu }{\sigma }\right)}^2\right]}{\left.x\right)\sigma \sqrt{2\pi }}$$f\left.x\right)=\frac{1}{2}+\frac{\mathit{erf}\left.\frac{In(x)-\mu }{\sqrt{2}\sigma }\right]}{2}$$0<x<+\mathrm{\infty }$11Laplace$f\left.x\right)=\frac{1}{2b}+\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-\frac{\left.x-\mu \right|}{b}\right)$$F\left.x\right)=\left.\begin{array}{l}\frac{1}{2}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.\frac{x-u}{b}\right)\mathrm{}\mathit{for}\mathrm{}x\le u\mathrm{}\\ 1-\frac{1}{2}\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-\frac{x-u}{b}\right)\mathit{for}x\ge u\end{array}\right\}$$x\in R$12Logistic$f\left.x\right)=\frac{\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-\frac{x-\mu }{s}\right)}{s{\left.1+\hspace{0.17em}\text{exp}\hspace{0.17em}\left.\frac{x-\mu }{s}\right)\right)}^2}$$F(x)=\frac{1}{1+\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-\frac{x-\mu }{s}\right)}$$-\mathrm{\infty }<x<+\mathrm{\infty }$13Gumbel$f\left.x\right)=\frac{1}{\beta }\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-(z+\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-z\right))\right]$$z=\left.\frac{x-\mu }{\beta }\right)$$\text{exp}\hspace{0.17em}\left.-\hspace{0.17em}\text{exp}\hspace{0.17em}\left.-\frac{x-\mu }{\beta }\right)\right)$$x\in R$14Rayleigh$f\left.x\right)=\frac{x}{{\sigma }^2}\mathrm{e}\mathrm{xp}\left.\frac{{-x}^2}{{(2\sigma }^2)}\right)$$F\left.x\right)=1-\exp \left.\frac{{-x}^2}{\left.{2\sigma }^2\right)}\right)$$0<x<+\mathrm{\infty }$154P Beta$f(x)=\mathrm{}\frac{{(x-\mathit{min})}^{\alpha -1}{(\mathit{max}-x)}^{\beta -1}}{\mathit{B}\left.\alpha ,\mathrm{}\beta \right){(\mathit{max}-\mathit{min})}^{\alpha +\beta -1}}$$F(x)=\mathrm{}\frac{\mathit{B}\left.\frac{x}{\alpha },\mathrm{}\beta \right)}{\mathit{B}\left.\alpha ,\mathrm{}\beta \right)}$$\mathit{min}<x<\mathit{max}$

Table 5.

AstorBunjiGupisIndexScalesDistributionBICDistributionBICDistributionBICFor SPI13P Weibull−1036.513P Weibull−1030.984P Beta−788.693Gamma−1279.05Gamma−824.87Gamma−1264.956Gamma−892.67Skewed-normal−1162.15Gumbel−1305.359Gamma−896.09Normal−649.05Johnson SU−1518.9612Cosine−913.33Laplace−688.10Johnson SU−937.6124Skewed-normal−1294.42Laplace−843.74Scaled/Shifted t−1407.98For SPEI1Trapezoidal−710.05Johnson SB−1248.44Johnson SB−977.623Trapezoidal−941.49Johnson SB−1323.78Johnson SB−1098.616Trapezoidal−1440.21Johnson SB−1094.44Trapezoidal−1482.599Triangular−1405.81Trapezoidal−976.73Trapezoidal−1513.1212Trapezoidal−1052.48Logistic−1158.15Laplace−1063.6624Johnson SU−1471.48Gumbel−1508.11Laplace−1474.12For SPTI13P Weilbull−483.523P Weibull−188.384P Beta−374.243Johnson SU−542.87Gumbel−190.69Chi-Square−432.186Log-normal−721.00Skewed-normal−300.34Johnson SU−410.049Gamma−725.22Rayleigh−380.83Johnson SU−463.2112Triangular−702.87Laplace−411.86Johnson SU−466.5724Laplace−564.00Trapezoidal−304.73Scaled/Shifted t−629.46

Table 6.

EDEWMDMWNDSDSWSumAstoreSPINA0.023090.092540.095830.676540.069490.042521.0Scale-1SPEINA0.030130.167160.086610.661130.021350.033611.0SPTINA0.019520.079950.118720.678870.053390.049541.0SPINA0.026740.096260.074870.76292NA0.039221.0Scale-3SPEINA0.017560.164750.119970.633950.037610.026161.0SPTINA0.015750.132150.109070.658430.012670.071931.0SPI0.019050.021370.103110.103300.666140.044280.042751.0Scale-6SPEINA0.021510.152330.114700.636200.032260.043011.0SPTINA0.016030.134010.097980.651980.043000.057001.0SPINA0.070270.021620.084680.77117NA0.052251.0Scale-9SPEI0.045730.001780.087880.153500.624730.054240.032131.0SPTI0.007270.017910.106500.085980.678920.049680.053741.0SPI0.011310.019740.106540.111280.667670.056540.026921.0Scale-12SPEI0.024340.023160.094800.098000.685970.057710.016041.0SPTINA0.021360.130680.096100.642460.064910.044491.0SPI0.011110.037040.070370.057410.738890.075930.009261.0Scale-24SPEI0.01667NA0.127780.105560.661110.064810.024071.0SPTINANA0.124070.111110.640740.074070.050001.0SPINA0.021310.284190.074600.57549NA0.044411.0Scale-1SPEINA0.005340.162010.088520.683650.046290.014191.0SPTINA0.012430.284190.094140.55773NA0.051511.0SPI0.012480.010700.055260.076650.778970.016040.049911.0Scale-3SPEINA0.007130.169340.146170.638150.032090.007131.0SPTINA0.026740.096260.074870.76292NA0.039221.0SPINA0.039430.091400.053760.77240NA0.043011.0Scale-6SPEINA0.007130.169340.146170.638150.032090.007131.0SPTINA0.026740.096260.074870.76292NA0.039221.0GupisSPINA0.061260.041440.082880.74955NA0.064861.0Scale-9SPEI0.016220.016220.118920.086490.688290.045050.028831.0SPTINA0.070270.021620.084680.77117NA0.052251.0SPINA0.067030.048910.072460.74457NA0.067031.0Scale-12SPEINA0.036230.103260.105070.695650.005430.054351.0SPTI0.039860.003620.074280.108700.697460.018120.057971.0SPI0.03704NA0.057410.131480.659260.044440.070371.0Scale-24SPEINA0.029630.109260.111110.67037NA0.079631.0SPTINA0.040740.057410.083330.73704NA0.081481.0SPINA0.014220.201210.086950.64801NA0.049611.0Scale-1SPEINA0.007060.185270.171280.593700.026730.015971.0SPTINA0.014240.204720.072750.66037NA0.047931.0SPINA0.026740.096260.074870.76292NA0.039221.0Scale-3SPEINA0.017560.164750.119970.633950.037610.026161.0SPTINA0.015750.132150.109070.658430.012670.071931.0SPI0.019050.021370.103110.103300.666140.044280.042751.0Scale-6SPEINA0.021510.152330.114700.636200.032260.043011.0SPTINA0.016030.134010.097980.651980.043000.057001.0BunjiSPINA0.070270.021620.084680.77117NA0.052251.0Scale-9SPEI0.045730.001780.087880.153500.624730.054240.032131.0SPTI0.007270.017910.106500.085980.678920.049680.053741.0SPI0.011310.019740.106540.111280.667670.056540.026921.0Scale-12SPEI0.024340.023160.094800.098000.685970.057710.016041.0SPTINA0.021360.130680.096100.642460.064910.044491.0SPINANA0.148150.127780.611110.048150.064811.0Scale-24SPEI0.05185NA0.061110.187040.646300.048150.005560.9SPTINANA0.144440.135190.611110.027780.081481.0

Table 7.

KappaP-valueStationScaleMethodSPISPEISPTIPWJADILAWJADCSPISPEISPTIPWJADILAWJADCScale-1LAWJADC0.28500.01760.22200.82501.00000.00000.60300.00000.00000.0000AstorePWJADI0.29700.09700.24301.00000.82500.00000.00480.00000.00000.0000Scale-3LAWJADC0.26500.10600.10900.48001.00000.00000.00020.00020.00000.0000PWJADI0.48400.03810.16301.00000.48000.00000.29200.00000.00000.0000Scale-6LAWJADC0.39000.17300.27900.55601.00000.00000.00000.00000.00000.0000PWJADI0.32100.40600.38701.00000.55600.00000.00000.00000.00000.0000Scale-9LAWJADC0.21800.02630.06150.63201.00000.00000.25800.02160.00000.0000PWJADI0.43200.01890.15901.00000.63200.00000.55700.00000.00000.0000Scale-12LAWJADC0.18500.00290.49600.39201.00000.00000.94000.00000.00000.0000PWJADI0.29600.36300.53101.00000.39200.00000.00000.00000.00000.0000Scale-24LAWJADC0.17300.32500.31400.06831.00000.00000.00000.00000.08800.0000PWJADI0.45000.06380.05151.0000−0.06830.00000.07170.21000.00000.0880Scale-1LAWJADC0.34300.11900.28500.90401.00000.00000.00020.00000.00000.0000BunjiPWJADI0.29800.17100.29901.00000.90400.00000.00000.00000.00000.0000Scale-3LAWJADC0.11600.06890.23300.61601.00000.00000.00470.00000.00000.0000PWJADI0.15300.15300.36401.00000.61600.00000.00000.00000.00000.0000Scale-6LAWJADC0.07270.13000.18400.17301.00000.00640.00000.00000.00000.0000PWJADI0.59900.03630.62701.00000.17300.00000.38200.00000.00000.0000Scale-9LAWJADC0.22400.20600.26700.57801.00000.00000.00000.00000.00000.0000PWJADI0.29300.25400.29501.00000.57800.00000.00000.00000.00000.0000Scale-12LAWJADC0.48900.50800.56700.67501.00000.00000.00000.00000.00000.0000PWJADI0.53800.56800.75601.00000.67500.00000.00000.00000.00000.0000Scale-24LAWJADC0.22000.60100.27700.73001.00000.00000.00000.00000.00000.0000PWJADI0.39200.55200.40701.00000.73000.00000.00000.00000.00000.0000Scale-1LAWJADC0.20600.12800.22500.91201.00000.00000.00040.00000.00000.0000GupisPWJADI0.21200.18300.24501.00000.91200.00000.00000.00000.00000.0000Scale-3LAWJADC0.1420−0.01090.20400.65901.00000.00000.66800.00000.00000.0000PWJADI0.20000.01450.40101.00000.65900.00000.65300.00000.00000.0000Scale-6LAWJADC0.33400.05810.28800.65601.00000.00000.04780.00000.00000.0000PWJADI0.55000.07390.07391.00000.65600.00000.04610.04610.00000.0000Scale-9LAWJADC0.40200.13900.49600.58501.00000.00000.00010.00000.00000.0000PWJADI0.60300.14900.73401.00000.58500.00000.00020.00000.00000.0000Scale-12LAWJADC0.75200.30600.07150.87801.00000.00000.00000.07820.00000.0000PWJADI0.71000.37900.02641.00000.87800.00000.00000.52000.00000.0000Scale-24LAWJADC0.58600.17500.08090.87701.00000.00000.00000.04000.00000.0000Scale-24PWJADI0.54300.19300.06581.00000.87700.00000.00000.11400.00000.0000