The Multivariate Percentile Power Method Transformation Dr. - PowerPoint PPT Presentation

The Multivariate Percentile Power Method Transformation Dr. Jennifer Koran Mathematics Colloquium Southern Illinois University Carbondale November 10, 2016

Power Method (PM) Transformation Headrick (2010): 𝑛 𝑑 𝑗 𝑎 𝑗− 1 𝑞 ( 𝑎 ) = 𝑗 =1 𝑎 𝑨 = 𝜒 𝑨 = 2 𝜌 − 1 2 exp − 𝑨 2 2 𝑔 𝑨 𝐺 𝑎 𝑨 = Φ 𝑨 = 𝜒 𝑣 𝑒𝑣 , −∞ < 𝑨 < + ∞ −∞

The conventional moment based Fleishman third-order power method Headrick (2010), based on Fleishman (1978): 𝛽 1 = 0 = 𝑑 1 + 𝑑 3 2 + 2 𝑑 3 2 + 6 𝑑 2 𝑑 4 + 15 𝑑 4 2 𝛽 2 = 1 = 𝑑 2 3 + 6 𝑑 2 2 𝑑 3 + 72 𝑑 2 𝑑 3 𝑑 4 + 270 𝑑 3 𝑑 4 2 𝛽 3 = 8 𝑑 3 ? 4 + 60 𝑑 2 2 + 60 𝑑 3 4 + 60 𝑑 2 2 𝑑 3 3 𝑑 4 + 936 𝑑 2 𝑑 3 2 𝑑 4 𝛽 4 = 3 𝑑 2 2 + 4500 𝑑 3 2 + 3780 𝑑 2 𝑑 4 2 𝑑 4 2 𝑑 4 3 + 630 𝑑 2 4 − 3. + 10395 𝑑 4

The percentile based power method uses four moment-like parameters Karian and Dudewicz (2011, pp. 172-173): Median: 𝛿 1 = 𝜄 0.50 Inter-decile range: 𝛿 2 = 𝜄 0.90 − 𝜄 0.10 𝜄 0.50 −𝜄 0.10 Left-right tail-weight ratio : 𝛿 3 = 𝜄 0.90 −𝜄 0.50 “percentile skew” 𝜄 0.75 −𝜄 0.25 “percentile kurtosis” Tail-weight factor: 𝛿 4 = 𝛿 2 Restrictions: −∞ < 𝛿 1 < + ∞ , 𝛿 2 ≥ 0, 𝛿 3 ≥ 0, 0 ≤ 𝛿 4 ≤ 1 A symmetric distribution implies that 𝛿 3 = 1.

Substitute the standard normal distribution percentiles ( 𝑨 𝑣 ) into parameter equations 𝛿 1 = 𝑞 ( 𝑨 0.50 ) = 𝑑 1 3 𝛿 2 = 𝑞 ( 𝑨 0.90 ) − 𝑞 ( 𝑨 0.10 ) = 2 𝑑 2 𝑨 0.90 + 2 𝑑 4 𝑨 0.90 𝛿 3 = 𝑞 ( 𝑨 0.50 ) − 𝑞 ( 𝑨 0.10 ) 2 𝑑 3 𝑨 0.90 𝑞 ( 𝑨 0.90 ) − 𝑞 ( 𝑨 0.50 ) = 1 − 2 𝑑 2 + 𝑑 3 𝑨 0.90 + 2 𝑑 4 𝑨 0.90 3 𝛿 4 = 𝑞 ( 𝑨 0.75 ) − 𝑞 ( 𝑨 0.25 ) = 2 𝑑 2 𝑨 0.75 + 2 𝑑 4 𝑨 0.75 3 𝛿 2 2 𝑑 2 𝑨 0.90 + 2 𝑑 4 𝑨 0.90 where 𝑨 0.50 = 0, 𝑨 0.90 = 1.281 ⋯ , 𝑨 0.75 = 0.6744 ⋯ from the standard normal distribution.

closed-form expressions for the Percentile PM coefficients 𝑑 1 = 𝛿 1 3 3 𝛿 2 𝛿 4 𝑨 0.90 − 𝑨 0.75 Boundary conditions 𝑑 2 = 3 3 2 𝑨 0.90 𝑨 0.75 − 2 𝑨 0.90 𝑨 0.75 for Percentile PM pdfs 𝛿 2 1 − 𝛿 3 𝑑 3 = 2 2 1 + 𝛿 3 𝑨 0.90 𝛿 2 𝛿 4 𝑨 0.90 − 𝑨 0.75 𝑑 4 = − 3 3 2 𝑨 0.90 𝑨 0.75 − 2 𝑨 0.90 𝑨 0.75

Univariate Percentile PM Transformation process 𝑑 1 = 𝛿 1 3 3 𝛿 2 𝛿 4 𝑨 0.90 − 𝑨 0.75 𝑑 2 = 3 3 2 𝑨 0.90 𝑨 0.75 − 2 𝑨 0.90 𝑨 0.75 𝛿 2 1 − 𝛿 3 𝑑 3 = 2 2 1 + 𝛿 3 𝑨 0.90 𝑛 𝑑 𝑗 𝑎 𝑗−1 𝑞 𝑎 = 𝛿 2 𝛿 4 𝑨 0.90 − 𝑨 0.75 𝑗=1 𝑑 4 = − 3 3 2 𝑨 0.90 𝑨 0.75 − 2 𝑨 0.90 𝑨 0.75

Simulating correlated data 𝑞 𝐾 𝑎 𝐾 = 𝐾 𝑞 𝐿 𝑎 𝐿 = 𝐿 ? 𝐷𝑝𝑠𝑠 𝑎 𝐾 , 𝑎 𝐿 ≠ 𝐷𝑝𝑠𝑠 J , 𝐿 𝐷𝑝𝑠𝑠 𝑎 𝐾 , 𝑎 𝐿 = 𝐷𝑝𝑠𝑠 J , 𝐿 Intermediate Specified correlation correlation

Multivariate Conventional PM Vale and Maurelli (1983) 𝜍 𝑘𝑙 = 𝐹 𝑞 𝑎 𝑘 𝑞 𝑎 𝑙 2 = 𝑠 𝑘𝑙 𝑑 𝑘2 𝑑 𝑙2 + 3𝑑 𝑘4 𝑑 𝑙2 + 3𝑑 𝑘2 𝑑 𝑙4 + 9𝑑 𝑘4 𝑑 𝑙4 + 2𝑑 𝑘1 𝑑 𝑙1 𝑠 𝑘𝑙 + 6𝑑 𝑘4 𝑑 𝑙4 𝑠 𝑘𝑙 Specified Correlation Matrix Intermediate Correlation Matrix 𝑆 Ρ 1 2 3 4 1 2 3 4 1 1 1 1 2 2 0.80 1 0.897 1 3 0.70 0.60 1 3 0.831 0.666 1 4 0.65 0.50 0.45 1 4 1 0.750 0.580 0.489

Multivariate Percentile PM with Spearman correlation 𝑜 − 1 sin −1 𝑠 𝜊 𝑘𝑙 = 6 𝑜 − 2 1 𝑘𝑙 𝑜 − 1 sin −1 𝑠 + 𝑘𝑙 𝜌 2 Specified Correlation Matrix Intermediate Correlation Matrix, n = 25 𝑆 Ξ 1 2 3 4 1 2 3 4 1 1 1 1 2 0.835 1 2 0.80 1 3 0.739 0.639 1 3 0.70 0.60 1 4 1 0.689 0.536 0.484 4 0.65 0.50 0.45 1

Multivariate Percentile PM Transformation process 1. Specify percentiles and obtain polynomial coefficients to transform each variable 2. Specify Spearman correlations for each pair of variables 3. Solve for intermediate Pearson correlations 4. Simulate random normal variates with the intermediate Pearson correlations 5. Substitute the random normal variates into the polynomial equations using the coefficients from Step 1

The Simulation and Monte Carlo Study • Fortran algorithm • generate 25,000 independent sample estimates for the specified parameters – conventional skew ( 𝛽 3 ) and kurtosis ( 𝛽 4 ) and – left-right tail-weight ratio 𝛿 3 and tail-weight factor 𝛿 4 • 𝑜 = 25 and 𝑜 = 750 • Bias-corrected accelerated bootstrapped median estimates, using 10,000 resamples [Spotfire S+]

Distribution 1 Figure 1. The power method (PM) pdf of Distribution 1. Conventional PM Percentile PM Percentiles 𝜄 𝑦 0.10 = − 0.7560 Skew: 𝛽 3 = 0 Left-right tail-weight ratio: 𝛿 3 = 1.0000 𝜄 𝑦 0.25 = − 0.2347 Kurtosis: 𝛽 4 = 25 Tail-weight factor : 𝛿 4 = 0.3105 𝜄 𝑦 0.50 = 0 𝜄 𝑦 0.75 = 0.2347 𝑑 1 = 0 𝑑 1 = 0 𝜄 𝑦 0.90 = 0.7560 𝑑 2 = 0.2553 𝑑 2 = 0.4327 𝑑 3 = 0 𝑑 3 = 0 𝑑 4 = 0.2038 𝑑 4 = 0.3454

Distribution 2 Figure 2. The power method (PM) pdf of Distribution 2. Conventional PM Percentile PM Percentiles 𝜄 𝑦 0.10 = − 0.6851 Skew: 𝛽 3 = 3 Left-right tail-weight ratio: 𝛿 3 = 0.3130 𝜄 𝑦 0.25 = − 4652 Kurtosis: 𝛽 4 = 21 Tail-weight factor : 𝛿 4 = 0.3335 𝜄 𝑦 0.50 = − 0.2523 𝜄 𝑦 0.75 = 0.1901 𝑑 1 = − 0.2523 𝑑 1 = − 0.3203 𝜄 𝑦 0.90 = 1.0092 𝑑 2 = 0.4186 𝑑 2 = 0.5315 𝑑 3 = 0.2523 𝑑 3 = 0.3203 𝑑 4 = 0.1476 𝑑 4 = 0.1874

Distribution 3 Figure 3. The power method (PM) pdf of Distribution 3. Conventional PM Percentile PM Percentiles 𝜄 𝑦 0.10 = − 0.9207 Skew: 𝛽 3 = 2 Left-right tail-weight ratio: 𝛿 3 = 0.2841 𝜄 𝑦 0.25 = − 0.6717 Kurtosis: 𝛽 4 = 7 Tail-weight factor : 𝛿 4 = 0.1894 𝜄 𝑦 0.50 = − 0.2600 𝜄 𝑦 0.75 = 0.3882 𝑑 1 = − 0.2600 𝑑 1 = − 0.2908 𝜄 𝑦 0.90 = 1.2547 𝑑 2 = 0.7616 𝑑 2 = 0.8516 𝑑 3 = 0.2600 𝑑 3 = 0.2908 𝑑 4 = 0.0531 𝑑 4 = 0.0593

Distribution 4 Figure 4. The power method (PM) pdf of Distribution 4. Conventional PM Percentile PM Percentiles 𝜄 𝑦 0.10 = − 1.2816 Skew: 𝛽 3 = 0 Left-right tail-weight ratio: 𝛿 3 = 0.0000 𝜄 𝑦 0.25 = − 0.6745 Kurtosis: 𝛽 4 = 0 Tail-weight factor : 𝛿 4 = 0.1226 𝜄 𝑦 0.50 = 0 𝜄 𝑦 0.75 = 0.6745 𝑑 1 = 0 𝑑 1 = 0 𝜄 𝑦 0.90 = 1.2816 𝑑 2 = 1 𝑑 2 = 1 𝑑 3 = 0 𝑑 3 = 0 𝑑 4 = 0 𝑑 4 = 0

Marginal Results n = 25 Skew ( 𝛽 3 ) and Kurtosis ( 𝛽 4 ) results for the Conventional PM. Dist Parameter Estimate 95% Bootstrap C.I. Standard Error Relative Bias % 1 𝛽 3 = 0 -0.0223 -0.0497,0.0045 0.013660 -- 𝛽 4 = 25 4.4560 4.4011,4.5261 0.030200 -82.18 2 𝛽 3 = 3 1.5750 1.5579,1.5911 0.008122 -47.50 𝛽 4 = 21 3.6960 3.6452,3.7525 0.027010 -82.40 3 𝛽 3 = 2 1.2780 1.2677,1.2893 0.005561 -36.10 𝛽 4 = 7 1.5230 1.4849,1.5662 0.020430 -78.24 4 𝛽 3 = 0 0.0034 -0.0038,0.0103 0.003626 -- 𝛽 4 = 0 -0.1786 -0.1906,-0.1678 0.005579 -- Left-right tail-weight ratio 𝛿 3 and tail-weight factor 𝛿 4 results for Percentiles PM Dist Parameter Estimate 95% Bootstrap C.I. Stand. Error Relative Bias % 1 𝛿 3 = 1.0000 1.0050 0.9942, 1.0154 0.005348 -- 𝛿 4 = 0.3105 0.3208 0.3191, 0.3227 0.000947 -- 2 𝛿 3 = 0.3430 0.3466 0.3438, 0.3497 0.001485 1.04 𝛿 4 = 0.3868 0.3972 0.3954, 0.3993 0.000983 2.70 3 𝛿 3 = 0.4361 0.4472 0.4444, 0.4501 0.001464 2.53 𝛿 4 = 0.4872 0.4960 0.4943, 0.4980 0.001003 1.80 4 𝛿 3 = 1.0000 0.9978 0.9912, 1.0045 0.003380 -- 𝛿 4 = 0.5263 0.5294 0.5279, 0.5310 0.000801 --

Correlation Results n = 25 Correlation results for the Conventional PM, 𝑜 = 25 Parameter Estimate 95% Bootstrap C.I. Standard Error RSE Relative Bias % ∗ = 0.80 𝜍 12 0.8275 0.8258 , 0.8290 0.002612 3.43 0.0032 ∗ = 0.70 𝜍 13 0.7358 0.7340 , 0.7376 0.001944 5.12 0.0026 ∗ = 0.65 𝜍 14 0.6959 0.6943 , 0.6976 0.001575 7.07 0.0023 ∗ = 0.60 𝜍 23 0.6209 0.6185 , 0.6236 0.002075 3.48 0.0033 ∗ = 0.50 𝜍 24 0.5376 0.5354 , 0.5400 0.001595 7.52 0.0030 ∗ = 0.45 𝜍 34 0.4677 0.4650 , 0.4700 0.001638 3.93 0.0035 Correlation results for the Percentiles PM, 𝑜 = 25 Parameter Estimate 95% Bootstrap C.I. Standard Error RSE Relative Bias % 𝜊 12 = 0.80 0.8141 0.8123 , 0.8146 0.002007 1.76 0.0025 𝜊 13 = 0.70 0.7138 0.7119 , 0.7162 0.002005 1.97 0.0028 𝜊 14 = 0.65 0.6658 0.6646 , 0.6685 0.001954 2.43 0.0029 𝜊 23 = 0.60 0.6142 0.6115 , 0.6154 0.001719 2.37 0.0028 𝜊 24 = 0.50 0.5154 0.5131 , 0.5177 0.001809 3.09 0.0035 𝜊 34 = 0.45 0.4646 0.4631 , 0.4685 0.001534 3.23 0.0033