Application of maximum statistical entropy in formulating a non-gaussian probability density function in flow uncertainty analysis with prior measurement knowledge | International Journal of Metrology and Quality Engineering (IJMQE)

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Table 1

Summary of proposed discretization scheme accuracies for sampling from a non-Gaussian PDF.

Shockling (n = 1000)		0th order		1st order		2nd order
Random value	root #, i	ξ_i	ϕ(ξ_i)	ξ_i	ϕ(ξ_i)	ξ_i	ϕ(ξ_i)
r = 0.09	i = 1	1.07485	−0.00122	1.08469	0.00042	1.08399	0.00034
r = 0.09	i = 2	1.10897	0.00014	1.10999	0.00003	1.11169	−0.00015
r = 0.09	i = 3	1.31370	−0.00285	1.32899	0.00000	1.32903	0.00000
r = 0.43	i = 1	2.30325	−0.00574	2.31354	0.00004	2.31349	0.00001
r = 0.43	i = 2	2.59329	0.00874	2.60730	0.00002	2.60738	−0.00002
r = 0.43	i = 3	2.76390	−0.01371	2.77486	−0.00081	2.77586	0.00050
r = 0.63	i = 1	2.88333	−0.00188	2.88584	0.00049	2.88552	0.00020
r = 0.63	i = 2	3.01982	0.00385	3.03578	0.00000	3.03576	0.00000
r = 0.63	i = 3	3.15631	−0.00229	3.16304	−0.00017	3.16389	0.000114
Shockling (n =5000)		0th order		1st order		2nd order
Random value	root #, i	ξ_i	ϕ(ξ_i)	ξ_i	ϕ(ξ_i)	ξ_i	ϕ(ξ_i)
r = 0.09	i = 1	1.08074	−0.00008	1.08135	0.00000	1.08132	0.00000
r = 0.09	i = 2	1.10894	0.00015	1.11033	0.00000	1.11034	0.00000
r = 0.09	i = 3	1.32788	−0.00021	1.32900	0.00000	1.32900	0.00000
r = 0.43	i = 1	2.31314	−0.00018	2.31347	0.00000	2.31347	0.00000
r = 0.43	i = 2	2.60672	0.00038	2.60734	0.00000	2.60734	0.00000
r = 0.43	i = 3	2.77425	−0.00161	2.77548	0.00000	2.77548	0.00000
r = 0.63	i = 1	2.88372	−0.00149	2.88530	0.00000	2.88530	0.00000
r = 0.63	i = 2	3.03466	0.00025	3.03574	0.00000	3.03574	0.00000
r = 0.63	i = 3	3.16238	−0.00038	3.16355	0.00000	3.16355	0.00000
Shockling(n =15000)		0th order		1st order		2nd order
Random value	root #, i	ξ_i	ϕ(ξ_i)	ξ_i	ϕ(ξ_i)	ξ_i	ϕ(ξ_i)
r = 0.09	i = 1	1.08115	−0.00002	1.08132	0.00000	1.08132	0.00000
r = 0.09	i = 2	1.10990	0.00004	1.11034	0.00000	1.11034	0.00000
r = 0.09	i = 3	1.32882	−0.00003	1.32900	0.00000	1.32900	0.00000
r = 0.43	i = 1	2.31339	−0.00004	2.31347	0.00000	2.31347	0.00000
r = 0.43	i = 2	2.60693	0.00025	2.60734	0.00000	2.60734	0.00000
r = 0.43	i = 3	2.77499	−0.00064	2.77548	0.00000	2.77548	0.00000
r = 0.63	i = 1	2.88500	−0.00027	2.88530	0.00000	2.88530	0.00000
r = 0.63	i = 2	3.03537	0.00008	3.03574	0.00000	3.03574	0.00000
r = 0.63	i = 3	3.16307	−0.00016	3.16355	0.00000	3.16355	0.00000

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