According to the quadrature rules based on interpolation trigonometric polynomials, Kim and Choi gave two quadrature formulas for evaluating (1.1) with (alpha=-2) in [7], in which the cosine transform of variables and trigonometric polynomial interpolation at the practical abscissa were used, where a three-term recurrence relation was used to evaluate the quadrature weights.
The new transformed set of variables (demand principal components) is then updated online from traffic counts in a novel reduced state space model for real time estimation of OD demand.
where f X *, f X 0 *, f X 1 *,..., f X k * are the Laplace transforms of random variables X, X0, X1,..., Xk, respectively.
The proofs depend on the harmonic analysis machinery we developed for integral transforms of two variables, without reference to the combinatorics of moments and bi-free cumulants.
where ( {mathrm{mathcal{L}}}_{I_{FB}}(s) ) and ( {mathrm{mathcal{L}}}_{I_{MB}}(s) ) are the Laplace transform of random variables I FB and I MB evaluated at s ( left(s=frac{m_{d,M}}{varOmega_{d,M}}frac{gamma {r_M}^{alpha }}{P_M}right) ), respectively.
Where ( {mathrm{mathcal{L}}}_{I_{FB}}(s) ) and ( {mathrm{mathcal{L}}}_{I_{MB}}(s) ) are the Laplace transform of random variables I FB and I MB evaluated at ( sleft(s=frac{m_{d,F}theta {r_F}^{alpha }}{varOmega_{d,F}{P}_F}right) ) respectively.
SBP, DBP, and PP were analysed in an ordinary linear regression model using the log-transform of these variables.
where (mathcal {L}(s)) is the Laplace transform of random variable I r evaluated at s conditioned on the distance of the nearest base station from the origin.
The Sumudu transform of one variable function f ( x ) is introduced as a new integral transform by Watugala in [1] and is given by S f ( t ) ( y ) = 1 y ∫ R + f ( t ) exp ( − t y ) d t, y ∈ ( − τ 1, τ 2 ).
10 Denoting the GLS transform of each variable with '*', we finally obtain the HT-estimator from the regressions (3) PSP it * = α 1 PGP it * + α 2 CSP it * + α 3 CGP it * + X i * δ + λ i * + ν it *, using the within average y ¯ it, y ¯ ¯ it = y it − y ¯ it and the level of the time-invariant but exogenous X i -variables as instruments, where y ∈ { PGP, CSP, CGP}.
