qrminfundtheorem | R Documentation |
This function minimizes the negative of log-likelihood for OR1 model with respect to cut-points δ using the fundamental theorem of calculus.
qrminfundtheorem(deltaIn, y, x, beta, cri0, cri1, stepsize, maxiter, h, dh, sw, p)
deltaIn |
initialization of cut-points. |
y |
observed ordinal outcomes, column vector of size (n x 1). |
x |
covariate matrix of size (n x k) including a column of ones with or without column names. |
beta |
β, a column vector of size (k x 1). |
cri0 |
initial criterion, cri0 = 1. |
cri1 |
criterion lies between (0.001 to 0.0001). |
stepsize |
learning rate lies between (0.1, 1). |
maxiter |
maximum number of iteration. |
h |
change in each value of δ, holding other δ constant for first derivatives. |
dh |
change in each value of δ, holding other δ constant for second derivaties. |
sw |
iteration to switch from BHHH to inv(-H) algorithm. |
p |
quantile level or skewness parameter, p in (0,1). |
First derivative from first principle
dy/dx=[f(x+h)-f(x-h)]/2h
Second derivative from first principle
f'(x-h)=(f(x)-f(x-h))/h
f''(x)= [{(f(x+h)-f(x))/h} - (f(x)-f(x-h))/h]/h
= [(f(x+h)+f(x-h)-2 f(x))]/h^2
cross partial derivatives
f(x) = [f(x+dh,y)-f(x-dh,y)]/2dh
f(x,y)=[{(f(x+dh,y+dh) - f(x+dh,y-dh))/2dh} - {(f(x-dh,y+dh) - f(x-dh,y-dh))/2dh}]/2dh
= 0.25* [{(f(x+dh,y+dh)-f(x+dh,y-dh))} -{(f(x-dh,y+dh)-f(x-dh,y-dh))}]/dh2
Returns a list with components
deltamin
: cutpoint vector that minimizes the log-likelihood function.
negsum
: negative sum of log-likelihood.
logl
: log-likelihood values.
G
: gradient vector, (n x k) matrix with i-th row as the score
for the i-th unit.
H
: Hessian matrix.
Rahman, M. A. (2016). “Bayesian Quantile Regression for Ordinal Models.” Bayesian Analysis, 11(1): 1-24. DOI: 10.1214/15-BA939
differential calculus, functional maximization, mldivide
set.seed(101) deltaIn <- c(-0.002570995, 1.044481071) data("data25j4") y <- data25j4$y xMat <- data25j4$x p <- 0.25 beta <- c(0.3990094, 0.8168991, 2.8034963) cri0 <- 1 cri1 <- 0.001 stepsize <- 1 maxiter <- 10 h <- 0.002 dh <- 0.0002 sw <- 20 output <- qrminfundtheorem(deltaIn, y, xMat, beta, cri0, cri1, stepsize, maxiter, h, dh, sw, p) # deltamin # 0.8266967 0.3635708 # negsum # 645.4911 # logl # -0.7136999 # -1.5340787 # -1.1072447 # -1.4423124 # -1.3944677 # -0.7941271 # -1.6544072 # -0.3246632 # -1.8582422 # -0.9220822 # -2.1117739 .. soon # G # 0.803892784 0.00000000 # -0.420190546 0.72908381 # -0.421776117 0.72908341 # -0.421776117 -0.60184063 # -0.421776117 -0.60184063 # 0.151489598 0.86175120 # 0.296995920 0.96329114 # -0.421776117 0.72908341 # -0.340103190 -0.48530164 # 0.000000000 0.00000000 # -0.421776117 -0.60184063.. soon # H # -338.21243 -41.10775 # -41.10775 -106.32758
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