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jacobian <- function(eq, x, paraHat){
## Jacobian matrix calculation
n <- length(x)
mpara <- length(paraHat)
Alpha <- paraHat[1]
Beta <- paraHat[2]
if (mpara == 3) Gamma <- paraHat[3]
if (mpara == 4) {Gamma <- paraHat[3]; Delta <- paraHat[4]}
if (mpara == 5) {Gamma <- paraHat[3]; Delta <- paraHat[4]; Epsilon <- paraHat[5]}
jac <- matrix(rep(0, n * mpara), n, mpara)
jacFun <- switch(eq,
Hill = c('-1 / (1 + (Alpha / x)^Beta)^2 * (Alpha / x)^Beta*Beta / Alpha', '-1 / (1 + (Alpha / x)^Beta)^2*(Alpha / x)^Beta * log(Alpha / x)'),
Hill_two = c('x / (Alpha + x)', '-Beta * x / (Alpha + x)^2'),
Hill_three = c('-Gamma / (1 + (Alpha / x)^Beta)^2 * (Alpha / x)^Beta * Beta / Alpha', '-Gamma / (1 + (Alpha / x)^Beta)^2 * (Alpha / x)^Beta * log(Alpha / x)', '1 / (1 + (Alpha / x)^Beta)'),
Hill_four = c('-(Gamma - Delta) / (1 + (Alpha / x)^Beta)^2 * (Alpha / x)^Beta * Beta / Alpha', '-(Gamma - Delta) / (1 + (Alpha / x)^Beta)^2 * (Alpha / x)^Beta * log(Alpha / x)',
'1 /(1 + (Alpha / x)^Beta)', '1 - 1 / (1 + (Alpha / x)^Beta)'),
# Delta for Alpha_one and Epsilon for Beta_one
Hill_five = c('(Gamma-1)/(1+(Alpha/x)^Beta)^2*(Alpha/x)^Beta*Beta/Alpha*(1-1/(1+(Delta/x)^Epsilon))', '(Gamma-1)/(1+(Alpha/x)^Beta)^2*(Alpha/x)^Beta*log(Alpha/x)*(1-1/(1+(Delta/x)^Epsilon))',
'-1/(1+(Alpha/x)^Beta)*(1-1/(1+(Delta/x)^Epsilon))', '-(1+(Gamma-1)/(1+(Alpha/x)^Beta))/(1+(Delta/x)^Epsilon)^2*(Delta/x)^Epsilon*Epsilon/Delta',
'-(1+(Gamma-1)/(1+(Alpha/x)^Beta))/(1+(Delta/x)^Epsilon)^2*(Delta/x)^Epsilon*log(Delta/x)'),
Weibull = c('exp(Alpha + Beta * log(x) / log(10)) * exp(-exp(Alpha + Beta * log(x) / log(10)))',
'log(x) / log(10) * exp(Alpha + Beta * log(x) / log(10)) * exp(-exp(Alpha + Beta * log(x) / log(10)))'),
Weibull_three = c('Gamma * exp(Alpha + Beta * log(x) / log(10)) * exp( -exp(Alpha + Beta * log(x) / log(10)))',
'Gamma * log(x) / log(10) * exp(Alpha + Beta * log(x) / log(10)) * exp( -exp(Alpha + Beta * log(x) / log(10)))', '1 - exp( -exp(Alpha + Beta * log(x) / log(10)))'),
Weibull_four = c(' -(Delta - Gamma) * exp(Alpha + Beta * log(x) / log(10)) * exp( -exp(Alpha + Beta * log(x) / log(10)))',
' -(Delta - Gamma) * log(x) / log(10) * exp(Alpha + Beta * log(x) / log(10)) * exp( -exp(Alpha + Beta * log(x) / log(10)))',
'1 -exp( -exp(Alpha + Beta * log(x) / log(10)))', 'exp( -exp(Alpha + Beta * log(x) / log(10)))'),
Logit = c('1 / (1 + exp(-Alpha - Beta * log(x) / log(10)))^2 * exp(-Alpha - Beta * log(x) / log(10))',
'1 / (1 + exp(-Alpha - Beta * log(x) / log(10)))^2 * log(x) / log(10) * exp(-Alpha - Beta * log(x) / log(10))'),
Logit_three = c('Gamma / (1 + exp( -Alpha - Beta * log(x) / log(10)))^2 * exp( -Alpha - Beta * log(x) / log(10))',
'Gamma / (1 + exp( -Alpha - Beta * log(x) / log(10)))^2 * log(x) / log(10) * exp( -Alpha - Beta * log(x) / log(10))',
'1 / (1 + exp( -Alpha - Beta * log(x) / log(10)))'),
Logit_four = c('( -Delta + Gamma) / (1 + exp( -Alpha - Beta * log(x) / log(10)))^2 * exp( -Alpha - Beta * log(x) / log(10))',
'( -Delta + Gamma) / (1 + exp( -Alpha - Beta * log(x) / log(10)))^2 * log(x) / log(10) * exp( -Alpha - Beta * log(x) / log(10))',
'1 / (1 + exp( -Alpha - Beta * log(x) / log(10)))', '1 - 1 / (1 + exp( -Alpha - Beta * log(x) / log(10)))'),
BCW = c('exp(Alpha + Beta * (x^Gamma - 1) / Gamma) * exp(-exp(Alpha + Beta * (x^Gamma - 1) / Gamma))',
'(x^Gamma - 1) / Gamma * exp(Alpha + Beta * (x^Gamma - 1) / Gamma) * exp(-exp(Alpha + Beta * (x^Gamma - 1) / Gamma))',
'(Beta * x^Gamma * log(x) / Gamma - Beta * (x^Gamma - 1) / Gamma^2) * exp(Alpha + Beta * (x^Gamma - 1) / Gamma) * exp(-exp(Alpha + Beta * (x^Gamma - 1) / Gamma))'),
BCL = c('1 /(1 + exp(-Alpha - Beta * (x^Gamma - 1) / Gamma))^2 * exp(-Alpha - Beta * (x^Gamma - 1) / Gamma)',
'1 / (1 + exp(-Alpha - Beta * (x^Gamma - 1) / Gamma))^2 * (x^Gamma - 1) / Gamma * exp(-Alpha - Beta * (x^Gamma - 1) / Gamma)',
'-1 / (1 + exp(-Alpha - Beta * (x^Gamma - 1) / Gamma)) ^ 2 * (-Beta * x^Gamma * log(x) / Gamma + Beta * (x^Gamma - 1) / Gamma^2) * exp(-Alpha - Beta * (x^Gamma - 1) / Gamma)'),
GL = c('1 / ((1 + exp(-Alpha - Beta * log(x) / log(10)))^Gamma) * Gamma * exp(-Alpha - Beta * log(x) / log(10)) / (1 + exp(-Alpha - Beta * log(x) / log(10)))',
'1 / ((1 + exp(-Alpha - Beta * log(x) / log(10)))^Gamma) * Gamma * log(x) / log(10) * exp(-Alpha - Beta * log(x) / log(10)) / (1 + exp(-Alpha - Beta * log(x) / log(10)))',
'-1 / ((1 + exp(-Alpha - Beta * log(x) / log(10)))^Gamma) * log(1 + exp(-Alpha - Beta * log(x) / log(10)))'),
Brain_Consens = c('-x / (1 + exp(Beta * Gamma) * x^Beta)',
'(1 + Alpha * x) / (1 + exp(Beta * Gamma) * x^Beta)^2 * (Gamma * exp(Beta * Gamma) * x^Beta + exp(Beta * Gamma) * x^Beta * log(x))',
'(1 + Alpha * x) / (1 + exp(Beta * Gamma) * x^Beta)^2 * Beta * exp(Beta * Gamma) * x^Beta'),
BCV = c('-(1 + Beta * x) / (1 + (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta)',
'-Alpha * x / (1 + (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta) - 2 * Alpha * (1 + Beta * x) / (1 + (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta)^2 * Gamma * (x / Gamma)^Delta',
'Alpha * (1 + Beta * x) / (1 + (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta)^2 * (2 * Beta * (x / Gamma)^Delta - (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta * Delta / Gamma)',
'Alpha * (1 + Beta * x) / (1 + (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta)^2 * (1 + 2 * Beta * Gamma) * (x / Gamma)^Delta * log(x / Gamma)'),
Cedegreen = c('-exp(-1 / (x^Beta)) / (1 + exp(Gamma * (log(x) - log(Delta))))',
'-Alpha / (x^Beta) * log(x) * exp(-1 / (x^Beta)) / (1 + exp(Gamma * (log(x) - log(Delta))))',
'(1 + Alpha * exp(-1 / (x^Beta))) / (1 + exp(Gamma * (log(x) - log(Delta))))^2 * (log(x) - log(Delta)) * exp(Gamma * (log(x) - log(Delta)))',
'-(1 + Alpha * exp(-1 / (x^Beta))) / (1 + exp(Gamma * (log(x) - log(Delta))))^2 * Gamma / Delta * exp(Gamma * (log(x) - log(Delta)))'),
Beckon = c('(1 - 1 / (1 + (Beta / x)^Gamma)) / (1 + (x / Delta)^Epsilon)',
'Alpha / (1 + (Beta / x)^Gamma)^2 * (Beta / x)^Gamma * Gamma / Beta / (1 + (x / Delta)^Epsilon)',
'Alpha / (1 + (Beta / x)^Gamma)^2 * (Beta / x)^Gamma * log(Beta / x) / (1 + (x / Delta)^Epsilon)',
'(Alpha + 1 - Alpha / (1 + (Beta / x)^Gamma)) / (1 + (x / Delta)^Epsilon)^2 * (x / Delta)^Epsilon * Epsilon / Delta',
'-(Alpha + 1 - Alpha / (1 + (Beta / x)^Gamma)) / (1 + (x / Delta)^Epsilon)^2 * (x / Delta)^Epsilon * log(x / Delta)'),
Biphasic = c('1 - 1 / (1 + 10^((x-Beta) * Gamma)) - 1 / (1 + 10^((Delta - x) * Epsilon))',
'-Alpha / (1 + 10^((x-Beta) * Gamma))^2 * 10^((x - Beta) * Gamma) * Gamma * log(10)',
'Alpha / (1 + 10^((x - Beta) * Gamma))^2 * 10^((x - Beta) * Gamma) * (x - Beta) * log(10)',
'-(1 - Alpha) / (1 + 10^((Delta - x) * Epsilon))^2 * 10^((Delta - x) * Epsilon) * Epsilon * log(10)',
'-(1 - Alpha) / (1 + 10^((Delta - x) * Epsilon))^2 * 10^((Delta - x) * Epsilon) * (Delta - x) * log(10)')
)
for (i in seq(mpara)) jac[, i] <- eval(parse(text = jacFun[i]))
return(jac)
}
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