kiener1: Symmetric Kiener Distribution K1
In FatTailsR: Kiener Distributions and Fat Tails in Finance and Neuroscience
kiener1 R Documentation
Symmetric Kiener Distribution K1
Description
Density, distribution function, quantile function, random generation,
value-at-risk, expected shortfall (+ signed left/right tail mean)
and additional formulae for symmetric Kiener distribution K1.
This distribution is similar to the power hyperbola logistic distribution
but with additional parameters for location (m) and scale (g).
Usage
dkiener1(x, m = 0, g = 1, k = 3.2, log = FALSE)
pkiener1(q, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE)
qkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE)
rkiener1(n, m = 0, g = 1, k = 3.2)
dpkiener1(p, m = 0, g = 1, k = 3.2, log = FALSE)
dqkiener1(p, m = 0, g = 1, k = 3.2, log = FALSE)
lkiener1(x, m = 0, g = 1, k = 3.2)
dlkiener1(lp, m = 0, g = 1, k = 3.2, log = FALSE)
qlkiener1(lp, m = 0, g = 1, k = 3.2, lower.tail = TRUE)
varkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
ltmkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
rtmkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
dtmqkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
eskiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE,
signedES = FALSE)
Arguments
x
vector of quantiles.
m
numeric. The median.
g
numeric. The scale parameter, preferably strictly positive.
k
numeric. The tail parameter, preferably strictly positive.
log
logical. If TRUE, densities are given in log scale.
q
vector of quantiles.
lower.tail
logical. If TRUE, use p. If FALSE, use 1-p.
log.p
logical. If TRUE, probabilities p are given as log(p).
p
vector of probabilities.
n
number of observations. If length(n) > 1, the length is
taken to be the number required.
lp
vector of logit of probabilities.
signedES
logical. FALSE (default) returns positive numbers for
left and right tails. TRUE returns negative number
(= ltmkiener1) for left tail and positive number
(= rtmkiener1) for right tail.
Details
Kiener distributions use the following parameters, some of them being redundant.
See aw2k and pk2pk for the formulas and
the conversion between parameters:
-
m (mu) is the median of the distribution,.
-
g (gamma) is the scale parameter.
-
a (alpha) is the left tail parameter.
-
k (kappa) is the harmonic mean of a and w
and describes a global tail parameter.
-
w (omega) is the right tail parameter.
-
d (delta) is the distortion parameter.
-
e (epsilon) is the eccentricity parameter.
Kiener distributions K1(m, g, k, ...) describe distributions
with symmetric left and right fat tails and with a tail parameter k.
This parameter is the power exponent mentionned in the Pareto formula and
Karamata theorems.
m is the median of the distribution. g is the scale parameter
and is linked for any value of k to the density at the
median through the relation
g * f(x=m, g=g) = \frac{\pi}{4\sqrt{3}} \approx 0.453
When k = Inf, g is very close to sd(x).
NOTE: In order to match this standard deviation, the value of g has
been updated from versions < 1.9.0 by a factor
\frac{2\pi}{\sqrt{3}}.
The functions dkiener1, pkiener1 and lkiener1 have an
explicit form (whereas dkiener2347, pkiener2347 and
lkiener2347 have no explicit forms).
dkiener1 function is defined for x in (-Inf, +Inf) by:
\begin{array}{l}
y = \frac{1}{k}\frac{\pi}{\sqrt{3}}\frac{(x-m)}{g} \\[4pt]
dkiener1(x,m,g,k) = \pi*\left[2\sqrt{3}\,g\,\sqrt{y^2 +1}
\left(1+\cosh(k*asinh(y))\right)\right]^{-1}
\end{array}
pkiener1 function is defined for q in (-Inf, +Inf) by:
\begin{array}{l}
y = \frac{1}{k}\frac{\pi}{\sqrt{3}}\frac{(x-m)}{g} \\[4pt]
pkiener1(q,m,g,k) = 1/(1+exp(-k*asinh(y)))
\end{array}
qkiener1 function is defined for p in (0, 1) by:
qkiener1(p,m,g,k) = m + \frac{\sqrt{3}}{\pi}*g*k*
\sinh\left(\frac{logit(p)}{k}\right)
rkiener1 generates n random quantiles.
In addition to the classical d, p, q, r functions, the prefixes
dp, dq, l, dl, ql are also provided.
dpkiener1 is the density function calculated from the probability p.
It is defined for p in (0, 1) by:
dpkiener1(p,m,g,k) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}
sech\left(\frac{logit(p)}{k}\right)
dqkiener1 is the derivate of the quantile function calculated from
the probability p. It is defined for p in (0, 1) by:
dqkiener1(p,m,g,k) = \frac{\sqrt{3}}{\pi}\frac{g}{p(1-p)}
\cosh\left(\frac{logit(p)}{k}\right)
lkiener1 function is equivalent to kashp function but with additional
parameters m and g. Being computed from the x (or q) vector,
it can be compared to the logit of the empirical probability logit(p)
through a nonlinear regression with ordinary or weighted least squares
to estimate the distribution parameters.
It is defined for x in (-Inf, +Inf) by:
\begin{array}{l}
y = \frac{1}{k}\frac{\pi}{\sqrt{3}}\frac{(x-m)}{g} \\[4pt]
lkiener1(q,m,g,k) = k*asinh(y)
\end{array}
dlkiener1 is the density function calculated from the logit of the
probability lp = logit(p). It is defined for lp in (-Inf, +Inf) by:
dlkiener1(lp,m,g,k) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}
sech\left(\frac{lp}{k}\right)
qlkiener1 is the quantile function calculated from the logit of the
probability lp = logit(p). It is defined for lp in (-Inf, +Inf) by:
qlkiener1(p,m,g,k) = m + \frac{\sqrt{3}}{\pi}*g*k*2* \sinh\left(\frac{lp}{k}\right)
varkiener1 designates the Value a-risk and turns negative numbers
into positive numbers with the following rule:
varkiener1 <- if\;(p <= 0.5)\;\; (- qkiener1)\;\; else\;\; (qkiener1)
Usual values in finance are p = 0.01, p = 0.05, p = 0.95 and
p = 0.99. lower.tail = FALSE uses 1-p rather than p.
ltmkiener1, rtmkiener1 and eskiener1 are respectively the
left tail mean, the right tail mean and the expected shortfall of the distribution
(sometimes called average VaR, conditional VaR or tail VaR).
Left tail mean is the integrale from -Inf to p of the quantile function
qkiener1 divided by p.
Right tail mean is the integrale from p to +Inf of the quantile function
qkiener1 divided by 1-p.
Expected shortfall turns negative numbers into positive numbers with the following rule:
eskiener1 <- if\;(p <= 0.5)\;\; (- ltmkiener1)\;\; else\;\; (rtmkiener1)
Usual values in finance are p = 0.01, p = 0.025, p = 0.975 and
p = 0.99. lower.tail = FALSE uses 1-p rather than p.
dtmqkiener1 is the difference between the left tail mean and the quantile
when (p <= 0.5) and the difference between the right tail mean and the quantile
when (p > 0.5). It is in quantile unit and is an indirect measure of the tail curvature.
References
P. Kiener, Explicit models for bilateral fat-tailed distributions and
applications in finance with the package FatTailsR, 8th R/Rmetrics Workshop
and Summer School, Paris, 27 June 2014. Download it from:
https://www.inmodelia.com/exemples/2014-0627-Rmetrics-Kiener-en.pdf
P. Kiener, Fat tail analysis and package FatTailsR,
9th R/Rmetrics Workshop and Summer School, Zurich, 27 June 2015.
Download it from:
https://www.inmodelia.com/exemples/2015-0627-Rmetrics-Kiener-en.pdf
C. Acerbi, D. Tasche, Expected shortfall: a natural coherent alternative to
Value at Risk, 9 May 2001. Download it from:
https://www.bis.org/bcbs/ca/acertasc.pdf
See Also
Standardized logistic distribution logisst,
asymmetric Kiener distributions K2, K3, K4 and K7
kiener2, kiener3, kiener4,
kiener7,
regression function regkienerLX.
Examples
require(graphics)
### EXAMPLE 1
x <- seq(-5, 5, by = 0.1) ; x
pkiener1(x, m=0, g=1, k=4)
dkiener1(x, m=0, g=1, k=4)
lkiener1(x, k=4)
plot( x, pkiener1(x, m=0, g=1, k=4), las=1)
lines(x, pkiener1(x, m=0, g=1, k=9999))
plot( x, lkiener1(x, m=0, g=1, k=4), las=1)
lines(x, lkiener1(x, m=0, g=1, k=9999))
p <- c(ppoints(11, a = 1), NA, NaN) ; p
qkiener1(p, k = 4)
dpkiener1(p, k = 4)
dqkiener1(p, k=4)
varkiener1(p=0.01, k=4)
ltmkiener1(p=0.01, k=4)
eskiener1(p=0.01, k=4) # VaR and ES should be positive
### END EXAMPLE 1
### PREPARE THE GRAPHICS FOR EXAMPLES 2 AND 3
xx <- c(-4,-2, 0, 2, 4)
lty <- c( 1, 2, 3, 4, 5, 1)
lwd <- c( 2, 1, 1, 1, 1, 1)
col <- c("black","green3","cyan3","dodgerblue2","purple2","brown3")
lat <- c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9)
lgt <- c("logit(0.999) = 6.9", "logit(0.99) = 4.6", "logit(0.95) = 2.9",
"logit(0.50) = 0", "logit(0.05) = -2.9", "logit(0.01) = -4.6",
"logit(0.001) = -6.9 ")
funleg <- function(xy, k) legend(xy, title = expression(kappa), legend = names(k),
lty = lty, col = col, lwd = lwd, inset = 0.02, cex = 0.8)
funlgt <- function(xy) legend(xy, title = "logit(p)", legend = lgt,
inset = 0.02, cex = 0.6)
### EXAMPLE 2
### PROBA, DENSITY, LOGIT-PROBA, LOG-DENSITY FROM x
x <- seq(-5, 5, by = 0.1) ; head(x, 10)
k <- c(9999, 9, 5, 3, 2, 1) ; names(k) <- k
mat11 <- outer(x, k, \(x,k) pkiener1(x, k=k)) ; head(mat11, 10)
mat12 <- outer(x, k, \(x,k) dkiener1(x, k=k)) ; mat12
mat13 <- outer(x, k, \(x,k) lkiener1(x, k=k)) ; mat13
mat14 <- outer(x, k, \(x,k) dkiener1(x, k=k, log=TRUE)) ; mat14
op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
matplot(x, mat11, type="l", lwd=lwd, lty=lty, col=col,
main="pkiener1(x, m=0, g=1, k=k)", xlab="", ylab="")
funleg("topleft", k)
matplot(x, mat12, type="l", lwd=lwd, lty=lty, col=col,
main="dkiener1", xlab="", ylab="")
funleg("topleft", k)
matplot(x, mat13, type="l", lwd=lwd, lty=lty, col=col, yaxt="n",
main="lkiener1", xlab="", ylab="")
axis(2, at=lat, las=1)
funleg("bottomright", k)
funlgt("topleft")
matplot(x, mat14, type="l", lwd=lwd, lty=lty, col=col,
main="log(dkiener1)", xlab="", ylab="")
funleg("bottom", k)
par(op)
### END EXAMPLE 2
### EXAMPLE 3
### QUANTILE, DIFF-QUANTILE, DENSITY, LOG-DENSITY FROM p
p <- ppoints(1999, a=0) ; head(p, n=10)
k <- c(9999, 9, 5, 3, 2, 1) ; names(k) <- k
mat15 <- outer(p, k, \(p,k) qkiener1(p, k=k)) ; head(mat15, 10)
mat16 <- outer(p, k, \(p,k) dqkiener1(p, k=k)) ; head(mat16, 10)
mat17 <- outer(p, k, \(p,k) dpkiener1(p, k=k)) ; head(mat17, 10)
op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
matplot(p, mat15, type="l", xlim=c(0,1), ylim=c(-5,5),
lwd=lwd, lty=lty, col=col, las=1,
main="qkiener1(p, m=0, g=1, k=k)", xlab="", ylab="")
funleg("topleft", k)
matplot(p, mat16, type="l", xlim=c(0,1), ylim=c(0,40),
lwd=lwd, lty=lty, col=col, las=1,
main="dqkiener1", xlab="", ylab="")
funleg("top", k)
plot(NA, NA, xlim=c(-5, 5), ylim=c(0, 0.5), las=1,
main="qkiener1, dpkiener1", xlab="", ylab="")
mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(mat17),
lwd=lwd, lty=1, col=col)
funleg("topright", k)
plot(NA, NA, xlim=c(-5, 5), ylim=c(-7, -0.5), las=1,
main="qkiener1, log(dpkiener1)", xlab="", ylab="")
mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(log(mat17)),
lwd=lwd, lty=lty, col=col)
funleg("bottom", k)
par(op)
### END EXAMPLE 3
### EXAMPLE 4: PROCESSUS: which processus look credible?
### PARAMETER k VARIES
### RUN SEED ii <- 1 THEN THE cairo_pdf CODE WITH THE 6 SEEDS
# cairo_pdf("K1-6x6-stocks-k.pdf")
# for (ii in c(1,2016,2018,2022,2023,2024)) {
ii <- 1
set.seed(ii)
p <- sample(ppoints(299, a=0), 299)
k <- c(9999, 6, 4, 3, 2, 1) ; names(k) <- k
mat18 <- outer(p, k, \(p,k) qkiener1(p=p, g=0.85, k=k))
mat19 <- apply(mat18, 2, cumsum)
title <- paste0(
"stock_", ii,
": k_left = c(", paste(k[1:3], collapse = ", "), ")",
", k_right = c(", paste(k[4:6], collapse = ", "), ")")
plot.ts(mat19, ann=FALSE, las=1,
mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
mtext(title, outer = TRUE, line=-1.5, font=2)
plot.ts(mat18, ann=FALSE, las=1,
mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
mtext(title, outer=TRUE, line=-1.5, font=2)
# }
# dev.off()
### END EXAMPLE 4
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| kiener1 | R Documentation |
Symmetric Kiener Distribution K1
Description
Density, distribution function, quantile function, random generation,
value-at-risk, expected shortfall (+ signed left/right tail mean)
and additional formulae for symmetric Kiener distribution K1.
This distribution is similar to the power hyperbola logistic distribution
but with additional parameters for location (m) and scale (g).
Usage
dkiener1(x, m = 0, g = 1, k = 3.2, log = FALSE)
pkiener1(q, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE)
qkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE)
rkiener1(n, m = 0, g = 1, k = 3.2)
dpkiener1(p, m = 0, g = 1, k = 3.2, log = FALSE)
dqkiener1(p, m = 0, g = 1, k = 3.2, log = FALSE)
lkiener1(x, m = 0, g = 1, k = 3.2)
dlkiener1(lp, m = 0, g = 1, k = 3.2, log = FALSE)
qlkiener1(lp, m = 0, g = 1, k = 3.2, lower.tail = TRUE)
varkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
ltmkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
rtmkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
dtmqkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE,
log.p = FALSE)
eskiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE,
signedES = FALSE)
Arguments
x |
vector of quantiles. |
m |
numeric. The median. |
g |
numeric. The scale parameter, preferably strictly positive. |
k |
numeric. The tail parameter, preferably strictly positive. |
log |
logical. If TRUE, densities are given in log scale. |
q |
vector of quantiles. |
lower.tail |
logical. If TRUE, use p. If FALSE, use 1-p. |
log.p |
logical. If TRUE, probabilities p are given as log(p). |
p |
vector of probabilities. |
n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
lp |
vector of logit of probabilities. |
signedES |
logical. FALSE (default) returns positive numbers for
left and right tails. TRUE returns negative number
(= |
Details
Kiener distributions use the following parameters, some of them being redundant.
See aw2k and pk2pk for the formulas and
the conversion between parameters:
-
m(mu) is the median of the distribution,. -
g(gamma) is the scale parameter. -
a(alpha) is the left tail parameter. -
k(kappa) is the harmonic mean ofaandwand describes a global tail parameter. -
w(omega) is the right tail parameter. -
d(delta) is the distortion parameter. -
e(epsilon) is the eccentricity parameter.
Kiener distributions K1(m, g, k, ...) describe distributions
with symmetric left and right fat tails and with a tail parameter k.
This parameter is the power exponent mentionned in the Pareto formula and
Karamata theorems.
m is the median of the distribution. g is the scale parameter
and is linked for any value of k to the density at the
median through the relation
g * f(x=m, g=g) = \frac{\pi}{4\sqrt{3}} \approx 0.453
When k = Inf, g is very close to sd(x).
NOTE: In order to match this standard deviation, the value of g has
been updated from versions < 1.9.0 by a factor
\frac{2\pi}{\sqrt{3}}.
The functions dkiener1, pkiener1 and lkiener1 have an
explicit form (whereas dkiener2347, pkiener2347 and
lkiener2347 have no explicit forms).
dkiener1 function is defined for x in (-Inf, +Inf) by:
\begin{array}{l}
y = \frac{1}{k}\frac{\pi}{\sqrt{3}}\frac{(x-m)}{g} \\[4pt]
dkiener1(x,m,g,k) = \pi*\left[2\sqrt{3}\,g\,\sqrt{y^2 +1}
\left(1+\cosh(k*asinh(y))\right)\right]^{-1}
\end{array}
pkiener1 function is defined for q in (-Inf, +Inf) by:
\begin{array}{l}
y = \frac{1}{k}\frac{\pi}{\sqrt{3}}\frac{(x-m)}{g} \\[4pt]
pkiener1(q,m,g,k) = 1/(1+exp(-k*asinh(y)))
\end{array}
qkiener1 function is defined for p in (0, 1) by:
qkiener1(p,m,g,k) = m + \frac{\sqrt{3}}{\pi}*g*k*
\sinh\left(\frac{logit(p)}{k}\right)
rkiener1 generates n random quantiles.
In addition to the classical d, p, q, r functions, the prefixes dp, dq, l, dl, ql are also provided.
dpkiener1 is the density function calculated from the probability p.
It is defined for p in (0, 1) by:
dpkiener1(p,m,g,k) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}
sech\left(\frac{logit(p)}{k}\right)
dqkiener1 is the derivate of the quantile function calculated from
the probability p. It is defined for p in (0, 1) by:
dqkiener1(p,m,g,k) = \frac{\sqrt{3}}{\pi}\frac{g}{p(1-p)}
\cosh\left(\frac{logit(p)}{k}\right)
lkiener1 function is equivalent to kashp function but with additional
parameters m and g. Being computed from the x (or q) vector,
it can be compared to the logit of the empirical probability logit(p)
through a nonlinear regression with ordinary or weighted least squares
to estimate the distribution parameters.
It is defined for x in (-Inf, +Inf) by:
\begin{array}{l}
y = \frac{1}{k}\frac{\pi}{\sqrt{3}}\frac{(x-m)}{g} \\[4pt]
lkiener1(q,m,g,k) = k*asinh(y)
\end{array}
dlkiener1 is the density function calculated from the logit of the
probability lp = logit(p). It is defined for lp in (-Inf, +Inf) by:
dlkiener1(lp,m,g,k) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}
sech\left(\frac{lp}{k}\right)
qlkiener1 is the quantile function calculated from the logit of the
probability lp = logit(p). It is defined for lp in (-Inf, +Inf) by:
qlkiener1(p,m,g,k) = m + \frac{\sqrt{3}}{\pi}*g*k*2* \sinh\left(\frac{lp}{k}\right)
varkiener1 designates the Value a-risk and turns negative numbers
into positive numbers with the following rule:
varkiener1 <- if\;(p <= 0.5)\;\; (- qkiener1)\;\; else\;\; (qkiener1)
Usual values in finance are p = 0.01, p = 0.05, p = 0.95 and
p = 0.99. lower.tail = FALSE uses 1-p rather than p.
ltmkiener1, rtmkiener1 and eskiener1 are respectively the
left tail mean, the right tail mean and the expected shortfall of the distribution
(sometimes called average VaR, conditional VaR or tail VaR).
Left tail mean is the integrale from -Inf to p of the quantile function
qkiener1 divided by p.
Right tail mean is the integrale from p to +Inf of the quantile function
qkiener1 divided by 1-p.
Expected shortfall turns negative numbers into positive numbers with the following rule:
eskiener1 <- if\;(p <= 0.5)\;\; (- ltmkiener1)\;\; else\;\; (rtmkiener1)
Usual values in finance are p = 0.01, p = 0.025, p = 0.975 and
p = 0.99. lower.tail = FALSE uses 1-p rather than p.
dtmqkiener1 is the difference between the left tail mean and the quantile
when (p <= 0.5) and the difference between the right tail mean and the quantile
when (p > 0.5). It is in quantile unit and is an indirect measure of the tail curvature.
References
P. Kiener, Explicit models for bilateral fat-tailed distributions and applications in finance with the package FatTailsR, 8th R/Rmetrics Workshop and Summer School, Paris, 27 June 2014. Download it from: https://www.inmodelia.com/exemples/2014-0627-Rmetrics-Kiener-en.pdf
P. Kiener, Fat tail analysis and package FatTailsR, 9th R/Rmetrics Workshop and Summer School, Zurich, 27 June 2015. Download it from: https://www.inmodelia.com/exemples/2015-0627-Rmetrics-Kiener-en.pdf
C. Acerbi, D. Tasche, Expected shortfall: a natural coherent alternative to Value at Risk, 9 May 2001. Download it from: https://www.bis.org/bcbs/ca/acertasc.pdf
See Also
Standardized logistic distribution logisst,
asymmetric Kiener distributions K2, K3, K4 and K7
kiener2, kiener3, kiener4,
kiener7,
regression function regkienerLX.
Examples
require(graphics)
### EXAMPLE 1
x <- seq(-5, 5, by = 0.1) ; x
pkiener1(x, m=0, g=1, k=4)
dkiener1(x, m=0, g=1, k=4)
lkiener1(x, k=4)
plot( x, pkiener1(x, m=0, g=1, k=4), las=1)
lines(x, pkiener1(x, m=0, g=1, k=9999))
plot( x, lkiener1(x, m=0, g=1, k=4), las=1)
lines(x, lkiener1(x, m=0, g=1, k=9999))
p <- c(ppoints(11, a = 1), NA, NaN) ; p
qkiener1(p, k = 4)
dpkiener1(p, k = 4)
dqkiener1(p, k=4)
varkiener1(p=0.01, k=4)
ltmkiener1(p=0.01, k=4)
eskiener1(p=0.01, k=4) # VaR and ES should be positive
### END EXAMPLE 1
### PREPARE THE GRAPHICS FOR EXAMPLES 2 AND 3
xx <- c(-4,-2, 0, 2, 4)
lty <- c( 1, 2, 3, 4, 5, 1)
lwd <- c( 2, 1, 1, 1, 1, 1)
col <- c("black","green3","cyan3","dodgerblue2","purple2","brown3")
lat <- c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9)
lgt <- c("logit(0.999) = 6.9", "logit(0.99) = 4.6", "logit(0.95) = 2.9",
"logit(0.50) = 0", "logit(0.05) = -2.9", "logit(0.01) = -4.6",
"logit(0.001) = -6.9 ")
funleg <- function(xy, k) legend(xy, title = expression(kappa), legend = names(k),
lty = lty, col = col, lwd = lwd, inset = 0.02, cex = 0.8)
funlgt <- function(xy) legend(xy, title = "logit(p)", legend = lgt,
inset = 0.02, cex = 0.6)
### EXAMPLE 2
### PROBA, DENSITY, LOGIT-PROBA, LOG-DENSITY FROM x
x <- seq(-5, 5, by = 0.1) ; head(x, 10)
k <- c(9999, 9, 5, 3, 2, 1) ; names(k) <- k
mat11 <- outer(x, k, \(x,k) pkiener1(x, k=k)) ; head(mat11, 10)
mat12 <- outer(x, k, \(x,k) dkiener1(x, k=k)) ; mat12
mat13 <- outer(x, k, \(x,k) lkiener1(x, k=k)) ; mat13
mat14 <- outer(x, k, \(x,k) dkiener1(x, k=k, log=TRUE)) ; mat14
op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
matplot(x, mat11, type="l", lwd=lwd, lty=lty, col=col,
main="pkiener1(x, m=0, g=1, k=k)", xlab="", ylab="")
funleg("topleft", k)
matplot(x, mat12, type="l", lwd=lwd, lty=lty, col=col,
main="dkiener1", xlab="", ylab="")
funleg("topleft", k)
matplot(x, mat13, type="l", lwd=lwd, lty=lty, col=col, yaxt="n",
main="lkiener1", xlab="", ylab="")
axis(2, at=lat, las=1)
funleg("bottomright", k)
funlgt("topleft")
matplot(x, mat14, type="l", lwd=lwd, lty=lty, col=col,
main="log(dkiener1)", xlab="", ylab="")
funleg("bottom", k)
par(op)
### END EXAMPLE 2
### EXAMPLE 3
### QUANTILE, DIFF-QUANTILE, DENSITY, LOG-DENSITY FROM p
p <- ppoints(1999, a=0) ; head(p, n=10)
k <- c(9999, 9, 5, 3, 2, 1) ; names(k) <- k
mat15 <- outer(p, k, \(p,k) qkiener1(p, k=k)) ; head(mat15, 10)
mat16 <- outer(p, k, \(p,k) dqkiener1(p, k=k)) ; head(mat16, 10)
mat17 <- outer(p, k, \(p,k) dpkiener1(p, k=k)) ; head(mat17, 10)
op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1)
matplot(p, mat15, type="l", xlim=c(0,1), ylim=c(-5,5),
lwd=lwd, lty=lty, col=col, las=1,
main="qkiener1(p, m=0, g=1, k=k)", xlab="", ylab="")
funleg("topleft", k)
matplot(p, mat16, type="l", xlim=c(0,1), ylim=c(0,40),
lwd=lwd, lty=lty, col=col, las=1,
main="dqkiener1", xlab="", ylab="")
funleg("top", k)
plot(NA, NA, xlim=c(-5, 5), ylim=c(0, 0.5), las=1,
main="qkiener1, dpkiener1", xlab="", ylab="")
mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(mat17),
lwd=lwd, lty=1, col=col)
funleg("topright", k)
plot(NA, NA, xlim=c(-5, 5), ylim=c(-7, -0.5), las=1,
main="qkiener1, log(dpkiener1)", xlab="", ylab="")
mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(log(mat17)),
lwd=lwd, lty=lty, col=col)
funleg("bottom", k)
par(op)
### END EXAMPLE 3
### EXAMPLE 4: PROCESSUS: which processus look credible?
### PARAMETER k VARIES
### RUN SEED ii <- 1 THEN THE cairo_pdf CODE WITH THE 6 SEEDS
# cairo_pdf("K1-6x6-stocks-k.pdf")
# for (ii in c(1,2016,2018,2022,2023,2024)) {
ii <- 1
set.seed(ii)
p <- sample(ppoints(299, a=0), 299)
k <- c(9999, 6, 4, 3, 2, 1) ; names(k) <- k
mat18 <- outer(p, k, \(p,k) qkiener1(p=p, g=0.85, k=k))
mat19 <- apply(mat18, 2, cumsum)
title <- paste0(
"stock_", ii,
": k_left = c(", paste(k[1:3], collapse = ", "), ")",
", k_right = c(", paste(k[4:6], collapse = ", "), ")")
plot.ts(mat19, ann=FALSE, las=1,
mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
mtext(title, outer = TRUE, line=-1.5, font=2)
plot.ts(mat18, ann=FALSE, las=1,
mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5))
mtext(title, outer=TRUE, line=-1.5, font=2)
# }
# dev.off()
### END EXAMPLE 4
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