regkienerLX: Regression Function for Kiener Distributions
In FatTailsR: Kiener Distributions and Fat Tails in Finance
Description
Usage
Arguments
Details
Value
See Also
Examples
Description
One function to estimate the parameters of Kiener distributions K1, K2,
K3 and K4 and display the results in a list with many data.frame
ready to use for plotting. This function performs an unweighted nonlinear
regression of the logit of the empirical probabilities logit(p) on
the quantiles X.
Usage
1
2
3
regkienerLX(X, model = "K4", pdgts = c(3, 3, 1, 1, 1, 3, 2, 4, 4, 2, 2),
maxk = 10, mink = 0.2, app = 0, probak = pprobs2, dgts = NULL,
exfitk = NULL)
Arguments
X
vector of quantiles.
model
the model used for the regression: "K1", "K2", "K3", "K4".
pdgts
vector of length 11. Control the rounding of output parameters.
maxk
numeric. The maximum value of tail parameter k.
mink
numeric. The minimum value of tail parameter k.
app
numeric. The parameter "a" in the function ppoints.
probak
vector of probabilities used in output regk$fitk.
For instance pprobs0.
dgts
rounding parameter applied globally to output regk$fitk.
exfitk
character. A vector of parameter names to subset regk$fitk.
For instance exfit0.
Details
This function is designed to estimate the parameters of Kiener distributions
for a given dataset. It encapsulates the four distributions described in
this package.
"K1" uses model lqkiener1, "K2" uses model lqkiener2,
"K3" uses model lqkiener3 and "K4" uses model lqkiener4.
A typical input is a numeric vector that describes the returns of a stock.
Conversion from a (possible) time series format to a sorted numeric vector
is done automatically and without any check of the initial format.
There is also no check of missing values, Na, NaN,
-Inf, +Inf.
Empirical probabilities of each point in the sorted dataset is calculated
with the function ppoints. The parameter app
corresponds to the parameter a in ppoints but has been
limited to the range (0, 0.5). Default value is 0 as large datasets are
very common in finance.
A nonlinear regression is performed with nlsLM
from the logit of the probabilities logit(p) over the quantiles X
with one of the functions lqkiener1234.
These functions have been selected as they
have an explicit form in the four types (this is unfortunately not the case
for dkiener234) and return satisfactory results with ordinary least
squares. The median is calculated before the regression and is injected
as a mandatory value in the regression function.
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.
Where:
c(m, g, k) of length 3 for distribution "K1".
c(m, g, a, w) of length 4 for distribution "K2".
c(m, g, k, d) of length 4 for distribution "K3".
c(m, g, k, e) of length 4 for distribution "K4".
c(m, g, a, k, w, d, e) of length 7 extracted from object of class
clregk like regkienerLX (typically "reg$coefk").
Model "K1" return results with 1+2=3 parameters and describes a
(assumed) symmetric distribution. Parameters d and e are set
to 0. Models "K2", "K3" and "K4" describe asymmetric
distributions. They return results with 1+3=4 parameters.
Model "K2" has a very clear parameter definition but unfortunately
parameters a and w are highly correlated.
Model "K3" has the least correlated parameters but the meaning of
the distortion parameter d, usually of order 1e-3, is not simple.
Model "K4" exhibits a reasonable correlation between each parameter
and should be the preferred intermediate model between "K1" and "K2" models.
The eccentricity parameter e is well defined and easy to understand:
e=(a-w)/(a+w), a=k/(1-e) and w=k/(1+e). It varies between
-1 and +1 and can be understood as a percentage (if times 100)
of eccentricty. e = -1 corresponds to w = infinity,
e = +1 corresponds to a = infinity and the model becomes a single
log-logistic funtion with a right / left stopping point and a left / right tail.
Tail parameter lower and upper values are controlled by maxk and
mink. An upper value maxk = 10 is appropriate for datasets
of low and medium size, less than 50.000 points. For larger datasets, the
upper limit can be extended up to maxk = 20. Such a limit returns
results which are very closed to the logistic distribution, an alternate
distribution which could be more appropriate. The lower limit mink
is intended to avoid the value k=0. Remind
that value k < 2 describes distribution with no stable variance and
k < 1 describes distribution with no stable mean.
The output is an object in a flat format of class clregk. It can be
listed with the function attributes.
First are the data.frames with the initial data and the estimated results.
Second is the result of the regression regk0 given by
nlsLM from which a few information
have been extracted and listed here.
Third are the regression parameters (without the median) in plain format
(no rounding), the variance-covariance matrix, the variance-covariance
matrix times 1e+6 and the correlation matrix in a rounded format.
Note that regk0, coefk0, coefk0tt, vcovk0,
mcork0 have a polymorphic format and changing parameters that
depend from the selected model: "K1", "K2", "K3", "K4". They should be
used with care in subsequent calculations.
Fourth are the distribution parameters tailored to every model "K1", "K2",
"K3", "K4" plus estimated quantiles at levels:
c(0.001, 0.005, 0.01, 0.05, 0.5, 0.95, 0.99, 0.995, 0.999).
They are intended to subsequent calculations.
-
Fifth are the same parameters presented in a more readable format thanks
to the vector pdgts which controls the rounding of the parameters in
the following order:
-
pdgts = c("m","g","a","k","w","d","e","vcovk0","vcovk0m","mcork0","quantr").
Sixth are some probabilities and the corresponding estimated quantiles
and estimated Expected Shortfall stored in a data.frame format.
Last is fitk which returns all parameters in the same format
than fitkienerX, eventually subsetted by exfitk.
IMPORTANT : if you need to subset fitk, always subset it by parameter names
and never subset it by rank number as new items may be added in the future.
Use for instance exfitk = exfit0, ..., exfit7.
Value
dfrXP
data.frame. X = initial quantiles. P = empirical probabilities.
dfrXL
data.frame. X = initial quantiles. L = logit of probabilities.
dfrXR
data.frame. X = initial quantiles. R = residuals after regression.
dfrEP
data.frame. E = estimated quantiles. P = probabilities.
dfrEL
data.frame. E = estimated quantiles. L = logit of probabilities.
dfrED
data.frame. E = estimated quantiles.
D = estimated density (from probabilities).
regk0
object of class nls extracted from
the regression function nlsLM.
coefk0
the regression parameters in plain format.
The median is out of the regression.
vcovk0
rounded variance-covariance matrix.
vcovk0m
rounded 1e+6 times variance-covariance matrix.
mcork0
rounded correlation matrix.
coefk
all parameters in plain format.
coefk1
parameters for model "K1".
coefk2
parameters for model "K2".
coefk3
parameters for model "K3".
coefk4
parameters for model "K4".
quantk
quantiles of interest.
coefr
all parameters in a rounded format.
coefr1
rounded parameters for model "K1".
coefr2
rounded parameters for model "K2".
coefr3
rounded parameters for model "K3".
coefr4
rounded parameters for model "K4".
quantr
quantiles of interest in a rounded format.
dfrQkPk
data.frame. Qk = Estimated quantiles of interest.
Pk = probabilities.
dfrQkLk
data.frame. Qk = Estimated quantiles of interest.
Lk = Logit of probabilities.
dfrESkPk
data.frame. ESk = Estimated Expected Shortfall.
Pk = probabilities.
dfrESkLk
data.frame. ESk = Estimated Expected Shortfall.
Lk = Logit of probabilities.
fitk
Parameters, quantiles, moments, VaR, ES and other parameters (not rounded).
Length of fitk depends on the choice applied to probak.
IMPORTANT : if you need to subset fitk, always subset it by parameter names
and never subset it by rank number as new items may be added in the future.
Use for instance exfit0, ..., exfit7.
See Also
nlsLM, laplacegaussnorm,
Kiener distributions K1, K2, K3 and K4: kiener1
kiener2, kiener3, kiener4.
Other estimation function: fitkienerX and its derivatives.
fitk subsetting: exfit0.
Examples
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require(graphics)
require(minpack.lm)
require(timeSeries)
### Load the datasets and select one number (1-16)
DS <- getDSdata()
j <- 5
### and run this block
X <- DS[[j]]
nameX <- names(DS)[j]
reg <- regkienerLX(X)
## Plotting
lleg <- 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 ")
pleg <- c( paste("m =", reg$coefr4[1]), paste("g =", reg$coefr4[2]),
paste("k =", reg$coefr4[3]), paste("e =", reg$coefr4[4]) )
op <- par(mfrow=c(2,2), mgp=c(1.5,0.8,0), mar=c(3,3,2,1))
plot(X, type="l", main = nameX)
plot(reg$dfrXL, main = nameX, yaxt = "n")
axis(2, las=1, at=c(-9.2, -6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9, 9.2))
abline(h = c(-4.6, 4.6), lty = 4)
abline(v = c(reg$quantk[5], reg$quantk[9]), lty = 4)
legend("topleft", legend = lleg, cex = 0.7, inset = 0.02, bg = "#FFFFFF")
lines(reg$dfrEL, col = 2, lwd = 2)
points(reg$dfrQkLk, pch = 3, col = 2, lwd = 2, cex = 1.5)
plot(reg$dfrXP, main = nameX)
legend("topleft", legend = pleg, cex = 0.9, inset = 0.02 )
lines(reg$dfrEP, col = 2, lwd = 2)
plot(density(X), main = nameX)
lines(reg$dfrED, col = 2, lwd = 2)
round(cbind("k" = kmoments(reg$coefk, lengthx = nrow(reg$dfrXL)), "X" = xmoments(X)), 2)
## Attributes
attributes(reg)
head(reg$dfrXP)
head(reg$dfrXL)
head(reg$dfrXR)
head(reg$dfrEP)
head(reg$dfrEL)
head(reg$dfrED)
reg$regk0
reg$coefk0
reg$vcovk0
reg$vcovk0m
reg$mcork0
reg$coefk
reg$coefk1
reg$coefk2
reg$coefk3
reg$coefk4
reg$quantk
reg$coefr
reg$coefr1
reg$coefr2
reg$coefr3
reg$coefr4
reg$quantr
reg$dfrQkPk
reg$dfrQkLk
reg$dfrESkPk
reg$dfrESkLk
reg$fitk
## subset fitk
names(reg$fitk)
reg$fitk[exfit6]
reg$fitk[c(exfit1, exfit4)]
### End block
FatTailsR documentation built on March 12, 2021, 9:06 a.m.
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Description Usage Arguments Details Value See Also Examples
Description
One function to estimate the parameters of Kiener distributions K1, K2, K3 and K4 and display the results in a list with many data.frame ready to use for plotting. This function performs an unweighted nonlinear regression of the logit of the empirical probabilities logit(p) on the quantiles X.
Usage
1 2 3 | regkienerLX(X, model = "K4", pdgts = c(3, 3, 1, 1, 1, 3, 2, 4, 4, 2, 2),
maxk = 10, mink = 0.2, app = 0, probak = pprobs2, dgts = NULL,
exfitk = NULL)
|
Arguments
X |
vector of quantiles. |
model |
the model used for the regression: "K1", "K2", "K3", "K4". |
pdgts |
vector of length 11. Control the rounding of output parameters. |
maxk |
numeric. The maximum value of tail parameter |
mink |
numeric. The minimum value of tail parameter |
app |
numeric. The parameter " |
probak |
vector of probabilities used in output regk$fitk.
For instance |
dgts |
rounding parameter applied globally to output regk$fitk. |
exfitk |
character. A vector of parameter names to subset regk$fitk.
For instance |
Details
This function is designed to estimate the parameters of Kiener distributions
for a given dataset. It encapsulates the four distributions described in
this package.
"K1" uses model lqkiener1, "K2" uses model lqkiener2,
"K3" uses model lqkiener3 and "K4" uses model lqkiener4.
A typical input is a numeric vector that describes the returns of a stock.
Conversion from a (possible) time series format to a sorted numeric vector
is done automatically and without any check of the initial format.
There is also no check of missing values, Na, NaN,
-Inf, +Inf.
Empirical probabilities of each point in the sorted dataset is calculated
with the function ppoints. The parameter app
corresponds to the parameter a in ppoints but has been
limited to the range (0, 0.5). Default value is 0 as large datasets are
very common in finance.
A nonlinear regression is performed with nlsLM
from the logit of the probabilities logit(p) over the quantiles X
with one of the functions lqkiener1234.
These functions have been selected as they
have an explicit form in the four types (this is unfortunately not the case
for dkiener234) and return satisfactory results with ordinary least
squares. The median is calculated before the regression and is injected
as a mandatory value in the regression function.
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.
Where:
c(m, g, k) of length 3 for distribution "K1".
c(m, g, a, w) of length 4 for distribution "K2".
c(m, g, k, d) of length 4 for distribution "K3".
c(m, g, k, e) of length 4 for distribution "K4".
c(m, g, a, k, w, d, e) of length 7 extracted from object of class
clregklikeregkienerLX(typically"reg$coefk").
Model "K1" return results with 1+2=3 parameters and describes a
(assumed) symmetric distribution. Parameters d and e are set
to 0. Models "K2", "K3" and "K4" describe asymmetric
distributions. They return results with 1+3=4 parameters.
Model "K2" has a very clear parameter definition but unfortunately
parameters a and w are highly correlated.
Model "K3" has the least correlated parameters but the meaning of
the distortion parameter d, usually of order 1e-3, is not simple.
Model "K4" exhibits a reasonable correlation between each parameter
and should be the preferred intermediate model between "K1" and "K2" models.
The eccentricity parameter e is well defined and easy to understand:
e=(a-w)/(a+w), a=k/(1-e) and w=k/(1+e). It varies between
-1 and +1 and can be understood as a percentage (if times 100)
of eccentricty. e = -1 corresponds to w = infinity,
e = +1 corresponds to a = infinity and the model becomes a single
log-logistic funtion with a right / left stopping point and a left / right tail.
Tail parameter lower and upper values are controlled by maxk and
mink. An upper value maxk = 10 is appropriate for datasets
of low and medium size, less than 50.000 points. For larger datasets, the
upper limit can be extended up to maxk = 20. Such a limit returns
results which are very closed to the logistic distribution, an alternate
distribution which could be more appropriate. The lower limit mink
is intended to avoid the value k=0. Remind
that value k < 2 describes distribution with no stable variance and
k < 1 describes distribution with no stable mean.
The output is an object in a flat format of class clregk. It can be
listed with the function attributes.
First are the data.frames with the initial data and the estimated results.
Second is the result of the regression
regk0given bynlsLMfrom which a few information have been extracted and listed here.Third are the regression parameters (without the median) in plain format (no rounding), the variance-covariance matrix, the variance-covariance matrix times 1e+6 and the correlation matrix in a rounded format. Note that
regk0,coefk0,coefk0tt,vcovk0,mcork0have a polymorphic format and changing parameters that depend from the selected model: "K1", "K2", "K3", "K4". They should be used with care in subsequent calculations.Fourth are the distribution parameters tailored to every model "K1", "K2", "K3", "K4" plus estimated quantiles at levels: c(0.001, 0.005, 0.01, 0.05, 0.5, 0.95, 0.99, 0.995, 0.999). They are intended to subsequent calculations.
-
Fifth are the same parameters presented in a more readable format thanks to the vector
pdgtswhich controls the rounding of the parameters in the following order: -
pdgts = c("m","g","a","k","w","d","e","vcovk0","vcovk0m","mcork0","quantr"). Sixth are some probabilities and the corresponding estimated quantiles and estimated Expected Shortfall stored in a data.frame format.
Last is
fitkwhich returns all parameters in the same format thanfitkienerX, eventually subsetted byexfitk. IMPORTANT : if you need to subsetfitk, always subset it by parameter names and never subset it by rank number as new items may be added in the future. Use for instanceexfitk =exfit0, ...,exfit7.
Value
dfrXP |
data.frame. X = initial quantiles. P = empirical probabilities. |
dfrXL |
data.frame. X = initial quantiles. L = logit of probabilities. |
dfrXR |
data.frame. X = initial quantiles. R = residuals after regression. |
dfrEP |
data.frame. E = estimated quantiles. P = probabilities. |
dfrEL |
data.frame. E = estimated quantiles. L = logit of probabilities. |
dfrED |
data.frame. E = estimated quantiles. D = estimated density (from probabilities). |
regk0 |
object of class |
coefk0 |
the regression parameters in plain format. The median is out of the regression. |
vcovk0 |
rounded variance-covariance matrix. |
vcovk0m |
rounded 1e+6 times variance-covariance matrix. |
mcork0 |
rounded correlation matrix. |
coefk |
all parameters in plain format. |
coefk1 |
parameters for model "K1". |
coefk2 |
parameters for model "K2". |
coefk3 |
parameters for model "K3". |
coefk4 |
parameters for model "K4". |
quantk |
quantiles of interest. |
coefr |
all parameters in a rounded format. |
coefr1 |
rounded parameters for model "K1". |
coefr2 |
rounded parameters for model "K2". |
coefr3 |
rounded parameters for model "K3". |
coefr4 |
rounded parameters for model "K4". |
quantr |
quantiles of interest in a rounded format. |
dfrQkPk |
data.frame. Qk = Estimated quantiles of interest. Pk = probabilities. |
dfrQkLk |
data.frame. Qk = Estimated quantiles of interest. Lk = Logit of probabilities. |
dfrESkPk |
data.frame. ESk = Estimated Expected Shortfall. Pk = probabilities. |
dfrESkLk |
data.frame. ESk = Estimated Expected Shortfall. Lk = Logit of probabilities. |
fitk |
Parameters, quantiles, moments, VaR, ES and other parameters (not rounded).
Length of |
See Also
nlsLM, laplacegaussnorm,
Kiener distributions K1, K2, K3 and K4: kiener1
kiener2, kiener3, kiener4.
Other estimation function: fitkienerX and its derivatives.
fitk subsetting: exfit0.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 |
require(graphics)
require(minpack.lm)
require(timeSeries)
### Load the datasets and select one number (1-16)
DS <- getDSdata()
j <- 5
### and run this block
X <- DS[[j]]
nameX <- names(DS)[j]
reg <- regkienerLX(X)
## Plotting
lleg <- 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 ")
pleg <- c( paste("m =", reg$coefr4[1]), paste("g =", reg$coefr4[2]),
paste("k =", reg$coefr4[3]), paste("e =", reg$coefr4[4]) )
op <- par(mfrow=c(2,2), mgp=c(1.5,0.8,0), mar=c(3,3,2,1))
plot(X, type="l", main = nameX)
plot(reg$dfrXL, main = nameX, yaxt = "n")
axis(2, las=1, at=c(-9.2, -6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9, 9.2))
abline(h = c(-4.6, 4.6), lty = 4)
abline(v = c(reg$quantk[5], reg$quantk[9]), lty = 4)
legend("topleft", legend = lleg, cex = 0.7, inset = 0.02, bg = "#FFFFFF")
lines(reg$dfrEL, col = 2, lwd = 2)
points(reg$dfrQkLk, pch = 3, col = 2, lwd = 2, cex = 1.5)
plot(reg$dfrXP, main = nameX)
legend("topleft", legend = pleg, cex = 0.9, inset = 0.02 )
lines(reg$dfrEP, col = 2, lwd = 2)
plot(density(X), main = nameX)
lines(reg$dfrED, col = 2, lwd = 2)
round(cbind("k" = kmoments(reg$coefk, lengthx = nrow(reg$dfrXL)), "X" = xmoments(X)), 2)
## Attributes
attributes(reg)
head(reg$dfrXP)
head(reg$dfrXL)
head(reg$dfrXR)
head(reg$dfrEP)
head(reg$dfrEL)
head(reg$dfrED)
reg$regk0
reg$coefk0
reg$vcovk0
reg$vcovk0m
reg$mcork0
reg$coefk
reg$coefk1
reg$coefk2
reg$coefk3
reg$coefk4
reg$quantk
reg$coefr
reg$coefr1
reg$coefr2
reg$coefr3
reg$coefr4
reg$quantr
reg$dfrQkPk
reg$dfrQkLk
reg$dfrESkPk
reg$dfrESkLk
reg$fitk
## subset fitk
names(reg$fitk)
reg$fitk[exfit6]
reg$fitk[c(exfit1, exfit4)]
### End block
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