Title:	Rmetrics - Markets and Basic Statistics
Version:	4041.97
Description:	Provides a collection of functions to explore and to investigate basic properties of financial returns and related quantities. The covered fields include techniques of explorative data analysis and the investigation of distributional properties, including parameter estimation and hypothesis testing. Even more there are several utility functions for data handling and management.
Depends:	R (≥ 2.15.1)
Imports:	timeDate, timeSeries (≥ 4021.105), stats, grDevices, graphics, methods, utils, MASS, spatial, gss, stabledist
Suggests:	interp, RUnit, tcltk
LazyData:	yes
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding:	UTF-8
URL:	https://geobosh.github.io/fBasicsDoc/ (doc), https://r-forge.r-project.org/scm/viewvc.php/pkg/fBasics/?root=rmetrics (devel), https://www.rmetrics.org
BugReports:	https://r-forge.r-project.org/projects/rmetrics
NeedsCompilation:	yes
Packaged:	2024-08-18 18:37:43 UTC; georgi
Author:	Diethelm Wuertz [aut] (original code), Tobias Setz [aut], Yohan Chalabi [aut], Martin Maechler [ctb], CRAN Team [ctb], Georgi N. Boshnakov [cre, aut]
Maintainer:	Georgi N. Boshnakov <georgi.boshnakov@manchester.ac.uk>
Repository:	CRAN
Date/Publication:	2024-08-19 10:00:02 UTC

Portfolio modelling, optimization and backtesting

Description

The Rmetrics fBasics package is a collection of functions to explore and to investigate basic properties of financial returns and related quantities.

The covered fields include techniques of explorative data analysis and the investigation of distributional properties, including parameter estimation and hypothesis testing. Evenmore there are several utility functions for data handling and management.

Details

Note by the maintainer (GNB): Some of the information on this overview page may be outdated. The documentation website https://geobosh.github.io/fBasicsDoc/ of the package (generated with pkgdown) provides up-to-date help pages, arranged by topic.

1 Introduction

The fBasics package contains basics tools often required in computational finance and financial engineering. The topics are: basic statistics functions, financial return distributions, hypothesis testing, plotting routines, matrix computations and linear algebra, and some usefule utility functions.

2 Basic Statistics Functions

Financial Return Statistics

 
    basicStats            Returns a basic statistics summary
    

Distribution Function of Maximum Drawdowns

    dmaxdd                Density function of mean Max-Drawdowns
    pmaxdd                Probability function of mean Max-Drawdowns
    rmaxdd                Random Variates of mean Max-Drawdowns
    maxddStats            Expectation of Drawdowns for BM with drift
    

Calculation of Sample Moments

 
    sampleLmoments        Computes sample L-moments
    sampleMED             Returns sample median
    sampleIQR             returns sample inter quartal range
    sampleSKEW            returns robust sample skewness
    sampleKURT            returns robust sample kurtosis
    

Bivariate Interpolation:

 
    akimaInterp           Interpolates irregularly spaced points
    akimaInterpp          Interpolates and smoothes pointwise
    krigeInterp           Kriges irregularly spaced data points
    linearInterp          Interpolates irregularly spaced points
    linearInterpp         Interpolates linearly pointwise
    

Utility Statistics Functions:

    colStats              Computes sample statistics by col
    colSums               Computes sums of values in each col
    colMeans              Computes means of values in each col
    colSds                Computes standard deviation of each col
    colVars               Computes sample variance by col
    colSkewness           Computes sample skewness by col
    colKurtosis           Computes sample kurtosis by col
    colMaxs               Computes maximum values in each col
    colMins               Computes minimum values in each col
    colProds              Computes product of values in each col
    colQuantiles          Computes product of values in each col
    

    rowStats              Computes sample statistics by row
    rowSums               Computes sums of values in each row
    rowMeans              Computes means of values in each row
    rowSds                Computes standard deviation of each row
    rowVars               Computes sample variance by row
    rowSkewness           Computes sample skewness by row
    rowKurtosis           Computes sample kurtosis by row
    rowMaxs               Computes maximum values in each row
    rowMins               Computes minimum values in each row
    rowProds              Computes product of values in each row
    rowQuantiles          Computes product of values in each row
    

3 Financial Return Distributions

Generalized Hyperbolic Distribution:

    dghReturns            Density for the GH distribution
    pghreturns            Probability for the GH distribution
    qghreturns            Quantiles for the GH distribution
    rghreturns            Random variates for the GH distribution
    ghFitFits             Fits parameters of the GH distribution
    ghMode                Computes mode of the GH distribution. 
    ghMean                Returns true mean of the GH distribution
    ghVar                 Returns true variance of the GH distribution
    ghSkew                Returns true skewness of the GH distribution
    ghKurt                Returns true kurtosis of the GH distribution
    ghMoments             Returns true n-th moment of the GH distribution
    ghMED                 Returns true median of te GH distribution
    ghIQR                 Returns true inter quartal range of te GH
    ghSKEW                Returns true robust skewness of te GH
    ghKURT                Returns true robust kurtosis of te GH
    

Hyperbolic Distribution:

    dhyp                  Returns density for the HYP distribution
    phyp                  Returns probability for the HYP distribution
    qhyp                  Returns quantiles for the HYP distribution
    rhyp                  Returns random variates for the HYP distribution
    hypFit                Fits parameters of the HYP distribution
    hypMode               Computes mode of the HYP distribution
    hypMean               Returns true mean of the HYP distribution
    hypVar R              Returns true variance of the HYP distribution
    hypSkew               Returns true skewness of the HYP distribution
    hypKurt               Returns true kurtosis of the HYP distribution
    hypMoments            Returns true n-th moment of the HYP distribution
    hypMED                Returns true median of the HYP distribution
    hypIQR                Returns true inter quartal range of the HYP
    hypSKEW               Returns true robust skewness of the HYP
    hypKURT               Returns true robust kurtosis of the HYP
    

Normal Inverse Gaussian:

    dnig                  Returns density for the NIG distribution
    pnig                  Returns probability for the NIG distribution
    qnig                  Returns quantiles for the NIG distribution
    rnig                  Returns random variates for the NIG distribution
      .pnigC              fast C Implementation of function pnig()
      .qnigC              fast CImplementation of function qnig()
    nigFit                Fits parameters of a NIG distribution
      .nigFit.mle         Uses max Log-likelihood estimation
      .nigFit.gmm         Uses generalized method of moments
      .nigFit.mps         Maximum product spacings estimation
      .nigFit.vmps        Minimum variance mps estimation
    nigMode               Computes mode of the NIG distribution
    nigMean               Returns true mean of the NIG distribution
    nigVar                Returns true variance of the NIG distribution
    nigSkew               Returns true skewness of the NIG distribution
    nigKurt               Returns true kurtosis of the NIG distribution
    nigMoments            Returns true n-th moment of the NIG distribution
    nigMED                Returns true median of the NIG distribution
    nigIQR                Returns true inter quartal range of the NIG
    nigSKEW               Returns true robust skewness of the NIG
    nigKURT               Returns true robust kurtosis of the NIG
    

Generalized Hyperbolic Student-t Distribution:

    dght                  Returns density for the GHT distribution
    pght                  Returns probability for the GHT distribution
    qght                  Returns quantiles for the GHT distribution
    rght                  Returns random variates for the GHT distribution
    ghtFit                Fits parameters of the GHT distribution
    ghtMode               Computes mode of the GHT distribution
    ghtMean               Returns true mean of the NIG distribution
    ghtVar                Returns true variance of the GHT distribution
    ghtSkew               Returns true skewness of the GHT distribution
    ghtKurt               Returns true kurtosis of the GHT distribution
    ghtMoments            Returns true n-th moment of the GHT distribution
    ghtMED                Returns true median of the GHT distribution
    ghtIQR                Returns true inter quartal range of the GHT
    ghtSKEW               Returns true robust skewness of the GHT
    ghtKURT               Returns true robust kurtosis of the GHT
    

Stable Distribution:

    dstable               Returns density for the stable distribution
    pstable               Returns probability for the stable distribution
    qstable               Returns quantiles for the stable distribution
    rstable               Returns random variates for the dtsble distribution
    stableFit             Fits parameters of a the stable distribution
      .phiStable          Creates contour table for McCulloch estimators
      .PhiStable          Contour table created by function .phiStable()
      .qStableFit         Estimates parameters by McCulloch's approach
      .mleStableFit       Estimates stable parameters by MLE approach
      .stablePlot         Plots results of stable parameter estimates
    stableMode            Computes mode of the stable distribution
    

Generalized Lambda Distribution:

    dgld                  Returns density for the GLD distribution
    pgld                  Returns probability for the GLD distribution
    qgld                  Returns quantiles for the GLD distribution
    rgld                  Returns random variates for the GLD distribution
    gldFit                Fits parameters of the GLD distribution
      .gldFit.mle         fits GLD using maximum log-likelihood
      .gldFit.mps         fits GLD using maximum product spacings 
      .gldFit.gof         fits GLD using Goodness of Fit statistics
                          
                          
                          
      .gldFit.hist        fits GLD using a histogram fit
      .gldFit.rob         fits GLD using robust moments fit
    gldMode               Computes mode of the GLD distribution.
    gldMED                Returns true median of the GLD distribution
    gldIQR                Returns true inter quartal range of the GLD
    gldSKEW               Returns true robust skewness of the GLD
    gldKURT               Returns true robust kurtosis of the GLD
    

Spline Smoothed Distribution:

    dssd                  Returns spline smoothed density function
    pssd                  Returns spline smoothed probability function
    qssd                  Returns spline smoothed quantile function
    rssd                  Returns spline smoothed random variates.
    ssdFit                Fits parameters for a spline smoothed distribution
    

4 Hypthesis Testing

One Sample Nornality Tests:

    ksnormTest            One sample Kolmogorov-Smirnov normality test
    shapiroTest           Shapiro-Wilk normality test
    jarqueberaTest        Jarque-Bera normality test
    normalTest            Normality tests S-Plus compatible call
    dagoTest              D'Agostino normality test
    adTest                Anderson-Darling normality test
    cvmTest               Cramer-von Mises normality test
    lillieTest            Lilliefors (KS) normality test 
    pchiTest              Pearson chi-square normality test 
    sfTest                Shapiro-Francia normality test     
    jbTest                Finite sample adjusted JB LM and ALM test
    

One Sample Location, Scale and variance Tests:

    locationTest          Performs locations tests on two samples
      .tTest              Unpaired t test for differences in mean
      .kw2Test            Kruskal-Wallis test for differences in locations
    scaleTest             Performs scale tests on two samples
      .ansariTest         Ansari-Bradley test for differences in scale
      .moodTest           Mood test for differences in scale
    varianceTest          Performs variance tests on two samples
      .varfTest           F test for differences in variances
      .bartlett2Test      Bartlett's test for differences in variances
      .fligner2Test       Fligner-Killeen test for differences in variances
    

Two Sample Tests:

    ks2Test               Performs a two sample Kolmogorov-Smirnov test
    correlationTest       Performs correlation tests on two samples
    pearsonTest           Pearson product moment correlation coefficient
    kendallTest           Kendall's tau correlation test
    spearmanTest          Spearman's rho correlation test
    

Test Utilities:

    'fHTEST'              S4 Class Representation
    show.fHTEST           S4 Print Method
      .jbALM              Jarque Bera Augmented Lagrange Multiplier Data
      .jbLM               Jarque-Bera Lagrange Multiplier Data
      .jbTable            Finite sample p values for the Jarque Bera test
      .jbPlot             Plots probabilities
      .pjb                Returns probabilities for JB given quantiles
      .qjb                Returns quantiles for JB given probabilities
    

5 Plotting Routines

Financial Time Series Plots:

    seriesPlot            Dispalys a time series plot
    cumulatedPlot         Displays cumulated series give returns
    returnPlot            Displays returns given cumulated series
    drawdownPlot          Displays drawdown series from returns
    

Correlation Plots:

    acfPlot               Displays tailored ACF plot
    pacfPlot              Displays tailored partial ACF plot
    teffectPlot           Displays the Taylor effect
    lacfPlot              Displays lagged autocorrelations
    

Distribution Plots:

    histPlot              Returns tailored histogram plot
    densityPlot           Returns tailored density plot 
    logDensityPlot        Returns tailored log density plot
    boxPlot               Returns side-by-side standard box plot
    boxPercentile         Plotreturns box-percentile plot
    qqnormPlot            Returns normal quantile-quantile plot
    qqnigPlot             Returns NIG quantile-quantile plot
    qqghtPlot             Rreturns  GHT quantile-quantile plot
    qqgldPlot             Returns GLD quantile-quantile plot
    

Time Series Aggregation Plots:

    scalinglawPlot        Displays scaling law behavior
    

5. Matrix Computations and Linear Algebra

Elementar Matrix Operation Addons:

    kron                  Returns the Kronecker product
    vec                   Stacks a matrix as column vector
    vech                  Stacks a lower triangle matrix
    pdl                   Returns regressor matrix for polynomial lags
    tslag                 Returns Lagged/leading vector/matrix
    

Linear Algebra Addons:

    inv                   Returns the inverse of a matrix
    norm                  Returns the norm of a matrix
    rk                    Returns the rank of a matrix
    tr                    Returns the trace of a matrix
    

General Matrix Utility Addons:

    isPositiveDefinite    Checks if a matrix is positive definite
    makePositiveDefinite  Forces a matrix to be positive definite
    colVec                Creates a column vector from a data vector
    rowVec                Creates a row vector from a data vector
    gridVector            Creates from two vectors rectangular grid
    triang                Extracs lower tridiagonal part from a matrix
    Triang                Extracs upper tridiagonal part from a matrix
    

Selected Matrix Examples:

    hilbert               Creates a Hilbert matrix
    pascal                Creates a Pascal matrix
    

6 Utility Functions

Color Utilities:

    colorLocator            Plots Rs 657 named colors for selection
    colorMatrix             Returns matrix of R's color names.
    colorTable              Table of Color Codes and Plot Colors itself
    rainbowPalette          Contiguous rainbow color palette
    heatPalette             Contiguous heat color palette
    terrainPalette          Contiguous terrain color palette
    topoPalette             Contiguous topo color palette
    cmPalette               Contiguous cm color palette
    greyPalette             R's gamma-corrected gray palette
    timPalette              Tim's Matlab like color palette
    rampPalette             Color ramp palettes
    seqPalette              Sequential color brewer palettes
    divPalette              Diverging color brewer palettes
    qualiPalette            Qualified color brewer palettes
    focusPalette            Red, green blue focus palettes
    monoPalette             Red, green blue mono palettes
    

Graphics Utilities:

    symbolTable             Shows a table of plot symbols
    characterTable          Shows a table of character codes
    decor                   Adds horizontal grid and L shaped box
    hgrid                   Adds horizontal grid lines
    vgrid                   Adds vertical grid lines
    boxL                    Adds L-shaped box
    box                     Adds unterlined box
      .xrug                 Adds rugs on x axis
      .yrug                 Adds rugs on y axis
    copyright               Adds copyright notice
    interactivePlot         Plots several graphs interactively
    

Special Function Utilities:

    Heaviside               Computes Heaviside unit step function
    Sign                    Another signum function
    Delta                   Computes delta function
    Boxcar                  Computes boxcar function
    Ramp                    Computes ramp function
    tsHessian               Computes Two Sided Hessian matrix
    

Other Utilities:

    .unirootNA              Computes zero of a function without error exit
    getModel                Extracts the model slot from a S4 object
    getTitle                Extracts the title slot from a S4 object
    getDescription          Extracts the description slot
    getSlot                 Extracts a specified slot from a S4 object
    

About Builtin Functions

Builtin functions are borrowed from contributed R packages and other sources. There are several reasons why we have modified and copied code from other sources and included in this package.

* The builtin code is not available on Debian, so that Linux users have no easy acces to this code.

* The original code conflicts with other code from this package or conflicts with Rmetrics design objectives.

* We only need a very small piece of functionality from the original package which may depend on other packages which are not needed.

* The package from which we builtin the code is under current development, so that the functions often change and thus leads to unexpectect behavior in the Rmetrics packages.

* The package may be incompatible since it uses other time date and time series classes than the 'timeDate' and 'timeSeries' objects and methods from Rmetrics.

We put the code in script files named builtin-funPackage.R where "fun" denotes the (optional) major function name, and "Package" the name of the contributed package from which we copied the original code.

Builtin functions include:

    gelGmm                gll function from gmm package
    gmmGMM                gmm function from gmm package
    kweightsSandwhich     kweights from sandwhich package
    glGld                 gl functions from gld package
    ssdenGss              ssden from the gss package
    hypHyperbolicDist     hyp from HyperbolicDist package
    

Compiled Fortran and C Code:

    gld.c                 source code from gld package
    nig.c                 source code from Kersti Aas
    gss.f                 source code fromsandwhich package
    

About Rmetrics:

The fBasics Rmetrics package is written for educational support in teaching "Computational Finance and Financial Engineering" and licensed under the GPL.

Author(s)

Diethelm Wuertz [aut] (original code), Tobias Setz [aut], Yohan Chalabi [aut], Martin Maechler [ctb] (<https://orcid.org/0000-0002-8685-9910>), Georgi N. Boshnakov [cre, ctb], CRAN Team [ctb]

Maintainer: Georgi N. Boshnakov <georgi.boshnakov@manchester.ac.uk>

Basic time series statistics

Description

Computes basic financial time series statistics.

Usage

 
basicStats(x, ci = 0.95)

Arguments

Details

Computes a number of sample statistics for each column of x. The statistics should be clear from the row names of the returned data frame.

"LCL" and "UCL" stand for lower/upper confidence limits, computed under the null hypothesis of i.i.d.

"Kurtosis" represents the excess kurtosis, so its theoretical value for the normal distribution is zero, not 3.

These statistics are often computed as a first step in the study of returns on financial assets. In that case any inference on these statistics (including the confidence intervals for the mean) should be considered exploratory, since returns are virtually never i.i.d.

Value

a data frame with one column for each column of x and the following rows:

Examples

## Simulated Monthly Return Data
tS <- timeSeries(matrix(rnorm(12)), timeDate::timeCalendar())
basicStats(tS)

Time series box plots

Description

Produce a box plot or a box percentile plot.

Usage

 
boxPlot(x, col = "steelblue", title = TRUE, ...)
boxPercentilePlot(x, col = "steelblue", title = TRUE, ...) 

Arguments

Details

boxPlot produces a side-by-side standard box plot,

boxPercentilePlot produces a side-by-side box-percentile plot.

Value

NULL, displays a time series plot

Examples

## data
data(LPP2005REC, package = "timeSeries")
LPP <- LPP2005REC[, 1:6]
plot(LPP, type = "l", col = "steelblue", main = "SP500")
abline(h = 0, col = "grey")
   
boxPlot(LPP) 

Fit normal, Student-t and stable distributions

Description

A collection of moment and maximum likelihood estimators to fit the parameters of a distribution.

The functions are:

Usage

nFit(x, doplot = TRUE, span = "auto", title = NULL, description = NULL, ...)

tFit(x, df = 4, doplot = TRUE, span = "auto", trace = FALSE, title = NULL, 
    description = NULL, ...)
    
stableFit(x, alpha = 1.75, beta = 0, gamma = 1, delta = 0, 
    type = c("q", "mle"), doplot = TRUE, control = list(),
    trace = FALSE, title = NULL, description = NULL) 

Arguments

Details

Stable Parameter Estimation:

Estimation techniques based on the quantiles of an empirical sample were first suggested by Fama and Roll [1971]. However their technique was limited to symmetric distributions and suffered from a small asymptotic bias. McCulloch [1986] developed a technique that uses five quantiles from a sample to estimate alpha and beta without asymptotic bias. Unfortunately, the estimators provided by McCulloch have restriction alpha>0.6.

Remark: The parameter estimation for the stable distribution via the maximum Log-Likelihood approach may take a quite long time.

Value

an object from class "fDISTFIT".

Slot fit has components estimate, minimum, code and gradient (but for nFit code is NA and gradient is missing).

Examples

set.seed(1953)
s <- rnorm(n = 1000, 0.5, 2) 

nFit(s, doplot = TRUE) 

Heaviside and related functions

Description

Functions which compute the Heaviside and related functions. These include the Heaviside function, the sign function, the delta function, the boxcar function, and the ramp function.

Usage

Heaviside(x, a = 0)
Sign(x, a = 0)
Delta(x, a = 0)
Boxcar(x, a = 0.5)
Ramp(x, a = 0)

Arguments

Details

Heaviside computes the Heaviside unit step function. Heaviside is 1 for x > a, 1/2 for x = a, and 0 for x < a.

Sign computes the sign function. Sign is 1 for x > a, 0 for x = a, and -1 for x < a.

Delta computes the delta function. Delta is defined as: Delta(x) = d/dx H(x-a).

Boxcar computes the boxcar function. Boxcar is defined as: Boxcar(x) = H(x+a) - H(x-a).

Ramp computes ramp function. The ramp function is defined as: Ramp(x) = (x-a) * H(x-a).

Value

numeric vector

Note

The Heaviside function is used in the implementation of the skew Normal, Student-t, and Generalized Error distributions, distributions functions which play an important role in modelling GARCH processes.

References

Weisstein W. (2004); http://mathworld.wolfram.com/HeavisideStepFunction.html, Mathworld.

See Also

GarchDistribution, GarchDistributionFits

Examples

x <- sort(round(c(-1, -0.5, 0, 0.5, 1, 5*rnorm(5)), 2))

h <- Heaviside(x)
s <- Sign(x)
d <- Delta(x)
Pi <- Boxcar(x)
r <- Ramp(x)

cbind(x = x, Step = h, Signum = s, Delta = d, Pi = Pi, R = r)        

Histogram and density plots

Description

Produce tailored histogram plots and kernel density/log-density estimate plots.

Usage

        
histPlot(x, labels = TRUE, col = "steelblue", fit = TRUE, 
    title = TRUE, grid = TRUE, rug = TRUE, skip = FALSE, ...) 
densityPlot(x, labels = TRUE, col = "steelblue", fit = TRUE, hist = TRUE, 
    title = TRUE, grid = TRUE, rug = TRUE, skip = FALSE, ...)    
logDensityPlot(x, labels = TRUE, col = "steelblue", robust = TRUE,  
    title = TRUE, grid = TRUE, rug = TRUE, skip = FALSE, ...)    

Arguments

Details

histPlot produces a tailored histogram plot.

densityPlot produces a tailored kernel density estimate plot.

logDensityPlot produces a tailored log kernel density estimate plot.

Value

NULL, invisibly. The functions are used for the side effect of producing a plot.

Examples

## data
data(LPP2005REC, package = "timeSeries")
SPI <- LPP2005REC[, "SPI"]
plot(SPI, type = "l", col = "steelblue", main = "SP500")
abline(h = 0, col = "grey")
   
histPlot(SPI) 
   
densityPlot(SPI) 

Set and retrieve column/row names

Description

Sets and retrieves column and row names. The functions are for compatibility with SPlus.

Usage

colIds(x, ...)
rowIds(x, ...)

Arguments

Details

Usually in R the functions colnames and rownames are used to retrieve and set the names of matrices. The functions rowIds and colIds, are S-Plus like synonyms.

Examples

## Create Pascal Matrix:
P <- pascal(3)
P
 
rownames(P) <- letters[1:3]
P   
   
colIds(P) <- as.character(1:3)
P                            

Tests for normality

Description

A collection of functions of one sample tests for testing normality of financial return series.

The functions for testing normality are:

Functions for high precision Jarque Bera LM and ALM tests:

Additional functions for testing normality from the 'nortest' package:

For SPlus/Finmetrics Compatibility:

Usage

ksnormTest(x, title = NULL, description = NULL)

jbTest(x, title = NULL, description = NULL)
shapiroTest(x, title = NULL, description = NULL)
normalTest(x, method = c("sw", "jb"), na.rm = FALSE)

jarqueberaTest(x, title = NULL, description = NULL)
dagoTest(x, title = NULL, description = NULL)

adTest(x, title = NULL, description = NULL)
cvmTest(x, title = NULL, description = NULL)
lillieTest(x, title = NULL, description = NULL)
pchiTest(x, title = NULL, description = NULL)
sfTest(x, title = NULL, description = NULL)

Arguments

Details

The hypothesis tests may be of interest for many financial and economic applications, especially for the investigation of univariate time series returns.

Several tests for testing if the records from a data set are normally distributed are available. The input to all these functions may be just a vector x or a univariate time series object x of class timeSeries.

First, there exists a wrapper function which allows to call one from two normal tests either the Shapiro–Wilks test or the Jarque–Bera test. This wrapper was introduced for compatibility with S-Plus' FinMetrics package.

Also available are the Kolmogorov–Smirnov one sample test and the D'Agostino normality test.

The remaining five normal tests are the Anderson–Darling test, the Cramer–von Mises test, the Lilliefors (Kolmogorov–Smirnov) test, the Pearson chi–square test, and the Shapiro–Francia test. They are calling functions from R's contributed package nortest. The difference to the original test functions implemented in R and from contributed R packages is that the Rmetrics functions accept time series objects as input and give a more detailed output report.

The Anderson-Darling test is used to test if a sample of data came from a population with a specific distribution, here the normal distribution. The adTest goodness-of-fit test can be considered as a modification of the Kolmogorov–Smirnov test which gives more weight to the tails than does the ksnormTest.

Note that jarqueBeraTest computes the asymptotic statistic and p-value, while jbTesT gives final sample approximations.

Value

an object from class fHTEST.

Slot test is a list containing the following (optionally empty) elements (in addition to those described in fHTEST):

ksnormTest

the 'D' statistic and p-values for the three alternatives 'two-sided, 'less' and 'greater'.

shapiroTest

the 'W' statistic and the p-value.

jarqueberaTest

no additional elements.

jbTest

the 'Chi-squared' statistic with 2 degrees of freedom and the asymptotic p-value. jbTest is the finite sample version of the Jarque Bera Lagrange multiplier, LM, and adjusted Lagrange multiplier test, ALM.

dagoTest

the 'Chi-squared', the 'Z3' (Skewness) and 'Z4' (Kurtosis) statistic together with the corresponding p values.

adTest

the 'A' statistic and the p-value.

cvmTest

the 'W' statistic and the p-value.

lillieTest

the 'D' statistic and the p-value.

pchiTest

the value for the 'P' statistic and the p-values for the adjusted and not adjusted test cases. In addition the number of classes is printed, taking the default value due to Moore (1986) computed from the expression n.classes = ceiling(2 * (n^(2/5))), where n is the number of observations.

sfTest

the 'W' statistic and the p-value.

Note

Some of the test implementations are selected from R's ctest and nortest packages.

Author(s)

R-core team for the tests from R's ctest package,
Adrian Trapletti for the runs test from R's tseries package,
Juergen Gross for the normal tests from R's nortest package,
James Filliben for the Fortran program producing the runs report,
Diethelm Wuertz and Helmut Katzgraber for the finite sample JB tests,
Diethelm Wuertz for the Rmetrics R-port.
Earlier versions of theses functions were based on Fortran code of Paul Johnson.

References

Anderson T.W., Darling D.A. (1954); A Test of Goodness of Fit, JASA 49:765–69.

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

D'Agostino R.B., Pearson E.S. (1973); Tests for Departure from Normality, Biometrika 60, 613–22.

D'Agostino R.B., Rosman B. (1974); The Power of Geary's Test of Normality, Biometrika 61, 181–84.

Durbin J. (1961); Some Methods of Constructing Exact Tests, Biometrika 48, 41–55.

Durbin,J. (1973); Distribution Theory Based on the Sample Distribution Function, SIAM, Philadelphia.

Geary R.C. (1947); Testing for Normality; Biometrika 36, 68–97.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

Linnet K. (1988); Testing Normality of Transformed Data, Applied Statistics 32, 180–186.

Moore, D.S. (1986); Tests of the chi-squared type, In: D'Agostino, R.B. and Stephens, M.A., eds., Goodness-of-Fit Techniques, Marcel Dekker, New York.

Shapiro S.S., Francia R.S. (1972); An Approximate Analysis of Variance Test for Normality, JASA 67, 215–216.

Shapiro S.S., Wilk M.B., Chen V. (1968); A Comparative Study of Various Tests for Normality, JASA 63, 1343–72.

Thode H.C. (2002); Testing for Normality, Marcel Dekker, New York.

Weiss M.S. (1978); Modification of the Kolmogorov-Smirnov Statistic for Use with Correlated Data, JASA 73, 872–75.

Wuertz D., Katzgraber H.G. (2005); Precise finite-sample quantiles of the Jarque-Bera adjusted Lagrange multiplier test, ETHZ Preprint.

Examples

x <- rnorm(100)

## Kolmogorov-Smirnov one-sampe test
ksnormTest(x)

## Shapiro-Wilk test
shapiroTest(x)

## Jarque-Bera Test
jarqueberaTest(x)
jbTest(x)

Quantile-Quantile plots

Description

Returns quantile-quantile plots for the normal, the normal inverse Gaussian, the generalized hyperbolic Student-t and the generalized lambda distribution.

Usage

     
qqnormPlot(x, labels = TRUE, col = "steelblue", pch = 19,
    title = TRUE, mtext = TRUE, grid = FALSE, rug = TRUE, 
    scale = TRUE, ...) 
qqnigPlot(x, labels = TRUE, col = "steelblue", pch = 19,
    title = TRUE, mtext = TRUE, grid = FALSE, rug = TRUE, 
    scale = TRUE, ...) 
qqghtPlot(x, labels = TRUE, col = "steelblue", pch = 19,
    title = TRUE, mtext = TRUE, grid = FALSE, rug = TRUE, 
    scale = TRUE, ...) 
qqgldPlot(x, labels = TRUE, col = "steelblue", pch = 19,
    title = TRUE, mtext = TRUE, grid = FALSE, rug = TRUE, 
    scale = TRUE, ...)

Arguments

Details

List of Functions:

Value

displays a quantile-quantile plot.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## data
data(LPP2005REC, package = "timeSeries")
SPI <- LPP2005REC[, "SPI"]
plot(SPI, type = "l", col = "steelblue", main = "SP500")
abline(h = 0, col = "grey")
   
qqnormPlot(SPI) 

Return series plots

Description

A graphical user interface to display finanical time series plots. returnSeriesGUI opens a GUI for return series plots.

Usage

 
returnSeriesGUI(x)

Arguments

Value

For returnSeriesGUI function, besides the graphical user interface no values are returned.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Scaling law behaviour

Description

Evaluates the scaling exponent of a financial return series and plots the scaling law.

Usage

scalinglawPlot(x, span = ceiling(log(length(x)/252)/log(2)), doplot = TRUE, 
    labels = TRUE, trace = TRUE, ...)

Arguments

Details

Scaling Behavior:

The function scalingPlot plots the scaling law of financial time series under aggregation and returns an estimate for the scaling exponent. The scaling behavior is a very striking effect of the foreign exchange market and also other markets expressing a regular structure for the volatility. Considering the average absolute return over individual data periods one finds a scaling power law which relates the mean volatility over given time intervals to the size of these intervals. The power law is in many cases valid over several orders of magnitude in time. Its exponent usually deviates significantly from a Gaussian random walk model which implies 1/2.

Value

a list with the following components:

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

References

Taylor S.J. (1986); Modeling Financial Time Series, John Wiley and Sons, Chichester.

Examples

## data
data(LPP2005REC, package = "timeSeries")
SPI <- LPP2005REC[, "SPI"]
plot(SPI, type = "l", col = "steelblue", main = "SP500")
abline(h = 0, col = "grey")

## Scaling Law Effect
scalinglawPlot(SPI)

Slider GUI for Stable Distribution

Description

The stableSlider() function provides interactive displays of density and probabilities of stable distributions.

Usage

stableSlider(col= "steelblue", col.med = "gray30")

Arguments

Value

The stableSlider() function displays densities and probabilities of the skew stable distribution, for educational purposes.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

References

see those in dstable, in package stabledist.

Examples

if(dev.interactive())
   stableSlider()

Financial time series plots

Description

Produces an index/price, a cumulated return, a return, or a drawdown plot.

Usage

 
seriesPlot(x, labels = TRUE, type = "l", col = "steelblue", 
    title = TRUE, grid = TRUE, box = TRUE, rug = TRUE, ...) 
cumulatedPlot(x, index = 100, labels = TRUE, type = "l", col = "steelblue", 
    title = TRUE, grid = TRUE, box = TRUE, rug = TRUE, ...) 
returnPlot(x, labels = TRUE, type = "l", col = "steelblue", 
    title = TRUE, grid = TRUE, box = TRUE, rug = TRUE, ...) 
drawdownPlot(x, labels = TRUE, type = "l", col = "steelblue", 
    title = TRUE, grid = TRUE, box = TRUE, rug = TRUE, ...) 

Arguments

Details

List of Functions:

The plot functions can be used to plot univariate and multivariate time series of class timeSeries.

The graphical parameters type and col can be set by the values specified through the argument list. In the case of multivariate time series col can be specified by the values returned by a color palette.

Automated titles including main title, x- and y-lables, grid lines, box style and rug represenatations cann be selected by setting these arguments to TRUE which is the default. If the title flag is unset, then the main title, x-, and y-labels are empty strings. This allows to set user defined labels with the function title after the plot is drawn.

Beside type, col, main, xlab and ylab, all other par arguments can be passed to the plot function.

If the labels flag is unset to FALSE, then no decorations will be added tothe plot, and the plot can be fully decorated by the user.

Value

displays a time series plot

Examples

data(LPP2005REC, package = "timeSeries")
tS <- as.timeSeries(LPP2005REC)

seriesPlot(tS)

Autocorrelation function plots

Description

Returns plots of autocorrelations including the autocorrelation function ACF, the partial ACF, the lagged ACF, and the Taylor effect plot.

The functions to display stylized facts are:

Usage

acfPlot(x, labels = TRUE, ...)
pacfPlot(x, labels = TRUE, ...) 

lacfPlot(x, n = 12, lag.max = 20, type = c("returns", "values"),
    labels = TRUE, ...)

teffectPlot(x, deltas = seq(from = 0.2, to = 3, by = 0.2), lag.max = 10, 
    ymax = NA, standardize = TRUE, labels = TRUE, ...)

Arguments

Details

Autocorrelation Functions:

The functions acfPlot and pacfPlot, plot and estimate autocorrelation and partial autocorrelation function. The functions allow to get a first view on correlations within the time series. The functions are synonym function calls for R's acf and pacf from the the ts package.

Taylor Effect:

The "Taylor Effect" describes the fact that absolute returns of speculative assets have significant serial correlation over long lags. Even more, autocorrelations of absolute returns are typically greater than those of squared returns. From these observations the Taylor effect states, that that the autocorrelations of absolute returns to the the power of delta, abs(x-mean(x))^delta reach their maximum at delta=1. The function teffect explores this behaviour. A plot is created which shows for each lag (from 1 to max.lag) the autocorrelations as a function of the exponent delta. In the case that the above formulated hypothesis is supported, all the curves should peak at the same value around delta=1.

Value

for acfPlot and pacfplot, an object of class "acf", see acf;

for teffectPlot, a numeric matrix of order deltas by max.lag with the values of the autocorrelations;

for lacfPlot, a list with the following two elements:

References

Taylor S.J. (1986); Modeling Financial Time Series, John Wiley and Sons, Chichester.

Ding Z., Granger C.W.J., Engle R.F. (1993); A long memory property of stock market returns and a new model, Journal of Empirical Finance 1, 83.

Examples

## data
data(LPP2005REC, package = "timeSeries")
SPI <- LPP2005REC[, "SPI"]
plot(SPI, type = "l", col = "steelblue", main = "SP500")
abline(h = 0, col = "grey")

## Taylor Effect:
teffectPlot(SPI)

Bivariate Spline Interpolation

Description

Interpolates bivariate data sets using Akima spline interpolation.

Usage

akimaInterp(x, y = NULL, z = NULL, gridPoints = 21,
    xo = seq(min(x), max(x), length = gridPoints),
    yo = seq(min(y), max(y), length = gridPoints), extrap = FALSE)

akimaInterpp(x, y = NULL, z = NULL, xo, yo, extrap = FALSE)

Arguments

Details

Two options are available: gridded and pointwise interpolation.

akimaInterp is a wrapper to interp provided by the contributed R package akima. The Fortran code of the Akima spline interpolation routine was written by H. Akima.

Linear surface fitting and krige surface fitting are provided by the functions linearInterp and krigeInterp.

Value

akimaInterp

returns a list with at least three entries, x, y and z. Note, that the returned values, can be directly used by the persp and contour 3D plotting methods.

akimaInterpp

returns a data.frame with columns "x", "y", and "z".

Note

Package akima is no longer needed. Equivalent functions from package interp are now called instead.

References

Akima H., 1978, A Method of Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points, ACM Transactions on Mathematical Software 4, 149-164.

Akima H., 1996, Algorithm 761: Scattered-Data Surface Fitting that has the Accuracy of a Cubic Polynomial, ACM Transactions on Mathematical Software 22, 362-371.

See Also

linearInterp, krigeInterp.

Examples

## Does not run for r-solaris-x86
## akimaInterp -- Akima Interpolation:
if (requireNamespace("interp")) {
  set.seed(1953)
  x <- runif(999) - 0.5
  y <- runif(999) - 0.5
  z <- cos(2*pi*(x^2+y^2))
  ans <- akimaInterp(x, y, z, gridPoints = 41, extrap = FALSE)
  persp(ans, theta = -40, phi = 30, col = "steelblue",
       xlab = "x", ylab = "y", zlab = "z")
  contour(ans)
}

## Use spatial as alternative on r-solaris-x86
## spatialInterp - Generate Kriged Grid Data:
if (requireNamespace("spatial")) {
  RNGkind(kind = "Marsaglia-Multicarry", normal.kind = "Inversion")
  set.seed(4711, kind = "Marsaglia-Multicarry")
  x <- runif(999)-0.5
  y <- runif(999)-0.5
  z <- cos(2*pi*(x^2+y^2))
  ans <- krigeInterp(x, y, z, extrap = FALSE)
  persp(ans)
  title(main = "Kriging")
  contour(ans)
  title(main = "Kriging")
}

Generic functions extensions

Description

Basic extensions which add and/or modify additional functionality which is not available in R's basic packages.

Usage

## Default S3 method:
stdev(x, na.rm = FALSE)

## Default S3 method:
termPlot(model, ...)

Arguments

Details

stdev and termPlot are generic functions with default methods stats::sd and stats::termplot, respectively.

See Also

sd, termplot

Table of characters

Description

Displays a table of numerical equivalents to Latin characters.

Usage

 
characterTable(font = 1, cex = 0.7)

Arguments

Value

displays a table with the characters of the requested font. The character on line "xy" and column "z" of the table has code "\xyz", e.g cat("\126") prints: V for font number 1. These codes can be used as any other characters.

Note

What happens with non-ASCII characters in plots is system dependent and depends on the graphics device, as well. Use of such characters is not recommended for portable code.

See Also

colorTable, symbolTable

points for use of characters in plotting

Examples

## Character Table for Font 1:
# characterTable(font = 1)

Column and row vectors

Description

Creates a column or row vector from a numeric vector.

Usage

colVec(x)
rowVec(x)

Arguments

Details

colVec and rowVec transform a vector into a column and row vector, respectively. A column vector is a matrix object with one column, and a row vector is a matrix object with one row.

Examples

x = rnorm(5)
   
colVec(x)
rowVec(x)                        

Named colors in R

Description

Displays R's 657 named colors for selection and returns optionally R's color names.

Usage

colorLocator(locator = FALSE, cex.axis = 0.7)
colorMatrix()

Arguments

Details

Color Locator:

The colorLocator function plots R's 657 named colors. If locator=TRUE then you can interactively point and click to select the colors for which you want names. To end selection, right click on the mouse and select 'Stop', then R returns the selected color names.

The functions used here are wrappers to the functions provided by Tomas Aragon in the contributed R package epitools.

Value

Color Locator:

colorsLocator() generates a plot with R colors and, when locator is true, returns matrix with graph coordinates and names of colors selected. colorsMatrix() quietly returns the matrix of names.

See Also

colorPalette, colorTable.

Examples

 colorLocator()

Color palettes

Description

Functions to create color palettes.

The functions are:

Usage

 
rainbowPalette(n = 64, ...) 
heatPalette(n = 64, ...) 
terrainPalette(n = 64, ...) 
topoPalette(n = 64, ...) 
cmPalette(n = 64, ...) 

greyPalette(n = 64, ...)
timPalette(n = 64)

rampPalette(n, name = c("blue2red", "green2red", "blue2green",     
    "purple2green", "blue2yellow", "cyan2magenta"))
    
seqPalette(n, name = c(
    "Blues", "BuGn", "BuPu", "GnBu", "Greens", "Greys", "Oranges", 
    "OrRd", "PuBu", "PuBuGn", "PuRd", "Purples", "RdPu", "Reds", 
    "YlGn", "YlGnBu", "YlOrBr", "YlOrRd"))
divPalette(n, name = c(
    "BrBG", "PiYG", "PRGn", "PuOr", "RdBu", "RdGy", "RdYlBu", "RdYlGn", 
    "Spectral"))   
qualiPalette(n, name = c(
    "Accent", "Dark2", "Paired", "Pastel1", "Pastel2", "Set1", "Set2", 
    "Set3")) 
    
focusPalette(n, name = c("redfocus", "greenfocus", "bluefocus"))
monoPalette(n, name = c("redmono", "greenmono", "bluemono"))

Arguments

Details

All Rmetrics' color sets are named as fooPalette, where the prefix foo denotes the name of the underlying color set.

R's Contiguous Color Palettes:

Palettes for n contiguous colors are implemented in the grDevices package. To conform with Rmetrics' naming convention for color palettes we have build wrappers around the underlying functions. These are the rainbowPalette, heatPalette, terrainPalette, topoPalette, and the cmPalette.

Conceptually, all of these functions actually use (parts of) a line cut out of the 3-dimensional color space, parametrized by the function hsv(h,s,v,gamma), where gamma=1 for the fooPalette function, and hence, equispaced hues in RGB space tend to cluster at the red, green and blue primaries.

Some applications, such as contouring, require a palette of colors which do not wrap around to give a final color close to the starting one. To pass additional arguments to the underlying functions see help(rainbow). With rainbow, the parameters start and end can be used to specify particular subranges of hues. Synonym function calls are rainbow, heat.colors, terrain.colors, topo.colors, and cm.colors.

R's Gamma-Corrected Gray Palette:

grayPalette chooses a series of n gamma-corrected gray levels. The range of the gray levels can be optionally monitored through the ... arguments, for details see help(gray.colors), which is a synonym function call in the grDevices package.

Tim's Matlab like Color Palette:

timPalette creates a color set ranging from blue to red, and passes through the colors cyan, yellow, and orange. It comes from the Matlab software, originally used in fluid dynamics simulations. The function here is a copy from R's contributed package fields doing a spline interpolation on n=64 color points.

Color Ramp Palettes:

rampPalette creates several color ramps. The function is implemented from Tim Keitt's contributed R package colorRamps. Supported through the argument name are the following color ramps: "blue2red", "green2red", "blue2green", "purple2green", "blue2yellow", "cyan2magenta".

Color Brewer Palettes:

The functions seqPalette, divPalette, and qualiPalette create color sets according to R's contributed RColorBrewer package. The first letter in the function name denotes the type of the color set: "s" for sequential palettes, "d" for diverging palettes, and "q" for qualitative palettes.

Sequential palettes are suited to ordered data that progress from low to high. Lightness steps dominate the look of these schemes, with light colors for low data values to dark colors for high data values. The sequential palettes names are: Blues, BuGn, BuPu, GnBu, Greens, Greys, Oranges, OrRd, PuBu, PuBuGn, PuRd, Purples, RdPu, Reds, YlGn, YlGnBu, YlOrBr, YlOrRd.

Diverging palettes put equal emphasis on mid-range critical values and extremes at both ends of the data range. The critical class or break in the middle of the legend is emphasized with light colors and low and high extremes are emphasized with dark colors that have contrasting hues. The diverging palettes names are: BrBG, PiYG, PRGn, PuOr, RdBu, RdGy, RdYlBu, RdYlGn, Spectral.

Qualitative palettes do not imply magnitude differences between legend classes, and hues are used to create the primary visual differences between classes. Qualitative schemes are best suited to representing nominal or categorical data. The qualitative palettes names are: Accent, Dark2, Paired, Pastel1, Pastel2, Set1, Set2, Set3.

In contrast to the original color brewer palettes, the palettes here are created by spline interpolation from the color variation with the most different values, i.e for the sequential palettes these are 9 values, for the diverging palettes these are 11 values, and for the qualitative palettes these are between 8 and 12 values dependeing on the color set.

Graph Color Palettes

The function perfanPalette creates color sets inspired by R's contributed package Performance Analytics. These color palettes have been designed to create readable, comparable line and bar graphs with specific objectives.

Focused Color Palettes

Color sets designed to provide focus to the data graphed as the first element. This palette is best used when there is clearly an important data set for the viewer to focus on, with the remaining data being secondary, tertiary, etc. Later elements graphed in diminishing values of gray.

Monochrome Color Palettes

These include color sets for monochrome color displays.

Value

a character string of color strings

Note

The palettes are wrapper functions provided in several contributed R packages. These include:

Cynthia Brewer and Mark Harrower for the brewer palettes,
Peter Carl and Brian G. Peterson for the "PerformanceAnalytics" package,
Tim Keitt for the "colorRamps" package,
Ross Ihaka for the "colorspace" package,
Tomas Aragon for the "epitools" package,
Doug Nychka for the "fields" package,
Erich Neuwirth for the "RColorBrewer" package.

Additional undocumented hidden functions:

Examples

greyPalette()

Table of colors

Description

Displays a table of color codes and plots the colors themselves.

Usage

 
colorTable(cex = 0.7)

Arguments

Value

a table of plot colors with the associated color numbers

See Also

characterTable, symbolTable

Examples

colorTable()

Correlation tests

Description

Tests if two series are correlated.

Usage

    
correlationTest(x, y, method = c("pearson", "kendall", "spearman"), 
    title = NULL, description = NULL)

pearsonTest(x, y, title = NULL, description = NULL) 
kendallTest(x, y, title = NULL, description = NULL)
spearmanTest(x, y, title = NULL, description = NULL)

Arguments

Details

These functions test for association/correlation between paired samples based on the Pearson's product moment correlation coefficient (a.k.a. sample correlation), Kendall's tau, and Spearman's rho coefficients.

pearsonTest, kendallTest, and spearmanTest are wrappers of base R's cor.test with simplified interface. They provide 'exact' and approximate p-values for all three alternatives (two-sided, less, and greater), as well as 95% confidence intervals. This is particularly convenient in interactive use.

Instead of calling the individual functions, one can use correlationTest and specify the required test with argument method.

Value

an object from class fHTEST

References

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

See Also

locationTest, scaleTest, varianceTest

Examples

x <- rnorm(50)
y <- rnorm(50)
  
correlationTest(x, y, "pearson")
correlationTest(x, y, "kendall")

spearmanTest(x, y)

Functions for decorating plots

Description

Functions for decorating plots.

Usage

decor()

hgrid(ny = NULL, ...) 
vgrid(nx = NULL, ...) 

boxL(col = "white") 
box_(col = c("white", "black")) 

copyright()

Arguments

Details

decor is equivalent to hgrid() followed by boxL().

hgrid creates horizontal grid lines.

vgrid creates vertical grid lines.

boxL creates an L-shaped box

box_ creates a bottom line box.

copyright adds Rmetrics copyright to a plot.

Examples

plot(x = rnorm(100), type = "l", col = "red", 
     xlab = "", ylab = "Variates", las = 1)
title("Normal Deviates", adj = 0)
hgrid()
boxL()
copyright()

Distribution check

Description

Tests properties of an R implementation of a distribution, i.e., of all four of its “dpqr” functions.

Usage

distCheck(fun = "norm", n = 1000, robust = TRUE, subdivisions = 100, ...)

Arguments

Examples

distCheck("norm", mean = 1, sd = 1)

distCheck("lnorm", meanlog = 0.5, sdlog = 2, robust=FALSE)
## here, true E(X) = exp(mu + 1/2 sigma^2) = exp(.5 + 2) = exp(2.5) = 12.182
## and      Var(X) = exp(2*mu + sigma^2)*(exp(sigma^2) - 1) =       7954.67

Deprecated functions in package fBasics

Description

These functions are provided for compatibility with older versions of the package only, and may be defunct as soon as of the next release.

Details

There are none currently.

dstable, etc., now are defunct, as they have been available from stabledist since early 2011.

See Also

Deprecated, Defunct

fBasics data sets

Description

The following data sets are part of this package:

Details

All datasets are data frames. A brief description is given below.

Capitalization:
Capitalization contains market capitalization of 13 domestic companies for 6 years (from 2003 to 2008) in USD millions. Each row contains the data for one company/stock exchange.

cars2:
cars2 contains columns rowNames (model), Price, Country, Reliability, Mileage, (Type), (Weight), Disp. (engine displacement) and HP (net horsepower) reprsenting the indicated properties of 60 car models.

DowJones30:
DowJones30 contains 2529 daily observations from the ‘Dow Jones 30’ Index series. The first row contains the dates (from 1990-12-31 to 2001-01-02). Each of the remaining thirty columns represents the closing price of a stock in the Index.

HedgeFund:
HedgeFund contains monthly percentage returns of 16 hedge fund strategies from Hennessee Group LLC for year 2005.

msft.dat:
msft.dat contains daily prices (open, high, low and close) and volumes for the Microsoft stocks. It is a data frame with column names "%Y-%m-%d", "Open", "High", "Low", "Close", "Volume".

Note: there is a dataset, MSFT, in package timeSeries which contains the same data but is of class "timeSeries".

nyse:
nyse contains daily records of the NYSE Composite Index from 1966-01-04 to 2002-12-31 (9311 observations). The data is in column "NYSE" (second column). The first column contains the dates.

PensionFund:
PensionFund is a daily data set of the Swiss pension fund benchmark LPP-2005. The data set ranges from 2005-11-01 to 2007-04-11. The columns are named: SBI, SPI, SII, LMI, MPI, ALT, LPP25, LPP40, LPP60.

swissEconomy:
swissEconomy contains the GDP per capita (GDPR), exports (EXPO), imports (IMPO), interest rates (INTR), inflation (INFL), unemployment (UNEM) and population (POPU) foryears 1964 to 1999 for Switzerland.

SWXLP:
SWXLP is a daily data set of the Swiss pension fund benchmark LPP-2000. The data set ranges from 2000-01-03 to 2007-05-08 (1917 observations). The first column contains the dates. The remaining columns are named: SBI, SPI, SII, LP25, LP40, LP60.

usdthb:
usdthb Tick data of US Dollar (USD) in Thailand Bhat (THB) collected from Reuters. The date is in the first column in YYYYMMDDhhmm format. The remaining columns contain: delay time (DELAY), contributor (CONTRIBUTOR), bid (BID) and ask (ASK) prices, and quality flag (FLAG). It covers the Asia FX crisis in June 1997.

References

Capitalization:
World Federation of Stock Exchanges, http://www.world-exchanges.org/statistics.

cars2:
Derived from the car90 dataset within the rpart package. The car90 dataset is based on the car.all dataset in S-PLUS. Original data comes from: April 1990, Consumer Reports Magazine, pages 235-255, 281-285 and 287-288.

DowJones30
https://www.yahoo.com.

HedgeFund:
http://www.hennesseegroup.com/indices/returns/year/2005.html.

msft.dat:
https://www.yahoo.com.

nyse:
https://www.nyse.com.

PensionFund:
SBI, SPI, SII: SIX (Swiss Exchange Zurich); LPP25, LPP40, LPP60: Banque Pictet Geneva; LMI, MPI, ALT: Recalculated from the indices and benchmarks.

swissEconomy:
https://www.oecd.org/ and https://www.imf.org/.

SWXLP:
SBI, SPI, SII: SIX (Swiss Exchange Zurich); LPP25, LPP40, LPP60: Banque Pictet Geneva.

usdthb:
Reuters Select Feed Terminal (1997).

Examples

## Plot DowJones30 Example Data Set
series <- timeSeries::as.timeSeries(DowJones30)
head(series)
plot(series[,1:6], type = "l")

## msft.dat contains (almost?) the same data as MSFT in package timeSeries
data(MSFT, package = "timeSeries")

m1 <- as.matrix(msft.dat[, -1]) # drop date stamps in column 1
m2 <- as.matrix(MSFT)
all.equal(m1, m2, check.attributes = FALSE) # TRUE
## compare the dates:
all.equal(format(msft.dat[ , 1]), format(time(MSFT))) # TRUE

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

lagged

as.timeSeries

kurtosis

kurtosis

skewness

skewness

Class "fDISTFIT"

Description

S4 class representing fitted distributions.

Objects from the Class

Objects can be created by calls of the form new("fDISTFIT", ...) but are typically created by functions fitting distributions.

Slots

call:

Object of class "call" ~~

model:

Object of class "character" ~~

data:

Object of class "data.frame" ~~

fit:

Object of class "list" ~~

title:

Object of class "character" ~~

description:

Object of class "character" ~~

Slot fit is a list. Its components depend on the fitting functions. Here is the meaning of some common ones:

estimate

the point at which the maximum value of the log liklihood function is obtained.

minimum

the value of the estimated maximum, i.e. the value of the log liklihood function.

code

an integer indicating why the optimization process terminated.
1: relative gradient is close to zero, current iterate is probably solution;
2: successive iterates within tolerance, current iterate is probably solution;
3: last global step failed to locate a point lower than estimate. Either estimate is an approximate local minimum of the function or steptol is too small;
4: iteration limit exceeded;
5: maximum step size stepmax exceeded five consecutive times. Either the function is unbounded below, becomes asymptotic to a finite value from above in some direction or stepmax is too small.

gradient

the gradient at the estimated maximum.

Methods

show

signature(object = "fDISTFIT"): ...

Examples

showClass("fDISTFIT")

Class "fHTEST"

Description

An S4 class representing the outcome of a statistical test.

Objects from the Class

Objects from this class are created by some statistical test functions.

Slots

call:

the function call.

data:

the data as specified by the input argument(s).

test:

a list whose elements contain the results from the statistical test. The information provided is similar to a list object of class "htest".

title:

a character string with the name of the test. This can be overwritten specifying a user defined input argument.

description:

a character string with an optional user defined description. By default just the current date when the test was applied will be returned.

Slot @test is an object of class "list" containing at least the following elements:

statistic

the value(s) of the test statistic.

p.value

the p-value(s) of the test.

parameters

a numeric value or vector of parameters.

estimate

a numeric value or vector of sample estimates.

conf.int

a numeric two row vector or matrix of 95% confidence levels.

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

Methods

show

signature(object = "fHTEST"): ...

See Also

for functions returning objects from class "fHTEST", see scaleTest, correlationTest, ks2Test, locationTest, NormalityTests, varianceTest scaleTest

Examples

showClass("fHTEST")

General S4 Class Extractor Functions

Description

A collection and description of functions to extract slots from S4 class objects.

The extractor functions are:

Since R version 2.14.0, a generic getCall() is part of R; for earlier versions, we had provided a simple version for S4 objects.

Usage

getModel(object)
getTitle(object)
getDescription(object)

getSlot(object, slotName)

getArgs(f, signature)

Arguments

Value

for getModel, getTitle, getDescription, and getSlot - the content of the corresponding slot.

for getArgs the names of the arguments.

Examples

## Example S4 Representation:
## Hyothesis Testing with Control Settings
setClass("hypTest",
         representation(
             call = "call",
             data = "numeric",
             test = "list",
             description = "character")
         )

## Shapiro Wilk Normaility Test
swTest = function(x, description = "") {
    ans = shapiro.test(x)
    class(ans) = "list"
    new("hypTest",
        call = match.call(),
        data = x,
        test = ans,
        description = description)
}
test = swTest(x = rnorm(500), description = "500 RVs")

## Extractor Functions:
isS4(test)
getCall(test)
getDescription(test)

## get arguments
args(returns)
getArgs(returns)
getArgs("returns")
getArgs(returns, "timeSeries")
getArgs("returns", "timeSeries")

Generalized Hyperbolic Distribution

Description

Density, distribution function, quantile function and random generation for the generalized hyperbolic distribution.

Usage

dgh(x, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2, log = FALSE)
pgh(q, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
qgh(p, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
rgh(n, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

Arguments

Details

dgh gives the density, pgh gives the distribution function, qgh gives the quantile function, and rgh generates random deviates.

The meanings of the parameters correspond to the first parameterization, pm=1, which is the default parameterization for this distribution.

In the second parameterization, pm=2, alpha and beta take the meaning of the shape parameters (usually named) zeta and rho.

In the third parameterization, pm=3, alpha and beta take the meaning of the shape parameters (usually named) xi and chi.

In the fourth parameterization, pm=4, alpha and beta take the meaning of the shape parameters (usually named) a.bar and b.bar.

The generator rgh is based on the GH algorithm given by Scott (2004).

Value

numeric vector

Author(s)

David Scott for code implemented from R's contributed package HyperbolicDist.

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## rgh -
   set.seed(1953)
   r = rgh(5000, alpha = 1, beta = 0.3, delta = 1)
   plot(r, type = "l", col = "steelblue",
     main = "gh: alpha=1 beta=0.3 delta=1")
 
## dgh - 
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   x = seq(-5, 5, 0.25)
   lines(x, dgh(x, alpha = 1, beta = 0.3, delta = 1))
 
## pgh -  
   # Plot df and compare with true df:
   plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
   lines(x, pgh(x, alpha = 1, beta = 0.3, delta = 1))
   
## qgh -
   # Compute Quantiles:
   qgh(pgh(seq(-5, 5, 1), alpha = 1, beta = 0.3, delta = 1), 
     alpha = 1, beta = 0.3, delta = 1) 

GH Distribution Fit

Description

Estimates the distrinbutional parameters for a generalized hyperbolic distribution.

Usage

ghFit(x, alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2, 
    scale = TRUE, doplot = TRUE, span = "auto", trace = TRUE, 
    title = NULL, description = NULL, ...) 

Arguments

Details

The meanings of the parameters correspond to the first parameterization, see gh for further details.

The function nlm is used to minimize the "negative" maximum log-likelihood function. nlm carries out a minimization using a Newton-type algorithm.

Value

an object from class "fDISTFIT". Slot fit is a list, currently with components estimate, minimum and code.

Examples

set.seed(1953)
s <- rgh(n = 1000, alpha = 1.5, beta = 0.3, delta = 0.5, mu = -1.0) 

ghFit(s, alpha = 1, beta = 0, delta = 1, mu = mean(s), doplot = TRUE, trace = FALSE) 

Generalized Hyperbolic Mode

Description

Computes the mode of the generalized hyperbolic function.

Usage

ghMode(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

Arguments

Details

The meanings of the parameters correspond to the first parameterization, see gh for further details.

Value

a numeric value, the mode of the generalized hyperbolic distribution

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## ghMode -
   ghMode()

Generalized Hyperbolic Distribution Moments

Description

Calculates moments of the generalized hyperbolic distribution.

Usage

ghMean(alpha=1, beta=0, delta=1, mu=0, lambda=-1/2)
ghVar(alpha=1, beta=0, delta=1, mu=0, lambda=-1/2)
ghSkew(alpha=1, beta=0, delta=1, mu=0, lambda=-1/2)
ghKurt(alpha=1, beta=0, delta=1, mu=0, lambda=-1/2)

ghMoments(order, type = c("raw", "central", "mu"),
    alpha = 1, beta=0, delta=1, mu=0, lambda=-1/2)

Arguments

Value

a named numerical value. The name is one of mean, var, skew, or kurt, obtained by dropping the nig prefix from the name of the corresponding function and lowercasing it.

for ghMoments, the name is obtained by paste0("m", order, type).

Author(s)

Diethelm Wuertz

References

Scott, D. J., Wuertz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

Examples

   
## ghMean -
   ghMean(alpha=1.1, beta=0.1, delta=0.8, mu=-0.3, lambda=1)
   
## ghKurt -
   ghKurt(alpha=1.1, beta=0.1, delta=0.8, mu=-0.3, lambda=1)
   
## ghMoments -
   ghMoments(4, 
     alpha=1.1, beta=0.1, delta=0.8, mu=-0.3, lambda=1)
   ghMoments(4, "central",
     alpha=1.1, beta=0.1, delta=0.8, mu=-0.3, lambda=1)

Robust Moments for the GH

Description

Computes the first four robust moments for the generalized hyperbolic distribution.

Usage

ghMED(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
ghIQR(alpha= 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
ghSKEW(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)
ghKURT(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

Arguments

Details

The meanings of the parameters correspond to the first parameterization, see gh for further details.

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the gh prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz.

Examples

## ghMED -
   # Median:
   ghMED(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

## ghIQR -
   # Inter-quartile Range:
   ghIQR(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

## ghSKEW -
   # Robust Skewness:
   ghSKEW(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

## ghKURT -
   # Robust Kurtosis:
   ghKURT(alpha = 1, beta = 0, delta = 1, mu = 0, lambda = -1/2)

Generalized Hyperbolic Distribution Slider

Description

Displays interactively the dependence of the generalized hyperbolic distribution on its parameters.

Usage

ghSlider()

Value

a tcl/tk based graphical user interface.

This is a nice display for educational purposes to investigate the densities and probabilities of the generalized hyperbolic distribution.

Examples

   
## ghSlider -
   # ghSlider()

Generalized Hyperbolic Student-t distribution

Description

Density, distribution function, quantile function and random generation for the generalized hyperbolic Student-t distribution.

Usage

dght(x, beta = 0.1, delta = 1, mu = 0, nu = 10, log = FALSE)
pght(q, beta = 0.1, delta = 1, mu = 0, nu = 10)
qght(p, beta = 0.1, delta = 1, mu = 0, nu = 10)
rght(n, beta = 0.1, delta = 1, mu = 0, nu = 10)

Arguments

Details

dght gives the density, pght gives the distribution function, qght gives the quantile function, and rght generates random deviates.

The parameters are as in the first parameterization.

Value

numeric vector

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## ght -
   #

GHT distribution fit

Description

Estimates the distributional parameters for a generalized hyperbolic Student-t distribution.

Usage

 
ghtFit(x, beta = 0.1, delta = 1, mu = 0, nu = 10, 
    scale = TRUE, doplot = TRUE, span = "auto", trace = TRUE, 
    title = NULL, description = NULL, ...) 

Arguments

Details

The function nlm is used to minimize the "negative" log-likelihood function. nlm carries out a minimization using a Newton-type algorithm.

Value

an object from class "fDISTFIT". Slot fit is a list, currently with components estimate, minimum and code.

Examples

## ghtFit -
   # Simulate Random Variates:
   set.seed(1953)
   
## ghtFit -  
   # Fit Parameters:

Generalized Hyperbolic Student-t Mode

Description

Computes the mode of the generalized hyperbolic Student-t distribution.

Usage

ghtMode(beta = 0.1, delta = 1, mu = 0, nu = 10)

Arguments

Details

These are the parameters in the first parameterization.

Value

a numeric value, the mode for the generalized hyperbolic Student-t distribution.

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## ghtMode -
   ghtMode()

Generalized Hyperbolic Student-t Moments

Description

Calculates moments of the generalized hyperbolic Student-t distribution.

Usage

ghtMean(beta=0.1, delta=1, mu=0, nu=10)
ghtVar(beta=0.1, delta=1, mu=0, nu=10)
ghtSkew(beta=0.1, delta=1, mu=0, nu=10)
ghtKurt(beta=0.1, delta=1, mu=0, nu=10)

ghtMoments(order, type = c("raw", "central", "mu"),
    beta=0.1, delta=1, mu=0, nu=10)

Arguments

Value

a named numerical value. The name is one of mean, var, skew, or kurt, obtained by dropping the nig prefix from the name of the corresponding function and lowercasing it.

for ghtMoments, the name is obtained by paste0("m", order, type).

Author(s)

Diethelm Wuertz

References

Scott, D.J., Wuertz, D. and Tran, T.T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

Examples

   
## ghtMean -
   ghtMean(beta=0.2, delta=1.2, mu=-0.5, nu=4)
   
## ghtKurt -
   ghtKurt(beta=0.2, delta=1.2, mu=-0.5, nu=4)
   
## ghtMoments -
   ghtMoments(4, 
     beta=0.2, delta=1.2, mu=-0.5, nu=4)
   ghtMoments(4, "central",
     beta=0.2, delta=1.2, mu=-0.5, nu=4)

Robust Moments for the GHT

Description

Computes the first four robust moments for the generalized hyperbolic Student-t.

Usage

ghtMED(beta = 0.1, delta = 1, mu = 0, nu = 10)
ghtIQR(beta = 0.1, delta = 1, mu = 0, nu = 10)
ghtSKEW(beta = 0.1, delta = 1, mu = 0, nu = 10)
ghtKURT(beta = 0.1, delta = 1, mu = 0, nu = 10)

Arguments

Details

The parameters are those of the first parameterization.

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the ght prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz.

Examples

## ghtMED -
   # Median:
   ghtMED(beta = 0.1, delta = 1, mu = 0, nu = 10)

## ghtIQR -
   # Inter-quartile Range:
   ghtIQR(beta = 0.1, delta = 1, mu = 0, nu = 10)

## ghtSKEW -
   # Robust Skewness:
   ghtSKEW(beta = 0.1, delta = 1, mu = 0, nu = 10)

## ghtKURT -
   # Robust Kurtosis:
   ghtKURT(beta = 0.1, delta = 1, mu = 0, nu = 10)

Generalized Lambda Distribution

Description

Density, distribution function, quantile function and random generation for the generalized lambda distribution.

Usage

dgld(x, lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8, log = FALSE)
pgld(q, lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
qgld(p, lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
rgld(n, lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)

Arguments

Details

dgld gives the density, pgld gives the distribution function, qgld gives the quantile function, and rgld generates random deviates.

Value

numeric vector

Author(s)

Chong Gu for code implemented from R's contributed package gld.

Examples

   
## rgld -
   set.seed(1953)
   r = rgld(500, 
     lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8)
   plot(r, type = "l", col = "steelblue",
     main = "gld: lambda1=0 lambda2=-1 lambda3/4=-1/8")
 
## dgld - 
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", 
     col = "steelblue")
   x = seq(-5, 5, 0.25)
   lines(x, dgld(x, 
     lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8))
 
## pgld -  
   # Plot df and compare with true df:
   plot(sort(r), ((1:500)-0.5)/500, main = "Probability", 
     col = "steelblue")
   lines(x, pgld(x, 
     lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8))
   
## qgld -
   # Compute Quantiles:
   qgld(pgld(seq(-5, 5, 1), 
     lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8), 
     lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8) 

GH Distribution Fit

Description

Estimates the distrinbutional parameters for a generalized lambda distribution.

Usage

gldFit(x, lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8, 
    method = c("mle", "mps", "gof", "hist", "rob"),
    scale = NA, doplot = TRUE, add = FALSE, span = "auto", trace = TRUE, 
    title = NULL, description = NULL, ...) 

Arguments

Details

The function nlminb is used to minimize the objective function. The following approaches have been implemented:

"mle", maximimum log likelihoo estimation.

"mps", maximum product spacing estimation.

"gof", goodness of fit approaches, type="ad" Anderson-Darling, type="cvm" Cramer-vonMise, type="ks" Kolmogorov-Smirnov.

"hist", histogram binning approaches, "fd" Freedman-Diaconis binning, "scott", Scott histogram binning, "sturges", Sturges histogram binning.

"rob", robust moment matching.

Value

an object from class "fDISTFIT". Slot fit is a list, currently with components estimate, minimum and code.

Examples

set.seed(1953)
s <- rgld(n = 1000, lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8) 

gldFit(s, lambda1=0, lambda2=-1, lambda3=-1/8, lambda4=-1/8, 
       doplot = TRUE, trace = FALSE) 

Generalized Lambda Distribution Mode

Description

Computes the mode of the generalized lambda distribution.

Usage

gldMode(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)

Arguments

Author(s)

Implemented by Diethelm Wuertz

Robust Moments for the GLD

Description

Computes the first four robust moments for the Generalized Lambda Distribution.

Usage

gldMED(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
gldIQR(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
gldSKEW(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
gldKURT(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)

Arguments

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the gld prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz.

Examples

   
## gldMED -
   # Median:
   gldMED(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
 
## gldIQR - 
   # Inter-quartile Range:
   gldIQR(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
 
## gldSKEW -  
   # Robust Skewness:
   gldSKEW(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)
   
## gldKURT -
   # Robust Kurtosis:
   gldKURT(lambda1 = 0, lambda2 = -1, lambda3 = -1/8, lambda4 = -1/8)

Grid vector coordinates

Description

Creates rectangular grid coordinates from two vectors.

Usage

gridVector(x, y = NULL)

Arguments

Details

The grid is obtained by pairing each element of y with all elements of x. The X and Y coordinates of the points are stored in separate vectors. This is convenient, for example, for plotting. It can be useful also for brute force optimisation or simulation.

If y is NULL, the default, then y = x.

Value

a list with two components, X and Y, giving the coordinates which span the bivariate grid.

See Also

expand.grid

Examples

## a small grid vector with row and col transformations
gridVector(0:2)
data.frame(gridVector(0:2))
do.call("rbind", gridVector(0:2))

gridVector(0:2, 0:3)

## grid over a unit square
gridVector((0:10)/10) # equivalently: gridVector((0:10)/10, (0:10)/10)

Hilbert matrix

Description

Creates a Hilbert matrix.

Usage

hilbert(n)

Arguments

Details

An n,n matrix with (i,j)th element equal to 1/(i+j-1) is said to be a Hilbert matrix of order n. Hilbert matrices are symmetric and positive definite.

They are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm condition number of a 5x5 Hilbert matrix above is about 4.8e5.

Value

a matrix

References

Hilbert D., Collected papers, vol. II, article 21.

Beckermann B, (2000); The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik 85, 553–577, 2000.

Choi, M.D., (1983); Tricks or Treats with the Hilbert Matrix, American Mathematical Monthly 90, 301–312, 1983.

Todd, J., (1954); The Condition Number of the Finite Segment of the Hilbert Matrix, National Bureau of Standards, Applied Mathematics Series 39, 109–116.

Wilf, H.S., (1970); Finite Sections of Some Classical Inequalities, Heidelberg, Springer.

Examples

## Create a Hilbert Matrix:
H = hilbert(5)
H                              

Hyperbolic distribution

Description

Density, distribution function, quantile function and random generation for the hyperbolic distribution.

Usage

dhyp(x, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1, log = FALSE)
phyp(q, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1, ...)
qhyp(p, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1, ...)
rhyp(n, alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1)

Arguments

Details

dhyp gives the density, phyp gives the distribution function, qhyp gives the quantile function, and rhyp generates random deviates.

The meaning of the parameters given above corresponds to the first parameterization, pm = 1, which is the default.

In the second parameterization, pm=2, alpha and beta take the meaning of the shape parameters (usually named) zeta and rho.

In the third parameterization, pm=3, alpha and beta take the meaning of the shape parameters (usually named) xi and chi.

In the fourth parameterization, pm=4, alpha and beta take the meaning of the shape parameters (usually named) a.bar and b.bar.

The generator rhyp is based on the HYP algorithm given by Atkinson (1982).

Value

numeric vector

Author(s)

David Scott for code implemented from R's contributed package HyperbolicDist.

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## hyp -
   set.seed(1953)
   r = rhyp(5000, alpha = 1, beta = 0.3, delta = 1)
   plot(r, type = "l", col = "steelblue",
     main = "hyp: alpha=1 beta=0.3 delta=1")
 
## hyp - 
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   x = seq(-5, 5, 0.25)
   lines(x, dhyp(x, alpha = 1, beta = 0.3, delta = 1))
 
## hyp -  
   # Plot df and compare with true df:
   plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
   lines(x, phyp(x, alpha = 1, beta = 0.3, delta = 1))
   
## hyp -
   # Compute Quantiles:
   qhyp(phyp(seq(-5, 5, 1), alpha = 1, beta = 0.3, delta = 1), 
     alpha = 1, beta = 0.3, delta = 1) 

Fit a hyperbolic distribution

Description

Estimates the parameters of a hyperbolic distribution.

Usage

    
hypFit(x, alpha = 1, beta = 0, delta = 1, mu = 0, 
    scale = TRUE, doplot = TRUE, span = "auto", trace = TRUE, 
    title = NULL, description = NULL, ...) 

Arguments

Details

The meaning of the parameters given above corresponds to the first parameterization, see dhyp for details.

The function nlm is used to minimize the "negative" maximum log-likelihood function. nlm carries out a minimization using a Newton-type algorithm.

Value

an object from class "fDISTFIT". Slot fit is a list, currently with components estimate, minimum and code.

Examples

set.seed(1953)
s <- rhyp(n = 1000, alpha = 1.5, beta = 0.3, delta = 0.5, mu = -1.0) 

hypFit(s, alpha = 1, beta = 0, delta = 1, mu = mean(s), doplot = TRUE,
       trace = FALSE) 

Hyperbolic mode

Description

Computes the mode of the hyperbolic distribution.

Usage

hypMode(alpha = 1, beta = 0, delta = 1, mu = 0, pm = 1)

Arguments

Value

a numeric value, the mode in the appropriate parameterization for the hyperbolic distribution.

Author(s)

David Scott for code implemented from R's contributed package HyperbolicDist.

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## hypMode -
   hypMode()

Hyperbolic distribution moments

Description

Calculates moments of the hyperbolic distribution function.

Usage

hypMean(alpha=1, beta=0, delta=1, mu=0)
hypVar(alpha=1, beta=0, delta=1, mu=0)
hypSkew(alpha=1, beta=0, delta=1, mu=0)
hypKurt(alpha=1, beta=0, delta=1, mu=0)

hypMoments(order, type = c("raw", "central", "mu"),
    alpha=1, beta=0, delta=1, mu=0)

Arguments

Value

a named numerical value. The name is one of mean, var, skew, or kurt, obtained by dropping the hyp prefix from the name of the corresponding function and lowercasing it.

for hypMoments, the name is obtained by paste0("m", order, type).

Author(s)

Diethelm Wuertz

References

Scott, D. J., Wuertz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

Examples

   
## hypMean -
   hypMean(alpha=1.1, beta=0.1, delta=0.8, mu=-0.3)
   
## ghKurt -
   hypKurt(alpha=1.1, beta=0.1, delta=0.8, mu=-0.3)
   
## hypMoments -
   hypMoments(4, alpha=1.1, beta=0.1, delta=0.8, mu=-0.3)
   hypMoments(4, "central", alpha=1.1, beta=0.1, delta=0.8, mu=-0.3)

Robust moments for the HYP

Description

Computes the first four robust moments for the hyperbolic distribution.

Usage

hypMED(alpha = 1, beta = 0, delta = 1, mu = 0)
hypIQR(alpha = 1, beta = 0, delta = 1, mu = 0)
hypSKEW(alpha = 1, beta = 0, delta = 1, mu = 0)
hypKURT(alpha = 1, beta = 0, delta = 1, mu = 0)

Arguments

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the hyp prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz

Examples

## hypMED -
   # Median:
   hypMED(alpha = 1, beta = 0, delta = 1, mu = 0)

## hypIQR -
   # Inter-quartile Range:
   hypIQR(alpha = 1, beta = 0, delta = 1, mu = 0)

## hypSKEW -
   # Robust Skewness:
   hypSKEW(alpha = 1, beta = 0, delta = 1, mu = 0)

## hypKURT -
   # Robust Kurtosis:
   hypKURT(alpha = 1, beta = 0, delta = 1, mu = 0)

Hyperbolic distribution slider

Description

Displays interactively the dependence of the hyperbolic distribution on its parameters.

Usage

hypSlider()

Value

a tcl/tk based graphical user interface.

This is a nice display for educational purposes to investigate the densities and probabilities of the hyperbolic distribution.

Examples

   
## hypSlider -
   #

Interactive Plot Utility

Description

Plots with emphasis on interactive plots.

Usage

interactivePlot(x, choices = paste("Plot", 1:9), 
    plotFUN = paste("plot.", 1:9, sep = ""), which = "all", ...)

Arguments

Details

If "which" is the character string "ask", then the user is presented with a menu to interactively choose which plot(s) to show. Argument choices is used for the choices in the menu, so they should be informative.

If "which" is equal to "all" all plots are drawn. If "which" is a logical vector, the indicate plots are displayed.

Note that if more plots are to be shown in one window, the arrangement should be made in advance (and cleaned up afterwards), see the examples.

Examples

## Test Plot Function
testPlot <- function(x, which = "all", ...) {   
    ## Plot Function and Addons
    plot.1 <<- function(x, ...) plot(x, ...)      
    plot.2 <<- function(x, ...) acf(x, ...)
    plot.3 <<- function(x, ...) hist(x, ...)      
    plot.4 <<- function(x, ...) qqnorm(x, ...)
    ## Plot
    interactivePlot(
        x,
        choices = c("Series Plot", "ACF", "Histogram", "QQ Plot"),
        plotFUN = c("plot.1", "plot.2", "plot.3", "plot.4"),
        which = which, ...)       
    ## Return Value
    invisible()
}

## Plot
## prepare the window and store its previous state
op <- par(mfrow = c(2, 2), cex = 0.7)
## produce the plot
testPlot(rnorm(500))            
## restore the previous state
par(op)

## Try:
## par(mfrow = c(1,1))
## testPlot(rnorm(500), which = "ask")

## similar to above but using functions for plotFUN
testPlot_2 <- function(x, which = "all", ...) {   
    interactivePlot(
        x,
        choices = c("Series Plot", "ACF", "Histogram", "QQ Plot"),
        plotFUN = c(plot.1 = function(x, ...) plot(x, ...),
                    plot.2 = function(x, ...) acf(x, ...),
                    plot.3 = function(x, ...) hist(x, ...),
                    plot.4 = function(x, ...) qqnorm(x, ...) ),
        which = which, ...)       
   
    ## Return Value:
    invisible()
}
## produce the plot
op <- par(mfrow = c(2, 2), cex = 0.7)
testPlot_2(rnorm(500))            
par(op)

The inverse of a matrix

Description

Computes the inverse of a matrix.

Usage

inv(x)

Arguments

Value

a matrix

References

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P = pascal(5)
P
         
## Compute the Inverse Matrix:
inv(P)
   
## Check:
inv(P) %*% P    
   
## Alternatives:
chol2inv(chol(P))
solve(P)                     

Bivariate Krige Interpolation

Description

Bivariate Krige Interpolation.

Usage

krigeInterp(x, y = NULL, z = NULL, gridPoints = 21,
    xo = seq(min(x), max(x), length = gridPoints),
    yo = seq(min(y), max(y), length = gridPoints),
    extrap = FALSE, polDegree = 6)

Arguments

Value

a list with at least three entries, x, y and z. The returned values can be used directly in persp and contour 3D plotting methods.

Note

krigeInterp() requires package spatial.

See Also

akimaInterp, linearInterp.

Examples

## The akima library is not auto-installed because of a different licence.
## krigeInterp -  Kriging:
set.seed(1953)
x = runif(999) - 0.5
y = runif(999) - 0.5
z = cos(2*pi*(x^2+y^2))
ans = krigeInterp(x, y, z, extrap = FALSE)
persp(ans, theta = -40, phi = 30, col = "steelblue",
    xlab = "x", ylab = "y", zlab = "z")
contour(ans)

Kronecker product

Description

Computes the Kronecker product of two matrices.

Usage

kron(x, y)

Arguments

Details

The Kronecker product can be computed using the operator %x% or alternatively using the function kron for SPlus compatibility.

Note

kron is a synonym to %x%.

References

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P = pascal(3)
P
   
## Return the Kronecker Product                     
kron(P, diag(3))
P %x% diag(3)                        

Two sample Kolmogorov-Smirnov test

Description

Tests if two series are distributionally equivalent using two sample Kolmogorov-Smirnov test.

Usage

ks2Test(x, y, title = NULL, description = NULL)

Arguments

Details

The test ks2Test performs a Kolmogorov-Smirnov two sample test that the two data samples, x and y, come from the same distribution, not necessarily a normal distribution. That means that it is not specified what that common distribution is.

ks2Test calls several times base R's ks.test p-values for all three alternatives (two-sided, less, and greater), as well as the exact p-value for the two-sided case.

Note that the p-values are computed under a hypothesis of i.i.d., which is rarely the case for time series. So, the results should be interpreted cautiosly if that is the case. The same applies when the data are residuals from fitted models.

Value

an object from class fHTEST

References

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

Examples

x <- rnorm(50)
y <- rnorm(50)
  
ks2Test(x, y)

Generator for Portable random innovations

Description

Functions to generate portable random innovations. The functions run under R and S-Plus and generate the same sequence of random numbers. Supported are uniform, normal and Student-t distributed random numbers.

The functions are:

Usage

set.lcgseed(seed = 4711)
get.lcgseed()

runif.lcg(n, min = 0, max = 1)
rnorm.lcg(n, mean = 0, sd = 1)
rt.lcg(n, df)   

Arguments

Details

A simple portable random number generator for use in R and SPlus. We recommend to use this generator only for comparisons of calculations in R and Splus.

The generator is a linear congruential generator with parameters LCG(a=13445, c=0, m=2^31-1, X=0). It is a simple random number generator which passes the bitwise randomness test.

Value

A vector of generated random innovations. The value of the current seed is stored in the variable lcg.seed.

References

Altman, N.S. (1988); Bitwise Behavior of Random Number Generators, SIAM J. Sci. Stat. Comput., 9(5), September, 941–949.

Examples

set.lcgseed(seed = 65890)
 
## runif.lcg, rnorm.lcg, rt.lcg
cbind(runif.lcg(10), rnorm.lcg(10), rt.lcg(10, df = 4))

get.lcgseed()  
   
## Note, to overwrite rnorm, use
   # rnorm = rnorm.lcg
   # Going back to rnorm
   # rm(rnorm)

Bivariate Linear Interpolation

Description

Bivariate Linear Interpolation. Options are available for gridded and pointwise interpolation.

Usage

 
linearInterp(x, y = NULL, z = NULL, gridPoints = 21,
    xo = seq(min(x), max(x), length = gridPoints),
    yo = seq(min(y), max(y), length = gridPoints))
    
linearInterpp(x, y = NULL, z = NULL, xo, yo)

Arguments

Value

for linearInterp, a list with at least three entries, x, y and z. The returned values, can be used directly in persp and contour 3D plotting methods.

for linearInterpp, a data.frame with columns "x", "y", and "z".

See Also

akimaInterp and krigeInterp

Examples

## Linear Interpolation:    
if (requireNamespace("interp")) {
    set.seed(1953)
    x <- runif(999) - 0.5
    y <- runif(999) - 0.5
    z <- cos(2 * pi * (x^2 + y^2))
    ans = linearInterp(x, y, z, gridPoints = 41)
    persp(ans, theta = -40, phi = 30, col = "steelblue",
          xlab = "x", ylab = "y", zlab = "z")
    contour(ans)
}

List exported functions in a package

Description

Utilities to list and count exported functions in a package, list the contents of the description file of a package, and

Prints the content of an index file for a package (a list of the objects exported by a package).

Usage

listFunctions(package, character.only = FALSE)
countFunctions(package, character.only = FALSE)

listIndex(package, character.only = FALSE)

Arguments

Value

for listFunctions, a character vector containing the names of the exported functions in a package,

for countFunctions, a named numeric value giving the number of the exported functions in a package.

listIndex doesn't return a useful value. It is used for the side effect of printing the description or index.

Note

Be aware that listFunctions and countFunctions attach the package to the search path.

See Also

packageDescription

Examples

listFunctions("fBasics")
   
countFunctions("fBasics")

Two sample location tests

Description

Tests if two series differ in their distributional location parameter.

Usage

locationTest(x, y, method = c("t", "kw2"), title = NULL,
    description = NULL)

Arguments

Details

The method = "t" can be used to determine if the two sample means are equal for unpaired data sets. Two variants are used, assuming equal or unequal variances.

The method = "kw2" performs a Kruskal-Wallis rank sum test of the null hypothesis that the central tendencies or medians of two samples are the same. The alternative is that they differ. Note, that it is not assumed that the two samples are drawn from the same distribution. It is also worth to know that the test assumes that the variables under consideration have underlying continuous distributions.

Value

an object from class fHTEST

Note

Some of the test implementations are selected from R's ctest package.

Author(s)

R-core team for hypothesis tests implemented from R's package ctest.

References

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

Examples

x <- rnorm(50)
y <- rnorm(50)
  
locationTest(x, y, "t")
locationTest(x, y, "kw2")

Drawdown statistics

Description

A collection of functions which compute drawdown statistics. Included are density, distribution function, and random generation for the maximum drawdown distribution. In addition the expectation of drawdowns for Brownian motion can be computed.

Usage

dmaxdd(x, sd = 1, horizon = 100, N = 1000) 
pmaxdd(q, sd = 1, horizon = 100, N = 1000) 
rmaxdd(n, mean = 0, sd = 1, horizon = 100)

maxddStats(mean = 0, sd = 1, horizon = 1000)

Arguments

Details

dmaxdd computes the density function of the maximum drawdown distribution. pmaxdd computes the distribution function. rmaxdd generates random numbers from that distribution. maxddStats computes the expectation of drawdowns.

dmaxdd returns for a trendless Brownian process mean=0 and standard deviation "sd" the density from the probability that the maximum drawdown "D" is larger or equal to "h" in the interval [0,T], where "T" denotes the time horizon of the investor.

pmaxdd returns for a trendless Brownian process mean=0 and standard deviation "sd" the probability that the maximum drawdown "D" is larger or equal to "h" in the interval [0,T], where "T" denotes the time horizon of the investor.

rmaxdd returns for a Brownian Motion process with mean mean and standard deviation sd random variates of maximum drawdowns.

maxddStats returns the expected value, E[D], of maximum drawdowns of Brownian Motion for a given drift mean, variance sd, and runtime horizon of the Brownian Motion process.

Note

Currrently, only the driftless case is implemented.

References

Magdon-Ismail M., Atiya A.F., Pratap A., Abu-Mostafa Y.S. (2003); On the Maximum Drawdown of a Brownian Motion, Preprint, CalTech, Pasadena USA, p. 24.

Examples

## rmaxdd
## Set a random seed
set.seed(1953)               
## horizon of the investor, time T
horizon <- 1000               
## number of MC samples, N -> infinity
samples <- 1000               
## Range of expected Drawdons
xlim <- c(0, 5) * sqrt(horizon) 

## Plot Histogram of Simulated Max Drawdowns:
r <- rmaxdd(n = samples, mean = 0, sd = 1, horizon = horizon)
hist(x = r, n = 40, probability = TRUE, xlim = xlim, 
     col = "steelblue4", border = "white", main = "Max. Drawdown Density") 
points(r, rep(0, samples), pch = 20, col = "orange", cex = 0.7)
   
## dmaxdd
x <- seq(0, xlim[2], length = 200)
d <- dmaxdd(x = x, sd = 1, horizon = horizon, N = 1000)
lines(x, d, lwd = 2)
    
## pmaxdd
## Count Frequencies of Drawdowns Greater or Equal to "h":
n <- 50
x <- seq(0, xlim[2], length = n)
g <- rep(0, times = n)
for (i in 1:n)
    g[i] <- length (r[r > x[i]]) / samples

plot(x, g, type ="h", lwd = 3,
     xlab = "q", main = "Max. Drawdown Probability")
## Compare with True Probability "G_D(h)":
x <- seq(0, xlim[2], length = 5*n)
p <- pmaxdd(q = x, sd = 1, horizon = horizon, N = 5000)
lines(x, p, lwd = 2, col="steelblue4") 
   
## maxddStats
## Compute expectation Value E[D]:
maxddStats(mean = -0.5, sd = 1, horizon = 10^(1:4))
maxddStats(mean =  0.0, sd = 1, horizon = 10^(1:4))
maxddStats(mean =  0.5, sd = 1, horizon = 10^(1:4))

Normal Inverse Gaussian Distribution

Description

Density, distribution function, quantile function and random generation for the normal inverse Gaussian distribution.

Usage

dnig(x, alpha = 1, beta = 0, delta = 1, mu = 0, log = FALSE)
pnig(q, alpha = 1, beta = 0, delta = 1, mu = 0)
qnig(p, alpha = 1, beta = 0, delta = 1, mu = 0)
rnig(n, alpha = 1, beta = 0, delta = 1, mu = 0)

Arguments

Details

dnig gives the density. pnig gives the distribution function. qnig gives the quantile function, and rnig generates random deviates.

The parameters alpha, beta, delta, mu are in the first parameterization of the distribution.

The random deviates are calculated with the method described by Raible (2000).

Value

numeric vector

Author(s)

David Scott for code implemented from R's contributed package HyperbolicDist.

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## nig -
   set.seed(1953)
   r = rnig(5000, alpha = 1, beta = 0.3, delta = 1)
   plot(r, type = "l", col = "steelblue",
     main = "nig: alpha=1 beta=0.3 delta=1")
 
## nig - 
   # Plot empirical density and compare with true density:
   hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
   x = seq(-5, 5, 0.25)
   lines(x, dnig(x, alpha = 1, beta = 0.3, delta = 1))
 
## nig -  
   # Plot df and compare with true df:
   plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
   lines(x, pnig(x, alpha = 1, beta = 0.3, delta = 1))
   
## nig -
   # Compute Quantiles:
   qnig(pnig(seq(-5, 5, 1), alpha = 1, beta = 0.3, delta = 1), 
     alpha = 1, beta = 0.3, delta = 1) 

Fit of a Normal Inverse Gaussian Distribution

Description

Estimates the parameters of a normal inverse Gaussian distribution.

Usage

   
nigFit(x, alpha = 1, beta = 0, delta = 1, mu = 0, 
    method = c("mle", "gmm", "mps", "vmps"), scale = TRUE, doplot = TRUE, 
    span = "auto", trace = TRUE, title = NULL, description = NULL, ...) 

Arguments

Value

an object from class "fDISTFIT".

Slot fit is a list, whose components depend on the method. See "fDISTFIT" for the meaning of the most common ones.

Here is an informal list of components for the various methods:

for mle: par, scale, estimate, minimum, code plus components from nlminb() plus additions from .distStandardErrors();

for gmm: only estimate;

for mps and vmps: estimate, minimum, error (s.e.'s), code.

Examples

## Simulate Random Variates
set.seed(1953)
s <- rnig(n = 1000, alpha = 1.5, beta = 0.3, delta = 0.5, mu = -1.0) 

nigFit(s, alpha = 1, beta = 0, delta = 1, mu = mean(s), doplot = TRUE,
       trace = FALSE) 

Normal Inverse Gaussian Mode

Description

Computes the mode of the norm inverse Gaussian distribution.

Usage

nigMode(alpha = 1, beta = 0, delta = 1, mu = 0)

Arguments

Value

a numeric value, the mode of the normal inverse Gaussian distribution

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## nigMode -
   nigMode()

Moments for the Normal Inverse Gaussian

Description

Computes the first four moments for the normal inverse Gaussian distribution.

Usage

nigMean(alpha = 1, beta = 0, delta = 1, mu = 0)
nigVar(alpha = 1, beta = 0, delta = 1, mu = 0)
nigSkew(alpha = 1, beta = 0, delta = 1, mu = 0)
nigKurt(alpha = 1, beta = 0, delta = 1, mu = 0)

Arguments

Value

a named numerical value. The name is one of mean, var, skew, or kurt, obtained by dropping the nig prefix from the name of the corresponding function and lowercasing it.

Author(s)

Diethelm Wuertz.

References

Scott, D. J., Wuertz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

Examples

   
## nigMean -
   # Median:
   nigMean(alpha = 1, beta = 0, delta = 1, mu = 0)
 
## nigVar - 
   # Inter-quartile Range:
   nigVar(alpha = 1, beta = 0, delta = 1, mu = 0)
 
## nigSKEW -  
   # Robust Skewness:
   nigSkew(alpha = 1, beta = 0, delta = 1, mu = 0)
   
## nigKurt -
   # Robust Kurtosis:
   nigKurt(alpha = 1, beta = 0, delta = 1, mu = 0)

Robust Moments for the NIG

Description

Computes the first four robust moments for the Normal Inverse Gaussian Distribution.

Usage

nigMED(alpha = 1, beta = 0, delta = 1, mu = 0)
nigIQR(alpha = 1, beta = 0, delta = 1, mu = 0)
nigSKEW(alpha = 1, beta = 0, delta = 1, mu = 0)
nigKURT(alpha = 1, beta = 0, delta = 1, mu = 0)

Arguments

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the nig prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz.

Examples

## nigMED -
   # Median:
   nigMED(alpha = 1, beta = 0, delta = 1, mu = 0)

## nigIQR -
   # Inter-quartile Range:
   nigIQR(alpha = 1, beta = 0, delta = 1, mu = 0)

## nigSKEW -
   # Robust Skewness:
   nigSKEW(alpha = 1, beta = 0, delta = 1, mu = 0)

## nigKURT -
   # Robust Kurtosis:
   nigKURT(alpha = 1, beta = 0, delta = 1, mu = 0)

NIG Shape Triangle

Description

Plots the normal inverse Gaussian Shape Triangle.

Usage

nigShapeTriangle(object, add = FALSE, labels = TRUE, ...)

Arguments

Value

displays the parameters of fitted distributions in the NIG shape triangle.

Author(s)

David Scott for code implemented from R's contributed package HyperbolicDist.

References

Atkinson, A.C. (1982); The simulation of generalized inverse Gaussian and hyperbolic random variables, SIAM J. Sci. Stat. Comput. 3, 502–515.

Barndorff-Nielsen O. (1977); Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen O., Blaesild, P. (1983); Hyperbolic distributions. In Encyclopedia of Statistical Sciences, Eds., Johnson N.L., Kotz S. and Read C.B., Vol. 3, pp. 700–707. New York: Wiley.

Raible S. (2000); Levy Processes in Finance: Theory, Numerics and Empirical Facts, PhD Thesis, University of Freiburg, Germany, 161 pages.

Examples

   
## nigShapeTriangle -
   #

nigerbolic Distribution Slider

Description

Displays interactively the dependence of the nigerbolic distribution on its parameters.

Usage

nigSlider()

Value

a tcl/tk based graphical user interface.

This is a nice display for educational purposes to investigate the densities and probabilities of the invetrse Gaussian distribution.

Examples

   
## nigSlider -
   # nigSlider()

Matrix norm

Description

Computes the norm of a matrix.

Usage

norm2(x, p = 2)

Arguments

Details

The function norm2 computes the norm of a matrix. Three choices are possible:

p=1

The maximum absolute column sum norm which is defined as the maximum of the sum of the absolute valued elements of columns of the matrix.

p=2

The spectral norm is "the norm" of a matrix X. This value is computed as the square root of the maximum eigenvalue of CX, where C is the conjugate transpose.

p=Inf

The maximum absolute row sum norm is defined as the maximum of the sum of the absolute valued elements of rows of the matrix.

Value

the requested norm of the matrix, a non-negative number

Note

Since base::norm() has become available in the R base environment, the function fBasics::norm() has become obsolete. To avoid conflicts with norm() we have renamed the fBasics' one to norm2.

References

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P <- pascal(5)
P                  
     
## Return the Norm of the Matrix:                      
norm2(P)                                                       

Robust moments for the Normal distribution

Description

Computes the first four robust moments for the Normal distribution.

Usage

normMED(mean = 0, sd = 1)
normIQR(mean = 0, sd = 1)
normSKEW(mean = 0, sd = 1)
normKURT(mean = 0, sd = 1)

Arguments

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the gh prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz

Examples

## normMED -
   # Median:
   normMED(mean = 0, sd = 1)

## normIQR -
   # Inter-quartile Range:
   normIQR(mean = 0, sd = 1)

## normSKEW -
   # Robust Skewness:
   normSKEW(mean = 0, sd = 1)

## normKURT -
   # Robust Kurtosis:
   normKURT(mean = 0, sd = 1)

Pascal matrix

Description

Creates a Pascal matrix.

Usage

pascal(n)

Arguments

Details

The function pascal generates a Pascal matrix of order n which is a symmetric positive definite matrix with integer entries made up from Pascal's triangle. The determinant of a Pascal matrix is 1. The inverse of a Pascal matrix has integer entries. If lambda is an eigenvalue of a Pascal matrix, then 1/lambda is also an eigenvalue of the matrix. Pascal matrices are ill-conditioned.

References

Call G.S., Velleman D.J., (1993); Pascal's matrices, American Mathematical Monthly 100, 372–376.

Edelman A., Strang G., (2004); Pascal Matrices, American Mathematical Monthly 111, 361–385.

Examples

## Create Pascal Matrix:
P = pascal(5)
P 
   
## Determinant
det(pascal(5)) 
det(pascal(10))   
det(pascal(15))   
det(pascal(20))                                       

Polynomial distributed lags

Description

Creates a regressor matrix for polynomial distributed lags.

Usage

pdl(x, d = 2, q = 3, trim = FALSE)

Arguments

See Also

tslag

Examples

## pdl -
   #                                      

Positive definite matrices

Description

Checks if a matrix is positive definite and/or forces a matrix to be positive definite.

Usage

isPositiveDefinite(x)
makePositiveDefinite(x)

Arguments

Details

The function isPositiveDefinite checks if a square matrix is positive definite.

The function makePositiveDefinite forces a matrix to be positive definite.

Author(s)

Korbinian Strimmer.

Examples

# the 3x3 Pascal Matrix is positive define 
isPositiveDefinite(pascal(3))                

Print control

Description

Unlists and prints a control object.

Usage

## S3 method for class 'control'
print(x, ...)

Arguments

Value

prints control

Examples

control <- list(n = 211, seed = 54, name = "generator")
print(control) 
class(control) <- "control"
print(control)  

The rank of a matrix

Description

Computes the rank of a matrix.

Usage

rk(x, method = c("qr", "chol"))

Arguments

Details

The function rk computes the rank of a matrix which is the dimension of the range of the matrix corresponding to the number of linearly independent rows or columns of the matrix, or to the number of nonzero singular values.

The rank of a matrix is also named linear map.

References

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P = pascal(5)
P
   
## Compute the Rank:
rk(P)
rk(P, "chol")                                

Row statistics

Description

Functions to compute row statistical properties of financial and economic time series data.

The functions are:

Usage

rowStats(x, FUN, ...) 


rowSds(x, ...)
rowVars(x, ...)
rowSkewness(x, ...)
rowKurtosis(x, ...)
rowMaxs(x, ...)
rowMins(x, ...)
rowProds(x, ...)
rowQuantiles(x, prob = 0.05, ...)

rowStdevs(x, ...)
rowAvgs(x, ...)

Arguments

Value

each function returns a numeric vector of the statistics

See Also

apply

Examples

## Simulated Return Data in Matrix Form:
x <- matrix(rnorm(10*10), nrow = 10)
     
rowStats(x, FUN = mean)
rowMaxs(x)  

Sample L-moments

Description

Computes L-moments from an empirical sample data set.

Usage

sampleLmoments(x, rmax = 4)

Arguments

Value

a named numeric vector of length rmax with names c("L1", "L2", ..., "L<rmax>")

Author(s)

Diethelm Wuertz

Examples

x <- rt(100, 4)
   
sampleLmoments(x)

Robust moments for the GLD

Description

Computes the first four robust moments for the Normal Inverse Gaussian Distribution.

Usage

sampleMED(x)
sampleIQR(x)
sampleSKEW(x)
sampleKURT(x)

Arguments

Value

a named numerical value. The name is one of MED, IQR, SKEW, or KURT, obtained by dropping the sample prefix from the name of the corresponding function.

Author(s)

Diethelm Wuertz

Examples

## Sample
x <- rt(100, 4)
   
## Median
sampleMED(x)
 
## Inter-quartile Range
sampleIQR(x)
 
## Robust Skewness
sampleSKEW(x)
   
## Robust Kurtosis
sampleKURT(x)

Two sample scale tests

Description

Tests if two series differ in their distributional scale parameter.

Usage

scaleTest(x, y, method = c("ansari", "mood"), 
    title = NULL, description = NULL)

Arguments

Details

The method="ansari" performs the Ansari-Bradley two-sample test for a difference in scale parameters. The test returns for any sizes of the series x and y the exact p value together with its asymptotic limit.

The method="mood", is another test which performs a two-sample test for a difference in scale parameters. The underlying model is that the two samples are drawn from f(x-l) and f((x-l)/s)/s, respectively, where l is a common location parameter and s is a scale parameter. The null hypothesis is s=1.

Value

an object from class fHTEST

Note

Some of the test implementations are selected from R's ctest package.

Author(s)

R-core team for hypothesis tests implemented from R's package ctest.

References

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

Moore, D.S. (1986); Tests of the chi-squared type, In: D'Agostino, R.B. and Stephens, M.A., eds., Goodness-of-Fit Techniques, Marcel Dekker, New York.

Examples

## Generate Series:
x = rnorm(50)
y = rnorm(50)
   
scaleTest(x, y, "ansari")
scaleTest(x, y, "mood")

Standardized Generalized Hyperbolic Distribution

Description

Density, distribution function, quantile function and random generation for the standardized generalized hyperbolic distribution.

Usage

dsgh(x, zeta = 1, rho = 0, lambda = 1, log = FALSE)
psgh(q, zeta = 1, rho = 0, lambda = 1)
qsgh(p, zeta = 1, rho = 0, lambda = 1)
rsgh(n, zeta = 1, rho = 0, lambda = 1)

Arguments

Details

dsgh gives the density, psgh gives the distribution function, qsgh gives the quantile function, and rsgh generates random deviates.

The generator rsgh is based on the GH algorithm given by Scott (2004).

Value

numeric vector

Author(s)

Diethelm Wuertz

Examples

   
## rsgh -
   set.seed(1953)
   r = rsgh(5000, zeta = 1, rho = 0.5, lambda = 1)
   plot(r, type = "l", col = "steelblue",
     main = "gh: zeta=1 rho=0.5 lambda=1")
 
## dsgh - 
   # Plot empirical density and compare with true density:
   hist(r, n = 50, probability = TRUE, border = "white", col = "steelblue",
     ylim = c(0, 0.6))
   x = seq(-5, 5, length = 501)
   lines(x, dsgh(x, zeta = 1, rho = 0.5, lambda = 1))
 
## psgh -  
   # Plot df and compare with true df:
   plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
   lines(x, psgh(x, zeta = 1, rho = 0.5, lambda = 1))
   
## qsgh -
   # Compute Quantiles:
   round(qsgh(psgh(seq(-5, 5, 1), zeta = 1, rho = 0.5), zeta = 1, rho = 0.5), 4)

Standardized GH distribution fit

Description

Estimates the distributional parameters for a standardized generalized hyperbolic distribution.

Usage

sghFit(x, zeta = 1, rho = 0, lambda = 1, include.lambda = TRUE,
    scale = TRUE, doplot = TRUE, span = "auto", trace = TRUE, 
    title = NULL, description = NULL, ...) 

Arguments

Value

an object from class "fDISTFIT". Slot fit is a list, currently with components estimate, minimum, code, param, mean (mean of the original data), var (variance of original data).

Examples

set.seed(1953)
s <- rsgh(n = 2000, zeta = 0.7, rho = 0.5, lambda = 0) 

sghFit(s, zeta = 1, rho = 0, lambda = 1, include.lambda = TRUE, 
       doplot = TRUE, trace = FALSE)

Standardized generalized hyperbolic Student-t Distribution

Description

Density, distribution function, quantile function and random generation for the standardized generalized hyperbolic Student-t distribution.

Usage

dsght(x, beta = 0.1, delta = 1, mu = 0, nu = 10, log = FALSE)
psght(q, beta = 0.1, delta = 1, mu = 0, nu = 10)
qsght(p, beta = 0.1, delta = 1, mu = 0, nu = 10)
rsght(n, beta = 0.1, delta = 1, mu = 0, nu = 10)

Arguments

Details

dsght gives the density, psght gives the distribution function, qsght gives the quantile function, and rsght generates random deviates.

These are the parameters in the first parameterization.

Value

numeric vector

Author(s)

Diethelm Wuertz

Examples

## rsght -
   set.seed(1953)
   r = rsght(5000, beta = 0.1, delta = 1, mu = 0, nu = 10)
   plot(r, type = "l", col = "steelblue",
     main = "gh: zeta=1 rho=0.5 lambda=1")

## dsght -
   # Plot empirical density and compare with true density:
   hist(r, n = 50, probability = TRUE, border = "white", col = "steelblue")
   x = seq(-5, 5, length = 501)
   lines(x, dsght(x, beta = 0.1, delta = 1, mu = 0, nu = 10))

## psght -
   # Plot df and compare with true df:
   plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
   lines(x, psght(x, beta = 0.1, delta = 1, mu = 0, nu = 10))

## qsght -
   # Compute Quantiles:
   round(qsght(psght(seq(-5, 5, 1), beta = 0.1, delta = 1, mu = 0, nu =10),
               beta = 0.1, delta = 1, mu = 0, nu = 10), 4)

Standardized Normal Inverse Gaussian Distribution

Description

Density, distribution function, quantile function and random generation for the standardized normal inverse Gaussian distribution.

Usage

dsnig(x, zeta = 1, rho = 0, log = FALSE)
psnig(q, zeta = 1, rho = 0)
qsnig(p, zeta = 1, rho = 0)
rsnig(n, zeta = 1, rho = 0)

Arguments

Details

dsnig gives the density, psnig gives the distribution function, qsnig gives the quantile function, and rsnig generates random deviates.

The random deviates are calculated with the method described by Raible (2000).

Value

numeric vector

Author(s)

Diethelm Wuertz

Examples

   
## snig -
   set.seed(1953)
   r = rsnig(5000, zeta = 1, rho = 0.5)
   plot(r, type = "l", col = "steelblue",
     main = "snig: zeta=1 rho=0.5")
 
## snig - 
   # Plot empirical density and compare with true density:
   hist(r, n = 50, probability = TRUE, border = "white", col = "steelblue")
   x = seq(-5, 5, length = 501)
   lines(x, dsnig(x, zeta = 1, rho = 0.5))
 
## snig -  
   # Plot df and compare with true df:
   plot(sort(r), (1:5000/5000), main = "Probability", col = "steelblue")
   lines(x, psnig(x, zeta = 1, rho = 0.5))
   
## snig -
   # Compute Quantiles:
   qsnig(psnig(seq(-5, 5, 1), zeta = 1, rho = 0.5), zeta = 1, rho = 0.5) 

Fit of a Standardized NIG Distribution

Description

Estimates the parameters of a standardized normal inverse Gaussian distribution.

Usage

   
snigFit(x, zeta = 1, rho = 0, scale = TRUE, doplot = TRUE, 
    span = "auto", trace = TRUE, title = NULL, description = NULL, ...) 

Arguments

Value

an object from class "fDISTFIT".

Slot fit is a list with the same components as the result from snigFit.

Examples

## Simulate Random Variates:
set.seed(1953)
s <- rsnig(n = 2000, zeta = 0.7, rho = 0.5) 

## snigFit -  
   # Fit Parameters:
   snigFit(s, zeta = 1, rho = 0, doplot = TRUE) 

Spline Smoothed Distribution

Description

Density, distribution function, quantile function and random generation from smoothing spline estimates.

Usage

dssd(x, param, log = FALSE)
pssd(q, param)
qssd(p, param)
rssd(n, param)

Arguments

Details

dssd gives the density, pssd gives the distribution function, qssd gives the quantile function, and rssd generates random deviates.

Value

numeric vector

Author(s)

Diethelm Wuertz, Chong Gu for the underlying gss package.

References

Gu, C. (2002), Smoothing Spline ANOVA Models, New York Springer–Verlag.

Gu, C. and Wang, J. (2003), Penalized likelihood density estimation: Direct cross-validation and scalable approximation, Statistica Sinica, 13, 811–826.

Examples

   
## ssdFit -
   set.seed(1953)
   r = rnorm(500)
   hist(r, breaks = "FD", probability = TRUE,
     col = "steelblue", border = "white")
 
## ssdFit - 
   param = ssdFit(r)
   
## dssd -  
   u = seq(min(r), max(r), len = 301)
   v = dssd(u, param)
   lines(u, v, col = "orange", lwd = 2)

Fit density using smoothing splines

Description

Estimates the parameters of a density function using smoothing splines.

Usage

   
ssdFit(x) 

Arguments

Value

for ssdFit, an object of class ssden. The returned object can be used to evaluate density, probabilities and quantiles.

Author(s)

Diethelm Wuertz, Chong Gu for the underlying gss package.

References

Gu, C. (2002), Smoothing Spline ANOVA Models, New York Springer–Verlag.

Gu, C. and Wang, J. (2003), Penalized likelihood density estimation: Direct cross-validation and scalable approximation, Statistica Sinica, 13, 811–826.

Examples

     
## ssdFit -
   set.seed(1953)
   r = rnorm(500)
   hist(r, breaks = "FD", probability = TRUE,
     col = "steelblue", border = "white")
 
## ssdFit - 
   param = ssdFit(r)
   
## dssd -  
   u = seq(min(r), max(r), len = 301)
   v = dssd(u, param)
   lines(u, v, col = "orange", lwd = 2)

Table of symbols

Description

Displays a table of plot characters and symbols.

Usage

 
symbolTable(font = par('font'), cex = 0.7)

Arguments

Value

displays a table with the plot characters and symbols numbered from 0 to 255 and returns invisibly the name of the font.

Note

Symbols with codes on the range 128-255 are not legitimate in some locales, most notably UTF-8. Moreover, what happens with non-ASCII characters in plots is system dependent and depends on the graphics device, as well. Use of such characters is not recommended for portable code.

From version 4031.95 of package fBasics, the characters are always defined as Latin1. In particular, in UTF8 locales the system converts them internally to UTF8. Still some symbols are not usable and non-ASCII symbols are not recommended, as pointed out above. For details, see the help page of points(), in particular the discussion of its argument pch.

See Also

characterTable, colorTable

pdf for discussion of encodings for the pdf device

Examples

  
# symbolTable()

Trace of a matrix

Description

Computes the trace of a matrix.

Usage

tr(x)

Arguments

Details

tr computes the trace of a square matrix, i.e., the sum of its diagonal elements.

If the matrix is not square, tr returns NA.

References

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P = pascal(3)
P
  
tr(P)                                  

Upper and lower triangular matrices

Description

Extracs the upper or lower triangular part from a matrix.

Usage

triang(x)
Triang(x)

Arguments

Details

triang and Triang transform a square matrix to a lower or upper triangular form. The functions just replace the remaining values with zeroes and work with non-square matrices, as well.

A triangular matrix is either an upper triangular matrix or lower triangular matrix. For the first case all matrix elements a[i,j] of matrix A are zero for i>j, whereas in the second case we have just the opposite situation. A lower triangular matrix is sometimes also called left triangular.

In fact, triangular matrices are so useful that much of computational linear algebra begins with factoring or decomposing a general matrix or matrices into triangular form. Some matrix factorization methods are the Cholesky factorization and the LU-factorization. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.

Triangular matrices have the following properties: the inverse of a triangular matrix is a triangular matrix, the product of two triangular matrices is a triangular matrix, the determinant of a triangular matrix is the product of the diagonal elements, the eigenvalues of a triangular matrix are the diagonal elements.

Value

a matrix of the same dimensions as x with the elements above or below the main diagonal set to zeroes

References

Higham, N.J., (2002); Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM.

Golub, van Loan, (1996); Matrix Computations, 3rd edition. Johns Hopkins University Press.

Examples

## Create Pascal Matrix:
P = pascal(3)
P
   
## Create lower triangle matrix
L = triang(P)
L                                

Two sided approximated Hessian

Description

Computes two sided (TS) approximated Hessian.

Usage

tsHessian(x, fun, ...)

Arguments

Author(s)

A function borrowed from Kevin Sheppard's Matlab garch toolbox as implemented by Alexios Ghalanos in his rgarch package.

Lagged or leading vector/matrix

Description

Creates a lagged or leading vector/matrix of selected order(s).

Usage

tslag(x, k = 1, trim = FALSE)

Arguments

See Also

pdl

Examples

## tslag -

Two sample variance tests

Description

Tests if two series differ in their distributional variance parameter.

Usage

 
varianceTest(x, y, method = c("varf", "bartlett", "fligner"), 
    title = NULL, description = NULL)

Arguments

Details

The method="varf" can be used to compare variances of two normal samples performing an F test. The null hypothesis is that the ratio of the variances of the populations from which they were drawn is equal to one.

The method="bartlett" performs the Bartlett test of the null hypothesis that the variances in each of the samples are the same. This fact of equal variances across samples is also called homogeneity of variances. Note, that Bartlett's test is sensitive to departures from normality. That is, if the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test (not yet implemented) is an alternative to the Bartlett test that is less sensitive to departures from normality.

The method="fligner" performs the Fligner-Killeen test of the null that the variances in each of the two samples are the same.

Value

an object from class fHTEST

Note

Some of the test implementations are selected from R's ctest package.

Author(s)

R-core team for hypothesis tests implemented from R's package ctest.

References

Conover, W. J. (1971); Practical nonparametric statistics, New York: John Wiley & Sons.

Lehmann E.L. (1986); Testing Statistical Hypotheses, John Wiley and Sons, New York.

Examples

x <- rnorm(50)
y <- rnorm(50)
   
varianceTest(x, y, "varf")
varianceTest(x, y, "bartlett")
varianceTest(x, y, "fligner")

Stacking vectors and matrices

Description

Stacks either a lower triangle matrix or a matrix.

Usage

vec(x)
vech(x)

Arguments

Details

The function vec implements the operator that stacks a matrix as a column vector, to be more precise in a matrix with one column. vec(X) = (X_{11}, X_{21}, ..., X_{N1}, X_{12}, X_{22}, ..., X_{NN}).

The function vech implements the operator that stacks the lower triangle of a NxN matrix as an N(N+1)/2x1 vector: vech(X) =(X_{11}, X_{21}, X_{22}, X_{31}, ..., X_{NN}), to be more precise in a matrix with one row.

Examples

## Create Pascal Matrix:
P = pascal(3)
   
## Stack the matrix
vec(P) 
   
## Stack the lower triangle
vech(P)                                        

Compute volatility

Description

Generic function for volatility computations.

Usage

volatility(object, ...)

## Default S3 method:
volatility(object, ...)

Arguments

Details

volatility is a generic function, whose default method centers and squares the values in object. Other packages can (and do) define methods for it.