Mode Estimation

Description

This package provides estimators of the mode of univariate unimodal (and sometimes multimodal) data, and values of the modes of usual probability distributions.

For a complete list of functions, use library(help = "modeest") or help.start().

References

• Parzen E. (1962). On estimation of a probability density function and mode. Ann. Math. Stat., 33(3):1065-1076.

• Chernoff H. (1964). Estimation of the mode. Ann. Inst. Statist. Math., 16:31-41.

• Huber P.J. (1964). Robust estimation of a location parameter. Ann. Math. Statist., 35:73-101.

• Dalenius T. (1965). The Mode - A Negleted Statistical Parameter. J. Royal Statist. Soc. A, 128:110-117.

• Grenander U. (1965). Some direct estimates of the mode. Ann. Math. Statist., 36:131-138.

• Venter J.H. (1967). On estimation of the mode. Ann. Math. Statist., 38(5):1446-1455.

• Lientz B.P. (1969). On estimating points of local maxima and minima of density functions. Nonparametric Techniques in Statistical Inference (ed. M.L. Puri, Cambridge University Press), p.275-282.

• Lientz B.P. (1970). Results on nonparametric modal intervals. SIAM J. Appl. Math., 19:356-366.

• Wegman E.J. (1971). A note on the estimation of the mode. Ann. Math. Statist., 42(6):1909-1915.

• Yamato H. (1971). Sequential estimation of a continuous probability density function and mode. Bull. Math. Statist., 14:1-12.

• Ekblom H. (1972). A Monte Carlo investigation of mode estimators in small samples. Applied Statistics, 21:177-184.

• Lientz B.P. (1972). Properties of modal intervals. SIAM J. Appl. Math., 23:1-5.

• Konakov V.D. (1973). On the asymptotic normality of the mode of multidimensional distributions. Theory Probab. Appl., 18:794-803.

• Robertson T. and Cryer J.D. (1974). An iterative procedure for estimating the mode. J. Amer. Statist. Assoc., 69(348):1012-1016.

• Kim B.K. and Van Ryzin J. (1975). Uniform consistency of a histogram density estimator and modal estimation. Commun. Statist., 4:303-315.

• Sager T.W. (1975). Consistency in nonparametric estimation of the mode. Ann. Statist., 3(3):698-706.

• Stone C.J. (1975). Adaptive maximum likelihood estimators of a location parameter. Ann. Statist., 3:267-284.

• Mizoguchi R. and Shimura M. (1976). Nonparametric Learning Without a Teacher Based on Mode Estimation. IEEE Transactions on Computers, C25(11):1109-1117.

• Adriano K.N., Gentle J.E. and Sposito V.A. (1977). On the asymptotic bias of Grenander's mode estimator. Commun. Statist.-Theor. Meth. A, 6:773-776.

• Asselin de Beauville J.-P. (1978). Estimation non parametrique de la densite et du mode, exemple de la distribution Gamma. Revue de Statistique Appliquee, 26(3):47-70.

• Sager T.W. (1978). Estimation of a multivariate mode. Ann. Statist., 6:802-812.

• Devroye L. (1979). Recursive estimation of the mode of a multivariate density. Canadian J. Statist., 7(2):159-167.

• Sager T.W. (1979). An iterative procedure for estimating a multivariate mode and isopleth. J. Amer. Statist. Assoc., 74(366):329-339.

• Eddy W.F. (1980). Optimum kernel estimators of the mode. Ann. Statist., 8(4):870-882.

• Eddy W.F. (1982). The Asymptotic Distributions of Kernel Estimators of the Mode. Z. Wahrsch. Verw. Gebiete, 59:279-290.

• Hall P. (1982). Asymptotic Theory of Grenander's Mode Estimator. Z. Wahrsch. Verw. Gebiete, 60:315-334.

• Sager T.W. (1983). Estimating modes and isopleths. Commun. Statist.-Theor. Meth., 12(5):529-557.

• Hartigan J.A. and Hartigan P.M. (1985). The Dip Test of Unimodality. Ann. Statist., 13:70-84.

• Hartigan P.M. (1985). Computation of the Dip Statistic to Test for Unimodality. Appl. Statist. (JRSS C), 34:320-325.

• Romano J.P. (1988). On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist., 16(2):629-647.

• Tsybakov A. (1990). Recursive estimation of the mode of a multivariate distribution. Probl. Inf. Transm., 26:31-37.

• Hyndman R.J. (1996). Computing and graphing highest density regions. Amer. Statist., 50(2):120-126.

• Vieu P. (1996). A note on density mode estimation. Statistics \& Probability Letters, 26:297–307.

• Leclerc J. (1997). Comportement limite fort de deux estimateurs du mode : le shorth et l'estimateur naif. C. R. Acad. Sci. Paris, Serie I, 325(11):1207-1210.

• Leclerc J. (2000). Strong limiting behavior of two estimates of the mode: the shorth and the naive estimator. Statistics and Decisions, 18(4).

• Shoung J.M. and Zhang C.H. (2001). Least squares estimators of the mode of a unimodal regression function. Ann. Statist., 29(3):648-665.

• Bickel D.R. (2002). Robust estimators of the mode and skewness of continuous data. Computational Statistics and Data Analysis, 39:153-163.

• Abraham C., Biau G. and Cadre B. (2003). Simple Estimation of the Mode of a Multivariate Density. Canad. J. Statist., 31(1):23-34.

• Bickel D.R. (2003). Robust and efficient estimation of the mode of continuous data: The mode as a viable measure of central tendency. J. Statist. Comput. Simul., 73:899-912.

• Djeddour K., Mokkadem A. et Pelletier M. (2003). Sur l'estimation recursive du mode et de la valeur modale d'une densite de probabilite. Technical report 105.

• Djeddour K., Mokkadem A. et Pelletier M. (2003). Application du principe de moyennisation a l'estimation recursive du mode et de la valeur modale d'une densite de probabilite. Technical report 106.

• Hedges S.B. and Shah P. (2003). Comparison of mode estimation methods and application in molecular clock analysis. BMC Bioinformatics, 4:31-41.

• Herrmann E. and Ziegler K. (2004). Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions. Statistics and Probability Letters, 68:359-368.

• Abraham C., Biau G. and Cadre B. (2004). On the Asymptotic Properties of a Simple Estimate of the Mode. ESAIM Probab. Stat., 8:1-11.

• Mokkadem A. and Pelletier M. (2005). Adaptive Estimation of the Mode of a Multivariate Density. J. Nonparametr. Statist., 17(1):83-105.

• Bickel D.R. and Fruehwirth R. (2006). On a Fast, Robust Estimator of the Mode: Comparisons to Other Robust Estimators with Applications. Computational Statistics and Data Analysis, 50(12):3500-3530.

See Also

mlv for general mode estimation.

The Asselin de Beauville mode estimator

Description

This mode estimator is based on the algorithm described in Asselin de Beauville (1978).

Usage

asselin(x, bw = NULL, ...)

Arguments

Value

A numeric value is returned, the mode estimate.

Note

The user may call asselin through mlv(x, method = "asselin", ...).

References

• Asselin de Beauville J.-P. (1978). Estimation non parametrique de la densite et du mode, exemple de la distribution Gamma. Revue de Statistique Appliquee, 26(3):47-70.

See Also

mlv for general mode estimation.

Examples

x <- rbeta(1000, shape1 = 2, shape2 = 5)

## True mode:
betaMode(shape1 = 2, shape2 = 5)

## Estimation:
asselin(x, bw = 1)
asselin(x, bw = 1/2)
mlv(x, method = "asselin")

Mode of some continuous and discrete distributions

Description

These functions return the mode of the main probability distributions implemented in R.

Usage

distrMode(x, ...)

betaMode(shape1, shape2, ncp = 0)

cauchyMode(location = 0, ...)

chisqMode(df, ncp = 0)

dagumMode(scale = 1, shape1.a, shape2.p)

expMode(...)

fMode(df1, df2)

fiskMode(scale = 1, shape1.a)

frechetMode(location = 0, scale = 1, shape = 1, ...)

gammaMode(shape, rate = 1, scale = 1/rate)

normMode(mean = 0, ...)

gevMode(location = 0, scale = 1, shape = 0, ...)

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

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

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

gompertzMode(scale = 1, shape)

gpdMode(location = 0, scale = 1, shape = 0)

gumbelMode(location = 0, ...)

hypMode(alpha = 1, beta = 0, delta = 1, mu = 0, pm = c(1, 2, 3, 4))

koenkerMode(location = 0, ...)

kumarMode(shape1, shape2)

laplaceMode(location = 0, ...)

logisMode(location = 0, ...)

lnormMode(meanlog = 0, sdlog = 1)

lomaxMode(...)

maxwellMode(rate)

mvnormMode(mean, ...)

nakaMode(scale = 1, shape)

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

paralogisticMode(scale = 1, shape1.a)

paretoMode(scale = 1, ...)

rayleighMode(scale = 1)

stableMode(alpha, beta, gamma = 1, delta = 0, pm = 0, ...)

stableMode2(loc, disp, skew, tail)

tMode(df, ncp)

unifMode(min = 0, max = 1)

weibullMode(shape, scale = 1)

yulesMode(...)

bernMode(prob)

binomMode(size, prob)

geomMode(...)

hyperMode(m, n, k, ...)

nbinomMode(size, prob, mu)

poisMode(lambda)

Arguments

Value

A numeric value is returned, the (true) mode of the distribution.

Note

Some functions like normMode or cauchyMode, which relate to symmetric distributions, are trivial, but are implemented for the sake of exhaustivity.

Author(s)

ghMode and ghtMode are from package fBasics; hypMode was written by David Scott; gldMode, nigMode and stableMode were written by Diethelm Wuertz.

See Also

mlv for the estimation of the mode; the documentation of the related distributions Beta, GammaDist, etc.

Examples

## Beta distribution
curve(dbeta(x, shape1 = 2, shape2 = 3.1), 
      xlim = c(0,1), ylab = "Beta density")
M <- betaMode(shape1 = 2, shape2 = 3.1)
abline(v = M, col = 2)
mlv("beta", shape1 = 2, shape2 = 3.1)

## Lognormal distribution
curve(stats::dlnorm(x, meanlog = 3, sdlog = 1.1), 
      xlim = c(0, 10), ylab = "Lognormal density")
M <- lnormMode(meanlog = 3, sdlog = 1.1)
abline(v = M, col = 2)
mlv("lnorm", meanlog = 3, sdlog = 1.1)

curve(VGAM::dpareto(x, scale = 1, shape = 1), xlim = c(0, 10))
abline(v = paretoMode(scale = 1), col = 2)

## Poisson distribution
poisMode(lambda = 6)
poisMode(lambda = 6.1)
mlv("poisson", lambda = 6.1)

The Grenander mode estimator

Description

This function computes the Grenander mode estimator.

Usage

grenander(x, bw = NULL, k, p, ...)

Arguments

Details

The Grenander estimate is defined by

\frac{ \sum_{j=1}^{n-k} \frac{(x_{j+k} + x_{j})}{2(x_{j+k} - x_{j})^p} } { \sum_{j=1}^{n-k} \frac{1}{(x_{j+k} - x_{j})^p} }

If p tends to infinity, this estimate tends to the Venter mode estimate; this justifies to call venter if p = Inf.

The user should either give the bandwidth bw or the argument k, k being taken equal to ceiling(bw*n) - 1 if missing.

Value

A numeric value is returned, the mode estimate. If p = Inf, the venter mode estimator is returned.

Note

The user may call grenander through mlv(x, method = "grenander", bw, k, p, ...).

Author(s)

D.R. Bickel for the original code, P. Poncet for the slight modifications introduced.

References

• Grenander U. (1965). Some direct estimates of the mode. Ann. Math. Statist., 36:131-138.

• Dalenius T. (1965). The Mode - A Negleted Statistical Parameter. J. Royal Statist. Soc. A, 128:110-117.

• Adriano K.N., Gentle J.E. and Sposito V.A. (1977). On the asymptotic bias of Grenander's mode estimator. Commun. Statist.-Theor. Meth. A, 6:773-776.

• Hall P. (1982). Asymptotic Theory of Grenander's Mode Estimator. Z. Wahrsch. Verw. Gebiete, 60:315-334.

See Also

mlv for general mode estimation; venter for the Venter mode estimate.

Examples

# Unimodal distribution
x <- rnorm(1000, mean = 23, sd = 0.5) 

## True mode
normMode(mean = 23, sd = 0.5) # (!)

## Parameter 'k'
k <- 5

## Many values of parameter 'p'
ps <- seq(0.1, 4, 0.01)

## Estimate of the mode with these parameters
M <- sapply(ps, function(p) grenander(x, p = p, k = k))

## Distribution obtained
plot(density(M), xlim = c(22.5, 23.5))

Bickel's half-range mode estimator

Description

SINCE THIS FUNCTION USED TO DEPEND ON THE BIOCONDUCTOR PACKAGE 'GENEFILTER', IT IS CURRENTLY DEFUNCT.

This function computes Bickel's half range mode estimator described in Bickel (2002). It is a wrapper around the function half.range.mode from package genefilter.

Usage

hrm(x, bw = NULL, ...)

Arguments

Details

The mode estimator is computed by iteratively identifying densest half ranges. A densest half range is an interval whose width equals half the current range, and which contains the maximal number of observations. The subset of observations falling in the selected densest half range is then used to compute a new range, and the procedure is iterated.

Value

A numeric value is returned, the mode estimate.

Note

The user may call hrm through mlv(x, method = "hrm", bw, ...).

Author(s)

The C and R code are due to Richard Bourgon bourgon@stat.berkeley.edu, see package genefilter. The algorithm is described in Bickel (2002).

References

• Bickel D.R. (2002). Robust estimators of the mode and skewness of continuous data. Computational Statistics and Data Analysis, 39:153-163.

• Hedges S.B. and Shah P. (2003). Comparison of mode estimation methods and application in molecular clock analysis. BMC Bioinformatics, 4:31-41.

• Bickel D.R. and Fruehwirth R. (2006). On a Fast, Robust Estimator of the Mode: Comparisons to Other Robust Estimators with Applications. Computational Statistics and Data Analysis, 50(12):3500-3530.

See Also

mlv() for general mode estimation; hsm() for the half sample mode; venter() for the Venter mode estimate.

Examples

## Not run: 
# Unimodal distribution 
x <- rgamma(1000, shape = 31.9)
## True mode
gammaMode(shape = 31.9)

## Estimate of the mode
hrm(x, bw = 0.4)
mlv(x, method = "hrm", bw = 0.4)

## End(Not run)

Half sample mode estimator

Description

This function computes the Robertson-Cryer mode estimator described in Robertson and Cryer (1974), also called half sample mode (if bw = 1/2) or fraction sample mode (for some other bw) by Bickel (2006).

Usage

hsm(x, bw = NULL, k, tie.action = "mean", tie.limit = 0.05, ...)

Arguments

Details

The modal interval, i.e. the shortest interval among intervals containing k+1 observations, is computed iteratively, until only one value is found, the mode estimate. At each step i, one takes k = ceiling(bw*n) - 1, where n is the length of the modal interval computed at step i-1. If bw is of class "function", then k = ceiling(bw(n)) - 1 instead.

Value

A numeric value is returned, the mode estimate.

Note

The user may call hsm through mlv(x, method = "hsm", ...).

Author(s)

D.R. Bickel for the original code, P. Poncet for the slight modifications introduced.

References

• Robertson T. and Cryer J.D. (1974). An iterative procedure for estimating the mode. J. Amer. Statist. Assoc., 69(348):1012-1016.

• Bickel D.R. and Fruehwirth R. (2006). On a Fast, Robust Estimator of the Mode: Comparisons to Other Robust Estimators with Applications. Computational Statistics and Data Analysis, 50(12):3500-3530.

See Also

mlv for general mode estimation; venter for the Venter mode estimate.

Examples

# Unimodal distribution
x <- rweibull(10000, shape = 3, scale = 0.9)

## True mode
weibullMode(shape = 3, scale = 0.9)

## Estimate of the mode
bandwidth <- function(n, alpha) {1/n^alpha}
hsm(x, bw = bandwidth, alpha = 2)
mlv(x, method = "hsm", bw = bandwidth, alpha = 2)

The empirical Lientz function and the Lientz mode estimator

Description

The Lientz mode estimator is nothing but the value minimizing the empirical Lientz function. A 'plot' and a 'print' methods are provided.

Usage

lientz(x, bw = NULL)

## S3 method for class 'lientz'
plot(x, zoom = FALSE, ...)

## S3 method for class 'lientz'
print(x, digits = NULL, ...)

## S3 method for class 'lientz'
mlv(x, bw = NULL, abc = FALSE, par = shorth(x), optim.method = "BFGS", ...)

Arguments

Details

The Lientz function is the smallest non-negative quantity S(x,\beta), where \beta = bw, such that

F(x+S(x,\beta)) - F(x-S(x,\beta)) \geq \beta.

Lientz (1970) provided a way to estimate S(x,\beta); this estimate is what we call the empirical Lientz function.

Value

lientz returns an object of class c("lientz", "function"); this is a function with additional attributes:

• x the x argument

• bw the bw argument

• call the call which produced the result

mlv.lientz returns a numeric value, the mode estimate. If abc = TRUE, the x value minimizing the Lientz empirical function is returned. Otherwise, the optim method is used to perform minimization, and the attributes: 'value', 'counts', 'convergence' and 'message', coming from the optim method, are added to the result.

Note

The user may call mlv.lientz through mlv(x, method = "lientz", ...).

References

• Lientz B.P. (1969). On estimating points of local maxima and minima of density functions. Nonparametric Techniques in Statistical Inference (ed. M.L. Puri, Cambridge University Press, p.275-282.

• Lientz B.P. (1970). Results on nonparametric modal intervals. SIAM J. Appl. Math., 19:356-366.

• Lientz B.P. (1972). Properties of modal intervals. SIAM J. Appl. Math., 23:1-5.

See Also

mlv for general mode estimation; shorth for the shorth estimate of the mode

Examples

# Unimodal distribution
x <- rbeta(1000,23,4)

## True mode
betaMode(23, 4)

## Lientz object
f <- lientz(x, 0.2)
print(f)
plot(f)

## Estimate of the mode
mlv(f)              # optim(shorth(x), fn = f)
mlv(f, abc = TRUE)  # x[which.min(f(x))]
mlv(x, method = "lientz", bw = 0.2)

# Bimodal distribution
x <- c(rnorm(1000,5,1), rnorm(1500, 22, 3))
f <- lientz(x, 0.1)
plot(f)

The Meanshift mode estimator

Description

The Meanshift mode estimator.

Usage

meanshift(
  x,
  bw = NULL,
  kernel = "gaussian",
  par = shorth(x),
  iter = 1000,
  tolerance = sqrt(.Machine$double.eps)
)

Arguments

Value

meanshift returns a numeric value, the mode estimate, with an attribute "iterations". The number of iterations can be less than iter if the stopping criteria specified by eps is reached.

Note

The user should preferentially call meanshift through mlv(x, method = "meanshift", ...).

References

• Fukunaga, K. and Hostetler, L. (1975). The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Transactions on Information Theory, 21(1):32–40.

See Also

mlv, tsybakov.

Examples

# Unimodal distribution
x <- rweibull(100, shape = 12, scale = 0.8)

## True mode
weibullMode(shape = 12, scale = 0.8)

## Estimate of the mode
mlv(x, method = "meanshift", par = mean(x))

Estimation of the Mode(s) or Most Likely Value(s)

Description

mlv is a generic function for estimating the mode of a univariate distribution. Different estimates (or methods) are provided:

• mfv, which returns the most frequent value(s) in a given numerical vector,

• the Lientz mode estimator, which is the value minimizing the Lientz function estimate,

• the Chernoff mode estimator, also called naive mode estimator, which is defined as the center of the interval of given length containing the most observations,

• the Venter mode estimator, including the shorth, i.e. the midpoint of the modal interval,

• the Grenander mode estimator,

• the half sample mode (HSM) and the half range mode (HRM), which are iterative versions of the Venter mode estimator,

• Parzen's kernel mode estimator, which is the value maximizing the kernel density estimate,

• the Tsybakov mode estimator, based on a gradient-like recursive algorithm,

• the Asselin de Beauville mode estimator, based on a algorithm detecting chains and holes in the sample,

• the Vieu mode estimator,

• the meanshift mode estimator.

mlv can also be used to compute the mode of a given distribution, with mlv.character.

Usage

mlv(x, ...)

## S3 method for class 'character'
mlv(x, na.rm = FALSE, ...)

## S3 method for class 'factor'
mlv(x, na.rm = FALSE, ...)

## S3 method for class 'logical'
mlv(x, na.rm = FALSE, ...)

## S3 method for class 'integer'
mlv(x, na.rm = FALSE, ...)

## Default S3 method:
mlv(x, bw = NULL, method, na.rm = FALSE, ...)

mlv1(x, ...)

Arguments

Details

For the default method of mlv, available methods are "lientz", "naive", "venter", "grenander", "hsm", "parzen", "tsybakov", "asselin", and "meanshift". See the description above and the associated links.

If x is of class "character" (with length > 1), "factor", or "integer", then the most frequent value found in x is returned using mfv from package statip.

If x is of class "character" (with length 1), x should be one of "beta", "cauchy", "gev", etc. i.e. a character for which a function *Mode exists (for instance betaMode, cauchyMode, etc.). See distrMode for the available functions. The mode of the corresponding distribution is returned.

If x is of class mlv.lientz, see Lientz for more details.

Value

A vector of the same type as x. Be aware that the length of this vector can be > 1.

References

See the references on mode estimation on the modeest-package's page.

See Also

mfv, parzen, venter, meanshift, grenander, hsm, lientz, naive, tsybakov, skewness

Examples

# Unimodal distribution
x <- rbeta(1000,23,4)

## True mode
betaMode(23, 4)
# or
mlv("beta", shape1 = 23, shape2 = 4)

## Be aware of this behaviour: 
mlv("norm") # returns 0, the mode of the standard normal distribution
mlv("normal") # returns 0 again, since "normal" is matched with "norm"
mlv("abnormal") # returns "abnormal", since the input vector "abrnormal" 
# is not recognized as a distribution name, hence is taken as a character 
# vector from which the most frequent value is requested. 

## Estimate of the mode
mlv(x, method = "lientz", bw = 0.2)
mlv(x, method = "naive", bw = 1/3)
mlv(x, method = "venter", type = "shorth")
mlv(x, method = "grenander", p = 4)
mlv(x, method = "hsm")
mlv(x, method = "parzen", kernel = "gaussian")
mlv(x, method = "tsybakov", kernel = "gaussian")
mlv(x, method = "asselin", bw = 2/3)
mlv(x, method = "vieu")
mlv(x, method = "meanshift")

The Chernoff or 'naive' mode estimator

Description

This estimator, also called the *naive* mode estimator, is defined as the center of the interval of given length containing the most observations. It is identical to Parzen's kernel mode estimator, when the kernel is chosen to be the uniform kernel.

Usage

naive(x, bw = 1/2)

Arguments

Value

A numeric vector is returned, the mode estimate, which is the center of the interval of length 2*bw containing the most observations.

Note

The user may call naive through mlv(x, method = "naive", bw).

References

• Chernoff H. (1964). Estimation of the mode. Ann. Inst. Statist. Math., 16:31-41.

• Leclerc J. (1997). Comportement limite fort de deux estimateurs du mode : le shorth et l'estimateur naif. C. R. Acad. Sci. Paris, Serie I, 325(11):1207-1210.

See Also

mlv for general mode estimation; parzen for Parzen's kernel mode estimation.

Examples

# Unimodal distribution
x <- rf(10000, df1 = 40, df2 = 30)

## True mode
fMode(df1 = 40, df2 = 30)

## Estimate of the mode
mean(naive(x, bw = 1/4))
mlv(x, method = "naive", bw = 1/4)

Parzen's Kernel mode estimator

Description

Parzen's kernel mode estimator is the value maximizing the kernel density estimate.

Usage

parzen(
  x,
  bw = NULL,
  kernel = "gaussian",
  abc = FALSE,
  tolerance = .Machine$double.eps^0.25,
  ...
)

Arguments

Details

If kernel = "uniform", the naive mode estimate is returned.

Value

parzen returns a numeric value, the mode estimate. If abc = TRUE, the x value maximizing the density estimate is returned. Otherwise, the optim method is used to perform maximization, and the attributes: 'value', 'counts', 'convergence' and 'message', coming from the optim method, are added to the result.

Note

The user may call parzen through mlv(x, method = "kernel", ...) or mlv(x, method = "parzen", ...).

Presently, parzen is quite slow.

References

• Parzen E. (1962). On estimation of a probability density function and mode. Ann. Math. Stat., 33(3):1065–1076.

• Konakov V.D. (1973). On the asymptotic normality of the mode of multidimensional distributions. Theory Probab. Appl., 18:794-803.

• Eddy W.F. (1980). Optimum kernel estimators of the mode. Ann. Statist., 8(4):870-882.

• Eddy W.F. (1982). The Asymptotic Distributions of Kernel Estimators of the Mode. Z. Wahrsch. Verw. Gebiete, 59:279-290.

• Romano J.P. (1988). On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist., 16(2):629-647.

• Abraham C., Biau G. and Cadre B. (2003). Simple Estimation of the Mode of a Multivariate Density. Canad. J. Statist., 31(1):23-34.

• Abraham C., Biau G. and Cadre B. (2004). On the Asymptotic Properties of a Simple Estimate of the Mode. ESAIM Probab. Stat., 8:1-11.

See Also

mlv, naive

Examples

# Unimodal distribution 
x <- rlnorm(10000, meanlog = 3.4, sdlog = 0.2) 

## True mode 
lnormMode(meanlog = 3.4, sdlog = 0.2) 

## Estimate of the mode 
mlv(x, method = "kernel", kernel = "gaussian", bw = 0.3, par = shorth(x)) 

Objects exported from other packages

Description

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

statip

mfv, mfv1

Skewness

Description

This function encodes different methods to calculate the skewness from a vector of observations.

Usage

skewness(x, na.rm = FALSE, method = c("moment", "fisher", "bickel"), M, ...)

Arguments

Value

skewness returns a numeric value. An attribute reports the method used.

Author(s)

Diethelm Wuertz and contributors for the original skewness function from package fBasics.

References

• Bickel D.R. (2002). Robust estimators of the mode and skewness of continuous data. Computational Statistics and Data Analysis, 39:153-163.

• Bickel D.R. et Fruehwirth R. (2006). On a Fast, Robust Estimator of the Mode: Comparisons to Other Robust Estimators with Applications. Computational Statistics and Data Analysis, 50(12):3500-3530.

See Also

mlv for general mode estimation; shorth for the shorth estimate of the mode

Examples

## Skewness = 0
x <- rnorm(1000)
skewness(x, method = "bickel", M = shorth(x))

## Skewness > 0 (left skewed case)
x <- rbeta(1000, 2, 5)
skewness(x, method = "bickel", M = betaMode(2, 5))

## Skewness < 0 (right skewed case)
x <- rbeta(1000, 7, 2)
skewness(x, method = "bickel", M = hsm(x, bw = 1/3))

The Tsybakov mode estimator

Description

This mode estimator is based on a gradient-like recursive algorithm, more adapted for online estimation. It includes the Mizoguchi-Shimura (1976) mode estimator, based on the window training procedure.

Usage

tsybakov(
  x,
  bw = NULL,
  a,
  alpha = 0.9,
  kernel = "triangular",
  dmp = TRUE,
  par = shorth(x)
)

Arguments

Details

If bw or a is missing, a default value advised by Djeddour et al (2003) is used: bw = (1:length(x))^(-1/7) and a = (1:length(x))^(-alpha). (with alpha = 0.9 if alpha is missing).

Value

A numeric value is returned, the mode estimate.

Warning

The Tsybakov mode estimate as it is presently computed does not work very well. The reasons of this inefficiency should be further investigated.

Note

The user may call tsybakov through mlv(x, method = "tsybakov", ...).

References

• Mizoguchi R. and Shimura M. (1976). Nonparametric Learning Without a Teacher Based on Mode Estimation. IEEE Transactions on Computers, C25(11):1109-1117.

• Tsybakov A. (1990). Recursive estimation of the mode of a multivariate distribution. Probl. Inf. Transm., 26:31-37.

• Djeddour K., Mokkadem A. et Pelletier M. (2003). Sur l'estimation recursive du mode et de la valeur modale d'une densite de probabilite. Technical report 105.

• Djeddour K., Mokkadem A. et Pelletier M. (2003). Application du principe de moyennisation a l'estimation recursive du mode et de la valeur modale d'une densite de probabilite. Technical report 106.

See Also

mlv for general mode estimation.

Examples

x <- rbeta(1000, shape1 = 2, shape2 = 5)

## True mode:
betaMode(shape1 = 2, shape2 = 5)

## Estimation:
tsybakov(x, kernel = "triangular")
tsybakov(x, kernel = "gaussian", alpha = 0.99)
mlv(x, method = "tsybakov", kernel = "gaussian", alpha = 0.99)

The Venter / Dalenius / LMS mode estimator

Description

This function computes the Venter mode estimator, also called the Dalenius, or LMS (Least Median Square) mode estimator.

Usage

venter(
  x,
  bw = NULL,
  k,
  iter = 1,
  type = 1,
  tie.action = "mean",
  tie.limit = 0.05,
  warn = FALSE
)

shorth(x, ...)

Arguments

Details

The modal interval, i.e. the shortest interval among intervals containing k+1 observations, is first computed. (In dimension > 1, this question is known as a 'k-enclosing problem'.) The user should either give the bandwidth bw or the argument k, k being taken equal to ceiling(bw*n) - 1 if missing, so bw can be seen as the fraction of the observations to be considered for the shortest interval.

If type = 1, the midpoint of the modal interval is returned. If type = 2, the floor((k+1)/2)th element of the modal interval is returned. If type = 3 or type = "dalenius", the median of the modal interval is returned. If type = 4 or type = "shorth", the mean of the modal interval is returned. If type = 5 or type = "ekblom", Ekblom's L_{-\infty} estimate is returned, see Ekblom (1972). If type = 6 or type = "hsm", the half sample mode (hsm) is computed, see hsm.

Value

A numeric value is returned, the mode estimate.

Note

The user may call venter through mlv(x, method = "venter", ...).

References

• Dalenius T. (1965). The Mode - A Negleted Statistical Parameter. J. Royal Statist. Soc. A, 128:110-117.

• Venter J.H. (1967). On estimation of the mode. Ann. Math. Statist., 38(5):1446-1455.

• Ekblom H. (1972). A Monte Carlo investigation of mode estimators in small samples. Applied Statistics, 21:177-184.

• Leclerc J. (1997). Comportement limite fort de deux estimateurs du mode : le shorth et l'estimateur naif. C. R. Acad. Sci. Paris, Serie I, 325(11):1207-1210.

See Also

mlv for general mode estimation, hsm for the half sample mode.

Examples

library(evd)

# Unimodal distribution
x <- rgev(1000, loc = 23, scale = 1.5, shape = 0)

## True mode
gevMode(loc = 23, scale = 1.5, shape = 0)

## Estimate of the mode
venter(x, bw = 1/3)
mlv(x, method = "venter", bw = 1/3)

Vieu's mode estimator

Description

Vieu's mode estimator is the value at which the kernel density derivative estimate is null.

Usage

vieu(x, bw = NULL, kernel = "gaussian", abc = FALSE, ...)

Arguments

Value

vieu returns a numeric value, the mode estimate. If abc = TRUE, the x value at which the density derivative estimate is null is returned. Otherwise, the uniroot method is used.

Note

The user may call vieu through mlv(x, method = "vieu", ...).

Presently, vieu is quite slow.

References

• Vieu P. (1996). A note on density mode estimation. Statistics \& Probability Letters, 26:297–307.

See Also

mlv, parzen.

Examples

# Unimodal distribution
x <- rlnorm(10000, meanlog = 3.4, sdlog = 0.2)

## True mode
lnormMode(meanlog = 3.4, sdlog = 0.2)

## Estimate of the mode
mlv(x, method = "vieu", kernel = "gaussian")