Package 'DepthProc'

Description

Add fitted line to a plot. This is overloaded function for robust regression methods from package depthproc.

Usage

## S4 method for signature 'RobReg'
abline(
  a = NULL,
  b = NULL,
  h = NULL,
  v = NULL,
  reg = NULL,
  coef = NULL,
  untf = FALSE,
  ...
)

Arguments

Description

Create a matrix from DepthCurve and DepthCurveList.

Usage

as.matrix(x, ...)

## S4 method for signature 'DepthCurveList'
as.matrix(x)

Arguments

Description

Produces an asymmetry curve estimated from given data.

Usage

asymmetryCurve(
  x,
  y = NULL,
  alpha = seq(0, 1, 0.01),
  movingmedian = FALSE,
  name = "X",
  name_y = "Y",
  depth_params = list(method = "Projection")
)

Arguments

Details

For sample depth function $D({x}, {{{Z}} ^ {n}})$, ${x} \in {{{R}} ^ {d}}$, $d \ge 2$, ${Z} ^ {n} = \{{{{z}}_{1}}, ..., {{{z}}_{n}}\} \subset {{{R}} ^ {d}}$, ${{D}_{\alpha}}({{{Z}} ^ {n}})$ denoting $\alpha$ — central region, we can define the asymmetry curve $AC(\alpha) = \left(\alpha, \left\| {{c} ^ {-1}}(\{{\bar{z}} - med|{{D}_{\alpha}}({{{Z}} ^ {n}})\}) \right\|\right) \subset {{{R}} ^ {2}}$, for $\alpha \in [0, 1]$ being nonparametric scale and asymmetry functional correspondingly, where $c$ — denotes constant, ${\bar{z}}$ — denotes mean vector, denotes multivariate median induced by depth function and $vol$ — denotes a volume.

Asymmetry curve takes uses function convhulln from package geometry for computing a volume of convex hull containing central region.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

References

Serfling R. J. Multivariate Symmetry and Asymmetry, Encyclopedia of Statistical Science, S Kotz, C.B. Read, N. Balakrishnan, B. Vidakovic (eds), 2nd, ed., John Wiley.

Liu, R.Y., Parelius, J.M. and Singh, K. (1999), Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 27, 783–858.

Chaudhuri, P. (1996), On a Geometric Notion of Quantiles for Multivariate Data, Journal of the American Statistical Association, 862–872.

Dyckerhoff, R. (2004), Data Depths Satisfying the Projection Property, Allgemeines Statistisches Archiv., 88, 163–190.

See Also

scaleCurve, depth

Examples

# EXAMPLE 1
library(sn)
xi <- c(0, 0)
alpha <- c(2, -5)
Omega <- diag(2) * 5

n <- 500
X <- mvrnorm(n, xi, Omega) # normal distribution
Y <- rmst(n, xi, Omega, alpha, nu = 1)
asymmetryCurve(X, Y, name = "NORM", name_y = "S_T(2, -5, 10)")

# EXAMPLE 2
data(under5.mort)
data(inf.mort)
data(maesles.imm)
data1990 <- cbind(under5.mort[, 1], inf.mort[, 1], maesles.imm[, 1])
data2011 <- cbind(under5.mort[, 22], inf.mort[, 22], maesles.imm[, 22])
as1990 <- asymmetryCurve(data1990, name = "scale curve 1990")
as2011 <- asymmetryCurve(data2011, name = "scale curve 2011")
figure <- getPlot(combineDepthCurves(as1990, as2011)) +
  ggtitle("Scale curves")
figure

Description

AsymmetryCurve is a class that stores results of asymmetryCurve function.

Details

The mechanism of creating plots with multiple curves is shown in DepthCurve-class (same mechanism is applied for ScaleCurve).

Description

Class that stores result of function binningDepth2D(...)

Slots

freq

Matrix with number of elements in certain bin.

mid_x

Middle values on x-axis.

mid_y

Middle values on y-axis.

breaks_x

Boundaries of bins.

breaks_y

Boundaries of bins.

input_data

Binned data.

max_depth_x

Point with maximum depth on x-axis.

max_depth_y

Point with maximum depth on y-axis.

Description

A robust method of decreasing a sample size and therefore a complexity of a statistical procedure. The method may be used within a kernel density or a predictive distribution estimation.

Usage

binningDepth2D(
  x,
  binmethod = "LocDepth",
  nbins = 8,
  k = 1,
  remove_borders = FALSE,
  depth_params = list(method = "LP")
)

Arguments

Details

Let us recall, that binning is a popular method of decreasing a sample size. To bin a window of $n$ points ${W}_{i, n} = \left\{ {X}_{i - n + 1}, ..., {X}_{i} \right\}$ to a grid ${{{X}'}_{1}}, ..., {{{X}'}_{m}}$ we simply assign each sample point ${{X}_{i}}$ to the nearest grid point ${{{X}'}_{j}}$. When binning is completed, each grid point ${{X}'}_{j}$ has an associated number ${c}_{i}$, which is the sum of all the points that have been assigned to ${{X}'}_{j}$. This procedure replaces the data ${W}_{i, n} = \left\{ {X}_{i - n + 1}, ..., {X}_{i} \right\}$ with the smaller set ${{W}'}_{j, m} = \left\{ {{X}'}_{j - m + 1}, ..., {{X}'}_{j} \right\}$. Although simple binning can speed up the computation, it is criticized for a lack of precise approximate control over the accuracy of the approximation. Robust binning however stresses properties of the majority of the data and decreases the computational complexity of the DSA at the same time.

For a 1D window ${W}_{i, n}$, let ${Z}_{i, n - k}$ denote a 2D window created basing on ${W}_{i, n}$ and consisted of $n - k$ pairs of observations and the $k$ lagged observations ${Z}_{i, n - k} = \left\{\left( {X}_{i - n - k}, {X}_{i - n + 1} \right)\right\}, 1\le i\le n - k$. Robust 2D binning of the ${Z}_{i, n - p}$ is a very useful technique in a context of robust estimation of the predictive distribution of a time series (see Kosiorowski:2013b).

Assume we analyze a data stream $\{{X}_{t}\}$ using a moving window of a fixed length $n$, i.e., ${W}_{i, n}$ and the derivative window ${Z}_{i, n - 1}$. In a first step we calculate the weighted sample $L ^ p$ depth for ${W}_{i, n}$. Next we choose equally spaced grid of points ${l}_{1}, ..., {l}_{m}$ in this way that $[{{l}_{1}}, {{l}_{m}}] \times [{{l}_{1}}, {{l}_{m}}]$ covers fraction of the $\beta$ central points of ${Z}_{i, n - 1}$ w.r.t. the calculated $L ^ p$ depth, i.e., it covers ${R} ^ {\beta}({Z}_{i, n - 1})$ for certain prefixed threshold $\beta \in (0, 1)$. For both ${X}_{t}$ and ${X}_{t - 1}$ we perform a simple binning using following bins: $(-\infty, {l}_{1})$, $({l}_{1}, {l}_{2})$, ..., $({l}_{m}, \infty)$. For robust binning we reject "border" classes and further use only midpoints and binned frequencies for classes $({l}_{1}, {l}_{2})$, $({l}_{2}, {l}_{3})$, ..., $({l}_{m - 1}, {l}_{m})$.

Value

freq: a matrix containing the binned frequencies

mid_x: mid points for x

mid_y: mid points for y

breaks_x: breaks for x

breaks_y: breaks for y

input_data: max_depth_x and max_depth_y:

Author(s)

Daniel Kosiorowski and Zygmunt Zawadzki from Cracow University of Economics.

References

Hall, P., Wand, M. P. (1996) On the Accuracy of Binned Kernel Density Estimators, Journal of Multivariate Analysis archive, Volume 56 Issue 2, 165–184

Holmstrom, L. (2000) The Accuracy and the Computational Complexity of a Multivariate Binned Kernel Density Estimator, Journal of Multivariate Analysis, Volume 72, Issue 2, 264–309, doi:10.1006/jmva.1999.1863. (https://www.sciencedirect.com/science/article/pii/S0047259X99918638)

See Also

depth

Examples

# EXAMPLE 1
Sigma1 <- matrix(c(10, 3, 3, 2), 2, 2)
X1 <- mvrnorm(n = 8500, mu = c(0, 0), Sigma1)
Sigma2 <- matrix(c(10, 0, 0, 2), 2, 2)
X2 <- mvrnorm(n = 1500, mu = c(-10, 6), Sigma2)
BALLOT <- rbind(X1, X2)
train <- sample(1:10000, 500)
data <- BALLOT[train, ]
plot(data)

b1 <- binningDepth2D(data, remove_borders = FALSE, nbins = 12, k = 1)
b2 <- binningDepth2D(data, nbins = 12, k = 1, remove_borders = TRUE)
plot(b1)
plot(b2)

# EXAMPLE 2
data(under5.mort)
data(maesles.imm)
data2011 <- cbind(under5.mort[, 22], maesles.imm[, 22])
plot(binningDepth2D(data2011, nbins = 8, k = 0.5, remove_borders = TRUE))

Description

Adds plots

Usage

combineDepthCurves(x, y, .list = NULL)

## S4 method for signature 'ANY,ANY,list'
combineDepthCurves(x, y, .list = NULL)

## S4 method for signature 'DepthCurveList,DepthCurve,ANY'
combineDepthCurves(x, y, .list = NULL)

## S4 method for signature 'DepthCurve,DepthCurveList,ANY'
combineDepthCurves(x, y, .list = NULL)

## S4 method for signature 'DepthCurve,DepthCurve,ANY'
combineDepthCurves(x, y, .list = NULL)

Arguments

Details

See DepthCurve-class for description.

Description

This class, derived from the virtual class "CovRobust" accomodates weighted by $L ^ p$ depth multivariate location and scatter estimator.

Details

See CovLP for the function used to calculate weighted by $L ^ p$ depth covariance matrix.

Description

Weighted by $L ^ p$ depth (outlyingness) multivariate location and scatter estimators.

Usage

CovLP(x, pdim = 2, la = 1, lb = 1)

Arguments

Details

Using depth function one can define a depth-weighted location and scatter estimators. In case of location estimator we have

$L(F) = {\int {{x}{{w}_{1}}(D({x}, F))dF({x})}} / {{{w}_{1}}(D({x}, F))dF({x})}$

Subsequently, a depth-weighted scatter estimator is defined as

$S(F) = \frac{ \int {({x} - L(F)){{({x} - L(F))} ^ {T}}{{w}_{2}}(D({x}, F))dF({x})} }{ \int {{{w}_{2}}(D({x}, F))dF({x})}},$

where ${{w}_{2}}(\cdot)$ is a suitable weight function that can be different from ${{w}_{1}}(\cdot)$.

The DepthProc package offers these estimators for weighted ${L} ^ {p}$ depth. Note that $L(\cdot)$ and $S(\cdot)$ include multivariate versions of trimmed means and covariance matrices. Their sample counterparts take the form

${{T}_{WD}}({{{X}} ^ {n}}) = {\sum\limits_{i = 1} ^ {n} {{{d}_{i}}{{X}_{i}}}} / {\sum\limits_{i = 1} ^ {n} {{{d}_{i}}}},$

$DIS({{{X}}^{n}}) = \frac{ \sum\limits_{i = 1} ^ {n} {{{d}_{i}}\left( {{{X}}_{i}} - {{T}_{WD}}({{{X}} ^ {n}}) \right){{\left( {{{X}}_{i}} - {{T}_{WD}}({{{X}} ^ {n}}) \right)} ^ {T}}} }{ \sum\limits_{i = 1} ^ {n} {{{d}_{i}}}},$

where ${{d}_{i}}$ are sample depth weights, ${{w}_{1}}(x) = {{w}_{2}}(x) = x$.

Value

loc: Robust Estimate of Location:

cov: Robust Estimate of Covariance:

Returns depth weighted covariance matrix.

Author(s)

Daniel Kosiorowski and Zygmunt Zawadzki from Cracow University of Economics.

See Also

depthContour and depthPersp for depth graphics.

Examples

# EXAMPLE 1
x <- mvrnorm(n = 100, mu = c(0, 0), Sigma = 3 * diag(2))
cov_x <- CovLP(x, 2, 1, 1)

# EXAMPLE 2
data(under5.mort, inf.mort, maesles.imm)
data1990 <- na.omit(cbind(under5.mort[, 1], inf.mort[, 1], maesles.imm[, 1]))
CovLP(data1990)

Description

Air pollution with PM10 in Cracow within day and night in December 2016

Usage

data("cracow.airpollution")

Format

data frame containing 744 rows.

References

1. Kosiorowski D, Rydlewski J P, Zawadzki Z (2017). 'Functional Outliers Detection By The Example Of Air Quality Monitoring.' submitted.

2. Kosiorowski D, Szlachtowska E (2017). 'K- local Median Algorithm for Functional Data in Empirical Analysis of Air Pollution Data.' Proceedings from the 11th Professor A. Zelias International Conference, pp. 153-162. Cracow University of Economics.

Description

Produces a normal DD plot of a multivariate dataset.

Usage

ddMvnorm(
  x,
  size = nrow(x),
  robust = FALSE,
  alpha = 0.05,
  title = "ddMvnorm",
  depth_params = list()
)

Arguments

Details

In the first step the location and scale of x are estimated and theoretical sample from normal distribution with those parameters is generated. The plot presents the depth of empirical points with respect to dataset x and with respect to the theoretical sample.

Value

Returns the normal depth versus depth plot of multivariate dataset x.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

References

Liu, R.Y., Parelius, J.M. and Singh, K. (1999), Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 27, 783–858.

Liu, R.Y., Singh K. (1993), A Quality Index Based on Data Depth and Multivariate Rank Test, Journal of the American Statistical Association vol. 88.

See Also

ddPlot to generate ddPlot to compare to datasets or to compare a dataset with other distributions.

Examples

# EXAMPLE 1
norm <- mvrnorm(1000, c(0, 0, 0), diag(3))
con <- mvrnorm(100, c(1, 2, 5), 3 * diag(3))
sample <- rbind(norm, con)
ddMvnorm(sample, robust = TRUE)

# EXAMPLE 2
data(under5.mort, inf.mort, maesles.imm)
data1990 <- na.omit(cbind(under5.mort[, 1], inf.mort[, 1], maesles.imm[, 1]))
ddMvnorm(data1990, robust = FALSE)

Description

Produces a DD plot which allows to compare two multivariate datasets or to compare a subject dataset with theoretical distribution.

Usage

ddPlot(
  x,
  y,
  scale = FALSE,
  location = FALSE,
  name = "X",
  name_y = "Y",
  title = "Depth vs. depth plot",
  depth_params = list()
)

Arguments

Details

For two probability distributions $F$ and $G$, both in ${{{R}} ^ {d}}$, we can define depth vs. depth plot being very useful generalization of the one dimensional quantile-quantile plot:

$DD(F, G) = \left\{\left( D({z}, F), D({z}, G) \right), {z} \in {{{R}} ^ {d}} \right\}$

Its sample counterpart calculated for two samples ${{{X}} ^ {n}} = \{{{X}_{1}}, ..., {{X}_{n}}\}$ from $F$, and ${{Y} ^ {m}} = \{{{Y}_{1}}, ..., {{Y}_{m}}\}$ from $G$ is defined as

$DD({{F}_{n}}, {{G}_{m}}) = \left\{\left( D({z}, {{F}_{n}}), D({z}, {{G}_{m}}) \right), {z} \in \{{{{X}} ^ {n}} \cup {{{Y}} ^ {m}}\} \right\}$

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

References

Liu, R.Y., Parelius, J.M. and Singh, K. (1999), Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 27, 822–831.

Liu, R.Y., Singh K. (1993), A Quality Index Based on Data Depth and Multivariate Rank Test, Journal of the American Statistical Association vol. 88.

Examples

library(sn)
library(mvtnorm)

# EXAMPLE 1: Location difference
standard <- mvrnorm(1000, c(0, 0), diag(2))
shift <- mvrnorm(1000, c(0.5, 0), diag(2))
ddPlot(x = standard, y = shift, title = "Difference in position")
ddPlot(x = standard, y = shift, location = TRUE, title = "Location aligned")

# EXAMPLE 2: Scale difference
standard <- mvrnorm(1000, c(0, 0), diag(2))
scale <- mvrnorm(1000, c(0, 0), 4 * diag(2))
ddPlot(x = standard, y = scale)
ddPlot(x = standard, y = scale, scale = TRUE)

Description

Class fro DDPlot

Slots

X

Object of class Depth-class.

Y

Object of class Depth-class.

title

title of a plot.

Description

This function calculates deepest regression estimator for simple regression.

Usage

deepReg2d(x, y)

Arguments

Details

Function originates from an original algorithm proposed by Rousseeuw and Hubert. Let ${Z} ^ {n} = {(x_1, y_1), ..., (x_n, y_n)} \subset {{R} ^ {d}}$ denotes a sample considered from a following semiparametric model: ${{y}_{l}} = {{a}_{0}} + {{a}_{1}}{{x}_{1l}} + ... + {{a}_{(d - 1)l}}{{x}_{(d - 1)l}} + {{\varepsilon }_{l}}, {l = 1, ..., n}$, we calculate a depth of a fit $\alpha = (a_{0}, ..., a_{d - 1})$ as $RD(\alpha, {{Z} ^ {n}}) = {u \ne 0}\min\sharp {l: \frac{ {{r}_{l}}(\alpha) }{ {{u} ^ {T}}{{x}_{l}} } < 0, l = 1, ..., n}$, where $r(\cdot)$ denotes the regression residual, $\alpha = (a_{0}, ..., a_{d - 1})$, ${u} ^ {T}{x}_{l} \ne 0$. The deepest regression estimator $DR(\alpha, {{Z} ^ {n}})$ is defined as $DR(\alpha, {{Z} ^ {n}}) = {\alpha \ne 0}\,\arg\max\,RD(\alpha, {{Z} ^ {n}})$

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

References

Rousseeuw J.P., Hubert M. (1998), Regression Depth, Journal of The American Statistical Association, vol.94.

Examples

# EXAMPLE 1
data(pension)
plot(pension)
abline(
  lm(Reserves ~ Income, data = pension),
  lty = 3,
  lwd = 2) # lm
abline(
  deepReg2d(pension[, 1], pension[, 2]),
  lwd = 2) # deepreg2d

# EXAMPLE 2
data(under5.mort)
data(inf.mort)
data(maesles.imm)
data2011 <- na.omit(
    cbind(under5.mort[, 22], inf.mort[, 22],
    maesles.imm[, 22]))

x <- data2011[, 3]
y <- data2011[, 2]
plot(
  x, y,
  cex = 1.2,
  ylab = "infant mortality rate per 1000 live birth",
  xlab = "against masles immunized percentage",
  main = "Projection Depth Trimmed vs. LS regressions"
)
abline(lm(x ~ y), lwd = 2, col = "black") # lm
abline(
  deepReg2d (x, y),
  lwd = 2, col = "red"
) # trimmed reg

legend(
  "bottomleft",
  c("LS", "DeepReg"),
  fill = c("black", "red"),
  cex = 1.4,
  bty = "n"
)

Description

Class for robust regression methods from depthproc package

Slots

coef

coefficients of fitted model

depth

regression depth of the fitted values

Description

Calculate depth functions.

Usage

depth(u, X, method = "Projection", threads = -1, ...)

Arguments

Details

The Mahalanobis depth

${D}_{MAH}(y, {X} ^ {n}) = \frac{ 1 }{ 1 + {{(y - \bar{x})} ^ {T}}{{S} ^ {-1}}(y - \bar{x}) },$

where $S$ denotes the sample covariance matrix ${X} ^ {n}$.

A symmetric projection depth $D\left( x, X\right)$ of a point $x \in {{{R}} ^ {d}}$, $d \ge 1$ is defined as

$D\left( x, X\right)_{PRO} = {{\left[ 1 + su{{p}_{\left\| u \right\| = 1}}\frac{ \left| {{u} ^ {T}}x - Med\left( {{u} ^ {T}}X\right)\right| }{ MAD\left( {{u} ^ {T}}X\right) }\right]} ^ {-1}},$

where Med denotes the univariate median, $MAD\left( Z \right)$ = $Med\left(\left| Z - Med\left( Z \right)\right|\right)$. Its sample version denoted by $D\left( x, {X} ^ {n} \right)$ or $D\left( x, {X} ^ {n} \right)$ is obtained by replacing $F$ by its empirical counterpart ${{F}_{n}}$ calculated from the sample ${X} ^ {n}$ .

Next interesting depth is the weighted ${L} ^ {p}$ depth. The weighted ${L} ^ {p}$ depth $D({x}, F)$ of a point ${x} \in {R} ^ {d}$, $d \ge 1$ generated by $d$ dimensional random vector ${X}$ with distribution $F$, is defined as $D({x}, F) = \frac{1 }{ 1 + Ew({{\left\| x - X \right\| }_{p}}) },$ where $w$ is a suitable weight function on $[0, \infty)$, and ${{\left\| \cdot \right\| }_{p}}$ stands for the ${L} ^ {p}$ norm (when p = 2 we have usual Euclidean norm). We assume that $w$ is non-decreasing and continuous on $[0, \infty)$ with $w(\infty-) = \infty$, and for $a, b \in {{{R}} ^ {d}}$ satisfying $w(\left\| a + b \right\|) \le w(\left\| a \right\|) + w(\left\| b \right\|)$. Examples of the weight functions are: $w(x) = a + bx$, $a, b > 0$ or $w(x) = {x} ^ {\alpha}$. The empirical version of the weighted ${L} ^ {p}$ depth is obtained by replacing distribution $F$ of ${X}$ in $Ew({{\left\| {x} - {X} \right\| }_{p}}) = \int {w({{\left\| x - t \right\| }_{p}})}dF(t)$ by its empirical counterpart calculated from the sample ${{{X}} ^ {n}}$...

The Projection and Tukey's depths are calculated using an approximate algorithm. Calculations of Mahalanobis, Euclidean and $L ^ p$ depths are exact. Returns the depth of multivariate point u with respect to data set X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

References

Liu, R.Y., Parelius, J.M. and Singh, K. (1999), Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 27, 783–858.

Mosler K (2013). Depth statistics. In C Becker, R Fried, K S (eds.), Robustness and Complex Data Structures, Festschrift in Honour of Ursula Gather, pp. 17–34. Springer.

Rousseeuw, P.J. and Struyf, A. (1998), Computing location depth and regression depth in higher dimensions, Stat. Comput., 8, 193–203.

Zuo, Y. and Serfling, R. (2000), General Notions of Statistical Depth Functions, Ann. Statist., 28, no. 2, 461–482.

See Also

depthContour and depthPersp for depth graphics.

Examples

library(robustbase)

# Calculation of Projection depth
data(starsCYG, package = "robustbase")
depth(t(colMeans(starsCYG)), starsCYG)

# Also for matrices
depth(starsCYG, starsCYG)

# Projection depth applied to a large bivariate data set
x <- matrix(rnorm(9999), nc = 3)
depth(x, x)

Description

Virtual class with structure for every depth class from depthproc package.

Slots

u

data set.

X

reference set.

method

depth type.

Description

Draws an approximate contours of depth for bivariate data.

Usage

depthContour(
  x,
  xlim = extendrange(x[, 1], f = 0.1),
  ylim = extendrange(x[, 2], f = 0.1),
  n = 50,
  pmean = TRUE,
  mcol = "blue",
  pdmedian = TRUE,
  mecol = "brown",
  legend = TRUE,
  points = FALSE,
  colors = heat_hcl,
  levels = 10,
  depth_params = list(),
  graph_params = list(),
  contour_method = c("auto", "convexhull", "contour")
)

Arguments

Details

The set of all points that have depth at least $\alpha$ is called $\alpha$-trimmed region. The $\alpha$-trimmed region w.r.t. $F$ is denoted by ${D}_{\alpha}(F)$, i.e.,

${D}_{\alpha}(F) = \left\{ z \in {{{R}} ^ {d}}:D(z, F) \ge \alpha\right\}.$

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

See Also

depthPersp

Examples

# EXAMPLE 1
set.seed(123)
x <- mvrnorm(1000, c(0, 0), diag(2))
depthContour(x, colors = gray.colors)
# with points
depthContour(x, points = TRUE)
depthContour(x, points = FALSE, levels = 10)

# EXAMPLE 2
data(inf.mort, maesles.imm)
data1990 <- na.omit(cbind(inf.mort[, 1], maesles.imm[, 1]))
depthContour(data1990, n = 50, pmean = TRUE, mcol = "blue",
             pdmedian = TRUE, mecol = "brown", legend = TRUE, points = TRUE,
             depth_params = list(method = "LP"),
             graph_params = list(
               xlab = "infant mortality rate per 1000 live birth",
               ylab = "against masles immunized percentage",
               main = "L2 depth, UN Fourth Goal 2011 year"))


#EXAMPLE 3
data("france")
depthContour(france,
    depth_params = list(method = "Tukey"),
    points = TRUE
)

Description

This page describes mechanism behavior of ScaleCurve and AsymmetryCurve

Details

DepthCurve is a virtual class that contains methods (getPlot(...) and plot(...)) for rendering single curve such as ScaleCurve or AsymmetryCurve. Such object can be combined by overloaded operator '

Slots

depth

object of Depth-class

name

name of dataset used on plot

title

title of a plot

alpha

central area values

Examples

library(mvtnorm)
x <- mvrnorm(n = 100, mu = c(0, 0), Sigma = 2 * diag(2))
y <- rmvt(n = 100, sigma = diag(2), df = 4)
s1 <- scaleCurve(x, depth_params = list(method = "Projection"))
s2 <- scaleCurve(y, depth_params = list(method = "Projection"), name = "Set2")

sc_list <- combineDepthCurves(s1, s2) # Add one curve to another

plot(sc_list) # Draw plot with two curves

z <- mvrnorm(n = 100, mu = c(0, 0), Sigma = 1 * diag(2))
s3 <- scaleCurve(z, depth_params = list(method = "Projection"))
plot(combineDepthCurves(sc_list, s3)) # Add third curve and draw a plot

Description

DepthCurveList is a special container for DepthCurve objects. See DepthCurve-class

Description

Experimental function used to fit depth weighted density estimator.

Usage

depthDensity(x, y, nx = 5, ny = 32, xg = NULL, yg = NULL, ...)

Arguments

References

Kosiorowski D. and Zawadzki Z. (2014) Notes on optimality of predictive distribution pseudo-estimators in the CHARME models and automatic trading strategies, FindEcon2014, submitted

Examples

## Not run: 
# .sampleData is special function for creating
# data for testing conditional denisty estimators
data <- DepthProc:::.sampleData(1:5, 100)
x <- data[, 1]
y <- data[, 2]
plot(x, y)
dep <- depthDensity(x, y)
plot(dep, type = "raw")
plot(dep, type = "depth")

## End(Not run)

Description

Class for depth based density estimator.

Details

depthDensity

Description

Computes the euclidean depth of a point or vectors of points with respect to a multivariate data set.

Usage

depthEuclid(u, X)

Arguments

Details

Calculation of Euclidean depth is exact.

Returns the depth of multivariate point u with respect to data set X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

x <- matrix(rnorm(9999), nc = 3)
depthEuclid(x, x)

Description

Computes local version of depth according to proposals of Paindaveine and Van Bever — see referencess.

Usage

depthLocal(
  u,
  X,
  beta = 0.5,
  depth_params1 = list(method = "Projection"),
  depth_params2 = depth_params1
)

Arguments

Details

A successful concept of local depth was proposed by Paindaveine and Van Bever (2012). For defining a neighbourhood of a point authors proposed using idea of symmetrisation of a distribution (a sample) with respect to a point in which depth is calculated. In their approach instead of a distribution ${P} ^ {X}$, a distribution ${{P}_{x}} = \frac{ 1 }{ 2 }{{P} ^ {X}} + \frac{ 1 }{ 2 }{{P} ^ {2x - X}}$ is used. For any $\beta \in [0, 1]$, let us introduce the smallest depth region bigger or equal to $\beta$,

${R} ^ {\beta}(F) = \bigcap\limits_{\alpha \in A(\beta)} {{{D}_{\alpha}}}(F),$

where $A(\beta) = \left\{ \alpha \ge 0:P\left[ {{D}_{\alpha}}(F)\right] \ge \beta\right\}$. Then for a locality parameter $\beta$ we can take a neighbourhood of a point $x$ as $R_{x} ^ {\beta}(P)$.

Formally, let $D(\cdot, P)$ be a depth function. Then the local depth with the locality parameter $\beta$ and w.r.t. a point $x$ is defined as

$L{{D} ^ {\beta}}(z, P):z \to D(z, P_{x} ^ {\beta}),$

where $P_{x} ^ {\beta}(\cdot) = P\left( \cdot |R_{x} ^ {\beta}(P)\right)$ is cond. distr. of $P$ conditioned on $R_{x} ^ {\beta}(P)$.

References

Paindaveine, D., Van Bever, G. (2013) From depth to local depth : a focus on centrality. Journal of the American Statistical Association 105, 1105–1119.

Examples

## Not run: 
# EXAMPLE 1
data <- mvrnorm(100, c(0, 5), diag(2) * 5)
# By default depth_params2 = depth_params1
depthLocal(data, data, depth_params1 = list(method = "LP"))
depthLocal(data, data, depth_params1 = list(method = "LP"),
           depth_params2 = list(method = "Projection"))
# Depth contour
depthContour(data, depth_params = list(method = "Local", depth_params1 = list(method = "LP")))

# EXAMPLE 2
data(inf.mort, maesles.imm)
data1990 <- na.omit(cbind(inf.mort[, 1], maesles.imm[, 1]))
depthContour(data1990,
             depth_params = list(
               method = "Local",
               depth_params1 = list(method = "LP"),
               beta = 0.3
             ))

# EXAMPLE 3
Sigma1 <- matrix(c(10, 3, 3, 2), 2, 2)
X1 <- mvrnorm(n = 8500, mu = c(0, 0), Sigma1)
Sigma2 <- matrix(c(10, 0, 0, 2), 2, 2)
X2 <- mvrnorm(n = 1500, mu = c(-10, 6), Sigma2)
BALLOT <- rbind(X1, X2)

train <- sample(1:10000, 100)
data <- BALLOT[train, ]
depthContour(data,
            depth_params = list(
              method = "Local",
              beta = 0.3,
              depth_params1 = list(method = "Projection")
            ))

## End(Not run)

Description

Computes the LP depth of a point or vectors of points with respect to a multivariate data set.

Usage

depthLP(u, X, pdim = 2, la = 1, lb = 1, threads = -1, func = NULL)

Arguments

Details

Returns the depth of multivariate point u with respect to data set X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

x <- matrix(rnorm(3000), ncol = 3)

# Same results
depthLP(x, x, pdim = 2)

Description

Computes the mahalanobis depth of a point or vectors of points with respect to a multivariate data set.

Usage

depthMah(u, X, cov = NULL, mean = NULL, threads = -1)

Arguments

Details

Calculation of Mahalanobis depth is exact.

Returns the depth of multivariate point u with respect to data set X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

x <- matrix(rnorm(9999), nc = 3)
depthMah(x, x)

Description

Return point with maximum depth function value. If multiple points have the same value, mean average of them will be returned.

Usage

depthMedian(x, depth_params = list(), convex = FALSE)

## S4 method for signature 'matrix'
depthMedian(x, depth_params = list(), convex = FALSE)

## S4 method for signature 'data.frame'
depthMedian(x, depth_params = list(), convex = FALSE)

## S4 method for signature 'Depth'
depthMedian(x, convex = FALSE)

Arguments

Examples

# depthMedian for matrix
x <- matrix(rnorm(600), nc = 3)
depthMedian(x)

# depthMedian works with object of class Depth
dp <- depth(x)
depthMedian(dp)

Description

Draws a perspective plot of depth function over x-y plane.

Usage

depthPersp(
  x,
  plot_method = "lattice",
  xlim = extendrange(x[, 1], f = 0.1),
  ylim = extendrange(x[, 2], f = 0.1),
  n = 50,
  xlab = "x",
  ylab = "y",
  plot_title = NULL,
  colors = heat_hcl,
  depth_params = list(),
  graph_params = list()
)

Arguments

Details

plot_method — rgl package is not in depends list beacuse it may cause problems when OpenGL is not supported. To use plot_method = "rgl" you must load this package on your own.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

# EXAMPLE 1
x <- mvrnorm(100, c(0, 0), diag(2))
depthPersp(x, depth_params = list(method = "Euclidean"))

# EXAMPLE 2
data(inf.mort, maesles.imm)
data1990 <- na.omit(cbind(inf.mort[, 1], maesles.imm[, 1]))

## Not run: 
library(rgl)
depthPersp(data1990, plot_method = "rgl",
           depth_params = list(method = "Projection"))

## End(Not run)

Description

Computes the Projection depth of a point or vectors of points with respect to a multivariate data set.

Usage

depthProjection(u, X, ndir = 1000, threads = -1)

Arguments

Details

Irrespective of dimension, Projection and Tukey's depth is obtained by approximate calculation.

Returns the depth of multivariate point u with respect to data set X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

x <- matrix(rnorm(3000), nc = 3)
a <- depthProjection(x, x, ndir = 2000)

Description

Computes the Tukey depth of a point or vectors of points with respect to a multivariate data set.

Usage

depthTukey(u, X, ndir = 1000, threads = -1, exact = FALSE)

Arguments

Details

Irrespective of dimension, Projection and Tukey's depth is obtained by approximate calculation.

Returns the depth of multivariate point u with respect to data set X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

## Not run: 
x <- matrix(rnorm(3000), nc = 3)
depthTukey(x, ndir = 2000)

## End(Not run)

# Exact algorithm in 2d
x <- matrix(rnorm(2000), nc = 2)
depthTukey(x, exact = TRUE)

Description

Functional boxplot based on Modified Band Depth

Usage

fncBoxPlot(u, X = NULL, bands = c(0, 0.5), method = "MBD", byrow = NULL, ...)

Arguments

Examples

# some data:
x <- matrix(rnorm(200), ncol = 10)

fncBoxPlot(x, bands = c(0, 0.5, 1), method = "FM")
fncBoxPlot(x, bands = c(0, 0.5, 1), method = "FM", byrow = FALSE)

colnames(x) <- paste0("f", 1:ncol(x))
fncBoxPlot(x, bands = c(0, 0.5, 1), method = "FM")

# fncBoxPlot handles zoo and xts objects
library(xts)
x <- matrix(rnorm(200), ncol = 10)
time <- as.POSIXct(1:ncol(x) * 86400, origin = "1970-01-01")
x_xts <- xts(t(x), order.by = time)
fncBoxPlot(x_xts, bands = c(0, 0.5, 1), method = "FM")

data("katowice.airpollution")
pl <- fncBoxPlot(katowice.airpollution, bands = c(0, 0.5, 1), method = "MBD")
pl + ggtitle("Air pollution in Katowice") + labs(y= "pollination ", x = "hour ")

Description

Calculates depth functions.

Usage

fncDepth(u, X = NULL, method = "MBD", byrow = NULL, ...)

## S3 method for class 'matrix'
fncDepth(u, X = NULL, method = "MBD", byrow = NULL, ...)

## S3 method for class 'zoo'
fncDepth(u, X = NULL, method = "MBD", byrow = NULL, ...)

Arguments

Examples

x <- matrix(rnorm(60), ncol = 20)
fncDepth(x, method = "FM", dep1d = "Mahalanobis")
fncDepth(x, byrow = FALSE)

# zoo and xts
library(xts)
data(sample_matrix)
sample.xts <- as.xts(sample_matrix, descr = "my new xts object")
fncDepth(sample.xts)

Description

Computes Frainman-Muniz depth for functional data.

Usage

fncDepthFM(u, X, dep1d_params = list(method = "Projection"))

Arguments

Examples

x <- matrix(rnorm(60), nc = 20)
fncDepthFM(x)

Description

Computes the modified band depth.

Usage

fncDepthMBD(u, X)

Arguments

Examples

x <- matrix(rnorm(60), nc = 20)
fncDepthMBD(x)
fncDepthMBD(x, x)

Description

Calculate functional median based on data depth.

Usage

fncDepthMedian(u, X = NULL, method = "MBD", byrow = NULL, unique = TRUE, ...)

Arguments

Examples

x <- matrix(rnorm(600), nc = 20)
md <- fncDepthMedian(x, method = "FM", dep1d = "Mahalanobis")

Description

Extract bands from functional depth object.

Usage

fncGetBand(obj, band = 0.5)

Arguments

Examples

x <- matrix(rnorm(600), nc = 20)
obj <- fncDepth(x, method = "FM", dep1d = "Mahalanobis")
fncGetBand(obj)

Description

Relation between minimum wage (MW) and unemployment rate (UR) in France.

Usage

data(france)

Format

data frame containing 17 rows and two column. MW is a minimum wage, and UR is an unemployment rate.

Description

Virtual class with structure for every functional depth class from depthproc package. Inherits from Depth-class.

Slots

index

numeric, or time-based object.

Description

Create an object of class ggplot from DepthCurve and DepthCurveList.

Usage

getPlot(object)

## S4 method for signature 'AsymmetryCurveList'
getPlot(object)

## S4 method for signature 'DDPlot'
getPlot(object)

## S4 method for signature 'ScaleCurveList'
getPlot(object)

Arguments

Description

Infant mortality rate (0–1 year) per 1,000 live births

Usage

data(inf.mort)

Format

A data frame with 654 rows and 4 variables

Source

http://mdgs.un.org/unsd/mdg/Data.aspx

Description

Internet view data

Usage

data(internet.users)

Format

data frame containing 17518 rows and 6 columns — 17518 working days of the Internet service considered with respect to variables: service, month, day, hour, unique users and page views.

References

Kosiorowski, Rydlewski, Snarska (2016), Detecting a Structural Change in Functional Time Series Using Local Wilcoxon Statistic, arXiv: 1604.03776v2

Description

Air pollution in Katowice city by hour.

Usage

data("katowice.airpollution")

Format

data frame containing 181 rows (days) and 24 columns. Each column is an air pollination for given hour.

Description

This function add one location-scale contour to the existing plot.

Usage

lsdAddContour(x, cont = NULL, ...)

## S4 method for signature 'LSDepthContour'
lsdAddContour(x, cont = NULL, ...)

Arguments

Examples

smp <- rf(100, 5, 10)
x <- lsdSampleDepthContours(smp)
plot(x)
lsdAddContour(x, 0.1, col = "grey50")
lsdAddContour(x, 0.3, col = "grey10", border = "red", lwd = 4)

Description

Class used to store maximum location-scale depth results.

Slots

max_depth

maximum Student depth value.

mu

location estimate in the deepest point.

sigma

scale estimate in the deepest point.

Description

Class used to store result of location-scale depth contours.

Slots

cont_depth

depth values used to calculate contours.

sample

original sample used to calculate depth contours.

.Data

list with estimated values of scale-depth contours.

Description

Get numeric values of the location-scale depth contour from existing object of LSDepthContour class.

Usage

lsdGetContour(x, cont)

## S4 method for signature 'LSDepthContour'
lsdGetContour(x, cont)

Arguments

Details

Calculations are based on lsdepth algorithm written by Ch. Muller.

Examples

dcont <- lsdSampleDepthContours(rf(200, 4, 7), depth = c(0.1, 0.2))

# get contour that is present in dcont object
lsdGetContour(dcont, 0.1)

# get contour that is not present in dcont
# it will be automatically calculated
lsdGetContour(dcont, 0.3)

Description

Calculate sample one-dimensional Mizera and Muller Student depth contours.

Usage

lsdSampleDepthContours(x, depth = c(0.1, 0.2, 0.3, 0.4), lengthmu = 1000)

Arguments

Details

Calculations are based on lsdepth algorithm written by Ch. Muller.

References

Mizera, I., Muller, C. H., 2004. Location-scale depth (with discussion). Journal of the American Statistical Association 99, 949–966.

Examples

# EXAMPLE 1
# F-distribution
dcont <- lsdSampleDepthContours(rf(200, 4, 7))
plot(dcont)

# EXAMPLE 2
# normal distribution - more contours calculated
dcont_norm <- lsdSampleDepthContours(rnorm(100), seq(0.05, 0.4, 0.05))
plot(dcont_norm)

Description

Calculates the maximum Student depth estimator of location and scale for one dimensional data (an alternative for MED and MAD or for the mean and standard deviation).

Usage

lsdSampleMaxDepth(x, iter = 100, eps = 1e-04, p_length = 10)

Arguments

Details

Calculations are based on lsdepth algorithm written by Ch. Muller.

References

Mizera, I., Muller, C. H., 2004. Location-scale depth (with discussion). Journal of the American Statistical Association 99, 949–966.

Examples

x <- rnorm(100)
lsdSampleMaxDepth(x)
y <- rf(100, 4, 10)
lsdSampleMaxDepth(y)

Description

Children 1 year old immunized against measles, percentage

Usage

data(maesles.imm)

Format

A data frame with 654 rows and 4 variables

Source

http://mdgs.un.org/unsd/mdg/Data.aspx

Description

Depth based multivariate Wilcoxon test for a scale difference.

Usage

mWilcoxonTest(x, y, alternative = "two.sided", depth_params = list())

Arguments

Details

Having two samples ${X} ^ {n}$ and ${Y} ^ {m}$ using any depth function, we can compute depth values in a combined sample ${Z} ^ {n + m} = {X} ^ {n} \cup {Y} ^ {m}$, assuming the empirical distribution calculated basing on all observations, or only on observations belonging to one of the samples ${X} ^ {n}$ or ${Y} ^ {m}$.

For example if we observe ${X}_{l}'s$ depths are more likely to cluster tightly around the center of the combined sample, while ${Y}_{l}'s$ depths are more likely to scatter outlying positions, then we conclude ${Y} ^ {m}$ was drawn from a distribution with larger scale.

Properties of the DD plot based statistics in the i.i.d setting were studied by Li & Liu (2004). Authors proposed several DD-plot based statistics and presented bootstrap arguments for their consistency and good effectiveness in comparison to Hotelling $T ^ 2$ and multivariate analogues of Ansari-Bradley and Tukey-Siegel statistics. Asymptotic distributions of depth based multivariate Wilcoxon rank-sum test statistic under the null and general alternative hypotheses were obtained by Zuo & He (2006). Several properties of the depth based rang test involving its unbiasedness was critically discussed by Jureckova & Kalina (2012). Basing on DD-plot object, which is available within the DepthProc it is possible to define several multivariate generalizations of one-dimensional rank and order statistics in an easy way. These generalizations cover well known Wilcoxon rang-sum statistic.

The depth based multivariate Wilcoxon rang sum test is especially useful for the multivariate scale changes detection and was introduced among other by Liu & Singh (2003) and intensively studied by Jureckowa & Kalina (2012) and Zuo & He (2006) in the i.i.d. setting.

For the samples ${{{X}} ^ {m}} = \{{{{X}}_{1}}, ..., {{{X}}_{m}}\}$, ${{{Y}} ^ {n}} = \{{{{Y}}_{1}}, ..., {{{Y}}_{n}}\}$, their $d_{1} ^ {X}, ..., d_{m} ^ {X}$, $d_{1} ^ {Y}, ..., d_{n} ^ {Y}$, depths w.r.t. a combined sample ${{Z}} = {{{X}} ^ {n}} \cup {{{Y}} ^ {m}}$ the Wilcoxon statistic is defined as $S = \sum\limits_{i = 1} ^ {m}{{{R}_{i}}}$, where ${R}_{i}$ denotes the rang of the i-th observation, $i = 1, ..., m$ in the combined sample $R({{{y}}_{l}}) = \sharp\left\{ {{{z}}_{j}} \in {{{Z}}}:D({{{z}}_{j}}, {{Z}}) \le D({{{y}}_{l}}, {{Z}}) \right\}, l = 1, ..., m$.

The distribution of $S$ is symmetric about $E(S) = \frac{ 1 }{ 2 }m(m + n + 1)$, its variance is ${{D} ^ {2}}(S) = \frac{ 1 }{ 12 }mn(m + n + 1)$.

References

Jureckova J, Kalina J (2012). Nonparametric multivariate rank tests and their unbiasedness. Bernoulli, 18(1), 229–251. Li J, Liu RY (2004). New nonparametric tests of multivariate locations and scales using data depth. Statistical Science, 19(4), 686–696. Liu RY, Singh K (1995). A quality index based on data depth and multivariate rank tests. Journal of American Statistical Association, 88, 252–260. Zuo Y, He X (2006). On the limiting distributions of multivariate depth-based rank sum statistics and related tests. The Annals of Statistics, 34, 2879–2896.

Examples

# EXAMPLE 1
x <- mvrnorm(100, c(0, 0), diag(2))
y <- mvrnorm(100, c(0, 0), diag(2) * 1.4)
mWilcoxonTest(x, y)
mWilcoxonTest(x, y, depth_params = list(method = "LP"))

# EXAMPLE 2
data(under5.mort)
data(inf.mort)
data(maesles.imm)
data2011 <- na.omit(cbind(under5.mort[, 22], inf.mort[, 22],
                          maesles.imm[, 22]))
data1990 <- na.omit(cbind(under5.mort[, 1], inf.mort[, 1], maesles.imm[, 1]))
mWilcoxonTest(data2011, data1990)

Description

Plot Depth curve

Usage

plot(x, y, ...)

## S4 method for signature 'DDPlot,ANY'
plot(x)

## S4 method for signature 'DepthCurve,ANY'
plot(x)

## S4 method for signature 'DepthCurveList,ANY'
plot(x)

Arguments

Examples

x <- mvrnorm(n = 100, mu = c(0, 0), Sigma = 3 * diag(2))
sc <- scaleCurve(x)
plot(sc)

Description

Binning 2d

Usage

## S4 method for signature 'BinnDepth2d,ANY'
plot(x, ..., alpha = 0.1, bg_col = "red", add_mid = TRUE)

Arguments

See Also

depth

Examples

tmp <- binningDepth2D(x = mvrnorm(100, rep(0, 2), diag(2)))
plot(tmp)

Description

Create plot for DepthDensity. See depthDensity for more information.

Usage

## S4 method for signature 'DepthDensity,ANY'
plot(x, type = "depth", ...)

Arguments

Description

Create location-scale depth plot. See lsdSampleDepthContours for more information.

Usage

## S4 method for signature 'LSDepthContour,ANY'
plot(
  x,
  cont = NULL,
  ratio = 1,
  mu_min = NULL,
  mu_max = NULL,
  col = NULL,
  border = NULL,
  ...
)

Arguments

Examples

smp <- rf(100, 5, 10)
x <- lsdSampleDepthContours(smp)
plot(x, col = paste0("grey", col = rev(seq(10, 40, 10))))

Description

Virtual class for robust regression methods from depthproc package

Slots

coef

coefficients of fitted model

Description

This function generates random numbers from p-dimensional unit sphere.

Usage

runifsphere(n, p = 2)

Arguments

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

Examples

x <- runifsphere(n = 100)
plot(x)

Description

Draws a scale curve: measure of dispersion.

Usage

scaleCurve(
  x,
  y = NULL,
  alpha = seq(0, 1, 0.01),
  name = "X",
  name_y = "Y",
  title = "Scale Curve",
  depth_params = list(method = "Projection")
)

Arguments

Details

For sample depth function $D({x}, {{{Z}} ^ {n}})$, ${x} \in {{{R}} ^ {d}}$, $d \ge 2$, ${Z} ^ {n} = \{{{{z}}_{1}}, ..., {{{z}}_{n}}\} \subset {{{R}} ^ {d}}$, ${{D}_{\alpha}}({{{Z}} ^ {n}})$ denoting $\alpha$ — central region, we can define the scale curve $SC(\alpha) = \left(\alpha, vol({{D}_{\alpha}}({{{Z}} ^ {n}})\right) \subset {{{R}} ^ {2}}$, for $\alpha \in [0, 1]$

The scale curve is a two-dimensional method of describing the dispersion of random vector around the depth induced median.

Function scalecurve for determining the volumes of the convex hull containing points from alpha central regions, uses function convhulln from geometry package.

The minimal dimension of data in X or Y is 2.

ggplot2 package is used to draw a plot.

Value

Returns the volume of the convex hull containing subsequent central points of X.

Author(s)

Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.

References

Liu, R.Y., Parelius, J.M. and Singh, K. (1999), Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 27, 783–858.

Chaudhuri, P. (1996), On a Geometric Notion of Quantiles for Multivariate Data, Journal of the American Statistical Association, 862–872.

Dyckerhoff, R. (2004), Data Depths Satisfying the Projection Property, Allgemeines Statistisches Archiv., 88, 163–190.

See Also

depthContour and depthPersp for depth graphics.

Examples

library(mvtnorm)
x <- mvrnorm(n = 100, mu = c(0, 0), Sigma = 3 * diag(2))
y <- rmvt(n = 100, sigma = diag(2), df = 2)
scaleCurve(x, y, depth_params = list(method = "Projection"))
# Comparing two scale curves
# normal distribution and mixture of normal distributions
x <- mvrnorm(100, c(0, 0), diag(2))
y <- mvrnorm(80, c(0, 0), diag(2))
z <- mvrnorm(20, c(5, 5), diag(2))
scaleCurve(x, rbind(y, z), name = "N", name_y = "Mixture of N",
           depth_params = list(method = "Projection"))