LogConcDEAD (version 1.4-0)

mlelcd: Compute the maximum likelihood estimator of a log-concave density

Description

Uses Shor's $r$-algorithm to compute the maximum likelihood estimator of a log-concave density based on an i.i.d. sample. The estimator is uniquely determined by its value at the data points. The output is an object of class "LogConcDEAD" which contains all the information needed to plot the estimator using the plot method, or to evaluate it using the function dlcd.

Usage

mlelcd(x, w=rep(1/nrow(x),nrow(x)), y=initialy(x),
  verbose=-1, alpha=5, c=1, sigmatol=10^-8, integraltol=10^-4,
  ytol=10^-4, stepscale=5.1, stepscale2=2, stepscale3=1.5,
  stepscale4=1.05, desiredsize=3.3, Jtol=0.001, chtol=10^-6)
lcd.mle(x, w=rep(1/nrow(x),nrow(x)), y=initialy(x),
  verbose=-1, alpha=5, c=1, sigmatol=10^-8, integraltol=10^-4,
  ytol=10^-4, stepscale=5.1, stepscale2=2, stepscale3=1.5,
  stepscale4=1.05, desiredsize=3.3, Jtol=0.001, chtol=10^-6)

Arguments

x

Data in $R^d$, in the form of an $n \times d$ numeric matrix

w

Vector of weights $w_i$ such that the computed estimator maximizes $$\sum_{i=1}^n w_i \log f(x_i)$$

y

Vector giving starting point for the $r$-algorithm. If none given, a kernel estimate is used.

verbose

• -1: (default) prints nothing\
• 0: prints warning messages
• $n>0$: prints summary information every$n$iterations

alpha

Scalar parameter for SolvOpt

c

Scalar giving starting step size

sigmatol

Real-valued scalar giving one of the stopping criteria: Relative change in $\sigma$ must be below sigmatol for algorithm to terminate. (See Details)

ytol

Real-valued scalar giving on of the stopping criteria: Relative change in $y$ must be below ytol for algorithm to terminate. (See Details)

integraltol

Real-valued scalar giving one of the stopping criteria: $| 1 - \exp(\bar{h}_y) |$ must be below integraltol for algorithm to terminate. (See Details)

stepscale, stepscale2, stepscale3, stepscale4, desiredsize

Scalar parameters for SolvOpt. Changing these is not recommended.

Jtol

Parameter controlling when Taylor expansion is used in computing the function $\sigma$

chtol

Parameter controlling convex hull computations

Value

• An object of class "LogConcDEAD", with the following components:
• x
  Data copied from input (may be reordered)
• w
  weights copied from input (may be reordered)
• logMLE
  vector of the log of the maximum likelihood estimate, evaluated at the observation points
• NumberOfEvaluations
  Vector containing the number of steps, number of function evaluations, and number of subgradient evaluations. If the SolvOpt algorithm fails, the first component will be an error code $(<0)$.< description="">
• MinSigma
  Real-valued scalar giving minimum value of the objective function
• b
  matrix (see Details)
• beta
  vector (see Details)
• triang
  matrix containing final triangulation of the convex hull of the data
• verts
  matrix containing details of triangulation for use in dlcd
• vertsoffset
  matrix containing details of triangulation for use in dlcd
• chull
  Vector containing vertices of faces of the convex hull of the data
• outnorm
  matrix where each row is an outward pointing normal vectors for the faces of the convex hull of the data. The number of vectors depends on the number of faces of the convex hull.
• outoffset
  matrix where each row is a point on a face of the convex hull of the data. The number of vectors depends on the number of faces of the convex hull.

Details

The log-concave maximum likelihood density estimator based on data $X_1, \ldots, X_n$ is the function that maximizes $$\sum_{i=1}^n w_i \log f(X_i)$$ subject to the constraint that $f$ is log-concave. For i.i.d.~data, the weights $w_i$ should be $\frac{1}{n}$ for each $i$. This is a function of the form $\bar{h}_y$ for some $y \in R^n$, where $$\bar{h}_y(x) = \inf \lbrace h(x) \colon h \textrm{ concave }, h(x_i) \geq y_i \textrm{ for } i = 1, \ldots, n \rbrace.$$

Functions of this form may equivalently be specified by dividing $C_n$, the convex hull of the data, into simplices $C_j$ for $j \in J$ (triangles in 2d, tetrahedra in 3d etc), and setting $$f(x) = \exp{b_j^T x - \beta_j}$$ for $x \in C_j$, and $f(x) = 0$ for $x \notin C_n$. This function uses Shor's $r$-algorithm (an iterative subgradient-based procedure) to minimize over vectors $y$ in $R^n$ the function $$\sigma(y) = -\frac{1}{n} \sum_{i=1}^n y_i + \int \exp(\bar{h}_y(x)) \, dx.$$ This is equivalent to finding the log-concave maximum likelihood estimator, as demonstrated in Cule, Samworth and Stewart (2008). An implementation of Shor's $r$-algorithm based on SolvOpt is used.

Computing $\sigma$ makes use of the qhull library, adapted from the Rimplementation in geometry. Code from this package is copied here as it is not currently possible to use compiled code from another package. For points not in general position, this requires a Taylor expansion of $\sigma$, discussed in Cule and D"umbgen (2008).

lcd.mle is deprecated, but retained for compatibility with previous versions.

References

Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T. (1996) The Quickhull algorithm for convex hulls ACM Trans. on Mathematical Software, 22(4) p.469-483 http://www.qhull.org

Cule, M. L. and D"umbgen, L. (2008) On an auxiliary function for log-density estimation, University of Bern technical report. http://arxiv.org/abs/0807.4719

Cule, M. L., Samworth, R. J., and Stewart, M. I. (2007) Maximum likelihood estimation of a log-concave density, Submitted.http://arxiv.org/abs/0804.3989

Kappel, F. and Kuntsevich, A. V. (2000) An implementation of Shor's r-algorithm Computational Optimization and Applications 15 http://www.uni-graz.at/imawww/kuntsevich/solvopt/

Shor, N. Z. (1985) Minimization methods for nondifferentiable functions Springer-Verlag

See Also

convhulln, logcondens, interplcd, plot.LogConcDEAD, interpmarglcd, rlcd, dlcd, dmarglcd

Examples

## Some simple normal data, and a few plots

x <- matrix(rnorm(200),ncol=2)
lcd <- mlelcd(x)
g <- interplcd(lcd)
par(mfrow=c(2,2), ask=TRUE)
plot(lcd, g=g, type="c")
plot(lcd, g=g, type="c", uselog=TRUE)
plot(lcd, g=g, type="i")
plot(lcd, g=g, type="i", uselog=TRUE)

## Some plots of marginal estimates
par(mfrow=c(1,1))
g.marg1 <- interpmarglcd(lcd, marg=1)
g.marg2 <- interpmarglcd(lcd, marg=2)
plot(lcd, marg=1, g.marg=g.marg1)
plot(lcd, marg=2, g.marg=g.marg2) 

## generate some points from the fitted density
generated <- rlcd(100, lcd)
genmean <- mean(generated)

## evaluate the fitted density
mypoint <- c(0, 0)
dlcd(mypoint, lcd, uselog=FALSE)
mypoint <- c(10, 0)
dlcd(mypoint, lcd, uselog=FALSE)

## evaluate the marginal density
dmarglcd(0, lcd, marg=1)
dmarglcd(1, lcd, marg=2)