activeSetLogCon {logcondens} | R Documentation |
Given a vector of observations {x} = (x_1, ..., x_m) with pairwise distinct entries and
a vector of weights {w}=(w_1, ..., w_m) s.t. sum_{i=1}^m w_i = 1, activeSetLogCon
computes a concave, piecewise
linear function widehat varphi_m on [x_1, x_m] with knots only in {x_1, ..., x_m} such that
L(varphi) = sum_{i=1}^m w_i varphi(x_i) - int_{-infty}^infty exp(varphi(t)) dt
is maximal. To accomplish this, an active set algorithm is used.
activeSetLogCon(x, w = NA, print = FALSE)
x |
Vector of independent and identically distributed numbers, with strictly increasing entries. |
w |
Optional vector of nonnegative weights corresponding to {x}, where w_1 > 0 and w_m > 0. These raw weights are normalized in order to sum to one. Default: w_i = 1 / m. |
print |
print = TRUE outputs log-likelihood in every loop, print = FALSE does not. Make sure to tell R to output (press CTRL+W). |
x |
Column vector with observations x_1, ..., x_m. |
phi |
Column vector with entries widehat varphi_m(x_i). |
IsKnot |
Column vector with entries IsKnot_i = 1{widehat varphi_m has a kink at x_i}. |
L |
The value L(widehat {varphi}_m) of the log-likelihood-function L at the maximum widehat {varphi}_m. |
F |
A vector (widehat F_{m,i})_{i=1}^m of the same size as {x} with entries
widehat F_{m,i} = int_{x_1}^{x_i} exp(widehat varphi_m(t)) dt. |
H |
Column vector (H_1, ..., H_m)' where H_i is the derivative of
t to L(varphi + tDelta_i)
|
Kaspar Rufibach, kaspar.rufibach@gmail.com
Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.staff.unibe.ch/duembgen
Duembgen, L, Huesler, A. and Rufibach, K. (2007) Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.
Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log–concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.
The following functions are used by activeSetLogCon
:
J00
, J10
, J11
, J20
,
Local_LL
, Local_LL_all
, LocalCoarsen
,
LocalConvexity
, LocalExtend
, LocalF
, LocalMLE
,
LocalNormalize
, MLE
Log concave density estimation via an iterative convex minorant algorithm can be performed using
icmaLogCon
.
## estimate gamma density set.seed(1977) x <- sort(rgamma(200, 2, 1)) res <- activeSetLogCon(x, w = NA, print = FALSE) ## plot resulting functions par(mfrow = c(2, 2), mar = c(3, 2, 1, 2)) plot(x, exp(res$phi), type = 'l'); rug(x) plot(x, res$phi, type = 'l'); rug(x) plot(x, res$Fhat, type = 'l'); rug(x) plot(x, res$H, type = 'l'); rug(x) ## compute and plot function values at an arbitrary point x0 <- (x[100] + x[101]) / 2 Fx0 <- evaluateLogConDens(x0, x, res$phi, res$Fhat, res$IsKnot)[3] plot(x, res$Fhat, type = 'l'); rug(x) abline(v = x0, lty = 3); abline(h = Fx0, lty = 3) ## compute and plot 0.9-quantile of Fhat q <- quantilesLogConDens(0.9, x, res$phi, res$Fhat) plot(x, res$Fhat, type = 'l'); rug(x) abline(h = 0.9, lty = 3); abline(v = q, lty = 3)