monomvn: Maximum Likelihood Estimation for Multivariate Normal Data...

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Description

Maximum likelihood estimation of the mean and covariance matrix of multivariate normal (MVN) distributed data with a monotone missingness pattern. Through the use of parsimonious/shrinkage regressions (e.g., plsr, pcr, ridge, lasso, etc.), where standard regressions fail, this function can handle an (almost) arbitrary amount of missing data

Usage

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monomvn(y, pre = TRUE, method = c("plsr", "pcr", "lasso", "lar",
        "forward.stagewise", "stepwise", "ridge", "factor"), p = 0.9,
        ncomp.max = Inf, batch = TRUE, validation = c("CV", "LOO", "Cp"),
        obs = FALSE, verb = 0, quiet = TRUE)

Arguments

y

data matrix were each row is interpreted as a random sample from a MVN distribution with missing values indicated by NA

pre

logical indicating whether pre-processing of the y is to be performed. This sorts the columns so that the number of NAs is non-decreasing with the column index

method

describes the type of parsimonious (or shrinkage) regression to be performed when standard least squares regression fails. From the pls package we have "plsr" (plsr, the default) for partial least squares and "pcr" (pcr) for standard principal component regression. From the lars package (see the "type" argument to lars) we have "lasso" for L1-constrained regression, "lar" for least angle regression, "forward.stagewise" and "stepwise" for fast implementations of classical forward selection of covariates. From the MASS package we have "ridge" as implemented by the lm.ridge function. The "factor" method treats the first p columns of y as known factors

p

when performing regressions, p is the proportion of the number of columns to rows in the design matrix before an alternative regression method (those above) is performed as if least-squares regression has “failed”. Least-squares regression is known to fail when the number of columns equals the number of rows, hence a default of p = 0.9 <= 1. Alternatively, setting p = 0 forces method to be used for every regression. Intermediate settings of p allow the user to control when least-squares regressions stop and the method ones start. When method = "factor" the p argument represents an integer (positive) number of initial columns of y to treat as known factors

ncomp.max

maximal number of (principal) components to include in a method—only meaningful for the "plsr" or "pcr" methods. Large settings can cause the execution to be slow as it drastically increases the cross-validation (CV) time

batch

indicates whether the columns with equal missingness should be processed together using a multi-response regression. This is more efficient if many OLS regressions are used, but can lead to slightly poorer, even unstable, fits when parsimonious regressions are used

validation

method for cross validation when applying a parsimonious regression method. The default setting of "CV" (randomized 10-fold cross-validation) is the faster method, but does not yield a deterministic result and does not apply for regressions on less than ten responses. "LOO" (leave-one-out cross-validation) is deterministic, always applicable, and applied automatically whenever "CV" cannot be used. When standard least squares is appropriate, the methods implemented the lars package (e.g. lasso) support model choice via the "Cp" statistic, which defaults to the "CV" method when least squares fails. This argument is ignored for the "ridge" method; see details below

obs

logical indicating whether or not to (additionally) compute a mean vector and covariance matrix based only on the observed data, without regressions. I.e., means are calculated as averages of each non-NA entry in the columns of y, and entries (a,b) of the covariance matrix are calculated by applying cov(ya,yb) to the jointly non-NA entries of columns a and b of y

verb

whether or not to print progress indicators. The default (verb = 0) keeps quiet, while any positive number causes brief statement about dimensions of each regression to print to the screen as it happens. verb = 2 causes each of the ML regression estimators to be printed along with the corresponding new entries of the mean and columns of the covariance matrix. verb = 3 requires that the RETURN key be pressed between each print statement

quiet

causes warnings about regressions to be silenced when TRUE

Details

If pre = TRUE then monomvn first re-arranges the columns of y into nondecreasing order with respect to the number of missing (NA) entries. Then (at least) the first column should be completely observed. The mean components and covariances between the first set of complete columns are obtained through the standard mean and cov routines.

Next each successive group of columns with the same missingness pattern is processed in sequence (assuming batch = TRUE). Suppose a total of j columns have been processed this way already. Let y2 represent the non-missing contingent of the next group of k columns of y with and identical missingness pattern, and let y1 be the previously processed j-1 columns of y containing only the rows corresponding to each non-NA entry in y2. I.e., nrow(y1) = nrow(y2). Note that y1 contains no NA entries since the missing data pattern is monotone. The k next entries (indices j:(j+k)) of the mean vector, and the j:(j+k) rows and columns of the covariance matrix are obtained by multivariate regression of y2 on y1. The regression method used (except in the case of method = "factor" depends on the number of rows and columns in y1 and on the p parameter. Whenever ncol(y1) < p*nrow(y1) least-squares regression is used, otherwise method = c("pcr", "plsr"). If ever a least-squares regression fails due to co-linearity then one of the other methods is tried. The "factor" method always involves an OLS regression on (a subset of) the first p columns of y.

All methods require a scheme for estimating the amount of variability explained by increasing the numbers of coefficients (or principal components) in the model. Towards this end, the pls and lars packages support 10-fold cross validation (CV) or leave-one-out (LOO) CV estimates of root mean squared error. See pls and lars for more details. monomvn uses CV in all cases except when nrow(y1) <= 10, in which case CV fails and LOO is used. Whenever nrow(y1) <= 3 pcr fails, so plsr is used instead. If quiet = FALSE then a warning is given whenever the first choice for a regression fails.

For pls methods, RMSEs are calculated for a number of components in 1:ncomp.max where a NULL value for ncomp.max it is replaced with

ncomp.max <- min(ncomp.max, ncol(y2), nrow(y1)-1)

which is the max allowed by the pls package.

Simple heuristics are used to select a small number of components (ncomp for pls), or number of coefficients (for lars), which explains a large amount of the variability (RMSE). The lars methods use a “one-standard error rule” outlined in Section 7.10, page 216 of HTF below. The pls package does not currently support the calculation of standard errors for CV estimates of RMSE, so a simple linear penalty for increasing ncomp is used instead. The ridge constant (lambda) for lm.ridge is set using the optimize function on the GCV output.

Based on the ML ncol(y1)+1 regression coefficients (including intercept) obtained for each of the columns of y2, and on the corresponding matrix of residual sum of squares, and on the previous j-1 means and rows/cols of the covariance matrix, the j:(j+k) entries and rows/cols can be filled in as described by Little and Rubin, section 7.4.3.

Once every column has been processed, the entries of the mean vector, and rows/cols of the covariance matrix are re-arranged into their original order.

Value

monomvn returns an object of class "monomvn", which is a list containing a subset of the components below.

call

a copy of the function call as used

mu

estimated mean vector with columns corresponding to the columns of y

S

estimated covariance matrix with rows and columns corresponding to the columns of y

na

when pre = TRUE this is a vector containing number of NA entries in each column of y

o

when pre = TRUE this is a vector containing the index of each column in the sorting of the columns of y obtained by o <- order(na)

method

method of regression used on each column, or "complete" indicating that no regression was necessary

ncomp

number of components in a plsr or pcr regression, or NA if such a method was not used. This field is used to record lambda when lm.ridge is used

lambda

if method is one of c("lasso", "forward.stagewise", "ridge"), then this field records the lambda penalty parameters used

mu.obs

when obs = TRUE this is the “observed” mean vector

S.obs

when obs = TRUE this is the “observed” covariance matrix, as described above. Note that S.obs is usually not positive definite

Note

The CV in plsr and lars are random in nature, and so can be dependent on the random seed. Use validation=LOO for deterministic (but slower) result.

When using method = "factor" in the current version of the package, the factors in the first p columns of y must also obey the monotone pattern, and, have no more NA entries than the other columns of y.

Be warned that the lars implementation of "forward.stagewise" can sometimes get stuck in (what seems like) an infinite loop. This is not a bug in the monomvn package; the bug has been reported to the authors of lars

Author(s)

Robert B. Gramacy rbg@vt.edu

References

Robert B. Gramacy, Joo Hee Lee, and Ricardo Silva (2007). On estimating covariances between many assets with histories of highly variable length.
Preprint available on arXiv:0710.5837: http://arxiv.org/abs/0710.5837

Roderick J.A. Little and Donald B. Rubin (2002). Statistical Analysis with Missing Data, Second Edition. Wilely.

Bjorn-Helge Mevik and Ron Wehrens (2007). The pls Package: Principal Component and Partial Least Squares Regression in R. Journal of Statistical Software 18(2)

Bradley Efron, Trevor Hastie, Ian Johnstone and Robert Tibshirani (2003). Least Angle Regression (with discussion). Annals of Statistics 32(2); see also
http://www-stat.stanford.edu/~hastie/Papers/LARS/LeastAngle_2002.pdf

Trevor Hastie, Robert Tibshirani and Jerome Friedman (2002). Elements of Statistical Learning. Springer, NY. [HTF]

Some of the code for monomvn, and its subroutines, was inspired by code found on the world wide web, written by Daniel Heitjan. Search for “fcn.q”

http://bobby.gramacy.com/r_packages/monomvn

See Also

bmonomvn, em.norm in the now defunct norm and mvnmle packages

Examples

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## standard usage, duplicating the results in
## Little and Rubin, section 7.4.3 -- try adding 
## verb=3 argument for a step-by-step breakdown
data(cement.miss)
out <- monomvn(cement.miss)
out
out$mu
out$S

##
## A bigger example, comparing the various methods
##

## generate N=100 samples from a 10-d random MVN
xmuS <- randmvn(100, 20)

## randomly impose monotone missingness
xmiss <- rmono(xmuS$x)

## plsr
oplsr <- monomvn(xmiss, obs=TRUE)
oplsr
Ellik.norm(oplsr$mu, oplsr$S, xmuS$mu, xmuS$S)

## calculate the complete and observed RMSEs
n <- nrow(xmiss) - max(oplsr$na)
x.c <- xmiss[1:n,]
mu.c <- apply(x.c, 2, mean)
S.c <- cov(x.c)*(n-1)/n
Ellik.norm(mu.c, S.c, xmuS$mu, xmuS$S)
Ellik.norm(oplsr$mu.obs, oplsr$S.obs, xmuS$mu, xmuS$S)

## plcr
opcr <- monomvn(xmiss, method="pcr")
Ellik.norm(opcr$mu, opcr$S, xmuS$mu, xmuS$S)

## ridge regression
oridge <- monomvn(xmiss, method="ridge")
Ellik.norm(oridge$mu, oridge$S, xmuS$mu, xmuS$S)

## lasso
olasso <- monomvn(xmiss, method="lasso")
Ellik.norm(olasso$mu, olasso$S, xmuS$mu, xmuS$S)

## lar
olar <- monomvn(xmiss, method="lar")
Ellik.norm(olar$mu, olar$S, xmuS$mu, xmuS$S)

## forward.stagewise
ofs <- monomvn(xmiss, method="forward.stagewise")
Ellik.norm(ofs$mu, ofs$S, xmuS$mu, xmuS$S)

## stepwise
ostep <- monomvn(xmiss, method="stepwise")
Ellik.norm(ostep$mu, ostep$S, xmuS$mu, xmuS$S)

Example output

Loading required package: pls

Attaching package: 'pls'

The following object is masked from 'package:stats':

    loadings

Loading required package: lars
Loaded lars 1.2

Loading required package: MASS

Call:
monomvn(y = cement.miss)

Methods used (p=0.9):
2	complete
3	lsr

        [,1]
x1  6.655166
x2 49.965259
x3 11.769231
x4 27.047089
y  95.423077
          x1         x2        x3         x4          y
x1  23.80287   22.26922 -26.97542  -12.28292   50.86578
x2  22.26922  258.86275 -17.13549 -273.80147  211.90393
x3 -26.97542  -17.13549  41.02564  -10.39915  -51.51923
x4 -12.28292 -273.80147 -10.39915  319.38777 -206.48170
y   50.86578  211.90393 -51.51923 -206.48170  226.31359

Call:
monomvn(y = xmiss, obs = TRUE)

Methods used (p=0.9):
1	complete
17	lsr
2	plsr-CV, ncomp range: [4,7]

[1] -24423.72
[1] NA
[1] NA
[1] -24421.08
[1] -24422.77
[1] -24947.67
[1] -24421.34
[1] -24421.53
[1] -24423.06

monomvn documentation built on Dec. 1, 2019, 1:10 a.m.