spregmix: EM-like Algorithm for Semiparametric Mixtures of Regressions

Description

Returns parameter estimates for finite mixtures of linear regressions with unspecified error structure. Based on Hunter and Young (2012).

Usage

spregmix(lmformula, bw = NULL, constbw = FALSE,
         bwmult = 0.9, z.hat = NULL, symm = TRUE, betamethod = "LS",
         m = ifelse(is.null(z.hat), 2, ncol(z.hat)),
         epsilon = 1e-04, maxit = 1000, verbose = FALSE, 
         ...)

Value

regmixEM returns a list of class npEM with items:

x: The set of predictors (which includes a column of 1's if addintercept = TRUE).
y: The response values.
lambda: The mixing proportions for every iteration in the form of a matrix with m columns and (#iterations) rows
beta: The final regression coefficients.
posterior: An nxm matrix of posterior probabilities for observations.
np.stdev: Nonparametric estimate of the standard deviation, as given in Hunter and Young (2012)
bandwidth: Final value of the bandwidth
density.x: Points at which the error density is estimated
density.y: Values of the error density at the points density.x
symmetric: Logical: Was the error density assumed symmetric?
loglik: A quantity similar to a log-likelihood, computed just like a standard loglikelihood would be, conditional on the component density functions being equal to the final density estimates.
ft: A character vector giving the name of the function.

Arguments

lmformula: Formula for a linear model, in the same format used by lm. Additional parameters may be passed to lm via the ... argument.
bw: Initial bandwidth value. If NULL, this will be chosen automatically by the algorithm.
constbw: Logical: If TRUE, the bandwidth is held constant throughout the algorithm; if FALSE, it adapts at each iteration according to the rules given in Hunter and Young (2012).
bwmult: Whenever it is updated automatically, the bandwidth is equal to bwmult divided by the fifth root of \(n\) times the smaller of s and IQR/1.34, where s and IQR are estimates of the standard deviation and interquartile range of the residuals, as explained in Hunter and Young (2012). The value of 0.9 gives the rule of Silverman (1986) and the value of 1.06 gives the rule of Scott (1992). Larger values lead to greater smoothing, whereas smaller values lead to less smoothing.
z.hat: Initial nxm matrix of posterior probabilities. If NULL, this is initialized randomly. As long as a parametric estimation method like least squares is used to estimate beta in each M-step, the z.hat values are the only values necessary to begin the EM iterations.
symm: Logical: If TRUE, the error density is assumed symmetric about zero. If FALSE, it is not. WARNING: If FALSE, the intercept parameter is not uniquely identifiable if it is included in the linear model.
betamethod: Method of calculating beta coefficients in the M-step. Current possible values are "LS" for least-squares; "L1" for least absolute deviation; "NP" for fully nonparametric; and "transition" for a transition from least squares to fully nonparametric. If something other than these four possibilities is used, then "NP" is assumed. For details of these methods, see Hunter and Young (2012).
m: Number of components in the mixture.
epsilon: Convergence is declared if the largest change in any lambda or beta coordinate is smaller than epsilon.
maxit: The maximum number of iterations; if convergence is never declared based on comparison with epsilon, then the algorithm stops after maxit iterations.
verbose: Logical: If TRUE, then various updates are printed during each iteration of the algorithm.
...: Additional parameters passed to the model.frame and model.matrix functions, which are used to obtain the response and predictor of the regression.

References

Hunter, D. R. and Young, D. S. (2012) Semi-parametric Mixtures of Regressions, Journal of Nonparametric Statistics 24(1): 19-38.

Scott, D. W. (1992) Multivariate Density Estimation, John Wiley & Sons Inc., New York.

Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman & Hall, London.

Examples

Run this code

data(tonedata)
## By default, the bandwidth will adapt and the error density is assumed symmetric
set.seed(100)
a=spregmix(tuned~stretchratio, bw=.2, data=tonedata, verb=TRUE)

## Look at the sp mixreg solution:
plot(tonedata)
abline(a=a$beta[1,1],b=a$beta[2,1], col=2)
abline(a=a$beta[1,2],b=a$beta[2,2], col=3)

## Look at the nonparametric KD-based estimate of the error density, 
## constrained to be zero-symmetric:
plot(xx<-a$density.x, yy<-a$density.y, type="l")
## Compare to a normal density with mean 0 and NP-estimated stdev:
z <- seq(min(xx), max(xx), len=200)
lines(z, dnorm(z, sd=sqrt((a$np.stdev)^2+a$bandwidth^2)), col=2, lty=2)
# Add bandwidth^2 to variance estimate to get estimated var of KDE

## Now add the sp mixreg estimate without assuming symmetric errors:
b=spregmix(tuned~stretchratio, bw=.2, , symm=FALSE, data=tonedata, verb=TRUE)
lines(b$density.x, b$density.y, col=3)