A-quick-tour-of-SNMoE

In meteorits: Mixture-of-Experts Modeling for Complex Non-Normal Distributions

library(knitr)
knitr::opts_chunk$set(
    fig.align = "center",
    fig.height = 5.5,
    fig.width = 6,
    warning = FALSE,
    collapse = TRUE,
    dev.args = list(pointsize = 10),
    out.width = "90%",
    par = TRUE
)
knit_hooks$set(par = function(before, options, envir)
  { if (before && options$fig.show != "none") 
       par(family = "sans", mar = c(4.1,4.1,1.1,1.1), mgp = c(3,1,0), tcl = -0.5)
})

library(meteorits)

Introduction

SNMoE (Skew-Normal Mixtures-of-Experts) provides a flexible modelling framework for heterogenous data with possibly skewed distributions to generalize the standard Normal mixture of expert model. SNMoE consists of a mixture of K skew-Normal expert regressors network (of degree p) gated by a softmax gating network (of degree q) and is represented by:

• The gating network parameters alpha's of the softmax net.
• The experts network parameters: The location parameters (regression coefficients) beta's, scale parameters sigma's, and the skewness parameters lambda's. SNMoE thus generalises mixtures of (normal, skew-normal) distributions and mixtures of regressions with these distributions. For example, when $q=0$, we retrieve mixtures of (skew-normal, or normal) regressions, and when both $p=0$ and $q=0$, it is a mixture of (skew-normal, or normal) distributions. It also reduces to the standard (normal, skew-normal) distribution when we only use a single expert ($K=1$).

Model estimation/learning is performed by a dedicated expectation conditional maximization (ECM) algorithm by maximizing the observed data log-likelihood. We provide simulated examples to illustrate the use of the model in model-based clustering of heterogeneous regression data and in fitting non-linear regression functions.

It was written in R Markdown, using the knitr package for production.

See help(package="meteorits") for further details and references provided by citation("meteorits").

Application to a simulated dataset

Generate sample

n <- 500 # Size of the sample
alphak <- matrix(c(0, 8), ncol = 1) # Parameters of the gating network
betak <- matrix(c(0, -2.5, 0, 2.5), ncol = 2) # Regression coefficients of the experts
lambdak <- c(3, 5) # Skewness parameters of the experts
sigmak <- c(1, 1) # Standard deviations of the experts
x <- seq.int(from = -1, to = 1, length.out = n) # Inputs (predictors)

# Generate sample of size n
sample <- sampleUnivSNMoE(alphak = alphak, betak = betak, sigmak = sigmak, 
                          lambdak = lambdak, x = x)
y <- sample$y

Set up SNMoE model parameters

K <- 2 # Number of regressors/experts
p <- 1 # Order of the polynomial regression (regressors/experts)
q <- 1 # Order of the logistic regression (gating network)

Set up EM parameters

n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE
verbose_IRLS <- FALSE

Estimation

snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter, 
                 threshold, verbose, verbose_IRLS)

Summary

snmoe$summary()

Plots

Mean curve

snmoe$plot(what = "meancurve")

Confidence regions

snmoe$plot(what = "confregions")

Clusters

snmoe$plot(what = "clusters")

Log-likelihood

snmoe$plot(what = "loglikelihood")

Application to a real dataset

Load data

data("tempanomalies")
x <- tempanomalies$Year
y <- tempanomalies$AnnualAnomaly

Set up SNMoE model parameters

K <- 2 # Number of regressors/experts
p <- 1 # Order of the polynomial regression (regressors/experts)
q <- 1 # Order of the logistic regression (gating network)

Set up EM parameters

n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE
verbose_IRLS <- FALSE

Estimation

snmoe <- emSNMoE(X = x, Y = y, K, p, q, n_tries, max_iter, 
                 threshold, verbose, verbose_IRLS)

Summary

snmoe$summary()

Plots

Mean curve

snmoe$plot(what = "meancurve")

Confidence regions

snmoe$plot(what = "confregions")

Clusters

snmoe$plot(what = "clusters")

Log-likelihood

snmoe$plot(what = "loglikelihood")

meteorits documentation built on Jan. 11, 2020, 9:13 a.m.