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Non-parametric and Semi-parametric models in R

author: Soroor Hediyehzadeh - Walter and Eliza Hall Institute of Medical Research date: December 2018 autosize: true

About #rstats lunch seminars

  • 45 - 50 minutes online sessions via Zoom
  • Aims to enhance the statistical knowledge and expertise of the RLadies community
  • Topics include statistical and statistical learning models

Recent achievements of RLadies Melbourne Members

Congradulations!

  • To Earo Wang for 2018 ASA Statistical Graphics Student Paper Award
  • And to our very own Alexandra Garnham for 2018 ABACBS Professional Bioinformatician Award

Non-parametric vs Parametric models

  • Strong distributional assumptions behind parametric models >> less flexibility
  • No assumptions about distribution of the data in non-parametric modelling. Models are more flexible, and more complex.
  • Parametric models are generalizable, while non-parametric models are not

  • Bias - variance trade-off: parametric estimates have smaller variance and large bias, non-parametric models have small bias, but large variance

Non-parametric Estimation

  • Non-parametric Density Estimation :
    • Histogram
    • Kernel Density Estimation
  • Non-parametric Regression :
    • Nadyara-Watson estimator
    • Local Polynomial estimators
    • splines

Non-parametric Density Estimation

  • Histogram
  • Kernel Density Estimation (KDE)

KDE_1

Kernel Density Estimation

KDE - choices for kernels

Non-parametric Regression

Typically used for trend estimation. Models conditional expectation of the response

N-W estimator (local constant regression)

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N-W estimator

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Local Polynomial Regression

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Splines

  • splines: Fix a number of data points. These points are called knots. Then, fit piecewise polynomials between knots. Typically local polynomial order 4 is used, which are called cubic splines.

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In the above expressions, epsilons are knots

Splines

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Smoothing Splines

  • Avoid selection of Knots
    • All distinct data points are taken as knots
  • Have control on the curvature (i.e. smoothness) of the fit

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Semi-parametric models

  • Non-parametric models require estimation of a large number of parameters, resulting in very complex models >> curse of dimensionality

  • Use a combination of parametric and non-parametric models to avoid estimation of large # of parameters >> semi-parametric models

Semi-parametric models include:

  • Additive Models

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* Partial linear models

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  • Generalized Partial Linear models

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  • Generalized additive model
  • Single Index model - link function unknown

Parkinson's Telemonitoring Dataset

  • Response variables:
  • motor_UPDRS
  • total_UPDRS
  • Covariates:
  • NHR,HNR
  • RPDE - A nonlinear dynamical complexity measure
  • DFA - Signal fractal scaling exponent
  • PPE - A nonlinear measure of fundamental frequency variation
  • Jitter(Abs)
  • Shimmer

Parkinson's Telemonitoring Dataset

  • 2-D KDE
# Generate 2-d density plots
contour(kde2d(Upd_avg[,param], Upd_avg$total_UPDRS, n=100),
				xlab = param, ylab = "total_UPDRS")

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Parkinson's Telemonitoring Dataset

  • Kernel regression
plot(ksmooth(x, y, kernel = "normal", bandwidth =bw))
points(x,y)

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Parkinson's Telemonitoring Dataset

Fitting GAM (Generalized Additive Models) with the mgcv CRAN package

  • Recall that the covariates in the dataset are a combination of linear and non-linear covariates
  • We use the gam() function to fit a Generalized Additive model with a Gaussian link function. The model consists of linear fit for linear covariates, and regression splines, denoted by s(), for non-linear terms. This is an extention to the Generalized Partial Linear model.

Parkinson's Telemonitoring Dataset

Fitting GAM (Generalized Additive Models) with the mgcv CRAN package

library(mgcv)
fit <- gam(motor_UPDRS ~ age + sex + Jitter + 
					 	JitterAbs + JitterRAP + 
					 	JitterDDP + Shimmer +
					 	ShimmerAPQ3 + 
					 	NHR + HNR + DFA + s(PPE) + s(RPDE)+
						s(subject,bs="re"), data = dat, method = "REML")

s(, bs="re") is a way to incorporate random effect terms to GAMs.

===================

summary(fit)

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====================

library(np)
fhat <- npcdens(motor_UPDRS~DFA,data=dat)
plot(fhat,view="fixed", theta = 310, phi=15,main="")

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Modeling interactions with splines

Tests for interactions between non-linear terms is done using te(), the tensor product smooths

fit <- gam(motor_UPDRS ~ age + sex + Jitter + 
					 	JitterAbs + JitterRAP + 
					 	JitterDDP + 
					 	NHR + HNR + DFA + s(PPE) + s(RPDE)+
						te(PPE, RPDE), data = dat, method = "REML")

Acknowledgments and resources

Definitions and figures in this presentation are taken from Liuhua Peng's material. The Parkinson's Telemonitoring data analysis is a joint work of Gavriel Olshansky and myself.

Suggested readings

  • Generalized Additive Models, Second Edition. Simon Wood.
  • RACINE, J. NONPARAMETRIC AND SEMIPARAMETRIC METHODS IN R
  • mgcv package vignette

Feedback time!

R-Ladies Melbourne will soon undergo substantial structural changes. Your feedback helps the committee to decide if #rstats lunch seminars should have a place in the upcoming changes.

Link to the Google Survey :

https://goo.gl/forms/YgWlrWZU8AlmajN82