Last updated: 2022-04-14

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Rmd b035721 Matthew Stephens 2022-04-14 workflowr::wflow_publish(“sudden_changes.Rmd”) 
library(smooth)
Warning: package 'smooth' was built under R version 4.1.1
Loading required package: greybox
Warning: package 'greybox' was built under R version 4.1.1
Package "greybox", v1.0.5 loaded.
This is package "smooth", v3.1.6
library(genlasso)
Loading required package: Matrix
Loading required package: igraph
Warning: package 'igraph' was built under R version 4.1.1

Attaching package: 'igraph'
The following objects are masked from 'package:stats':

    decompose, spectrum
The following object is masked from 'package:base':

    union
library(smashr)

Introduction

I wanted to illustrate the idea that some smoothing techniques are better than others at dealing with changes that occur on different “scales” (eg a combination of smooth changes as sudden changes).

Simulated data

Here I set up \(E(y)=\mu\) as a function that varies smoothly except for a sudden change in the middle, and simulate some data with that mean:

t = seq(0,1,length=1024)
mu = (t-0.5)^2 - 0.2*(t>0.5)

y = rnorm(1024,mu,0.05)
plot(t,mu, type="l", col=2, lwd=2)
points(t,y)

Compare some smoothers

Simple moving average (sma)

I tried a sliding window of sizes 10 and 50:

y.sma.10 = sma(y,order=10) 
y.sma.50 = sma(y,order=50)
plot(t,mu, type="l", col=2, lwd=2, main="sma, order 10 (green) and 50 (blue)")

lines(t,y.sma.10$fitted,col=3,lwd=2)
lines(t,y.sma.50$fitted,col=4,lwd=2)

Trend-filtering

y.tf.0 = trendfilter(y,t,ord=0)
y.tf.0.cv = cv.trendfilter(y.tf.0) # performs 5-fold CV
Fold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ... 
y.tf.1 = trendfilter(y,t,ord=1)
y.tf.1.cv = cv.trendfilter(y.tf.1) # performs 5-fold CV
Fold 1 ... Fold 2 ... Fold 3 ... Fold 4 ... Fold 5 ... 
plot(t,mu, type="l", col=2, lwd=2, main="trendfilter (order 0 = green; order 1=blue)")

lines(t,predict(y.tf.0, y.tf.0.cv$lambda.min)$fit,col=3,lwd=2)
lines(t,predict(y.tf.1, y.tf.1.cv$lambda.min)$fit,col=4,lwd=2)

Wavelet with EB shrinkage

y.smash = smash(y,filter.number=1) 
plot(t,mu, type="l", col=2, lwd=2, main="Wavelet (haar)")
lines(t,y.smash,col=3,lwd=2)


sessionInfo()
R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.2

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] smashr_1.2-9  genlasso_1.5  igraph_1.3.0  Matrix_1.3-4  smooth_3.1.6 
[6] greybox_1.0.5

loaded via a namespace (and not attached):
 [1] wavethresh_4.6.8  statmod_1.4.36    zoo_1.8-9         xfun_0.28        
 [5] ashr_2.2-54       lattice_0.20-45   vctrs_0.3.8       generics_0.1.2   
 [9] htmltools_0.5.2   yaml_2.2.1        utf8_1.2.2        rlang_0.4.12     
[13] pracma_2.3.8      mixsqp_0.3-43     jquerylib_0.1.4   nloptr_1.2.2.3   
[17] later_1.3.0       pillar_1.6.4      glue_1.5.0        lifecycle_1.0.1  
[21] stringr_1.4.0     workflowr_1.7.0   caTools_1.18.2    evaluate_0.14    
[25] knitr_1.36        fastmap_1.1.0     httpuv_1.6.3      invgamma_1.1     
[29] irlba_2.3.3       fansi_0.5.0       highr_0.9         Rcpp_1.0.7       
[33] promises_1.2.0.1  truncnorm_1.0-8   fs_1.5.0          texreg_1.38.6    
[37] digest_0.6.28     stringi_1.7.5     rprojroot_2.0.2   grid_4.1.0       
[41] bitops_1.0-7      tools_4.1.0       magrittr_2.0.2    tibble_3.1.6     
[45] crayon_1.4.2      whisker_0.4       pkgconfig_2.0.3   ellipsis_0.3.2   
[49] MASS_7.3-54       data.table_1.14.2 SQUAREM_2021.1    rmarkdown_2.11   
[53] httr_1.4.2        R6_2.5.1          git2r_0.29.0      compiler_4.1.0