Smoothed PMD

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Last updated: 2019-10-08

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Knit directory: SMF/

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File	Version	Author	Date	Message
html	8c12025	Dongyue Xie	2019-07-23	Build site.
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Introduction

Penalized matrix decomposition finds sparse \(L,F\) such that \(Y \approx LF^T\). It also has the option to give smooth estimate of factors in \(F\) using fused lasso. Here, we compare PMD-smooth, WaveletPMD and WaveletEBMF.

library(wavethresh)
library(flashr)
library(PMA)
# wavelet-based matrix factorization
#'@ y: observed matrix
#'@ k: number of factors
#'@ filter.number, family: wavelet type

WaveEBMF = function(y,k,filter.number = 1,family = 'DaubExPhase'){
  N=nrow(y)
  p=ncol(y)
  W = GenW(n=p,filter.number = filter.number,family = family)
  y_tilde = y%*%W
  f2 = flash(y_tilde,Kmax=k,var_type = 'constant',backfit = T,verbose=F)
  f2_fitted = flash_get_ldf(f2)
  f_hat = (W%*%f2_fitted$f)
  return(list(f=f_hat,l=f2_fitted$l))
}

WavePMD = function(y,k,filter.number = 1,family = 'DaubExPhase'){
  N=nrow(y)
  p=ncol(y)
  W = GenW(n=p,filter.number = filter.number,family = family)
  y_tilde = y%*%W
  param = PMD.cv(y_tilde,type='standard')
  f2 = PMD(y_tilde,type='standard',sumabs = param$bestsumabs,K=k)
  f_hat = (W%*%f2$v)
  return(list(f=f_hat,l=f2$u))
}

A single factor example

Simulate \(N=200\) and \(p=256\) under single-factor model \[l_i\sim \pi_0\delta_0+(1-\pi_0)\sum_{m=1}^5\frac{1}{5}N(0,\sigma^2_m)\]

Step function factor

\(f\) is a step function.

rmse = function(x1,x2){sqrt(mean((x1-x2)^2))}
set.seed(12345)
N = 200
p = 256
pi0 = 0.3
f = c(rep(2,p/4), rep(5, p/4), rep(6, p/4), rep(2, p/4))
l = c()
for (i in 1:N) {
  r = rbinom(1,1,pi0)
  if(r==1){
    l[i]=0
  }else{
      l[i] = mean(c(rnorm(1,0,sqrt(0.25)),rnorm(1,0,sqrt(0.5)),rnorm(1,0,sqrt(1)),
                    rnorm(1,0,sqrt(2)),rnorm(1,0,sqrt(4))))
      }
}
y = l%*%t(f)+matrix(rnorm(N*p,0,1),ncol=p)

# apply flash directly
f1 = WaveEBMF(y,k=1)
f2 = WavePMD(y,k=1)

 Fold  1  out of  5 
 Fold  2  out of  5 
 Fold  3  out of  5 
 Fold  4  out of  5 
 Fold  5  out of  5 

1

f3 = PMD.cv(y,type='ordered')

 Fold  1  out of  5 
12345678910 Fold  2  out of  5 
12345678910 Fold  3  out of  5 
12345678910 Fold  4  out of  5 
12345678910 Fold  5  out of  5 
12345678910

f3 = PMD(y,'ordered',sumabsu = f3$bestsumabsu, K=1)

1234

plot(f/norm(f,'2'),col='grey80',xlab='',ylab='',main='Estimate of factors',ylim=c(0,0.11))
lines(f1$f,col=4)
lines(-f2$f,col=2)
lines(f3$v,col=3)
legend('topleft',c('true f','WaveEBMF','WavePMD','PMD-smooth'),col=c('grey80',4,2,3),lty=c(1,1,1,1))

Past versions of unnamed-chunk-2-1.png

Version	Author	Date
8c12025	Dongyue Xie	2019-07-23

HeavySine function factor

f=DJ.EX(p,signal = 2)$heavi
y = l%*%t(f)+matrix(rnorm(N*p,0,1),ncol=p)

# apply flash directly
f1 = WaveEBMF(y,k=1,filter.number = 10,family = 'DaubLeAsymm')
f2 = WavePMD(y,k=1,filter.number = 10,family = 'DaubLeAsymm')

 Fold  1  out of  5 
 Fold  2  out of  5 
 Fold  3  out of  5 
 Fold  4  out of  5 
 Fold  5  out of  5 

1

f3 = PMD.cv(y,type='ordered')

 Fold  1  out of  5 
12345678910 Fold  2  out of  5 
12345678910 Fold  3  out of  5 
12345678910 Fold  4  out of  5 
12345678910 Fold  5  out of  5 
12345678910

f3 = PMD(y,'ordered',sumabsu = f3$bestsumabsu, K=1)

12345

plot(f/norm(f,'2'),col='grey80',xlab='',ylab='',main='Estimate of factors',ylim=c(-0.15,0.11))
lines(f1$f,col=4)
lines(f2$f,col=2)
lines(f3$v,col=3)
legend('topright',c('true f','WaveEBMF','WavePMD','PMD-smooth'),col=c('grey80',4,2,3),lty=c(1,1,1,1))

Past versions of unnamed-chunk-3-1.png

Version	Author	Date
8c12025	Dongyue Xie	2019-07-23

Spike function factor

spike.f = function(x) (0.75 * exp(-500 * (x - 0.23)^2) + 1.5 * exp(-2000 * (x - 0.33)^2) + 3 * exp(-8000 * (x - 0.47)^2) + 2.25 * exp(-16000 * 
    (x - 0.69)^2) + 0.5 * exp(-32000 * (x - 0.83)^2))

t = 1:p/p
f = 2*spike.f(t)

y = l%*%t(f)+matrix(rnorm(N*p,0,1),ncol=p)

# apply flash directly
f1 = WaveEBMF(y,k=1,filter.number = 10,family = 'DaubLeAsymm')
f2 = WavePMD(y,k=1,filter.number = 10,family = 'DaubLeAsymm')

 Fold  1  out of  5 
 Fold  2  out of  5 
 Fold  3  out of  5 
 Fold  4  out of  5 
 Fold  5  out of  5 

1

f3 = PMD.cv(y,type='ordered')

 Fold  1  out of  5 
12345678910 Fold  2  out of  5 
12345678910 Fold  3  out of  5 
12345678910 Fold  4  out of  5 
12345678910 Fold  5  out of  5 
12345678910

f3 = PMD(y,'ordered',sumabsu = f3$bestsumabsu, K=1)

12345

plot(f/norm(f,'2'),col='grey80',xlab='',ylab='',main='Estimate of factors',ylim=c(-0.05,0.5))
lines(f1$f,col=4)
lines(-f2$f,col=2)
lines(-f3$v,col=3)
legend('topright',c('true f','WaveEBMF','WavePMD','PMD-smooth'),col=c('grey80',4,2,3),lty=c(1,1,1,1))

Past versions of unnamed-chunk-4-1.png

Version	Author	Date
8c12025	Dongyue Xie	2019-07-23

Session information

sessionInfo()

R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] PMA_1.1          flashr_0.6-6     wavethresh_4.6.8 MASS_7.3-51.4   

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2        plyr_1.8.4        pillar_1.4.2     
 [4] compiler_3.6.1    git2r_0.26.1      workflowr_1.4.0  
 [7] iterators_1.0.12  tools_3.6.1       digest_0.6.21    
[10] tibble_2.1.3      evaluate_0.14     gtable_0.3.0     
[13] lattice_0.20-38   pkgconfig_2.0.3   rlang_0.4.0      
[16] Matrix_1.2-17     foreach_1.4.7     yaml_2.2.0       
[19] parallel_3.6.1    xfun_0.10         dplyr_0.8.3      
[22] stringr_1.4.0     knitr_1.25        fs_1.3.1         
[25] tidyselect_0.2.5  rprojroot_1.3-2   grid_3.6.1       
[28] impute_1.58.0     glue_1.3.1        R6_2.4.0         
[31] rmarkdown_1.16    mixsqp_0.1-97     reshape2_1.4.3   
[34] purrr_0.3.2       ggplot2_3.2.1     ashr_2.2-38      
[37] magrittr_1.5      whisker_0.4       backports_1.1.5  
[40] scales_1.0.0      codetools_0.2-16  htmltools_0.4.0  
[43] assertthat_0.2.1  softImpute_1.4    colorspace_1.4-1 
[46] stringi_1.4.3     lazyeval_0.2.2    doParallel_1.0.15
[49] pscl_1.5.2        munsell_0.5.0     truncnorm_1.0-8  
[52] SQUAREM_2017.10-1 crayon_1.3.4