flashier_stocks

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Last updated: 2021-05-23

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These are the previous versions of the repository in which changes were made to the R Markdown (analysis/flashier_stocks.Rmd) and HTML (docs/flashier_stocks.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

File	Version	Author	Date	Message
Rmd	749540f	Matthew Stephens	2021-05-23	workflowr::wflow_publish(“flashier_stocks.Rmd”)
html	1cfd374	Matthew Stephens	2021-05-22	Build site.
Rmd	083b740	Matthew Stephens	2021-05-22	../data/prices.csv

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Introduction

In preparing homework for my class I ran flashier on some stock data. (Note that flash errored out here, and in any case flashier was quite a bit faster…) It got me thinking about appropriate application of flash/flashier for time series, so I’m reporting the results here.

The data were downloaded as in https://stephens999.github.io/stat34800/stocks.html We probably could do with looking at a bigger dataset if we want to take this seriously, but I do that for now.

# AAPL: Apple
# NFLX: Netflix
# AMZN: Amazon
# MMM: 3M
# K: Kellogs
# O: Realty Income Corp
# NSRGY: Nestle
# LDSVF: Lindt
# JPM: JP Morgan Chase
# JNJ: Johnson and Johnson
# TSLA: Tesla
# V: Visa
# PFE: Pfizer

prices = read.csv("../data/prices.csv")
log_prices = log(prices)
log_returns = apply(log_prices,2, diff)

You can see some structure in the correlation matrix: the tech companys are correlated, as are the PFE/JNJ and the financial companies (V,JPM).

S = cor(log_returns)
heatmap(S, xlab = names(prices), symm=TRUE)

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Version	Author	Date
1cfd374	Matthew Stephens	2021-05-22

Flashier on the raw data

Note I tried to use column-specific residual variances but it errored out. So I switched to constant variances. Backfitting seemed to clean up the factors so I did that.

#library("flashr")
library("ebnm")
library("flashier")
#fit.f = flashr::flash(as.matrix(log_returns),ebnm_fn = "ebnm_pl") ## errors out
#fit.f = flashier::flash(as.matrix(log_returns),prior.family = prior.point.laplace(), var.type = 2) # this produces an error
fit.f = flashier::flash(as.matrix(log_returns),prior.family = prior.point.laplace(), var.type = 0, backfit=TRUE)

Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Adding factor 6 to flash object...
Adding factor 7 to flash object...
Adding factor 8 to flash object...
Adding factor 9 to flash object...
Adding factor 10 to flash object...
Adding factor 11 to flash object...
Adding factor 12 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 11 factors (tolerance: 5.17e-04)...
  Difference between iterations is within 1.0e+03...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
Wrapping up...
Done.
Nullchecking 11 factors...
Done.

Plot the factors.

for(i in 1:11){
  barplot(fit.f$loadings.pm[[2]][,i], names.arg=names(prices), horiz=TRUE,las=2, main=paste0("Factor ",i))
}

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for(i in 1:11){
  plot(fit.f$loadings.pm[[1]][,i], main=paste0("Factor ",i))
}

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Flashier on the correlation matrix

Try factor analysis on the correlation matrix. We see a lot fewer factors. I am interested why this is. I didn’t really expect such a big difference when the factors are dense, as they seem to be here (in time space).

One possibility is that although the inferred factors are dense, they are long tailed and the inferred factors are driven by the outliers. It might be worth doing some simulations with dense long-tailed factors and comparing inference from correlation vs full data.

Another possibility is that many of the raw data results are driven by stock-specific factors. Here stock-specific factors are not represented because I remove the diagonal. And the 2 factors identified could be thought of as 3 factors if we went non-negative… in that case the differences between the results do not look so stark.

I did try running this without removing diagonal, but the stock-specific factors don’t get picked up; I think this may be due to problems converging to them (eg svd would not initialize near them…) That may suggest it could also be difficult to find other sparse factors of course (eg involving pairs of stocks).

Smiss = S
diag(Smiss) <- NA
S.f = flashier::flash(Smiss,prior.family = prior.point.laplace(), var.type = 0, backfit=TRUE)

Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 2 factors (tolerance: 2.52e-06)...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
  Difference between iterations is within 1.0e-04...
  Difference between iterations is within 1.0e-05...
Wrapping up...
Done.
Nullchecking 2 factors...
Done.

for(i in 1:2){
  barplot(S.f$loadings.pm[[2]][,i], names.arg=names(prices), horiz=TRUE,las=2, main=paste0("Factor ",i))
}

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Version	Author	Date
1cfd374	Matthew Stephens	2021-05-22

Flashier on the covariance matrix

Try factor analysis on the covariance matrix. We still see a lot fewer factors.

S2 = cov(log_returns)
S2.f = flashier::flash(S2,prior.family = prior.point.laplace(), var.type = 0, backfit = TRUE)

Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 4 factors (tolerance: 2.52e-06)...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
  Difference between iterations is within 1.0e-04...
  Difference between iterations is within 1.0e-05...
Wrapping up...
Done.
Nullchecking 4 factors...
Done.

for(i in 1:4){
  barplot(S2.f$loadings.pm[[2]][,i], names.arg=names(prices), horiz=TRUE,las=2, main=paste0("Factor ",i))
}

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Flashier on the standardized raw data

Here I tried rerunning on the standardized raw data to see how that affects things.

fit.f3 = flashier::flash(scale(log_returns),prior.family = prior.point.laplace(), var.type = 0, backfit = TRUE)

Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Adding factor 6 to flash object...
Adding factor 7 to flash object...
Adding factor 8 to flash object...
Adding factor 9 to flash object...
Adding factor 10 to flash object...
Adding factor 11 to flash object...
Adding factor 12 to flash object...
Adding factor 13 to flash object...
Adding factor 14 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 13 factors (tolerance: 5.17e-04)...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
Wrapping up...
Done.
Nullchecking 13 factors...
Done.

for(i in 1:13){
  barplot(fit.f3$loadings.pm[[2]][,i], names.arg=names(prices), horiz=TRUE,las=2, main=paste0("Factor ",i))
}

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for(i in 1:13){
  plot(fit.f3$loadings.pm[[1]][,i], main=paste0("Factor ",i))
}

Wavelet transformed data

Since these are time series it would be nice to try wavelet transforming them before applying flashier. Here I explore some of these ideas.

First define haar transform functions

haar = function(x,scale= sqrt(2)){
  if(length(x)==1){
    return(x)
  }
  else{
    x = matrix(x,nrow=2)
    diff = (x[1,]-x[2,])/scale
    sum = (x[1,]+x[2,])/scale
    return(c(diff, haar(sum)))
  }
}

haar_inv = function(x,scale=sqrt(2)){
  n=length(x)
  if(n==1){
    return(x)
  }
  x = matrix(scale*x,nrow=2,byrow=TRUE)
  smoothed = haar_inv(x[2,]) 
  return(as.vector(rbind(smoothed+x[1,], smoothed-x[1,]))/2)
}

Now I plot the above fitted factors after haar transform. There isn’t an obvious decrease in sparsity (not surpising since there was not an obvious spatiol component.)

for(i in 1:13){
  plot(haar(fit.f3$loadings.pm[[1]][1:2048,i]), main=paste0("Factor ",i, " (transformed space)"))
}

Compute the haar transform of log returns:

lp.h = log_returns[1:2048,]
# do haar wavelet decomposition on log-returns and save in lp.h
for(i in 1:ncol(log_prices)){
  lp.h[,i] = haar(log_returns[1:2048,i])
}

Quick look at correlations of the transformed data.

S.h = cor(lp.h)
heatmap(S.h, xlab = names(prices), symm=TRUE)

Maybe it makes sense just to do the higher scales?

low_res = 2048-(0:255)
S.h = cor(lp.h[low_res,])
heatmap(S.h, xlab = names(prices), symm=TRUE)

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Version	Author	Date
1cfd374	Matthew Stephens	2021-05-22

I’m not quite sure of the right way to proceed here… I’m just going to apply flash to the wavelet transformed data, even though that does not really seem quite right (the iid prior on wavelet coefficients at different scales does not really seem sensible.)

lp.h.f = flashier::flash(lp.h, prior.family=prior.point.laplace(), backfit=TRUE)

Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Adding factor 6 to flash object...
Adding factor 7 to flash object...
Adding factor 8 to flash object...
Adding factor 9 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 8 factors (tolerance: 3.97e-04)...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
  Difference between iterations is within 1.0e-04...
Wrapping up...
Done.
Nullchecking 8 factors...
Done.

for(i in 1:8){
  barplot(lp.h.f$loadings.pm[[2]][,i], names.arg=names(prices), horiz=TRUE,las=2, main=paste0("Factor ",i))
}

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Version	Author	Date
1cfd374	Matthew Stephens	2021-05-22

for(i in 1:8){
  plot(lp.h.f$loadings.pm[[1]][,i], main=paste0("Factor ",i," (transformed space)"))
}

for(i in 1:8){
  plot(haar_inv(lp.h.f$loadings.pm[[1]][,i]), main=paste0("Factor ",i),type="l")
}

Session information

sessionInfo()

R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.16

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] flashier_0.2.7 ebnm_0.1-24   

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.6       pillar_1.4.6     compiler_3.6.0   later_1.1.0.1   
 [5] git2r_0.27.1     workflowr_1.6.2  tools_3.6.0      digest_0.6.27   
 [9] evaluate_0.14    lifecycle_1.0.0  tibble_3.0.4     lattice_0.20-41 
[13] pkgconfig_2.0.3  rlang_0.4.10     Matrix_1.2-18    rstudioapi_0.13 
[17] parallel_3.6.0   yaml_2.2.1       xfun_0.16        invgamma_1.1    
[21] stringr_1.4.0    knitr_1.29       fs_1.5.0         vctrs_0.3.8     
[25] rprojroot_1.3-2  grid_3.6.0       glue_1.4.2       R6_2.4.1        
[29] rmarkdown_2.3    mixsqp_0.3-43    irlba_2.3.3      ashr_2.2-51     
[33] magrittr_1.5     whisker_0.4      backports_1.1.10 promises_1.1.1  
[37] ellipsis_0.3.1   htmltools_0.5.0  httpuv_1.5.4     stringi_1.4.6   
[41] truncnorm_1.0-8  SQUAREM_2020.3   crayon_1.3.4