Last updated: 2022-06-06

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Rmd 06b638b Dongyue Xie 2022-06-06 Publish the initial files for myproject

Introduction

Plot log of density function

\[g(x) = \sum_k w_k \text{Laplace}(x;0,\sigma^2_k).\]

A concave function \(f(\cdot)\) satisfies for any \(\alpha\in[0,1]\),

\[f((1-\alpha)x+\alpha y)\geq (1-\alpha)f(x)+\alpha f(y)\]

#'density of laplace distribution x~Laplace(mu,b)
dlap = function(x,mu,b){
  return(1/2/b*exp(-abs(x-mu)/b))
}

#'density of log g(x)
#'x is a scalar
lg = function(x,w,s2){
  return(log(sum(w*dlap(x,0,s2))))
}

#'density of log g(x)
#'x is a vector
lg_vec = function(x,w,s2){
  n = length(x)
  out = rep(0,n)
  for(i in 1:n){
    out[i] = lg(x[i],w,s2)
  }
  return(out)
}

plot_lg = function(x,w,s2){
  plot(x,lg_vec(x,w,s2),type='l',xlab='x',ylab='log(g)',
       main=paste('w=(',paste(w,collapse = ", "),'); s2=(',paste(s2,collapse = ", "),')',sep=''))
}
lg_p = expression(log(w1*1/2/b1*exp(-x/b1)+w2*1/2/b2*exp(-x/b2)))
lg_n = expression(log(w1*1/2/b1*exp(x/b1)+w2*1/2/b2*exp(x/b2)))
lg2_p = D(D(lg_p,'x'),'x')
lg2_n = D(D(lg_n,'x'),'x')

plot_lg_d2 = function(r,w1,
                    b1 = 0.1,
                    b2 = 2){
  w2 = 1-w1
  xx = c()
  x = seq(-r,0,length.out = 500)
  xx = c(xx,x)
  y_n = eval(lg2_n)
  x = seq(0,r,length.out = 500)
  xx = c(xx,x)
  y_p = eval(lg2_p)
  plot(xx,c(y_n,y_p),type='l',ylab="(log g)''",xlab='x')
  print(range(c(y_n,y_p)))
}
x = seq(-5,5,length.out = 1000)
w = c(0.9,0.1)
s2 = c(0.1,3)
plot_lg(x,w,s2)

plot_lg_d2(5,w[1],s2[1],s2[2])

[1] 1.387779e-17 2.335889e+01
x = seq(-5,5,length.out = 1000)
w = c(0.1,0.9)
s2 = c(0.1,3)
plot_lg(x,w,s2)

plot_lg_d2(5,w[1],s2[1],s2[2])

[1] -2.775558e-17  2.335098e+01
#'softmax
softmax = function(a){
  exp(a-max(a))/sum(exp(a-max(a)))
}
s2 = exp(seq(-8,5,by=log(2)))
w = softmax(log(1/s2))
#plot(w,type='l')
#plot(s2,type='l')
plot_lg(x,w,s2)

more cases, let mixture components be close

x = seq(-20,20,length.out = 1000)
w = c(0.5,0.5)
s2 = c(2,3)
plot_lg(x,w,s2)

plot_lg_d2(20,w[1],s2[1],s2[2])

[1] 0.001339252 0.006944437
x = seq(-20,20,length.out = 1000)
w = c(0.1,0.9)
s2 = c(2,3)
plot_lg(x,w,s2)

plot_lg_d2(20,w[1],s2[1],s2[2])

[1] 0.0001632108 0.0034013605
x = seq(-20,20,length.out = 1000)
w = c(0.5,0.5)
s2 = c(2.9,3)
plot_lg(x,w,s2)

plot_lg_d2(20,w[1],s2[1],s2[2])

[1] 3.271432e-05 3.302946e-05

sessionInfo()
R version 4.2.0 (2022-04-22)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur/Monterey 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/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] workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.8.3     highr_0.9        bslib_0.3.1      compiler_4.2.0  
 [5] pillar_1.7.0     later_1.3.0      git2r_0.30.1     jquerylib_0.1.4 
 [9] tools_4.2.0      getPass_0.2-2    digest_0.6.29    jsonlite_1.8.0  
[13] evaluate_0.15    tibble_3.1.6     lifecycle_1.0.1  pkgconfig_2.0.3 
[17] rlang_1.0.2      cli_3.3.0        rstudioapi_0.13  yaml_2.3.5      
[21] xfun_0.30        fastmap_1.1.0    httr_1.4.2       stringr_1.4.0   
[25] knitr_1.38       sass_0.4.1       fs_1.5.2         vctrs_0.4.1     
[29] rprojroot_2.0.3  glue_1.6.2       R6_2.5.1         processx_3.5.3  
[33] fansi_1.0.3      rmarkdown_2.13   callr_3.7.0      magrittr_2.0.3  
[37] whisker_0.4      ps_1.7.0         promises_1.2.0.1 htmltools_0.5.2 
[41] ellipsis_0.3.2   httpuv_1.6.5     utf8_1.2.2       stringi_1.7.6   
[45] crayon_1.5.1