Curvature Vignette

Stan Model

Stan code for \(t\) distribution.

model_code <- "
  data {
    int<lower=1> nu;
    int<lower=1> K;
    vector[K] mu;
    cov_matrix[K] Sigma;
  } 
  parameters {
    vector[K] y;
  } 
  model {
    y ~ multi_student_t(nu, mu, Sigma);
  } 
"
model = stan_model(model_code = model_code)

Simulation

Prepare parameters for simulations.

d = 100 # dimension of distribution
mu = rep(0,d) # mean
Sigma = diag(1,d) # covariance
nu = 50 # degrees of freedom
T0 = 1000 # burn-in
T = 100 # after burn-in
noOfSecs = 1000
data = list(nu = nu,
            mu = mu, 
            Sigma = Sigma, 
            K = d)
fit = sampling(model, 
               data = data,
               chains = 1, 
               iter = T+T0, 
               warmup = T0,
               algorithm = "HMC",
               control = list(metric = "unit_e"))

## 
## SAMPLING FOR MODEL 'ef5e7059909d18d29ab9ddcad2812d10' NOW (CHAIN 1).
## 
## Gradient evaluation took 0.000111 seconds
## 1000 transitions using 10 leapfrog steps per transition would take 1.11 seconds.
## Adjust your expectations accordingly!
## 
## 
## Iteration:    1 / 1100 [  0%]  (Warmup)
## Iteration:  110 / 1100 [ 10%]  (Warmup)
## Iteration:  220 / 1100 [ 20%]  (Warmup)
## Iteration:  330 / 1100 [ 30%]  (Warmup)
## Iteration:  440 / 1100 [ 40%]  (Warmup)
## Iteration:  550 / 1100 [ 50%]  (Warmup)
## Iteration:  660 / 1100 [ 60%]  (Warmup)
## Iteration:  770 / 1100 [ 70%]  (Warmup)
## Iteration:  880 / 1100 [ 80%]  (Warmup)
## Iteration:  990 / 1100 [ 90%]  (Warmup)
## Iteration: 1001 / 1100 [ 91%]  (Sampling)
## Iteration: 1100 / 1100 [100%]  (Sampling)
## 
##  Elapsed Time: 0.744495 seconds (Warm-up)
##                0.078869 seconds (Sampling)
##                0.823364 seconds (Total)

res = curvature(fit = fit)

## computing gradients will take approx. 3 secs.

## computing Hessians will take approx. 148 secs.

plot(res)

summary(res)

## sectional curvature stats:

## # A tibble: 1 x 5
##           mean       median           sd          min          max
##          <dbl>        <dbl>        <dbl>        <dbl>        <dbl>
## 1 0.0001536921 0.0001470814 2.604606e-05 9.870614e-05 0.0002172657

Session Info

sessionInfo()

## R version 3.4.1 (2017-06-30)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS Sierra 10.12.6
## 
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] C/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] curvature_0.1.0      rstan_2.16.2         StanHeaders_2.16.0-1
## [4] ggplot2_2.2.1       
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_0.12.12      knitr_1.17        magrittr_1.5     
##  [4] munsell_0.4.3     colorspace_1.3-2  rlang_0.1.2      
##  [7] stringr_1.2.0     plyr_1.8.4        tools_3.4.1      
## [10] parallel_3.4.1    grid_3.4.1        gtable_0.2.0     
## [13] htmltools_0.3.6   yaml_2.1.14       lazyeval_0.2.0   
## [16] rprojroot_1.2     digest_0.6.12     tibble_1.3.4     
## [19] numDeriv_2016.8-1 gridExtra_2.2.1   codetools_0.2-15 
## [22] inline_0.3.14     evaluate_0.10.1   rmarkdown_1.6    
## [25] labeling_0.3      stringi_1.1.5     compiler_3.4.1   
## [28] scales_0.5.0      backports_1.1.0   stats4_3.4.1