Modeling censored information with tfprobability

Nothing’s ever good, and information isn’t both. One kind of “imperfection” is lacking information, the place some options are unobserved for some topics. (A subject for one more publish.) One other is censored information, the place an occasion whose traits we wish to measure doesn’t happen within the commentary interval. The instance in Richard McElreath’s Statistical Rethinking is time to adoption of cats in an animal shelter. If we repair an interval and observe wait instances for these cats that really did get adopted, our estimate will find yourself too optimistic: We don’t have in mind these cats who weren’t adopted throughout this interval and thus, would have contributed wait instances of size longer than the entire interval.

On this publish, we use a barely much less emotional instance which nonetheless could also be of curiosity, particularly to R package deal builders: time to completion of R CMD verify, collected from CRAN and supplied by the parsnip package deal as check_times. Right here, the censored portion are these checks that errored out for no matter cause, i.e., for which the verify didn’t full.

Why can we care in regards to the censored portion? Within the cat adoption state of affairs, that is fairly apparent: We would like to have the ability to get a sensible estimate for any unknown cat, not simply these cats that can grow to be “fortunate”. How about check_times? Nicely, in case your submission is a type of that errored out, you continue to care about how lengthy you wait, so despite the fact that their share is low (< 1%) we don’t wish to merely exclude them. Additionally, there’s the likelihood that the failing ones would have taken longer, had they run to completion, because of some intrinsic distinction between each teams. Conversely, if failures have been random, the longer-running checks would have a better likelihood to get hit by an error. So right here too, exluding the censored information might end in bias.

How can we mannequin durations for that censored portion, the place the “true length” is unknown? Taking one step again, how can we mannequin durations normally? Making as few assumptions as doable, the most entropy distribution for displacements (in house or time) is the exponential. Thus, for the checks that really did full, durations are assumed to be exponentially distributed.

For the others, all we all know is that in a digital world the place the verify accomplished, it will take at the least as lengthy because the given length. This amount might be modeled by the exponential complementary cumulative distribution perform (CCDF). Why? A cumulative distribution perform (CDF) signifies the likelihood {that a} worth decrease or equal to some reference level was reached; e.g., “the likelihood of durations <= 255 is 0.9”. Its complement, 1 – CDF, then offers the likelihood {that a} worth will exceed than that reference level.

Let’s see this in motion.

The information

The next code works with the present secure releases of TensorFlow and TensorFlow Likelihood, that are 1.14 and 0.7, respectively. When you don’t have tfprobability put in, get it from Github:

These are the libraries we want. As of TensorFlow 1.14, we name tf$compat$v2$enable_v2_behavior() to run with keen execution.

Apart from the verify durations we wish to mannequin, check_times reviews numerous options of the package deal in query, comparable to variety of imported packages, variety of dependencies, measurement of code and documentation information, and so on. The standing variable signifies whether or not the verify accomplished or errored out.

df <- check_times %>% choose(-package deal)
glimpse(df)

Observations: 13,626
Variables: 24
$ authors        <int> 1, 1, 1, 1, 5, 3, 2, 1, 4, 6, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,…
$ imports        <dbl> 0, 6, 0, 0, 3, 1, 0, 4, 0, 7, 0, 0, 0, 0, 3, 2, 14, 2, 2, 0…
$ suggests       <dbl> 2, 4, 0, 0, 2, 0, 2, 2, 0, 0, 2, 8, 0, 0, 2, 0, 1, 3, 0, 0,…
$ relies upon        <dbl> 3, 1, 6, 1, 1, 1, 5, 0, 1, 1, 6, 5, 0, 0, 0, 1, 1, 5, 0, 2,…
$ Roxygen        <dbl> 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0,…
$ gh             <dbl> 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0,…
$ rforge         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ descr          <int> 217, 313, 269, 63, 223, 1031, 135, 344, 204, 335, 104, 163,…
$ r_count        <int> 2, 20, 8, 0, 10, 10, 16, 3, 6, 14, 16, 4, 1, 1, 11, 5, 7, 1…
$ r_size         <dbl> 0.029053, 0.046336, 0.078374, 0.000000, 0.019080, 0.032607,…
$ ns_import      <dbl> 3, 15, 6, 0, 4, 5, 0, 4, 2, 10, 5, 6, 1, 0, 2, 2, 1, 11, 0,…
$ ns_export      <dbl> 0, 19, 0, 0, 10, 0, 0, 2, 0, 9, 3, 4, 0, 1, 10, 0, 16, 0, 2…
$ s3_methods     <dbl> 3, 0, 11, 0, 0, 0, 0, 2, 0, 23, 0, 0, 2, 5, 0, 4, 0, 0, 0, …
$ s4_methods     <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ doc_count      <int> 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,…
$ doc_size       <dbl> 0.000000, 0.019757, 0.038281, 0.000000, 0.007874, 0.000000,…
$ src_count      <int> 0, 0, 0, 0, 0, 0, 0, 2, 0, 5, 3, 0, 0, 0, 0, 0, 0, 54, 0, 0…
$ src_size       <dbl> 0.000000, 0.000000, 0.000000, 0.000000, 0.000000, 0.000000,…
$ data_count     <int> 2, 0, 0, 3, 3, 1, 10, 0, 4, 2, 2, 146, 0, 0, 0, 0, 0, 10, 0…
$ data_size      <dbl> 0.025292, 0.000000, 0.000000, 4.885864, 4.595504, 0.006500,…
$ testthat_count <int> 0, 8, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0,…
$ testthat_size  <dbl> 0.000000, 0.002496, 0.000000, 0.000000, 0.000000, 0.000000,…
$ check_time     <dbl> 49, 101, 292, 21, 103, 46, 78, 91, 47, 196, 200, 169, 45, 2…
$ standing         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…

Of those 13,626 observations, simply 103 are censored:

0     1 
103 13523 

For higher readability, we’ll work with a subset of the columns. We use surv_reg to assist us discover a helpful and fascinating subset of predictors:

survreg_fit <-
  surv_reg(dist = "exponential") %>% 
  set_engine("survreg") %>% 
  match(Surv(check_time, standing) ~ ., 
      information = df)
tidy(survreg_fit) 

# A tibble: 23 x 7
   time period             estimate std.error statistic  p.worth conf.low conf.excessive
   <chr>               <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
 1 (Intercept)     3.86      0.0219     176.     0.             NA        NA
 2 authors         0.0139    0.00580      2.40   1.65e- 2       NA        NA
 3 imports         0.0606    0.00290     20.9    7.49e-97       NA        NA
 4 suggests        0.0332    0.00358      9.28   1.73e-20       NA        NA
 5 relies upon         0.118     0.00617     19.1    5.66e-81       NA        NA
 6 Roxygen         0.0702    0.0209       3.36   7.87e- 4       NA        NA
 7 gh              0.00898   0.0217       0.414  6.79e- 1       NA        NA
 8 rforge          0.0232    0.0662       0.351  7.26e- 1       NA        NA
 9 descr           0.000138  0.0000337    4.10   4.18e- 5       NA        NA
10 r_count         0.00209   0.000525     3.98   7.03e- 5       NA        NA
11 r_size          0.481     0.0819       5.87   4.28e- 9       NA        NA
12 ns_import       0.00352   0.000896     3.93   8.48e- 5       NA        NA
13 ns_export      -0.00161   0.000308    -5.24   1.57e- 7       NA        NA
14 s3_methods      0.000449  0.000421     1.06   2.87e- 1       NA        NA
15 s4_methods     -0.00154   0.00206     -0.745  4.56e- 1       NA        NA
16 doc_count       0.0739    0.0117       6.33   2.44e-10       NA        NA
17 doc_size        2.86      0.517        5.54   3.08e- 8       NA        NA
18 src_count       0.0122    0.00127      9.58   9.96e-22       NA        NA
19 src_size       -0.0242    0.0181      -1.34   1.82e- 1       NA        NA
20 data_count      0.0000415 0.000980     0.0423 9.66e- 1       NA        NA
21 data_size       0.0217    0.0135       1.61   1.08e- 1       NA        NA
22 testthat_count -0.000128  0.00127     -0.101  9.20e- 1       NA        NA
23 testthat_size   0.0108    0.0139       0.774  4.39e- 1       NA        NA

Evidently if we select imports, relies upon, r_size, doc_size, ns_import and ns_export we find yourself with a mixture of (comparatively) highly effective predictors from totally different semantic areas and of various scales.

Earlier than pruning the dataframe, we save away the goal variable. In our mannequin and coaching setup, it’s handy to have censored and uncensored information saved individually, so right here we create two goal matrices as a substitute of 1:

# verify instances for failed checks
# _c stands for censored
check_time_c <- df %>%
  filter(standing == 0) %>%
  choose(check_time) %>%
  as.matrix()

# verify instances for profitable checks 
check_time_nc <- df %>%
  filter(standing == 1) %>%
  choose(check_time) %>%
  as.matrix()

Now we will zoom in on the variables of curiosity, organising one dataframe for the censored information and one for the uncensored information every. All predictors are normalized to keep away from overflow throughout sampling. We add a column of 1s to be used as an intercept.

df <- df %>% choose(standing,
                    relies upon,
                    imports,
                    doc_size,
                    r_size,
                    ns_import,
                    ns_export) %>%
  mutate_at(.vars = 2:7, .funs = perform(x) (x - min(x))/(max(x)-min(x))) %>%
  add_column(intercept = rep(1, nrow(df)), .earlier than = 1)

# dataframe of predictors for censored information  
df_c <- df %>% filter(standing == 0) %>% choose(-standing)
# dataframe of predictors for non-censored information 
df_nc <- df %>% filter(standing == 1) %>% choose(-standing)

That’s it for preparations. However after all we’re curious. Do verify instances look totally different? Do predictors – those we selected – look totally different?

Evaluating a couple of significant percentiles for each lessons, we see that durations for uncompleted checks are increased than these for accomplished checks all through, other than the 100% percentile. It’s not stunning that given the large distinction in pattern measurement, most length is increased for accomplished checks. In any other case although, doesn’t it appear like the errored-out package deal checks “have been going to take longer”?

How in regards to the predictors? We don’t see any variations for relies upon, the variety of package deal dependencies (other than, once more, the upper most reached for packages whose verify accomplished):

However for all others, we see the identical sample as reported above for check_time. Variety of packages imported is increased for censored information in any respect percentiles apart from the utmost:

Identical for ns_export, the estimated variety of exported features or strategies:

In addition to for ns_import, the estimated variety of imported features or strategies:

Identical sample for r_size, the scale on disk of information within the R listing:

And at last, we see it for doc_size too, the place doc_size is the scale of .Rmd and .Rnw information:

Given our activity at hand – mannequin verify durations making an allowance for uncensored in addition to censored information – we gained’t dwell on variations between each teams any longer; nonetheless we thought it fascinating to narrate these numbers.

So now, again to work. We have to create a mannequin.

The mannequin

As defined within the introduction, for accomplished checks length is modeled utilizing an exponential PDF. That is as simple as including tfd_exponential() to the mannequin perform, tfd_joint_distribution_sequential(). For the censored portion, we want the exponential CCDF. This one shouldn’t be, as of in the present day, simply added to the mannequin. What we will do although is calculate its worth ourselves and add it to the “most important” mannequin chance. We’ll see this under when discussing sampling; for now it means the mannequin definition finally ends up simple because it solely covers the non-censored information. It’s made from simply the mentioned exponential PDF and priors for the regression parameters.

As for the latter, we use 0-centered, Gaussian priors for all parameters. Commonplace deviations of 1 turned out to work nicely. Because the priors are all the identical, as a substitute of itemizing a bunch of tfd_normals, we will create them all of sudden as

tfd_sample_distribution(tfd_normal(0, 1), sample_shape = 7)

Imply verify time is modeled as an affine mixture of the six predictors and the intercept. Right here then is the entire mannequin, instantiated utilizing the uncensored information solely:

mannequin <- perform(information) {
  tfd_joint_distribution_sequential(
    checklist(
      tfd_sample_distribution(tfd_normal(0, 1), sample_shape = 7),
      perform(betas)
        tfd_independent(
          tfd_exponential(
            price = 1 / tf$math$exp(tf$transpose(
              tf$matmul(tf$solid(information, betas$dtype), tf$transpose(betas))))),
          reinterpreted_batch_ndims = 1)))
}

m <- mannequin(df_nc %>% as.matrix())

All the time, we take a look at if samples from that mannequin have the anticipated shapes:

samples <- m %>% tfd_sample(2)
samples

[[1]]
tf.Tensor(
[[ 1.4184642   0.17583323 -0.06547955 -0.2512014   0.1862184  -1.2662812
   1.0231884 ]
 [-0.52142304 -1.0036682   2.2664437   1.29737     1.1123234   0.3810004
   0.1663677 ]], form=(2, 7), dtype=float32)

[[2]]
tf.Tensor(
[[4.4954767  7.865639   1.8388556  ... 7.914391   2.8485563  3.859719  ]
 [1.549662   0.77833986 0.10015647 ... 0.40323067 3.42171    0.69368565]], form=(2, 13523), dtype=float32)

This appears to be like tremendous: We have now an inventory of size two, one ingredient for every distribution within the mannequin. For each tensors, dimension 1 displays the batch measurement (which we arbitrarily set to 2 on this take a look at), whereas dimension 2 is 7 for the variety of regular priors and 13523 for the variety of durations predicted.

How probably are these samples?

m %>% tfd_log_prob(samples)

tf.Tensor([-32464.521   -7693.4023], form=(2,), dtype=float32)

Right here too, the form is appropriate, and the values look affordable.

The subsequent factor to do is outline the goal we wish to optimize.

Optimization goal

Abstractly, the factor to maximise is the log probility of the info – that’s, the measured durations – beneath the mannequin.
Now right here the info is available in two elements, and the goal does as nicely. First, we’ve got the non-censored information, for which

m %>% tfd_log_prob(checklist(betas, tf$solid(target_nc, betas$dtype)))

will calculate the log likelihood. Second, to acquire log likelihood for the censored information we write a customized perform that calculates the log of the exponential CCDF:

get_exponential_lccdf <- perform(betas, information, goal) {
  e <-  tfd_independent(tfd_exponential(price = 1 / tf$math$exp(tf$transpose(tf$matmul(
    tf$solid(information, betas$dtype), tf$transpose(betas)
  )))),
  reinterpreted_batch_ndims = 1)
  cum_prob <- e %>% tfd_cdf(tf$solid(goal, betas$dtype))
  tf$math$log(1 - cum_prob)
}

Each elements are mixed in a little bit wrapper perform that enables us to check coaching together with and excluding the censored information. We gained’t try this on this publish, however you is likely to be to do it with your personal information, particularly if the ratio of censored and uncensored elements is rather less imbalanced.

get_log_prob <-
  perform(target_nc,
           censored_data = NULL,
           target_c = NULL) {
    log_prob <- perform(betas) {
      log_prob <-
        m %>% tfd_log_prob(checklist(betas, tf$solid(target_nc, betas$dtype)))
      potential <-
        if (!is.null(censored_data) && !is.null(target_c))
          get_exponential_lccdf(betas, censored_data, target_c)
      else
        0
      log_prob + potential
    }
    log_prob
  }

log_prob <-
  get_log_prob(
    check_time_nc %>% tf$transpose(),
    df_c %>% as.matrix(),
    check_time_c %>% tf$transpose()
  )

Sampling

With mannequin and goal outlined, we’re able to do sampling.

n_chains <- 4
n_burnin <- 1000
n_steps <- 1000

# maintain observe of some diagnostic output, acceptance and step measurement
trace_fn <- perform(state, pkr) {
  checklist(
    pkr$inner_results$is_accepted,
    pkr$inner_results$accepted_results$step_size
  )
}

# get form of preliminary values 
# to start out sampling with out producing NaNs, we are going to feed the algorithm
# tf$zeros_like(initial_betas)
# as a substitute 
initial_betas <- (m %>% tfd_sample(n_chains))[[1]]

For the variety of leapfrog steps and the step measurement, experimentation confirmed {that a} mixture of 64 / 0.1 yielded affordable outcomes:

hmc <- mcmc_hamiltonian_monte_carlo(
  target_log_prob_fn = log_prob,
  num_leapfrog_steps = 64,
  step_size = 0.1
) %>%
  mcmc_simple_step_size_adaptation(target_accept_prob = 0.8,
                                   num_adaptation_steps = n_burnin)

run_mcmc <- perform(kernel) {
  kernel %>% mcmc_sample_chain(
    num_results = n_steps,
    num_burnin_steps = n_burnin,
    current_state = tf$ones_like(initial_betas),
    trace_fn = trace_fn
  )
}

# vital for efficiency: run HMC in graph mode
run_mcmc <- tf_function(run_mcmc)

res <- hmc %>% run_mcmc()
samples <- res$all_states

Outcomes

Earlier than we examine the chains, here’s a fast take a look at the proportion of accepted steps and the per-parameter imply step measurement:

0.995

0.004953894

We additionally retailer away efficient pattern sizes and the rhat metrics for later addition to the synopsis.

effective_sample_size <- mcmc_effective_sample_size(samples) %>%
  as.matrix() %>%
  apply(2, imply)
potential_scale_reduction <- mcmc_potential_scale_reduction(samples) %>%
  as.numeric()

We then convert the samples tensor to an R array to be used in postprocessing.

# 2-item checklist, the place every merchandise has dim (1000, 4)
samples <- as.array(samples) %>% array_branch(margin = 3)

How nicely did the sampling work? The chains combine nicely, however for some parameters, autocorrelation continues to be fairly excessive.

prep_tibble <- perform(samples) {
  as_tibble(samples,
            .name_repair = ~ c("chain_1", "chain_2", "chain_3", "chain_4")) %>%
    add_column(pattern = 1:n_steps) %>%
    collect(key = "chain", worth = "worth",-pattern)
}

plot_trace <- perform(samples) {
  prep_tibble(samples) %>%
    ggplot(aes(x = pattern, y = worth, coloration = chain)) +
    geom_line() +
    theme_light() +
    theme(
      legend.place = "none",
      axis.title = element_blank(),
      axis.textual content = element_blank(),
      axis.ticks = element_blank()
    )
}

plot_traces <- perform(samples) {
  plots <- purrr::map(samples, plot_trace)
  do.name(grid.prepare, plots)
}

plot_traces(samples)

Determine 1: Hint plots for the 7 parameters.

Now for a synopsis of posterior parameter statistics, together with the standard per-parameter sampling indicators efficient pattern measurement and rhat.

all_samples <- map(samples, as.vector)

means <- map_dbl(all_samples, imply)

sds <- map_dbl(all_samples, sd)

hpdis <- map(all_samples, ~ hdi(.x) %>% t() %>% as_tibble())

abstract <- tibble(
  imply = means,
  sd = sds,
  hpdi = hpdis
) %>% unnest() %>%
  add_column(param = colnames(df_c), .after = FALSE) %>%
  add_column(
    n_effective = effective_sample_size,
    rhat = potential_scale_reduction
  )

abstract

# A tibble: 7 x 7
  param       imply     sd  decrease higher n_effective  rhat
  <chr>      <dbl>  <dbl>  <dbl> <dbl>       <dbl> <dbl>
1 intercept  4.05  0.0158  4.02   4.08       508.   1.17
2 relies upon    1.34  0.0732  1.18   1.47      1000    1.00
3 imports    2.89  0.121   2.65   3.12      1000    1.00
4 doc_size   6.18  0.394   5.40   6.94       177.   1.01
5 r_size     2.93  0.266   2.42   3.46       289.   1.00
6 ns_import  1.54  0.274   0.987  2.06       387.   1.00
7 ns_export -0.237 0.675  -1.53   1.10        66.8  1.01

Determine 2: Posterior means and HPDIs.

From the diagnostics and hint plots, the mannequin appears to work moderately nicely, however as there isn’t any simple error metric concerned, it’s arduous to know if precise predictions would even land in an applicable vary.

To ensure they do, we examine predictions from our mannequin in addition to from surv_reg.
This time, we additionally cut up the info into coaching and take a look at units. Right here first are the predictions from surv_reg:

train_test_split <- initial_split(check_times, strata = "standing")
check_time_train <- coaching(train_test_split)
check_time_test <- testing(train_test_split)

survreg_fit <-
  surv_reg(dist = "exponential") %>% 
  set_engine("survreg") %>% 
  match(Surv(check_time, standing) ~ relies upon + imports + doc_size + r_size + 
        ns_import + ns_export, 
      information = check_time_train)
survreg_fit(sr_fit)

# A tibble: 7 x 7
  time period         estimate std.error statistic  p.worth conf.low conf.excessive
  <chr>           <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
1 (Intercept)  4.05      0.0174     234.    0.             NA        NA
2 relies upon      0.108     0.00701     15.4   3.40e-53       NA        NA
3 imports      0.0660    0.00327     20.2   1.09e-90       NA        NA
4 doc_size     7.76      0.543       14.3   2.24e-46       NA        NA
5 r_size       0.812     0.0889       9.13  6.94e-20       NA        NA
6 ns_import    0.00501   0.00103      4.85  1.22e- 6       NA        NA
7 ns_export   -0.000212  0.000375    -0.566 5.71e- 1       NA        NA

survreg_pred <- 
  predict(survreg_fit, check_time_test) %>% 
  bind_cols(check_time_test %>% choose(check_time, standing))  

ggplot(survreg_pred, aes(x = check_time, y = .pred, coloration = issue(standing))) +
  geom_point() + 
  coord_cartesian(ylim = c(0, 1400))

Determine 3: Check set predictions from surv_reg. One outlier (of worth 160421) is excluded through coord_cartesian() to keep away from distorting the plot.

For the MCMC mannequin, we re-train on simply the coaching set and acquire the parameter abstract. The code is analogous to the above and never proven right here.

We will now predict on the take a look at set, for simplicity simply utilizing the posterior means:

df <- check_time_test %>% choose(
                    relies upon,
                    imports,
                    doc_size,
                    r_size,
                    ns_import,
                    ns_export) %>%
  add_column(intercept = rep(1, nrow(check_time_test)), .earlier than = 1)

mcmc_pred <- df %>% as.matrix() %*% abstract$imply %>% exp() %>% as.numeric()
mcmc_pred <- check_time_test %>% choose(check_time, standing) %>%
  add_column(.pred = mcmc_pred)

ggplot(mcmc_pred, aes(x = check_time, y = .pred, coloration = issue(standing))) +
  geom_point() + 
  coord_cartesian(ylim = c(0, 1400)) 

Determine 4: Check set predictions from the mcmc mannequin. No outliers, simply utilizing identical scale as above for comparability.

This appears to be like good!

Wrapup

We’ve proven methods to mannequin censored information – or fairly, a frequent subtype thereof involving durations – utilizing tfprobability. The check_times information from parsnip have been a enjoyable selection, however this modeling approach could also be much more helpful when censoring is extra substantial. Hopefully his publish has supplied some steerage on methods to deal with censored information in your personal work. Thanks for studying!