Modeling censored knowledge with tfprobability

Nothing’s ever excellent, and knowledge isn’t both. One kind of “imperfection” is lacking knowledge, the place some options are unobserved for some topics. (A subject for one more publish.) One other is censored knowledge, the place an occasion whose traits we wish to measure doesn’t happen within the statement 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 occasions for these cats that really did get adopted, our estimate will find yourself too optimistic: We don’t bear in mind these cats who weren’t adopted throughout this interval and thus, would have contributed wait occasions of size longer than the whole 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 test, collected from CRAN and offered 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 test didn’t full.

Why will we care concerning 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 change into “fortunate”. How about check_times? Nicely, in case your submission is a kind of that errored out, you continue to care about how lengthy you wait, so regardless that their proportion is low (< 1%) we don’t wish to merely exclude them. Additionally, there’s the chance that the failing ones would have taken longer, had they run to completion, resulting from some intrinsic distinction between each teams. Conversely, if failures had been random, the longer-running checks would have a larger likelihood to get hit by an error. So right here too, exluding the censored knowledge might lead to 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 generally? Making as few assumptions as doable, the most entropy distribution for displacements (in area 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 test accomplished, it could take a minimum of 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 Chance, that are 1.14 and 0.7, respectively. In case 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.

Moreover the test durations we wish to mannequin, check_times studies numerous options of the package deal in query, akin to variety of imported packages, variety of dependencies, measurement of code and documentation information, and so forth. The standing variable signifies whether or not the test 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 attention-grabbing subset of predictors:

survreg_fit <-
  surv_reg(dist = "exponential") %>% 
  set_engine("survreg") %>% 
  match(Surv(check_time, standing) ~ ., 
      knowledge = 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 completely 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 knowledge saved individually, so right here we create two goal matrices as an alternative of 1:

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

# test occasions 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, establishing one dataframe for the censored knowledge and one for the uncensored knowledge 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 knowledge  
df_c <- df %>% filter(standing == 0) %>% choose(-standing)
# dataframe of predictors for non-censored knowledge 
df_nc <- df %>% filter(standing == 1) %>% choose(-standing)

That’s it for preparations. However after all we’re curious. Do test occasions look completely different? Do predictors – those we selected – look completely different?

Evaluating a couple of significant percentiles for each courses, 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 “had been going to take longer”?

How concerning 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 test accomplished):

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

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

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

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

And eventually, 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 test durations bearing in mind uncensored in addition to censored knowledge – we gained’t dwell on variations between each teams any longer; nonetheless we thought it attention-grabbing 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 easy as including tfd_exponential() to the mannequin perform, tfd_joint_distribution_sequential(). For the censored portion, we want the exponential CCDF. This one will not be, as of at this time, simply added to the mannequin. What we will do although is calculate its worth ourselves and add it to the “essential” mannequin chance. We’ll see this beneath when discussing sampling; for now it means the mannequin definition finally ends up easy because it solely covers the non-censored knowledge. It’s manufactured from simply the stated exponential PDF and priors for the regression parameters.

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

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

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

mannequin <- perform(knowledge) {
  tfd_joint_distribution_sequential(
    record(
      tfd_sample_distribution(tfd_normal(0, 1), sample_shape = 7),
      perform(betas)
        tfd_independent(
          tfd_exponential(
            charge = 1 / tf$math$exp(tf$transpose(
              tf$matmul(tf$forged(knowledge, betas$dtype), tf$transpose(betas))))),
          reinterpreted_batch_ndims = 1)))
}

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

At all times, we check 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 seems to be superb: We’ve an inventory of size two, one factor for every distribution within the mannequin. For each tensors, dimension 1 displays the batch measurement (which we arbitrarily set to 2 on this check), whereas dimension 2 is 7 for the variety of regular priors and 13523 for the variety of durations predicted.

How doubtless 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 cheap.

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 – underneath the mannequin.
Now right here the info is available in two elements, and the goal does as effectively. First, we’ve got the non-censored knowledge, for which

m %>% tfd_log_prob(record(betas, tf$forged(target_nc, betas$dtype)))

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

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

Each elements are mixed in a little bit wrapper perform that enables us to match coaching together with and excluding the censored knowledge. We gained’t do this on this publish, however you may be to do it with your personal knowledge, 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(record(betas, tf$forged(target_nc, betas$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) {
  record(
    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'll feed the algorithm
# tf$zeros_like(initial_betas)
# as an alternative 
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 cheap 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
  )
}

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

How effectively did the sampling work? The chains combine effectively, however for some parameters, autocorrelation remains 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, colour = 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 same old 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 effectively, however as there isn’t a easy error metric concerned, it’s exhausting to know if precise predictions would even land in an acceptable vary.

To ensure they do, we examine predictions from our mannequin in addition to from surv_reg.
This time, we additionally break up the info into coaching and check 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, 
      knowledge = 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, colour = 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 check 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, colour = 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 seems to be good!

Wrapup

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