Modeling censored information with tfprobability

Nothing’s ever excellent, and information isn’t both. One sort of “imperfection” is lacking information, the place some options are unobserved for some topics. (A subject for one more submit.) 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 occasions for these cats that really did get adopted, our estimate will find yourself too optimistic: We don’t take note of 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 submit, 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 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 test didn’t full.

Why will we care concerning the censored portion? Within the cat adoption situation, 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 become “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 although 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, because of some intrinsic distinction between each teams. Conversely, if failures have been random, the longer-running checks would have a larger probability to get hit by an error. So right here too, exluding the censored information 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 normally? Making as few assumptions as attainable, 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 test accomplished, it could take at the least as lengthy because the given length. This amount may be modeled by the exponential complementary cumulative distribution operate (CCDF). Why? A cumulative distribution operate (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 provides 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 steady releases of TensorFlow and TensorFlow Likelihood, 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 reviews varied options of the package deal in query, resembling variety of imported packages, variety of dependencies, dimension of code and documentation information, and many others. 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 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 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 information 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 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 = operate(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 test occasions look completely different? Do predictors – those we selected – look completely different?

Evaluating a number of significant percentiles for each courses, we see that durations for uncompleted checks are increased than these for accomplished checks all through, aside from the 100% percentile. It’s not shocking that given the large distinction in pattern dimension, 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 concerning the predictors? We don’t see any variations for relies upon, the variety of package deal dependencies (aside from, 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 information in any respect percentiles apart from the utmost:

Similar 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:

Similar 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 test durations considering uncensored in addition to censored information – we received’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 easy as including tfd_exponential() to the mannequin operate, tfd_joint_distribution_sequential(). For the censored portion, we want the exponential CCDF. This one will not be, as of right this moment, simply added to the mannequin. What we will do although is calculate its worth ourselves and add it to the “most important” mannequin probability. 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 information. It’s made 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 nicely. 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 information solely:

mannequin <- operate(information) {
  tfd_joint_distribution_sequential(
    listing(
      tfd_sample_distribution(tfd_normal(0, 1), sample_shape = 7),
      operate(betas)
        tfd_independent(
          tfd_exponential(
            price = 1 / tf$math$exp(tf$transpose(
              tf$matmul(tf$forged(information, betas$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 positive: Now we have an inventory of size two, one component for every distribution within the mannequin. For each tensors, dimension 1 displays the batch dimension (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 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 information – that’s, the measured durations – beneath the mannequin. Now right here the information is available in two components, and the goal does as nicely. First, we’ve the non-censored information, for which

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

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

get_exponential_lccdf <- operate(betas, information, goal) {
  e <-  tfd_independent(tfd_exponential(price = 1 / tf$math$exp(tf$transpose(tf$matmul(
    tf$forged(information, betas$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 components are mixed in a little bit wrapper operate that permits us to match coaching together with and excluding the censored information. We received’t do this on this submit, however you may be to do it with your individual information, particularly if the ratio of censored and uncensored components is rather less imbalanced.

get_log_prob <-
  operate(target_nc,
           censored_data = NULL,
           target_c = NULL) {
    log_prob <- operate(betas) {
      log_prob <-
        m %>% tfd_log_prob(listing(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

# preserve monitor of some diagnostic output, acceptance and step dimension
trace_fn <- operate(state, pkr) {
  listing(
    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 dimension, 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 <- operate(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 dimension:

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 listing, 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 remains to be fairly excessive.

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

plot_traces <- operate(samples) {
  plots <- purrr::map(samples, plot_trace)
  do.name(grid.organize, 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 dimension 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 a easy error metric concerned, it’s laborious 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 information 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, shade = issue(standing))) +
  geom_point() + 
  coord_cartesian(ylim = c(0, 1400))

Determine 3: Take a look at set predictions from surv_reg. One outlier (of worth 160421) is excluded by way of 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, shade = issue(standing))) +
  geom_point() + 
  coord_cartesian(ylim = c(0, 1400)) 

Determine 4: Take a look at set predictions from the mcmc mannequin. No outliers, simply utilizing similar scale as above for comparability.

This appears to be like good!

Wrapup

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