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 put up.) One other is censored information, the place an occasion whose traits we need 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 truly did get adopted, our estimate will find yourself too optimistic: We don’t consider 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 put up, 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 purpose, 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 practical estimate for any unknown cat, not simply these cats that may transform “fortunate”. How about check_times? Properly, in case your submission is a kind of that errored out, you continue to care about how lengthy you wait, so regardless that their share is low (< 1%) we don’t need to merely exclude them. Additionally, there’s the likelihood that the failing ones would have taken longer, had they run to completion, on account of some intrinsic distinction between each teams. Conversely, if failures have been random, the longer-running checks would have a higher probability to get hit by an error. So right here too, exluding the censored information could 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 truly 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 could take a minimum of as lengthy because the given length. This amount will be modeled by the exponential complementary cumulative distribution operate (CCDF). Why? A cumulative distribution operate (CDF) signifies the chance {that a} worth decrease or equal to some reference level was reached; e.g., “the chance of durations <= 255 is 0.9”. Its complement, 1 – CDF, then provides the chance {that a} worth will exceed than that reference level.

Let’s see this in motion.

The info

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 need to mannequin, check_times experiences numerous options of the package deal in query, reminiscent of variety of imported packages, variety of dependencies, measurement of code and documentation information, and so forth. 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

It appears that 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 an alternative of 1:

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

# verify 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, 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 = 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 in fact we’re curious. Do verify occasions 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 greater than these for accomplished checks all through, other than the 100% percentile. It’s not stunning that given the big distinction in pattern measurement, most length is greater for accomplished checks. In any other case although, doesn’t it appear to be 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 greater for censored information in any respect percentiles in addition to 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 dimensions on disk of information within the R listing:

And eventually, we see it for doc_size too, the place doc_size is the dimensions of .Rmd and .Rnw information:

Given our process at hand – mannequin verify durations bearing in mind 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 simple as including tfd_exponential() to the mannequin operate, tfd_joint_distribution_sequential(). For the censored portion, we want the exponential CCDF. This one is just not, as of right now, simply added to the mannequin. What we will do although is calculate its worth ourselves and add it to the “predominant” 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 manufactured 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 properly. Because the priors are all the identical, as an alternative 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 whole mannequin, instantiated utilizing the uncensored information solely:

mannequin <- operate(information) {
  tfd_joint_distribution_sequential(
    checklist(
      tfd_sample_distribution(tfd_normal(0, 1), sample_shape = 7),
      operate(betas)
        tfd_independent(
          tfd_exponential(
            charge = 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 seems to be wonderful: We’ve an inventory of size two, one aspect 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 possible are these samples?

m %>% tfd_log_prob(samples)

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

Right here too, the form is right, and the values look cheap.

The following factor to do is outline the goal we need to optimize.

Optimization goal

Abstractly, the factor to maximise is the log probility of the information – that’s, the measured durations – underneath the mannequin.
Now right here the information is available in two components, and the goal does as properly. First, we now have the non-censored information, for which

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

will calculate the log chance. Second, to acquire log chance 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(charge = 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 bit of wrapper operate that permits us to match coaching together with and excluding the censored information. We received’t try this on this put up, however you is likely to 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(checklist(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 <- operate(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 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 <- 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 have 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 properly did the sampling work? The chains combine properly, however for some parameters, autocorrelation continues 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, 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 <- 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 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 fairly properly, however as there is no such thing as a simple error metric concerned, it’s onerous 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 cut 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, 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 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 procure 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, colour = issue(standing))) +
  geom_point() + 
  coord_cartesian(ylim = c(0, 1400)) 

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

This seems to be good!

Wrapup

We’ve proven the best way to mannequin censored information – or reasonably, 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 put up has supplied some steering on the best way to deal with censored information in your individual work. Thanks for studying!