Random Jump Forests

1.1 Motivation

• Transition-wise Cox models require specifying each covariate effect and interaction and impose proportional hazards; combining transitions also relies on a Markov assumption.Duration in a state can affect its exit rate, making that assumption questionable in some applications.Source: §1.1 · p. 2
• Landmarking targets probabilities conditional on a state at a fixed landmark, whereas this paper targets occupation probabilities conditional on baseline covariates.It replaces kernel smoothing in this setup with adaptive tree partitions shared across transitions.Source: §1.1 · p. 2

1.2 Contributions

• A single forest handles all transitions, with terminal nodes storing Nelson–Aalen rate estimates and initial distributions; product integration recovers occupation probabilities without a Markov assumption.The paper also describes out-of-bag prediction assessment and permutation variable importance for multi-state models.Source: §1.2 · p. 3
• With finitely many covariate values and suitable splitting conditions, the forest estimators are uniformly consistent, and cumulative-rate estimators have a functional Gaussian limit.In the Markov case, a consistent variance estimator supports pointwise confidence intervals and simultaneous bands.Source: §1.2 · p. 3
• Studies assess prediction under threshold interactions, irrelevant covariates, and non-proportional hazards, alongside four biomedical applications; they also examine proportional hazards, variable importance, and interval coverage.The reported studies show adaptation to structures the fitted Cox models cannot represent.Source: §1.2 · p. 3
• The methods are implemented in the R package JumpForests.Source: §1.2 · p. 3

• A multi-state process is represented by states and transition counts, with conditional occupation probabilities and cumulative transition rates defined given covariates.The state-occupation vector gives the probabilities of occupying each state at a given time.Source: §2 · p. 4
• The product-integral relation recovers occupation probabilities from initial probabilities and cumulative transition rates, including when the process is not Markov.For non-Markov processes, the rates are population rates: expected transitions out of a state per unit time divided by the fraction currently in that state.Source: §2 · p. 5
• Identifying rates from censored observations requires conditional random censoring: censoring may depend on covariates but cannot carry further information about the trajectory.Under this assumption, the relevant occupation and transition quantities can be estimated from right-censored data.Source: §2 · pp. 5–6
• Figure 1 depicts splits on X1 and X2 yielding terminal-node rate and initial-distribution estimates, then averages the corresponding node quantities across trees for a covariate value x.Source: Figure 1 · p. 6

3 The jump forest

• A single forest is grown for all transitions, and terminal nodes estimate cumulative transition rates nonparametrically from observed event histories.Trees are grown on subsamples and stop when no admissible split satisfies the minimum observed-jump constraints for each non-absorbing state.Source: §3 · pp. 6–7

3.1 Estimators and prediction

• Each terminal node stores Nelson–Aalen estimates for every transition and an empirical initial distribution; forest predictions average these quantities across trees and recover occupation probabilities by product integration.Out-of-bag predictions average only trees whose bootstrap samples exclude the observation, providing built-in error assessment without refitting validation subsets.Source: §3 · p. 7, §3.1 · p. 8

3.2 Log-rank splitting

• The default split score sums signed, standardized transition-specific log-rank statistics, rewarding splits that separate child nodes consistently across transitions.When effects on different transition rates point in opposite directions, these statistics can partially cancel; a squared-statistic alternative is noted but not pursued.Source: §3.2 · pp. 8–9

3.3 Prediction assessment, variable importance and extensions

• Censoring-adjusted Brier and logarithmic scores assess occupation-probability predictions, while permutation importance measures covariate contributions to those predictions.Out-of-bag evaluation supports tuning without reserving a separate validation sample; weighted and approximate log-rank splitting alternatives are also developed.Source: §3.3 · p. 9

4.1 Assumptions and scope

• Under finite covariate support and splitting conditions, forests converge to fully stratified Nelson–Aalen estimation, with bootstrap aggregation preserving its Gaussian limit and, for proportional subsamples, its √n rate.The asymptotic scope also requires non-vanishing occupation probabilities; the authors note that this can be relaxed by starting inference at a later t0.Source: §4.1 · pp. 10–11

4.2 Uniform consistency

• With finite second moments and subsample sizes tending to infinity, forests consistently estimate occupation probabilities; the paper obtains these from forest-averaged rates and the initial distribution.With finite fourth moments and the stated summability condition, both convergence statements hold almost surely.Source: §4.2 · p. 11

4.3 Gaussian limits for the forest

• Finite-sample pooling of covariate values with different rates can introduce bias not represented by the Gaussian approximation.The limit is centered at the rate for a fixed covariate value.Source: §4.3 · p. 12
• Under the stated independent-subsampling regimes, forest rate estimators have an unconditional Gaussian limit whose covariance separates full-data estimation error from Monte Carlo variation.Sampling without replacement reduces the Monte Carlo contribution by a finite-population factor; averaging trees does not remove the common full-data error.Source: §4.3 · p. 12

4.4 Confidence intervals and simultaneous bands

• In the Markov case with continuous rates, the results provide pointwise intervals and constant-width bands simultaneous over time for a fixed transition and covariate value.The variance estimator is consistent, and the intervals and bands require no additional model fitting or resampling.Source: §4.4 · p. 13
• Coverage is close to nominal in several settings, but pooling heterogeneous rates can cause bias and undercoverage despite improving precision.Pooling substantially improves coverage over fully stratified estimation for sparse transitions, though it does not always restore nominal coverage.Source: §4.4 · p. 13

5 Numerical studies

• The numerical studies cover threshold interactions, irrelevant covariates, non-proportional hazards, and four biomedical applications; complementary studies assess flexibility, variable importance, and interval coverage.All simulated processes are Markov conditionally on covariates.Source: §5 · p. 13

5.1 Design and evaluation

• Prediction is assessed by held-out normalised integrated Brier score over 100 generated-data splits or 100 splits of each biomedical data set, with relative differences reported against JF.The Cox, RSF and AJ Null comparators use transition-wise models or estimates; occupations are computed by product integration of transition-specific cumulative hazards where described.Source: §5.1 · p. 14

5.2 Proportional hazards with a threshold interaction

• For n ≥1000, the three Cox specifications have nearly identical median relative differences of 3.5–4.1 percent in favour of JF; RSF differences decline from 15.0 percent at n = 500 to 8.1 percent at n = 5000.The comparison evaluates complete prediction procedures and does not isolate the contribution of sharing splits across transitions.Source: Figure 2 · p. 15, §5.2 · p. 15
• With 0–50 added noise covariates at n = 1000, JF's median relative differences against every Cox specification, including lasso, range from 1.7 to 5.6 percent, and every lower quartile exceeds 1 percent.The added covariates are independently drawn from the standard normal distribution.Source: §5.2 · p. 15, Figure 3 · p. 16

5.3 Prediction under non-proportional hazards

• Adding interactions improves Cox prediction, but JF remains favored at every sample size; for n ≥2000, median paired differences favor JF by 14.2–15.8% against Cox and 6.9–8.9% against Cox interactions.Under correctly specified proportional hazards, Cox has median relative difference −1.6 percent without noise.Source: Figure 4 · p. 17, §5.3 · p. 17

5.4 Biomedical applications

• Without added noise, JF and Cox have median relative differences of −0.1 percent on ebmt3 and 0.1 percent on bmt; Cox is favored by 5.0 percent on cphholter, while prova results are more variable.On cphholter, fewer than ten observations occur for each of two transitions, where regression structure may stabilize estimation.Source: §5.4 · p. 18, Figure 5 · p. 19
• With added noise, JF generally improves on unpenalised Cox; its median differences favor JF in all twelve lasso settings by 0.2–4.7 percent, while AJ Null has higher median paired relative IBS in all sixteen settings.Several lasso interquartile ranges include zero; the AJ Null comparison suggests predictive information in the covariates.Source: §5.4 · p. 18, Figure 5 · p. 19

6 Discussion

• The forest improves prediction over misspecified Cox benchmarks under threshold interactions, including with irrelevant covariates, and non-proportional hazards.Under correctly specified proportional hazards, it incurs only a small loss relative to Cox regression; biomedical-data performance varies across data sets.Source: §6 · p. 20
• Near-nominal confidence-band coverage occurs in several settings, but pooling heterogeneous rates can produce substantial undercoverage when its bias dominates reported uncertainty.Coverage gains over stratified estimation are reported for sparse transitions.Source: §6 · p. 20
• Prediction studies use the log-rank split rule and minimum node size ⌈√n⌉; OOB IBS enables tuning without a separate validation sample.Optimizing prediction error does not ensure valid confidence intervals at a given covariate value.Source: §6 · p. 20
• Fitted rate curves can help explore departures from Cox specifications, but are not formal evidence of interaction.Source: §6 · p. 20
• Continuous-covariate theory requires controlling pooling error within data-dependent nodes; honest splitting is one possible starting point.Smaller nodes and more exhaustive split searches warrant study for reducing pooling bias, with coverage assessed separately from prediction error.Source: §6 · p. 20

7 Software and reproducibility

• JumpForests provides fitting, prediction, OOB and test-set error estimation, and permutation importance for regression, classification, survival, and multi-state forests.For multi-state forests, it includes all five splitting rules; the C++ implementation uses Rcpp and parallelizes over trees.Source: §7 · p. 20
• Scripts for all paper and supplement numerical results are available in the JumpForests/manuscript folder.The ebmt3 data come from mstate, while bmt, cphholter, and prova are available from Andersen and Ravn’s companion website.Source: §7 · p. 20

S1.1 Alternative splitting rules

• Weighted log-rank splitting includes classical test weights and an approximate rule that avoids summing exact variance contributions over event times.The log-rank weight has optimal power against proportional hazards alternatives, while Gehan, Tarone–Ware, and Peto–Prentice weights emphasize early-time differences.Source: S1.1 Alternative splitting rules · p. 28, Table S1.1 · p. 28

S1.2 Error estimation

• Because right-censoring prevents direct scoring of out-of-bag predictions, the method uses IPCW for occupation-probability Brier and Kullback–Leibler scores.The normalized integrated versions summarize prediction error over time.Source: S1.2 Error estimation · p. 29

S1.3 Variable importance

• Permutation importance measures the increase in integrated prediction risk after replacing one covariate with an independent draw from its marginal distribution.The two sample versions use held-out data or out-of-bag observations; trees that never split on that covariate contribute zero to internal importance. With dependent covariates, the replacement can fall outside the support of X, comparing predictions at covariate values that do not occur.Source: S1.3 Variable importance · pp. 30–31
• The importance measures changes in occupation probabilities jointly across states, so they need not detect transition-rate changes that leave those probabilities unchanged.Uniform state weights with Brier loss are the implementation default; averaging independent permutations can reduce Monte Carlo variability.Source: S1.3 Variable importance · pp. 30–31

S2.1 Single-tree and bootstrap consistency

• Bootstrap-tree cumulative-rate estimates are consistent when the subsample size grows, and averaging a fixed number of bootstrap trees yields the forest result.Under an additional fourth-moment condition and summability condition, convergence is almost sure; a sufficient condition for Assumption 3 is also provided.Source: S2.1 Single-tree and bootstrap… · pp. 32–33

S2.2 Single-tree and conditional bootstrap limits

• Single trees and fixed-size forests have conditional Gaussian fluctuation limits under the stated subsampling schemes, describing variation around the full-data estimator rather than its error around the population target.With the data fixed, bootstrap weights are the source of limiting fluctuation as trees reduce to stratified estimators with conditional probability tending to one.Source: S2.2 Single-tree and conditional… · p. 34

S2.3 Variance in the Markov case

• When the process is Markov conditionally on X, the supplement provides an asymptotic variance formula and, under continuity of the cumulative transition rate, a plug-in estimator.Source: S2.3 Variance in the Markov case · p. 35

S2.4 Transition probabilities in the Markov case

• Under the conditional Markov assumption, transition-probability estimates are uniformly consistent over both time endpoints and have a weak-convergence result.The weak limit is a tight mean-zero Gaussian process defined through integration by parts.Source: S2.4 Transition probabilities in the… · p. 35

S3.1 Proof of Theorem 4.1

• Uniform consistency of tree rate estimators and initial-state distributions carries through averaging and the product integral to forest occupation probabilities.The result holds in probability, or almost surely under additional stated conditions.Source: S3.1 Proof of Theorem 4.1 · p. 36

S3.2 Proof of Theorem 4.2

• Bootstrap replicates converge conditionally to the same tight Gaussian limit as the original rate estimator.The proof treats the replicates as evaluations of the same map using independent bootstrap weights.Source: S3.2 Proof of Theorem 4.2 · p. 37
• For fixed B, continuous mapping yields a tight mean-zero Gaussian limit for the forest estimator, with covariance accounting for the subsampling scheme.The covariance limits are shown to agree across multinomial subsampling and subsampling without replacement.Source: S3.2 Proof of Theorem 4.2 · p. 37

S3.3 Proof of Corollary 4.3

• Studentization gives asymptotically standard-normal pointwise inference, while supremum calibration yields simultaneous bands with c0.95 = 2.2414.The band calibration uses the 95th-percentile threshold for the supremum of the absolute standard Brownian motion on [0, 1].Source: S3.3 Proof of Corollary 4.3 · p. 38

S4.1 Consistency of trees and bootstrap trees

• Almost surely, terminal nodes eventually isolate each covariate value, yielding uniform convergence of cumulative-rate estimators and convergence of occupation probabilities.The cumulative-rate convergence is uniform on [0, t]; occupation-probability convergence is almost sure.Source: S4.1 Consistency of trees and… · pp. 39–40
• Bootstrap-tree cumulative-rate estimators are weakly consistent when m_n → ∞ and strongly consistent when m_n^-2 is summable.The proof establishes that bootstrap terminal nodes eventually contain a single covariate value, then transfers consistency from the node estimator.Source: S4.1 Consistency of trees and… · p. 42

S4.2 Single-tree Gaussian limits

• Under finite second moments for transition counts, the single-tree components converge jointly to a tight, mean-zero Gaussian process with the stated covariance structure.The proposition combines the component limits into a joint weak-convergence result.Source: S4.2 Single-tree Gaussian limits · p. 46
• For single-tree cumulative-rate estimators, the proof obtains the Gaussian limit by combining a strong-law reduction with Proposition S4.3 and the functional delta method.Slutsky’s lemma completes the argument.Source: S4.2 Single-tree Gaussian limits · p. 46

S4.3 Conditional bootstrap limits

• The conditional bootstrap limits hold for multinomial subsampling and subsampling without replacement, while independent tree weights yield conditional Gaussian limits for normalized forest sums.The without-replacement limit includes the factor σα; the forest argument uses conditional Slutsky convergence and the continuous mapping theorem.Source: S4.3 Conditional bootstrap limits · pp. 48–49

S4.4 Markov variance and transition probabilities

• Under the Markov assumption, the limiting covariance is identified through martingale predictable variation, and the stated variance expression simplifies when cumulative rates are continuous.The martingale argument uses the enlarged filtration under Assumption 1.Source: S4.4 Markov variance and transition… · pp. 49–50
• The matrix-valued cumulative-rate estimator converges weakly, and the product-integral map transfers this limit to transition probabilities.The argument applies the functional delta method jointly across transition pairs and uses bounded total variation of the estimated cumulative rates.Source: S4.4 Markov variance and transition… · p. 50

S5.1 Bootstrap averages

• Bootstrap averages are weakly consistent under S1, while strong consistency requires stronger weight conditions and nonnegative iid observations with finite fourth moment.The two subsampling schemes satisfy the stated requirements under conditions on their sample sizes.Source: S5.1 Bootstrap averages · pp. 52–54

S5.2 Uniform convergence

• Pointwise convergence of non-decreasing càdlàg functions and their left limits implies uniform convergence in the corresponding mode of convergence.The conclusion applies to almost-sure convergence or convergence in probability.Source: S5.2 Uniform convergence · p. 57

S5.3 Counting-process decomposition

• Because occupation indicators are not monotone, the proof decomposes them into monotone components using the inflow–outflow identity and a censoring contribution.Uniform consistency then follows from uniform consistency of the component counting processes.Source: S5.3 Counting-process decomposition · p. 58

S5.4 Empirical processes and differentiability

• Uniform Hadamard differentiability on bounded-variation domains supplies the functional delta method for transferring weak convergence to rate estimators.The Nelson–Aalen functional is included as integration against the left limit of the at-risk process.Source: S5.4 Empirical processes and… · pp. 59–60
• The Nelson–Aalen map uses the reciprocal at-risk process, so the differentiability argument requires its values to remain bounded away from zero.In the applications, the lower bound follows from Assumption 5 and choosing t < TR,x.Source: S5.4 Empirical processes and… · p. 60

S6.1 Proportional hazards: noise covariates and variable importance

• With 50 noise covariates, the median paired relative IBS difference was 7.6% for Cox, shifting from Cox’s −1.6% advantage without noise; lasso differences stayed between −0.2% and 0.4%.Positive relative differences favor JF; lasso interquartile ranges contained zero throughout.Source: S6.1 Proportional hazards: noise… · p. 62, Figure S6.1 · p. 64
• Internal forest importance recovered X1 and X2 in 90–96% of replications, but at 50 noise covariates lasso more often recovered X3 and all three signals jointly.Lasso recovered X3 and all three signals in 40% and 31% of replications, versus 27% and 22% for internal importance; the comparison does not establish uniformly better variable selection.Source: Table S6.2 · p. 64, S6.1 Proportional hazards: noise… · p. 65

S6.2 Supporting results for the threshold interaction

• In the threshold-interaction study, Cox approximated the low-intensity group but was biased for the high-intensity group; JF captured their separation more closely, while overestimating early state-2 occupation.RSF underestimated absorption more markedly; the curves complement replicated score comparisons.Source: S6.2 Supporting results for the… · p. 65, Figure S6.3 · p. 66

S6.3 Non-proportional hazards

• For n ≥ 2000, median JF IBS was 0.100–0.103, while Cox interactions remained at 0.107–0.111 for n ≥ 500.At n = 500, median JF IBS was 0.104.Source: S6.3 Non-proportional hazards · p. 67, Figure S6.4 · p. 67

S6.4 Coverage study

• With noise covariates and n = 500, JF coverage for 1 →3 in group X1 = 1 was 88.9%, versus 8.3% for NA, which had no observed transition in 91.7% of replication–covariate pairs.Pooling substantially improved coverage in this sparse setting, although JF remained below nominal.Source: S6.4 Coverage study · p. 68
• For 1 →2 in group X1 = 1 without noise, JF coverage was 41.1–65.0%; at n = 5000 and t = 4, its group-averaged bias was −0.71 versus −0.09 for NA.JF’s mean interval was less than half as wide (0.84 versus 1.76); pooling heterogeneous rates can bias estimates, and pointwise coverage deteriorated after t = 1.2.Source: S6.4 Coverage study · p. 68

S6.5 Biomedical data

• The biomedical supplement compares normalized test-set IBS across four data sets and four noise-covariate counts, but the supplied figure caption reports no winning method.It also presents the state structures and observed transition counts.Source: S6.5 Biomedical data · p. 80, Figure S6.12 · p. 81

S6.6 Covariate interactions

• In bmt, transplant-to-relapse rate orderings reverse across levels of all and bmonly, but comparable overall IBS leaves the fitted patterns exploratory; in ebmt3, T-cell-depletion differences vary by age while age-group ordering is preserved.Differences in absolute spacing alone do not establish interaction on the multiplicative hazard scale.Source: S6.6 Covariate interactions · pp. 82–83