Random Jump Forests

Martin Bladt; Rasmus Frigaard Lemvig

stat.MEOct 23, 2026

Random Jump Forests

Authors: Martin Bladt, Rasmus Frigaard Lemvig

Organizations: Department of Mathematical Sciences, University of Copenhagen

Abstract

Random jump forests are introduced as a nonparametric tool for estimating conditional cumulative transition rates and occupation probabilities in multi-state models. The main idea is that a single tree partitions the covariate space for all transitions simultaneously, using partially observed trajectories, and with Nelson--Aalen estimators at the terminal nodes. Occupation probabilities are recovered through the product integral without requiring the Markov assumption. Uniform consistency and a functional central limit theorem for the estimators of cumulative rates can be established under the assumption that the covariate vector takes finitely many values, and under suitable splitting conditions. In particular, pointwise confidence intervals and simultaneous bands are available in the Markov case. Simulation studies show a predictive accuracy close to the correctly specified benchmark models and marked improvements when benchmarks are misspecified, with further illustration on four biomedical data sets. In the coverage study, confidence bands attain near-nominal coverage in several settings, with pooling of heterogeneous rates being the main source of possible undercoverage. The method, including prediction assessment and variable importance, is implemented in the R package JumpForests.

Figures & tables

Panel of Figure 1 — Figure 1: Jump-tree construction and forest aggregation. (a) Splits on $X_{1}$ and $X_{2}$ produce three terminal nodes. Each node $h$ provides Nelson–Aalen estimates $\mathbb{\Lambda}^{h}$ of the cumulative transition rates and an empirical initial distribution $\mathbb{p}^{h}(0)$ . (b) For a covariate value $x$ , these quantities are averaged over the corresponding terminal nodes of $B$ trees grown on independent subsamples.

Figure 2: Threshold-interaction study,

100

replications. Relative difference (percent) in test-set IBS of the transition-wise Cox models (Cox, Cox X1:X2 and Cox interactions) and random survival forest (RSF) with respect to the jump forest (JF), as a function of sample size

n

. The vertical axis uses the transformation

y\mapsto\operatorname{arsinh}(y/4)

. Positive values favour JF.

Figure 3: Threshold-interaction study,

n=1000

100

replications. Relative difference (percent) in test-set IBS of the transition-wise Cox models (Cox, Cox X1:X2, Cox interactions and Cox Lasso) with respect to the jump forest (JF), as a function of the number of added noise covariates. Positive values favour JF.

Figure 4: Non-proportional hazards study,

100

replications. Relative difference (percent) in test-set IBS of the transition-wise Cox models (Cox and Cox interactions) with respect to the jump forest (JF), as a function of sample size

n

. Positive values favour JF.

Observed transitions
	ebmt3	bmt	cphholter	prova
States	3	4	4	3
Individuals	2204	2009	678	286
Covariates	4	5	10	7
Not absorbed	1363 (62%)	1272 (63%)	417 (62%)	211 (74%)
$1\to 2$	1169	976	70	50

Table 1: The four data sets and the settings of the jump forest. Observed transitions are counted from the available data using the state definitions as depicted in the Supplementary Figure S6.11 ; a dash marks a transition that the corresponding model does not contain. “Not absorbed” is the number (in percentage) of individuals censored before reaching the absorbing state. All covariates counted in the table enter every model except for bmt , where the 255-level categorical variable team is excluded. The settings follow Subsection 5.1 ; no tuning parameter is selected from the data.

Figure 5: Biomedical data,

100

random

80/20

training–test splits. Relative difference (percent) in test-set IBS of the Aalen–Johansen estimator without covariates (AJ Null), the transition-wise Cox model (Cox) and its lasso-penalised version (Cox Lasso) with respect to the jump forest (JF). Rows correspond to the four data sets; columns to

0

10

25

and

50

added noise covariates. Positive values favour JF. Symbols inside the boxes mark the means.

Splitting rule	Weight $w_{m}^{j}$
Log-rank ( Mantel, 1966 ; Cox, 1972 ; Peto and Peto, 1972 )	$1$
Gehan ( Gehan, 1965 )	$Y_{m}^{j}$
Tarone–Ware ( Tarone and Ware, 1977 )	$\sqrt{Y_{m}^{j}}$
Peto–Prentice ( Peto and Peto, 1972 ; Prentice, 1978 )	$\mathbb{p}_{j}^{h}(t_{m}-)\,Y_{m}^{j}/(1+Y_{m}^{j})$

Table S1.1 : Weights

w_{m}^{j}

in the splitting statistic ( S1.1 ). Here

\mathbb{p}_{j}^{h}(t_{m}-)

denotes the occupation probability of state

j

estimated within the node

h

just before time

t_{m}

, that is, the

j

th entry of

\mathbb{p}^{h}(0)\prodi\nolimits_{0}^{t_{m}-}(\text{Id}+\mathbb{\Lambda}^{h}(\mathrm{d}s))

, see ( 2.1 ).

	Proportional hazards	Threshold interaction	Non-proportional hazards	Coverage
Transitions	$1\to 2$ , $1\to 3$ , $2\to 3$	$1\to 2$ , $2\to 1$ , $1\to 3$ , $2\to 3$	$1\to 2$ , $2\to 1$ , $1\to 3$ , $2\to 3$	$1\to 2$ , $2\to 1$ , $1\to 3$ , $2\to 3$
Proportional hazards	yes	yes	no	no
Covariate effect	log-linear	threshold interaction	piecewise constant in time, with interactions	time-varying effects and interactions
Covariates	$X_{1}$ binary; $X_{2}$ , $X_{3}$ continuous	$X_{1}$ binary; $X_{2}$ , $X_{3}$ continuous	$X_{1}$ , $X_{3}$ binary; $X_{2}\in\{1,\dots,4\}$	$16$ signal combinations; $96$ with two noise covariates
Added noise covariates	$0,5,\dots,50$ , standard normal	$0,5,\dots,50$ at $n=1000$ , standard normal	none	none, or $X_{4}$ and $X_{5}$
Initial distribution	$(0.40,0.60,0)$	$(1,0,0)$	$(1,0,0)$	$(1/2,1/2,0)$

Table S6.1 : Overview of the simulation studies. All processes are Markov conditionally on the covariates, and every study uses

100

replications.

Figure S6.1 : Proportional hazards study with log-linear effects,

n=1000

100

replications. Relative difference (percent) in test-set IBS of the transition-wise Cox model (Cox) and its lasso-penalised version (Cox Lasso) with respect to the jump forest (JF), as a function of the number of added standard normal noise covariates. Positive values favour JF.

	$X_{1}$			$X_{2}$			$X_{3}$			All three
Noise covariates	10	25	50	10	25	50	10	25	50	10	25	50
JF, internal	93	90	91	96	94	94	44	36	27	35	26	22
JF, held-out	85	85	83	82	83	81	42	33	27	26	19	13
Cox, held-out	88	82	69	86	81	70	41	24	18	24	11	5
Cox lasso, held-out	88	80	74	83	76	71	51	46	40	42	38	31

Table S6.2 : Variable importance in the proportional hazards study of Supplementary Section S6.1 ,

n=1000

100

replications. Percentage of replications in which each signal covariate, and all three jointly, are ranked among the three most important covariates, for

10

25

and

50

added noise covariates. The internal importance ( S1.5 ) of the jump forest uses out-of-bag observations; the held-out importances ( S1.4 ) use the test data. Monte Carlo standard errors of the percentages are at most

5

percentage points.

Figure S6.2 : Threshold-interaction study,

100

replications. Normalised test-set IBS of the jump forest (JF), the transition-wise Cox models (Cox, Cox X1:X2 and Cox interactions) and random survival forest (RSF), as a function of sample size

n

Figure S6.3 : Threshold-interaction study,

n=1000

. True occupation probabilities (Truth) and estimates from the jump forest (JF), transition-wise Cox model (Cox) and random survival forest (RSF). Rows correspond to the three states; columns to four covariate values with

X_{3}=0

. The two right-hand columns have

c(x)=4

Figure S6.4 : Non-proportional hazards study,

100

replications. Normalised test-set IBS of the jump forest (JF) and transition-wise Cox models (Cox and Cox interactions), as a function of sample size

n

	Without noise covariates			With noise covariates
$n$	Splitting rule	Min. node size	OOB IBS	Splitting rule	Min. node size	OOB IBS
500	Peto–Prentice	50	0.1191	Peto–Prentice	50	0.1190
1000	Gehan	90	0.1133	Gehan	50	0.1129
2000	Peto–Prentice	150	0.1085	Gehan	125	0.1084
3000	Gehan	175	0.1081	Gehan	200	0.1079
4000	Gehan	200	0.1074	Gehan	175	0.1073

Table S6.3 : Tuning parameters of the jump forest in the coverage study. For each sample size, the splitting rule, chosen among the five rules described in Subsection 3.2 and Supplementary Section S1.1 , and the minimum node size, chosen from a candidate grid depending on the number of observations, minimise the normalised out-of-bag integrated Brier score (OOB IBS) on an independent pilot data set, which is shared by the two covariate settings; they are held fixed over the

100

evaluation data sets.

Covariates	Group	Transition	Model	$n=500$	$1000$	$2000$	$3000$	$4000$	$5000$
Without noise	$X_{1}=0$	$1\to 2$	JF	92.1	97.8	95.9	96.5	97.4	96.2
			NA	93.9	95.0	95.5	95.1	96.2	95.2
		$1\to 3$	JF	91.2	94.8	93.8	92.0	91.4	93.5
			NA	67.1	88.5	89.6	91.4	91.6	92.5
		$2\to 1$	JF	93.4	95.4	93.6	94.0	94.8	96.0
			NA	81.0	89.0	90.8	91.6	93.4	94.9

Table S6.4 : Simultaneous coverage (%) of the nominal

95

percent confidence bands on

[0,4]

for the jump forest (JF) and the fully stratified Nelson–Aalen estimator (NA), for all sample sizes, without noise covariates (

X_{1},X_{2},X_{3}

;

16

covariate values) and with noise covariates (

X_{1},\dots,X_{5}

;

96

covariate values). Coverage is averaged with equal weights over the covariate values within each group; intervals that cannot be computed count as non-coverage;

100

replications per sample size.

Covariates	Group	Transition	Model	$n=500$	$1000$	$2000$	$3000$	$4000$	$5000$
Without noise	$X_{1}=0$	$1\to 2$	JF	80.4	84.9	85.0	86.8	89.9	90.6
			NA	45.6	65.0	75.4	81.3	85.1	86.1
		$1\to 3$	JF	78.2	86.3	83.8	85.8	88.3	90.2
			NA	42.8	63.8	74.9	80.7	83.9	86.6
		$2\to 1$	JF	88.2	90.6	90.3	91.2	92.1	93.3
			NA	63.7	78.2	85.0	87.8	89.8	90.5

Table S6.5 : Pointwise coverage (%) of the nominal

95

percent confidence intervals of Corollary 4.3 (i), averaged over a grid of reporting times consisting of

250

common event times on

(0,4]

and the change points

1.2

2.3

and

3.3

, for the jump forest (JF) and the fully stratified Nelson–Aalen estimator (NA). Coverage is averaged with equal weights over the covariate values within each group;

100

replications per sample size.

			$t=1.2$			$t=2$			$t=3.3$			$t=4$
Group	Transition	Model	$n=500$	$2000$	$5000$	$n=500$	$2000$	$5000$	$n=500$	$2000$	$5000$	$n=500$	$2000$	$5000$
$X_{1}=0$	$1\to 2$	JF	78.8	85.9	92.2	83.1	89.1	93.1	82.8	91.2	94.8	91.8	94.6	95.5
		NA	27.8	74.9	88.8	44.8	88.9	91.2	63.1	86.6	92.2	93.0	94.2	95.4
	$1\to 3$	JF	75.9	86.1	91.2	89.5	89.2	93.9	92.6	90.6	94.4	88.9	91.0	91.8
		NA	30.5	75.8	90.2	46.8	91.2	92.8	64.5	88.9	93.4	67.1	87.0	90.4
	$2\to 1$	JF	91.5	93.0	93.2	92.4	91.9	93.8	91.4	92.8	96.2	91.9	93.2	95.8

Table S6.6 : Pointwise coverage (%) of the nominal

95

percent confidence intervals at the times

t\in\{1.2,2,3.3,4\}

for

n\in\{500,2000,5000\}

, without noise covariates, for the jump forest (JF) and the fully stratified Nelson–Aalen estimator (NA). Coverage is averaged with equal weights over the covariate values within each group;

100

replications per sample size.

			$t=1.2$			$t=2$			$t=3.3$			$t=4$
Group	Transition	Model	$n=500$	$2000$	$5000$	$n=500$	$2000$	$5000$	$n=500$	$2000$	$5000$	$n=500$	$2000$	$5000$
$X_{1}=0$	$1\to 2$	JF	82.1	89.8	95.6	84.2	93.5	96.7	82.9	94.3	96.9	92.1	97.0	98.2
		NA	5.3	21.2	46.5	9.3	33.4	64.3	15.2	48.4	81.8	76.3	90.4	93.6
	$1\to 3$	JF	77.4	88.8	94.2	90.3	92.0	95.5	93.1	90.6	95.9	89.9	92.8	94.5
		NA	6.1	21.2	46.5	10.2	32.7	64.7	16.2	48.2	81.5	17.1	51.5	83.3
	$2\to 1$	JF	92.8	95.7	95.9	93.4	94.9	97.0	92.5	95.4	98.0	92.5	95.8	98.3

Table S6.7 : Pointwise coverage (%) of the nominal

95

percent confidence intervals at the times

t\in\{1.2,2,3.3,4\}

for

n\in\{500,2000,5000\}

, with noise covariates, for the jump forest (JF) and the fully stratified Nelson–Aalen estimator (NA). Coverage is averaged with equal weights over the covariate values within each group;

100

replications per sample size.

			Bias				RMSE				Mean interval width
Group	Transition	Model	$t=1.2$	$2$	$3.3$	$4$	$t=1.2$	$2$	$3.3$	$4$	$t=1.2$	$2$	$3.3$	$4$
$X_{1}=0$	$1\to 2$	JF	0.000	0.000	0.000	$+$ 0.001	0.009	0.011	0.014	0.154	0.033	0.043	0.056	0.595
		NA	0.000	0.000	0.000	$+$ 0.002	0.013	0.016	0.021	0.227	0.044	0.060	0.079	0.843
	$1\to 3$	JF	0.000	0.000	0.000	$-$ 0.001	0.009	0.011	0.014	0.019	0.033	0.044	0.056	0.071
		NA	0.000	0.000	0.000	$-$ 0.001	0.012	0.015	0.021	0.027	0.045	0.061	0.079	0.098
	$2\to 1$	JF	$+$ 0.001	0.000	$+$ 0.003	$+$ 0.003	0.017	0.023	0.036	0.038	0.065	0.091	0.143	0.154

Table S6.8 : Bias and root mean squared error (RMSE) of the point estimators and mean width of the nominal

95

percent pointwise confidence intervals at the times

t\in\{1.2,2,3.3,4\}

, for

n=5000

without noise covariates, for the jump forest (JF) and the fully stratified Nelson–Aalen estimator (NA);

100

replications. The quantities are averaged with equal weights over the covariate values within each group, so that biases of opposite signs for different covariate values cancel in the average, whereas the RMSE (computed over all replications and covariate values of the group) does not. Entries

0.000

denote values smaller than

5\times 10^{-4}

in absolute value.

Figure S6.5 : Coverage study without noise covariates,

X_{1}=0

100

replications per sample size. Pointwise coverage of nominal

95

percent intervals for the jump forest (JF) and stratified Nelson–Aalen estimator (NA), averaged equally over covariate values within the group. Columns correspond to the four transitions; rows to

n=500

1000

2000

3000

4000

and

5000

Figure S6.6 : Coverage study without noise covariates,

X_{1}=1

100

replications per sample size. Pointwise coverage of nominal

95

percent intervals for the jump forest (JF) and stratified Nelson–Aalen estimator (NA), averaged equally over covariate values within the group. Columns correspond to the four transitions; rows to

n=500

1000

2000

3000

4000

and

5000

Figure S6.7 : Coverage study with noise covariates,

X_{1}=0

100

replications per sample size. Pointwise coverage of nominal

95

percent intervals for the jump forest (JF) and stratified Nelson–Aalen estimator (NA), averaged equally over covariate values within the group. Columns correspond to the four transitions; rows to

n=500

1000

2000

3000

4000

and

5000

Figure S6.8 : Coverage study with noise covariates,

X_{1}=1

100

replications per sample size. Pointwise coverage of nominal

95

percent intervals for the jump forest (JF) and stratified Nelson–Aalen estimator (NA), averaged equally over covariate values within the group. Columns correspond to the four transitions; rows to

n=500

1000

2000

3000

4000

and

5000

Figure S6.9 : Coverage study without noise covariates, $n=5000$ , transition $1\to 2$ . Cumulative transition rates for all $16$ covariate values. Grey curves show jump forest estimates from $100$ replications; black curves show the true rates. Blue curves show the estimate, nominal $95$ percent pointwise intervals and simultaneous band from the first replication.

Figure S6.10 : Coverage study without noise covariates, $n=5000$ , transition $2\to 3$ . Cumulative transition rates for all $16$ covariate values. Grey curves show jump forest estimates from $100$ replications; black curves show the true rates. Blue curves show the estimate, nominal $95$ percent pointwise intervals and simultaneous band from the first replication.

Panel of Figure 26 — Figure S6.11 : Multi-state models for the four biomedical data sets. Arrows indicate possible transitions and are labelled with the observed transition counts. Absorbing states have a heavier border.

Figure S6.12 : Biomedical data,

100

random

80/20

training–test splits. Normalised test-set IBS of the Aalen–Johansen estimator without covariates (AJ Null), the transition-wise Cox model (Cox), its lasso-penalised version (Cox Lasso) and the jump forest (JF). Rows correspond to the four data sets; columns to

0

10

25

and

50

added noise covariates. Points outside the whiskers mark individual replications.

Panel of Figure 28 — Figure S6.13 : Covariate combinations in the bmt data. Cumulative transition rates predicted by the jump forest for transplant to relapse ( $1\to 3$ , left) and GvHD to relapse ( $2\to 3$ , right). Top row: ages $19$ (blue), $32$ (grey) and $44$ years (orange), with sex and bmonly held fixed. Bottom row: bmonly $=1$ (blue) and $0$ (grey), with sex held fixed and age set to $32$ years. Solid and dashed curves denote all $=1$ and $0$ , respectively.

Panel of Figure 29 — Figure S6.14 : Covariate combinations in the ebmt3 data. Cumulative transition rates predicted by the jump forest for transplant to relapse or death ( $1\to 3$ , left) and platelet recovery to relapse or death ( $2\to 3$ , right). Colours denote age above $40$ years (blue), $20$ – $40$ years (grey) and at most $20$ years (orange). Top row, dissub : AML (solid), ALL (dashed) and CML (dash-dotted). Bottom row, tcd : no T-cell depletion (solid) and T-cell depletion (dashed). The remaining covariates are held fixed at their most frequent levels.

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