KESurv: A Kernel Ensemble Method for Patient-Specific Survival Prediction

Rahul Goswami

stat.MLOct 5, 2026

KESurv: A Kernel Ensemble Method for Patient-Specific Survival Prediction

Authors: Rahul Goswami

Organizations: Department of Mathematics, Indian Institute of Technology Guwahati

Abstract

Predicting patient-specific survival functions is crucial for clinicians in making informed decisions about patient care and treatment strategies. Among the various models available, the Survival Forest has demonstrated significant effectiveness in numerous scenarios. In this work, we propose an ensemble method that leverages the strengths of the Survival Forest as the master model, complemented by several base models. This ensemble incorporates the Beran estimator, a type of kernel estimator, to enhance predictions of patient-specific survival curves. We evaluated the performance of our proposed model using four distinct healthcare datasets. The results highlight the superiority of our ensemble method over baseline models in both calibration and ranking across most datasets. The findings suggest that our approach offers a more accurate and reliable estimation of patient-specific survival functions, providing a valuable tool for clinical decision-making.

Figures & tables

Panel of Figure 1 — Figure 1 : Overview of the proposed KESurv framework. Base survival learners compare subjects in survival-curve space, while the survival forest provides adaptive nearest-neighbor weights.

Dataset	$\bm{n}$	$\bm{p}$	Cens. (%)	Five-number summary of $\bm{Y}$					Ref.
				Min	$\bm{Q_{1}}$	Median	$\bm{Q_{3}}$	Max
Flchain	6524	8	69.93	0.00	2907.50	4303.00	4771.00	5166.00	a
Metabric	1904	9	42.07	0.00	60.82	114.90	184.47	355.20	b
Support	8873	14	31.97	3.00	26.00	231.00	763.00	2029.00	c
RoGBSG	2232	7	43.23	0.26	19.12	40.23	71.10	87.36	d

Table 1 : Dataset characteristics. Cens. denotes the percentage of censored observations

(\delta_{i}=0)

Dataset	Lasso	Net	Cox	Ridge	NN	Tree	XGBoost	Forest	KESurv
Flchain	0.7838	0.7840	0.7835	0.7835	0.7480	0.7672	0.7825	0.7688	0.7814
Flchain	( $\pm$ 0.0101)	( $\pm$ 0.0101)	( $\pm$ 0.0102)	( $\pm$ 0.0102)	( $\pm$ 0.0114)	( $\pm$ 0.0151)	( $\pm$ 0.0104)	( $\pm$ 0.0128)	( $\pm$ 0.0128)
Metabric	0.6281	0.6283	0.6276	0.6278	0.6378	0.6208	0.6407	0.6490	0.6599
Metabric	( $\pm$ 0.0198)	( $\pm$ 0.0190)	( $\pm$ 0.0196)	( $\pm$ 0.0196)	( $\pm$ 0.0188)	( $\pm$ 0.0238)	( $\pm$ 0.0145)	( $\pm$ 0.0226)	( $\pm$ 0.0172)
RoGBSG	0.6222	0.6192	0.6172	0.6172	0.6526	0.6423	0.6219	0.6501	0.6560
RoGBSG	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)

Table 2 : Time-dependent concordance results across four benchmark survival datasets. The best-performing method for each dataset is highlighted in bold.

Dataset	Lasso	Net	Cox	Ridge	NN	Tree	XGBoost	Forest	KESurv
Flchain	0.1036	0.1035	0.1034	0.1034	0.1067	0.1056	0.1333	0.1033	0.1027
Flchain	( $\pm$ 0.0040)	( $\pm$ 0.0036)	( $\pm$ 0.0039)	( $\pm$ 0.0039)	( $\pm$ 0.0055)	( $\pm$ 0.0046)	( $\pm$ 0.0047)	( $\pm$ 0.0047)	( $\pm$ 0.0045)
Metabric	0.1658	0.1658	0.1659	0.1658	0.1672	0.1730	0.1749	0.1690	0.1637
Metabric	( $\pm$ 0.0099)	( $\pm$ 0.0099)	( $\pm$ 0.0102)	( $\pm$ 0.0102)	( $\pm$ 0.0057)	( $\pm$ 0.0091)	( $\pm$ 0.0071)	( $\pm$ 0.0054)	( $\pm$ 0.0063)
RoGBSG	0.1763	0.1758	0.1756	0.1756	0.1717	0.1734	0.1975	0.1723	0.1679
RoGBSG	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)	( $\pm$ 0.0000)

Table 3 : Integrated Brier Score (IBS) results across four benchmark survival datasets. The best-performing method for each dataset is highlighted in bold.

Figure 2 : KESurv and Forest performance on the Support dataset across different maximum tree depths and numbers of trees. The top row reports IBS, where lower is better, and the bottom row reports time-dependent concordance, where higher is better.

Model	Search Space
Lasso	$\alpha_{\min\_\mathrm{ratio}}\in\{0.01,0.02,0.05,0.07\}$
Net	$\mathrm{l1\_ratio}$ linearly spaced in $[0,1]$ with 20 values
Cox	$\mathrm{ties}\in\{\mathrm{efron},\mathrm{breslow}\}$ , $n_{\mathrm{iter}}\in\{100,200,300,400\}$
Ridge	$\ell_{2}$ regularizer linearly spaced in $[0,1]$ with 20 values
NN	$k_{\mathrm{NN}}\in\{10,15,20,25,\ldots,60\}$ , distance $\in\{\mathrm{Manhattan},\mathrm{Euclidean}\}$
Tree	$\mathrm{max\_features}\in\{\sqrt{p},\log_{2}(p),3,4,5,6,7,8\}$ , $\mathrm{min\_samples\_leaf}\in\{8,32,128\}$

Table 4: Hyperparameter search space optimized using Bayesian optimization with Hyperopt-TPE.

Dataset	Lasso	Net	Cox	Ridge	NN	Tree	XGBoost	Forest	KESurv
Flchain	0.8468	0.8460	0.8466	0.8466	0.9308	0.8880	0.9682	0.8756	0.8480
Metabric	1.4070	1.4066	1.4084	1.4076	1.3932	1.4504	1.4182	1.3780	1.3350
RoGBSG	1.4608	1.4648	1.4680	1.4680	1.3816	1.4090	1.5462	1.3890	1.3596
Support	1.5870	1.5872	1.5874	1.5874	1.6042	1.6060	1.6640	1.5210	1.4846

Table 5 : Composite criterion

\mathcal{J}(\theta)=2\,(1-\mathrm{Concordance}(\theta))+4\,\mathrm{IBS}(\theta)

across four benchmark survival datasets. Lower is better; the best method per dataset is in bold.

Figure 3 : Nemenyi multiple-comparison-with-the-best plot based on the composite score

\mathcal{J}(\theta)=2(1-\mathrm{Concordance}(\theta))+4\,\mathrm{IBS}(\theta)

. Lower ranks indicate better overall performance. The Friedman test rejects equal average ranks across methods (

p=0.024

), and KESurv obtains the lowest mean rank.

Models	Flchain	Metabric	RoGBSG	Support
Lasso	1.0000	0.7318	0.3881	0.0033
Net	0.9999	0.7520	0.4058	0.0048
Cox	1.0000	0.3968	0.3533	0.0034
Ridge	1.0000	0.4267	0.3533	0.0032
NN	0.4030	0.9658	0.4378	$<0.0001$
Tree	0.9993	0.8271	0.4277	0.0109

Table 6 : Aggregated D-calibration

p

-values across experiments with different random seeds, combined using Fisher’s method Fisher (1925) . Fisher’s method computes the test statistic

\chi^{2}=-2\sum_{i=1}^{k}\ln(p_{i})

, where

k

represents the number of tests. The test did not reject D-calibration at the 5% level. Red text identifies models that fail to meet this threshold (

p\leq 0.05

	Distances	Flchain	Metabric	RoGBSG	Support
Con.	Median	0.7786 $\pm$ 0.0157	0.6544 $\pm$ 0.0148	0.6509	0.6409 $\pm$ 0.009
Con.	Max	0.7821 $\pm$ 0.0131	0.6558 $\pm$ 0.0198	0.6636	0.6415 $\pm$ 0.0098
IBS	Median	0.1029 $\pm$ 0.0048	0.1648 $\pm$ 0.0056	0.1665	0.1919 $\pm$ 0.0051
IBS	Max	0.1028 $\pm$ 0.0046	0.1658 $\pm$ 0.009	0.1673	0.1926 $\pm$ 0.0049

Table 7 : The proposed model with performance for different distances

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