Adaptive Shrinkage Estimation for Personalized Deep Kernel Regression in Modeling Brain Trajectories
Authors: Vasiliki Tassopoulou
Haochang Shou
Christos Davatzikos
Venue: ICLR 2025 Poster (2025)
Keywords: Deep Kernel Regression
Personalization
Posterior Correction
Longitudinal Biomarker Prediction
Primary area: applications to neuroscience & cognitive science

Abstract: Longitudinal biomedical studies monitor individuals over time to capture dynamics in brain development, disease progression, and treatment effects. However, estimating trajectories of brain biomarkers is challenging due to biological variability, inconsistencies in measurement protocols (e.g., differences in MRI scanners) as well as scarcity and irregularity in longitudinal measurements. Herein,
we introduce a novel personalized deep kernel regression framework for forecasting brain biomarkers, with application to regional volumetric measurements. Our approach integrates two key components: a population model that captures brain trajectories from a large and diverse cohort, and a subject-specific model that captures individual trajectories. To optimally combine these, we propose Adaptive Shrinkage Estimation, which effectively balances population and subject-specific models. We assess our model’s performance through predictive accuracy metrics, uncertainty quantification, and validation against external clinical studies. Benchmarking against state-of-the-art statistical and machine learning models—including linear mixed effects models, generalized additive models, and deep learning methods—demonstrates the superior predictive performance of our approach. Additionally, we apply our method to predict trajectories of composite neuroimaging biomarkers, which highlights the versatility of our approach in modeling the progression of longitudinal neuroimaging biomarkers. Furthermore, validation on three external neuroimaging studies confirms the robustness of our method across different clinical contexts. We make the code available at https://github.com/vatass/AdaptiveShrinkageDKGP.

Published as a conference paper at ICLR 2025
ADAPTIVE SHRINKAGE ESTIMATION FOR PERSON- ALIZED DEEP KERNEL REGRESSION IN MODELING BRAIN TRAJECTORIES
Vasiliki Tassopoulou 1.2, Haochang Shou and Christos Davatzikos
Hacchann Center for Al and Data Science for Integrated Diagnostics, University of Pennsylvania Department of Bioengineering, University of Pennsylvania
a "Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania
(vasilike tassopoulou, hshou, christos
www.nneeepppeeee..mmm edu
ABSTRACT
Longitudinal biomedical studies monitor individuals over time to capture dynam- ies in brain development, disease progression, and treatment effects. However, estimating trajectories of brain biomarkers challenging due to biological vari- ability, inconsistencies in measurement protocols (e.g., differences in MRI scan- ners) as well as scarcity and irregularity in longitudinal measurements. Herein, we introduce novel personalized deep kernel regression framework for fore- casting brain biomarkers, with application to regional volumetric measurements. Our approach integrates two key components: population model that captures brain trajectories from a large and diverse cohort, and subject-specifie model that captures individual trajectories. To optimally combine these, we propose Adap- tive Shrinkage Estimation, which effectively balances population and subject- specific models. We assess our model's performance through predictive accu- racy metrics, uncertainty quantification, and validation against external clinical studies. Benchmarking against state-of-the-ar statistical and machine learning models-innllding linear mixed effects models, generalized additive models, and deep learning sethhods-dmmonsteern the superior predictive performance of our approach. Additionally, we apply method predict trajectories of compos- ite neuroimaging biomarkers, which highlights the versatility of our approach in modeling the progression of longitudinal neuroimaging biomarkers. Further- more, validation on three external neuroimaging studies confirms the robustness of our method across different clinical contexts. We make the code available inkaghrDiKn
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INTRODUCTION
Accurately predicting the temporal progression of brain biomarkers is essential for monitoring dis- ease progression and determining optimal intervention points (Maheux al. 2023). However, challenges such as biological variability among individuals, limited longitudinal data, and irregu- lar observation intervals make model development particularly difficult. Since accurate and reliable predictions are imperative, models must dynamically adapt as new subject-specific data become available, ensuring personalized predictions.
Several predictive models have been proposed to model the progression of biomarkers in the field of neuroimaging (Marinescu et al., 2018). Traditional methods, such as linear mixed effects models (Lindstrom & Bates) 1988), often struggle to handle high -itb-diiensionall multivariate data effectively and are predominantly used for statistical inference (Bsrnal-Rusiel et al. 2013] Xie et al.) 2023). Additionally, mixed-effect regression modeling is commonly employed to address longitudinal pre- dictions by fitting biomarker progression to linear or sigmoidal curves (Sabuneu et al. 2014 Koval al. 2021 I). However, this approach may be limited by its reliance on predefined trajectory shapes. More recently, Hong et al. (209)) and Gruffaz et al. (02T) explored manifold learning techniques to capture biomarker trajectories requiring subjects with least two acquisitions for inference. Ad- ditionally, Lorenzi et al. (2019) introduced a Gaussian based disease progression model capable of predicting biomarkers like cognitive scores and volumetric measurements, but it relies

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on specific design assumptions regarding the number of observations per subject and also uses low- dimensional input (i.e., five biomarkers). In the same spectrum, Koval et al. (2021) presented a Bayesian mixed effects model for estimating biomarker trajectories from low-dimensional inputs. Abi Nader et al. (2020) proposed a method for spatiotemporal progression of biomarkers without adapting subject's ss follow-up. [Tassopoulou et al. (2022) proposed a deep kernel regression method to infer biomarker trajectories from high-dimensional multivariate imaging features, though it does not utilize individual subject trajectories to refine predictions. In a related direction, Rudovic et al. (2019) developed meta- weea-weighting scheme combining two personalized Gaussian process models to forecast ADAS- ADAS-Cog13 (Mohs al. 1997) scores up to two years ahead. Similarly, Chung et al. (2019) introduced deep mixed effects framework for personalization in electronic health record time-series data, employing a long short-term memory network (Hochreiter & Schmidhuber 1997) to model population trends while using Gaussian process to capture subject-specific deviations.
In this paper, we address the above limitations by proposing Deep Kernel Regression with Adap- five Shrinkage Estimation, composite framework for predicting longitudinal brain trajectories leveraging all the available observations of the test subject, either single acquisition or multiple randomll-timme timoolyyyyeedddeee acquisitions. Unlike previous approaches that predict biomarkers within predeter- mined time intervals (Rudovic et al.] 2019). our method is designed to forecast over a practically unbounded future time horizon while simultaneously refining past observations by reducing noise in subject-specific observations. This dual capability enhances measurement reliability and pre- serves the global progression trend from the initial observation to any unseen future time point. Moreover, our framework naturally handles randomly-timed and temporally unaligned longitudinal observations without requiring imputation, thereby leveraging all available data. By extending the shrinkage estimator concept from Bayesian statistics and penalized inference (James & Stein 1961 Shou et al.) 2014), our method learns weights to combine population and sbbject-specific deep ker- nel model through adaptive shrinkage estimator, while accounting for both observation time and predictive uncertainty.
Contributions. 1) We propose novel deep kernel regression framework for predicting biomarker trajectories from sparse longitudinal observations, that maps nigh-dimensional, imaging and clinical features into dimer-dimensional latent space predictive of biomarker progression. Our approach naturally accommodates randomly-timen and temporally unaligned observations without requiring imputation. 2) We introduce Adaptive Shrinkage Estimation that fuses the population and subject- specific models. This framework enables incremental updates to personalized predictions as new data arrive and it also refines historical observations reduce noise while preserving the overall progression trend from the first observation to any future time. Importantly, the Adaptive Shrinkage estimator is interpretable. offering insights into the relative contributions of population and subject- specific model. 3) We showcase the versatility of our method to be applied for the prediction of two additional composite neuroimaging biomarkers from digeenimmnsi multivariate imaging data and clinical covariates. 4) We demonstrate the generalizability of our method in different clinical contexts, showing its ability to generalize in three external clinical studies.
2
METHOD
2.1 PROBLEM FORMULATION
We address the problem of predicting biomarker trajectories, modeled as a one-dimensional signal spanning multiple years. Formally, biomarker progression is described by the function f U - Y, where U E R and Y E RR. The input is represented as U = (X, M, T), where X de- notes the imaging features, M denotes the clinical covariates at subject's first visit, and T rep- resents the temporal variable, indicating time in months from the first visit. The biomarker tra- jectory is denoted as Y (yo, (yo,31.... vI....... Ma), corresponding the biomarker values time points T = -to. .to.tt.......... Our goal is to learn smooth functions biomarker progression using imag- ing and clinical data. To achieve this, we employ Deep Kernel Learning (DKL) (Wilson et al.) 2015). The deep kernel integrates imaging and clinical covariates, learning lower-dimensional representation informative for biomarker progression, while a Gaussian Process (GP) models the temporal dependencies. The backbone model, Deep Kernel Gaussian Process (DKGP), is defined as: f(U) ~ (P(,, K(O(U), (UU)), where & is a transformation function.
2.2 POPULATION DEEP KERNEL MODEL (P-DKGP)
The population model leverages data from the population dataset Dp = {U,, Y,). comprising subjects with longitudinal observations. It applies the transformation Φuu; W. b), Multi-Layer
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Wp
gP (M, En,)
Personalization through Adaptive Shriniage
A
S Wy
A
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Subject Specitic
d W
+Yo, Ve
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Y, V.
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R Rsst
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Dajk Raxx
Forward Pass
Correction
Figure 1: Overview of the proposed framework. In Figure 1a, illustrate the training process of the two models, p-DKGP. The population dataset D, contains multiple longitudinal acquisitions of subjects, where N is the total number of samples across all subjects, and L is the latent dimen- sion obtained from transformation 4. Different shades of green in the population dataset indicate different subjects in Dy- Figure 1b, we illustrate the training process of the ss-DKGP. We de- note the observed trajectory of subject ý with samples as D DaylA- These samples are utilized to train the ss-DKGP. During the training of the ss-DKGP, the transformation + is fixed, and only the subject-specific Gaussian process is optimized. In Figure IC, we visualize the personalization pro- cess through the adaptive shrinkage parameter a. For subject j. we extrapolate biomarker values over time using both the p-DKGP and ss- ss-DKGP models. These extrapolated values are then used to infer the adaptive shrinkage a for posterior correction, yielding the personalized posterior predictive mean Y. variance V. of the subject's trajectory.
Perceptron (MLP), that maps the input data Up = (X, M, into a latent representation:
Zp @UU p: b.
A GP, subsequently, models the biomarker progression function f using Radial Basis Function (RBF) kernel as the covariance function and a zero mean: f(zp) ~ GP (o, (Zp Z,)).
The population parameters = ( pr bp, Lp. o oh include both the transformation parameters of & and the Gaussian Process (GP) hyperparameters: the lengthscale 4p. signal variance of. and noise variance o These parameters are jointly learned by maximizing the Marginal Log Likeli- hood (MLL) of the GP (Wilson et al. 2015 Rasmussen & Williams) 2006).
For a test subject j with input Uj = (v,, mj, (), we denote the transformed input as 2j = 4uyy W. b,).
The posterior predictive distribution of the biomarker function at point = (xz, mj, () is:
SP: (Zps Y pp (fp,, cov cov(Gp,))
The mean and variance of the predictive posterior distribution provide the predictions and their uncertainties, respectively, and are calculated as follows:
Fr, Eff. Zp, Y 25] K K(2, [ Zp. Zp) o 'y Var(f,,)
(3)
2,) K(2,, Z,) Zp)) + + I K K(p 27);
(4)
where o2 is the additive independent identically distributed Gaussian noise e.
For simplicity, the predictive mean and variance a biomarker for test subject from the p- DKGP are denoted as yp and Up. respectively. By prompting the p-DKGP model with different time intervals f, yields the predicted trajectory and predictive uncertainty across time, represented as Yp (Upr Upa.-... )pr) and Vp (1p,:
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2.3 SUBJECT-SPECIRE DEEP KERNEL MODEL (sS-DKGP)
For new test subject, let h denote the number of observations and Tobs the time of observation from the initial acquisition. The observed data for the subject represented as D2 {(X,, M,, Tg), Y-)- The ss-DKGP model is trained on Dg to capture the subject-specifie trajectory. The transformation ④(; p. bp), learned via the p-DKGP, initializes the deep kernel of the subject-specific model.
We initialize new GP with RBF kernel and a zero mean. During the training of the ss-DKGP, only the observed trajectory of the subject is used. Specifically, we update the GP hyperparame- ters, which include the lengthscale i and the signal variance o. while keeping the weights of the function ( W p. bp) frozen during backpropagation. The subject-specific GP hyperparameters
% = Ule oi oi are jointly learned by maximizing the MLL of the GP.
For subject with input új (z,, m,, (), we denote their transformation 2; &u,;; Wp. bp).
The posterior predictive distribution of the biomarker progression function at time point / is:
Saz (Zs; Y)), (fa,, cov (fv((s
where: 2j = (uy; Wp. bp))
The predictive mean and variance, representing the predictions and their associated uncertainties respectively, are computed as follows:
f,y = Ef,, 2s Y s: 2z] K(z,, Z.) Z.) + o 1 Var f..)) = (z) - K(z,, Z.) (K. Z.) + +o, 1 1K-KK( (Z., 2j).
(7)
where o. is the additive independent identically distributed Gaussian noise e.
For simplicity, the predictive mean and predictive variance of the ss-DKGP are denoted Vsj and Mays respectively. By querying the ss-DKGP model at different time intervals f we reconstruct the biomarker trajectory of subject j. yielding the predicted trajectory Y. (Vs, Voz...... ... : and predictive uncertainty V, = (Us) ,,7.
2.4 PREDICTIVE POSTERIOR CORRECTION
Given predictions yy and ys from the p-DKGP and ss-DKGP models, the personalized prediction is expressed linear combination:
Vc ay, (1 aayys.
where, is the shrinkage parameter reflecting the relative confidence in each model. Assuming independence between the models, the combined prediction ye retains Gaussian properties, and its variance is given by:
tc = a2u + - v)
In Supplementary Section
2.4 we address the independence assumption and its impact.
The weights a and a quantify the credibility of each model, yielding a new posterior predictive mean Y, and variance V.. Values of close to indicate higher confidence in D-DKGP model, while values close to 0 reflect greater trust in ss-DKGP model. We refer to a as the shrinkage parameter.
2.4.1 ACQUIRING THE ORACLE SHRINKAGE
Estimating the oracle shrinkage parameter a is crucial for constructing the personalized posterior predictive means and variances of the biomarker trajectory. To estimate a, of subjects with known trajectories, unseen by the population model. Predictions for these subjects are generated using the P-DKGP model. For each subject, the ss-DKGP component is trained by progressively increasing the length of the observed trajectory.
held-out set
The entire biomarker trajectory is reconstructed from the baseline time (1 0) to the subject's last time point t, Using both models, P-DKGP and ss- DK-D we obtain two estimates of the biomarker trajectory along with their predictive variances. Let Y, and V, denote the P-DKGP predictive mean and variance, and Y, and V, denote the ss-DKGI model predictive mean and variance. Let Y represent the ground truth biomarker values over time. The oracle a is estimated by minimizing the following criterion:
(a) =
Σ (vr (a yr. + +1--a) ai) y), 1=0
(10)
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The notation Jajh reflects that this optimization is performed for subject 8, given h observed acquisitions. The algorithm for calculating the oracle shrinkage estimates on the validation set is outlined in Algorithm [ Each subject's data is processed individually, applying the optimization to each sequence of observations. This process is repeated for every subject in the validation set.
Algorithm Oracle Shrinkage Estimation
Require: Validation set V = {(U*, Y())) s S}. where y )Tm is the ground truth trajectory for subject
Ensure: Optimal shrinkage parameters ús,A for each E S and h E H
1: for each 8 E S
2: Initialize list L +-[]
3: for each h E do
4: Obtain P-DKGP trajectory: Y (s.h) (n.h)
st=1
5: Obtain ss-DKGP trajectory: Y (o.h) (v..0h) =1=1
6:
Define objective function:
(an =
11
(ay((sh)))
(a.h) + +(1 a)ye, (8.1)) )
1=
7:
Compute:
urhh arg min J., (a) aE(0,1]
8: Append da,, to ((e)
9:
end for
10:
Store list (+) for subject
end for
2.4.2 LEARNING THE ADAPTIVE SHRINKAGE
The shrinkage parameter represents the trust factor between the two components (p- (p-DKGP and ss- ss-DKGP). We model a as a function the imput variables q = {ppp Ya. Up. Ue, Tots J. where q e R and Tok. represents the time of observation. Using oracle shrinkage obtained from Section 2.4.1 on the validation set, our objective to learn a mapping function 9a that transforms the input space q E R the output space of adaptive shrinkage E IR as á 9. (g: 0)-
We employ XGBoost regression to learn the function g that minimizes the difference between the predicted à and the oracle a. The learned function is denoted 9o- In Supplementary Section we provide results from additional non-linear functions we experiment with, demonstrating that XGBoost achieves the best performance for estimating the shrinkage a.
2.5 PERSONALIZATION THROOGHI ADAPTIVE SHRINKAGE ESTIMATION
For test subject with observations and Tobs the observation time (measured from the sub- ject's first visit), we train the ss-DKGP model as described in Section 2.3 The posterior-corrent predictive distribution, referred as pers-DKGP, is computed using the following algorithm:
Algorithm 2 Personalization through Adaptive Shrinkage Estimation
Require: P-DKGP model, ss-DKGP model, and learned function Jo
Ensure: Adapted predictive mean and variance: Y,, V.
Compute Y,, Vp (predictive mean and variance) from the p-DKGP model.
2: Compute Y., V, (predictive mean and variance) from the Ss-DKGP model.
3: Adapted Shrinkage Estimation: O (Y, (Y,,, Y., Vp, V, Tom.).
4: Compute the personalized predictive mean: Y, án. Yp + (1 (n)) - Y.,
5: Compute the personalized predictive variance: V. = ai Vo + +11 an)² 2. V..
6: return Y., V..
The personalization process through Adaptive Shrinkage Estimation is described in Algorithm
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ROI Volume Blomarkers
b
SPARE-AD
SPARE-BA
DKGP
LMM
CAM
DeepRegr
DME
Mean AE with Progresiin Status
Progression Status
Mean AE with Progression Status
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0,8
Mean MAE
0.5
Mean Ar
0.
y C.6
0.3
0.4
2
30.4
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0.1
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Progressc
Healthy Contral
MCI Progresee
MC Stable
UKN
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AD Prograse
AD
Healthy Control
MC Progresscs
MCI Stable
UKN
c.
AD Progressor
AD
Healthy Contral
MCI Progresor
MO Stuble
UKN
AE with Time
Time
AE Time
pers DKGP
2.00
DWGP
2.0
Pos-DKEP
1.75
LMM
Mean AB
Deeating
DIM
GAM
# 1.05
1.50
DME
GAM
DeeyPeng
Deepling
DME
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1.00
1.0
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0:75
0.4
0.2
10 12
10
12
2
13
32
Time from last observation (In ysars)
Time from last olsrrvation (in years)
Time from last absaroation (in yrars)
We compare the mean MAE per subject stratified by the progression status (top) and the AE with time from the last observation (bottom) of our method with the baselines for (a) the ROI Volume biomarkers, (b) SPARE-AD score and (c) SPARE-BA. Error bars, in the top row, denote the 95th percentile of the MAE across all subjects. Our method is denoted as pers-DKGP.
Figure 2:
3 EXPERIMENTS
3.1 PREDICTION OF REGIONAL VOLUMETRIC TRAJECTORIES
In this section, we apply deep kernel regression with Adaptive Shrinkage Estimation to predict tra- jectories of seven volumetric Regions of Interest (ROI): Hippocampus R, Hippocampus L. Thalamus Proper R, Amygdala R, Amygdala L, Parahippocampal Gyrus R and Lateral Ventricle R. For each ROI Volume model we use a dataset of 2, 200 subjects with U; = (X; Mr, T;) from subject i, where X, are volumetric measures from 145 brain regions collected at subject's first visit, M, are the co- variates of diagnosis at subject's first visit, sex, age, education, APOE4 Alleles, genetic variant related to AD and T; is the time from subject's first visit. For each ROI Volume biomarker, the P-DKGP model is trained on population cohort of 600 subjects, while the adaptive shrinkage estimator is trained on a held-out set of 200 subjects. Predictive performance is evaluated on 440 test subjects. For details the architecture and training of the ROI Volume deep kernel models, P-DKGP and ss-DKGP, see Section B.I
We combine preprocessed and harmonized neuroimaging measures from two well-known longitu- dinal studies: the Alzheimer's Disease Neuroimaging Initiative (ADNI) (Weiner et al.) 2017) and the Baltimore Longitudinal Study of Aging (BLSA) (Ferrueci, 2008). which focus on Alzheimer's Disease and Brain Aging, respectively. Further details on the studies and preprocessing pipelines
are provided in Supplementary Section A
We benchmark our method against several baselines and state-of-the-art predictors: Linear Mixed Model (LMM) (Lindstrom & Bates 1988), Generalized Additive Model (GAM) (Hastie & Tibshi- rani 1986). Deep Neural Network Regression, and the Deep Mixed Effects (DME) (Chung et al.] 2019). Further details on the architectural design and training of baselines are provided in Sup-
plementary B.3 Model performance is evaluated from two perspectives: predictive accuracy and uncertainty quantification (UQ), Predictive accuracy is measured using Absolute Error (AE) and Mean Absolute Error (MAE) per subject. UQ is assessed by interval width (the range between #2 standard deviations from the predictive mean) and coverage (the proportion of true biomarker values within that range). Importantly, these metrics are computed over the entire unseen trajectory of test subjects, providing a comprehensive evaluation of model performance over time. We refer to our method as personalized-DKGGr or shortly DKGS.D

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Healthy Control
Healthy Control
AD Progressor
pers-DKGP
LMM
4 Observations
Observations
Observations
7
Observations
Hippocampus (saadardized)
Thalanus Proper R (standardized)
Latera Ventricle R. (standandized
0.0
Time from baseline months)
baseline (in months)
Time from baseline months)
Figure 3: We present personalized ROI volume trajectories for three test subjects observations increase from 4 to 7 acquisitions. The dashed lines represent the prediction using LMM. The first two panels visualize the Hippocampus R and Thalamus Proper R Volume trajectories of Healthy Control subject. Last panel shows the Lateral Ventricle Volume for an AD Progressor. The shaded bands represent the predictive uncertainty over time.
For each predictor, Figure presents a comparative study of the predictive performance with spect to progression status and time from the last known acquisition. Progression status is defined by the subject's initial and final diagnoses, categorized as follows: AD refers to subjects diagnosed with AD at their first visit; AD Progressor includes subjects initially diagnosed as Cognitively Normal (CN) Mild Cognitively Impaired (MCI) who progress to AD; Healthy Controls are subjects who remain CN throughout all visits; MCI Progressor refers to subjects who progress from CN to MCI; MCI Stable includes subjects who remain MCI throughout their trajectory; and Unknown (UKN) corresponds to cases involving misdiagnosis.
Building on this categorization, Figure 2 shows the mean MAE across progression status for the seven volumetric ROIs Notably, the largest mean MAE differences between our method and base- lines occur participants with AD and AD Progressors, who exhibit non-linear and steeper trends that competing baselines fail to capture. Specifically, the Linear Mixed Model (LMM), constrained to linear patterns in ROI volumes, shows significant percentage mean MAE differences in AD (177.66%) and AD progressors (22.05%). Even in healthy controls, LMM exhibits a 29.78% MAE difference, highlighting its inability to capture trajectories even in cases of relatively stable volume trajectories. Further quantitative comparisons, including eтror stratification by covariates such as sex, APOE4 Alleles, and education years, are provided in Supplementary Sections D.I
In addition evaluating performance with respect progression status, we also assess the model's ability to predict long-term longitudinal trajectories. In Figure 24, we visualize the mean AE across different lengths of observed trajectories. with errors plotted relative to the time from the last obser- vation. Our method achieves progressively lower mean AE over time, indicating improved precision in both long-term and short-term predictions. This demonstrates the model's ability in capturing temporal trends and adapting to varying observation lengths.
To further highlight the strengths of our model, we provide a qualitative evaluation of the predicted trajectories in Figure 3 For the Volume ROIs of Hippocampus R, Thalamus Proper R, and Lateral Ventricle R. our model successfully adapts to the observations of test subjects, resulting in more accurate long- population prediction deviates from the actual trajectory. However, the number of observations increases, the pers-DKGP trajectory shifts toward the observed trajectory, effectively adapting to the subject-specific trend. Similarly, the third subject, an AD Progressor in Figure [3,, exhibits an abrupt increase in ventricular volume. This trend is captured with few observations by the pers- DKGP model, while the LMM underestimates the ventricular volume in the long term. Additional qualitative examples of trajectories are provided in Supplementary Section D33
long-term term predictions. For instance, in the Healthy Control subject shown in Figure the
Overall, the LMM exhibits limited flexibility in capturing non-linear patterns in ROI volumes, rendering it inadequate for long-term biomarker prediction. While it performs reasonably well in short-term forecasts and lower-dimensional settings, its expressiveness falls short for complex, high-dimensional inputs. Deep regression, though capable of learning from observed data, often
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yields non-smooth non-monotonic trajectories that deviate from biologically plausible biomarker progression trends. The DME model, which combines a shared deep mean function with subject- specific GP, struggles to achieve personalization in nigh-dimensional input spaces, resulting in per- sistent errors across time and diagnostic categories. These issues stem from the limitations of the RBF kernel in managing multivariate, high-dimensional data. In contrast, our method effectively approximates non-linear mixed effects models, demonstrating flexibility in handling multivariate, high-dimensional data and capturing diverse temporal patterns.
3.2 APPLICATION TO NEUROIMAGING BIOMARKERS: SPARE SCORES
Having demonstrated our framework's ability to personalize longitudinal predictions of volumet- ric ROIs as subject observations increase. we now show its versatility by applying it to composite neuroimaging biomarkers: the SPARE-AD (Davatzikos et al. (209) and SPARE-BA (Habes et al., 2016) scores. SPARE-AD quantifies the risk of AD progression, while SPARE- -BAE-BAAAA represents pre- dicted brain age. For both SPARE models use dataset of 2,200 subjects with U. = (X; M,, T.) from subject i, where X; are volumetric measures from 145 brain regions collected at subject's 's first visit, M; are the covariates of diagnosis at subject's first visit, sex, age, education, APOE4 Alleles, the SPARE-AD and SPARE-BA values at the first visit and T; is the time from subject's first visit. The P-DKGP model is trained on 1600 subjects, the adaptive shrinkage estimator trained on a held-out set of 200 subjects. The evaluation of the predictive performance is performed on the 440 test subjects. For details on the architectural design and training of the SPARE-AD and SPARE-BA deep kernel models (p-DKGP and ss-DKGP) see Section B.2
Our model demonstrates strong performance in predicting long-term longitudinal trajectories for both SPARE-AD and SPARE-BA biomarkers, as illustrated in Figure and 2 Notably, the model achieves progressively lower mean AE over time, indicating improved precision forecasting long- term outcomes. For SPARE-BA, model performance differences are minimal in stable subjects and healthy controls, but more pronounced in AD subjects, where SPARE-BA BA exhibits steeper progres- sion trends due to accelerated brain aging. For the SPARE-AD biomarker, we also visualize absolute error with the numher of observations. This highlights how our model adapts with increasing obser- vations, starting with single scan using the P-DKGP model (a 1) and transitioning to adapted shrinkage estimation for personalization as follow-up observations increase. Evidence is provided in Table 5 and Figure 7 in Supplementary Section D.2
3.3 APPLICATION TO EXTERNAL NEUROIMAGING STUDIES
In this section, we demonstrate the generalizability of our method to previously unseen neuroimag- ing datasets. After training the PPD and adaptive shrinkage estimator on the population and val- idation datasets from the ADNI and BLSA cohorts, we personalize starting from the first follow-up point for each subject and predict the remaining trajectory. This process is repeated for all follow-up points, with the very last follow-up reserved for testing.
We test the performance of our framework on subjects from three independent clinical studies: OA- SIS (Mareus et al., 2010). AIBL (Ellis et al. 22009). and PreventAD (Tremblay-Merciec Mercier al.) 2021). These datasets differ from the training population in terms of demographics, diagnosis composition, and follow-up intervals, presenting challenging test of the model's is generalizability across diverse populations. In Supplementary Section A we present details on the demographic and clinical char- acteristics of these studies.
The three external studies exhibit notable differences in demographics and follow-up intervals: AIBL: Includes 82 individuals with a mean age of 75 years, which is close the mean age of the joint cohort of ADNI and BLSA. It is predominantly composed of AD patients followed by MCI and Healthy Controls. On average, each subject has approximately 3 follow-up visits, with a mean interval of 24 months between visits. OASIS: Includes 559 individuals younger on average (67.8 years) compared to both ADNI and BLSA. It is primarily composed of healthy controls, with smaller representations of MCI and AD cases. The average number of follow-ups is ~ 3 per subject, with mean interval of 32 months. PreventAD: Includes 271 individuals and focuses on pre-symptomatic early detection of AD in healthier and younger population (mean age 65.3 years) with an average of 4 follow-up visits per subject and a shorter mean interval of 10 months.
Our method outperforms baseline predictors across three independent clinical studies AIBL, OA-
SIS, and PreventAD underscoring its effectiveness in diverse, real-world scenarios (Figure 4 The model achieves lower MAE compared to baselines, with narrow confidence intervals reflecting its
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stability. In the AIBL study, pers-DKGP achiives a Mean AE of 0.197 0.009, substantially outper- forming the baseline methods. A similar trend is observed in the OASIS study, where pers-DKGE attains Mean AE of 0.259 + 0.006. Notably, in the PreventAD study, method achieves the lowest Mean AE of 0.139 = 0.004, outperforming LMM and GAM. The narrow Cls of the AE as- sociated with pers-DKGP across all datasets highlight its reliability and consistent precision, even in the presence of data variability. Interestingly, the lowest error observed in the PreventAD study, along with the reduced disparity between pers-DKGP and statistical models like LMM and GAM, is attributed to the younger population and shorter follow-up intervals in this dataset. Predicting Vol- ume ROIs in a younger, healthier control population, as in PreventAD, is inherently less challenging compared to the older, partially demented populations in OASIS and AIBL.
Collectively, these results position our model as a robust and reliable framework for personalized forecasting of neuroimaging biomarkers, offering potential for application in clinical trials and neu- roimaging studies.
Mean Across Clinical Studies
0.6
Model
DKS
1.5
LMM
Mean AE
0.0
DeepRegr
0:3
DME
0.2
6.1
Preven:AD
Figure 4: We evaluate the mean absolute error for the seven ROI Volume biomarkers across three external neuroimaging studies. Error bars denote the 95th percentile the absolute error. Notice that the pers-DKGP achieves the lowest error across all external studies, in comparison with the competing baselines.
3.4 EXPLAINING ADAPTIVE SHRINKAGE: AN ABLATION STUDY ON THE a ESTIMATOR
In this section, we demonstrate the effectiveness and interpretability of the Adaptive Shrinkage esti- mator. We first compare it to alternative posterior correction approaches and then use explainability analysis to illustrate how Adaptive Shrinkage estimator learns to balance the two posterior predictive distributions data-driven manner, making its decision makinn process intuitive.
We explore various strategies for selecting the shrinkage parameter a. First, we experiment with constant a c, where e E (0, 1). representing an uninformative approach to posterior correction. Next, we employ semi-informative (deterministic) approach, where the for each test subject determined by optimizing the objective in Equation [T0 using only subject's observed trajectory. Finally, use Adaptive Shrinkage estimator to determine We conduct this experiment for seven ROI Volume biomarkers: Hippocampus R/L, Lateral Ventricle, Thalamus Proper, Amygdala R/L, and the Parahippocampal Gyrus R. Here, present results for Hippocampus R, Lateral Ventricle, and Thalamus Proper under the constant a and Adaptive Shrinkage. Results for the remaining Volume ROIs and the deterministic approach are provided in Table 6 of Supplementary Section D4.1
The deterministic approach (Table results in the worst outcomes in terms of both predictive per- formance and uncertainty quantification, suggesting that the observed trajectory alone is insufficient determine the a for future predictions. Per-subject optimized can overfit the noise gle subject's 's limited data, leading to poorer generalization, whereas the learned adaptive shrinkage generalizes better across subjects. Additionally, in the constant a section of Table [ we present the performance of the best constant a values. This demonstrates that optimal performance is not achieved through simple averaging and that the optimal a varies significantly across ROIs. For ex- ample. the best is 0.5 for Hippocampus, 0.3 for Lateral Ventricle, and 0.7 for Thalamus Proper. These results highlight the inadequacy of a one-size aits-all approach and underscore the necessity for more sophisticated method. The evidence suggests that Adaptive Shrinkage provides more informed approach for determining the ideal a, leading to improved predictive performance and uncertainty quantification.
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Table 1: Ablation study the shrinkage parameter We report the Mean AE along with its 95%
percentile CI, Mean Coverage, and Mean Interval Width
Best Constant
Adaptive Shrinkage
ROI
Mean AE (CI) Mean Cov.
Mean Int.
Mean AE (CI)
Mean Cov.
Mean Int.
Hippocampus R
0.5 0.257 (#0,007)
0.808
0,843
0.243 (#0.003)
0.795
0.902
Lateral Ventricle R
0.3
0.143 (#0.006)
0.853
0.507
0.131 (=0.002)
0.855
0.626
Thalamus Proper R
0.7
0.241 (#0.007)
0.934
1.127
0.219 (=0.003)
0.849
0.911
Following, elucidate the decision-making process of Adaptive Shrinkage with explainability analysis. We focus on the impact of each input variable-Y,, Y,, Vp. Vs, and actions on the prediction of the adaptive shrinkage parameter Specifically, aim to understand how the deviation between the population and subject-specific predictive means (6, = Yp Y)) and the observation time Toss influence the model's predictions.
their inter-
We employ SHAP (SHapley Additive exPlanations) values (Lundberg & Lee 2017) to interpret the contribution of each feature individual predictions. Figure [3 Supplementary Section D.4 reveals that Tobs is the most influential variable in the decision -makingakinng. process. This is further validated by the observation that the distribution of adaptive shrinkage a decreases as the number of follow-up visits (and thus Ton)) increases. Figure 12 in Supplementary Section D.4 demonstrates the distribution of a with the number of observations for the seven ROIS and SPARE scores, as well as the adaptive shrinkage obtained from external neuroimaging studies. The consistent trend of decreasing a as the number of observations increases highlights the biomarker-agnostic ability of Adaptive Shrinkage to optimally combine population and subject-specific trends. This behavior is also consistent across external neuroimaging studies, further validating the generalizability of the approach. Additional qualitative results demonstrating the decision-make making process of Adaptive Shrinkage are provided in Supplementary Section D.3] Figures 10 and
Furthermore, correlation analysis (Supplementary Section Table reveals consistent negative relationship between Tob. and the predicted a when the deviation by large. This indicates that, in the presence of significant deviations between the two predictors, Adaptive Shrinkage reduces the weight assigned to the population-level model (p-DKGP) for longer observation periods. This aligns with the intuition that as more follow-up observations available, greater trust is placed in the subject-speciff predictive distribution.
4 DISCUSSION
In this paper, we introduce deep kernel regression with Adaptive Shrinkage Estimation for predict- ing personalized biomarker trajectories via posterior correction. Our method learns the adaptive shrinkage parameter that effectively combines two posterior predictive distributions, enabling the predictive trajectory to adapt to each subject's follow-up acquisitions. Additionally, versatile, effectively modeling the progression of longitudinal biomarkers using multivariate imag- ing data and clinical covariates. Examples of such biomarkers are the cognitive scores (e.g., MMSE, IDAS-Cog13) and blood biomarkers (e.g., Amyloid-3, Tau protein). Importantly, our approach ex- hibits generalization capabilities when applied to external neuroimaging studies with diverse demo- graphics and follow-up intervals, which is particularly valuable for real-world applications, where models must perform robustly across heterogeneous populations.
method is
This property is particularly important as the use of predictive models in healthcare is increasingly critical for both patient management and drug development. Cummings et al. (2019) emphasize the need for Al-informed clinical trials, referred to as precision trial design, while Maheux et al. (2023) evaluate predictive models for biomarker trajectories Alzheimer's Disease, where derived mea- sures-such as the rate of change- serve as quantitative indicators of disease progression during clinical trials. These measures inform decisions on subject inclusion and treatment efficacy, under- scoring the importance of reliable and interpretable predictive tools. Our method's adaptive and intuitive design positions it as a valuable tool for clinical trial design, disease progression modeling, treatment effect estimation, and neuroimaging research. By leveraging personalized predicted ROI Volume and neuroimaging biomarkers, such as SPARE-AD, as endpoints for selecting trial subjects, application.
framework showcases its potential for
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At the same time, acknowledge limitations in approach, particularly the independence sumption between the a parameter and posterior distributions in the posterior correction step (Sup- plementary Section 2.4 While this simplification impacts uncertainty quantification, it does not affect the posterior-corrected predictive mean, ensuring accurate predictions. Further discussion on this assumption, including its theoretical justification well way to tackle its limitation, is provided in Supplementary Section C
Future work will explore extending Adaptive Shrinkage Estimation multivariate biomarker trajectories and improving uncertainty quantification in per- sonalized trajectories to address this aforementioned limitation.
Acknowledgments. This research is supported by the NIH U24NS130411 RF1A4404409 grant and the NIA contract ZIA-AG000191.
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APPENDIX
A DATASETS AND PREPROCESSING
We use the iSTAGING consortium (Habes al. 2021 that consolidated and harmonized imag- ing and clinical data from multiple cohorts. Our real data consists of neuroimaging and demo- graphic measures taken from subjects in the iSTAGING consortium. Specifically, the neuroimag- ing measures are the 145 anatomical brain ROI volumes (119 ROIs in gray matter, 20 ROIs in white matter and 6 ROIs in ventricles) extracted using a multi-atlas label fusion method (Doshi er al., (0066). Phase-level harmonization was applied on these 145 ROI volumes to remove site ef- fects (Pomponio et al., (020). Specifically, we use the Alzheimer's 's Disease Neuroimaging Initiative DOON/hhtp//wwwwaddmiee inrgo/////./....)....m.// which is a aublic-private collaborative longitudinal cohort study and has recruited participants categorized as Cognitively Normal (CN), Mild Cognitive Impairment (MCI) and diagnosed with Alzheimer's Disease (AD) through 4 phases (ADNII, ADNIGO and ADNI2) (Weiner et al.] 2017). We also use Baltimore Longitudinal Study of Aging (BLSA) (Fer- rucci 2008), which has been following participants who cognitively normal enrollment with imaging and cognitive exams since 1993.
We also extracted additional studies from the iSTAGING cohort, including the OASIS dataset Mar- cus et al. 2210), the Australian Imaging, Biomarker, and Lifestyle (AIBL) study (Ellis et al 2009). and the PreventAD study [Tremblay-Merrits as held-out datasets for evaluating our method on external neuroimaging data.
al. 202T). These studies were exclusively reserved
Our analysis incorporates subjects across all identified progression trajectories: Cognitively Nor- mal (CN) stables, individuals with Mild Cognitive Impairment (MCI), and those progressing to Alzheimer's Disease (AD) from either CN or MCI stages. For the clinical variables, we utilize Age at Baseline, Sex, Years of Education, and APOE4 Allele status, the latter being known risk factor for Alzheimer's Disease (AD). Diagnostic categories were designated as Cognitively Normal (CN), Mild Cognitive Impairment (MCI, and Alzheimer's ' Disease (AD). Subjects diagnosed with alterna- tive forms dementia, such as Lewy Body Dementia and Frontotemporal Dementia, were excluded minimal and did not significantly impact the overall sample
from the study. These exclusions size. Missing diagnostic information was classified as unknown (UKN). Furthermore, Years Ed- ucation was dichotomized: subjects with those with 16 years or fewer were coded as '0'. Detailed demographic and clinical characteristics of the diverse cohort are presented in Table 2
than 16 years of education were coded '1', while
Table 2: Summary of longitudinal studies with demographic and clinical Information. OASIS, AIBL, and PreventAD studies are used as held-out neuroimaging studies. For age, the mean and the standard deviation are reported. For sex, the number of males and the percentage is presented.
Study
Subjects Obs./Subject #Obs.
Age
Male (%)
Diagnosis ( (%)
CN
MCI
AD
ADNI
1616
5.0+2.0
7867
73.647.0
55.5
44.7
34.6
20.6
BLSA
584
3,0+1.0
1843
74.9+11.1
45.7
95.8
2.8
1.4
OASIS
548
3.0#1.0
1562
67.8+9.0
42.4
88.9
1.9
12.2
AIBL
82
3,0+1.0
247
75+7.7
56.14
33.74
28.81
37.45
PreventAD
271
4.2+1.4
1141
65.3+5.5
28.5
98.6
1.4
0.0
B ARCHITECTURAL DESIGN AND TRAINING
B. ROI VOLUME MODELS
For each ROI Volume biomarker, we build separate deep kernel regression model with adaptive shrinkage. The deep kernel models (p-DKGP and ss-DKGP) take as input 145 volumetric ROIs along with the following covariates: Age at Baseline, Sex, Diagnosis Baseline. APOE4 Alleles, Education Years, and Time. The transformation function & is implemented as a multilayer percep- tron (MIP) composed of sequence of linear layers. + reduces the input dimensionality from 151
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(145 imaging features 5 covariates and Time) 64, Based on empirical validation, further reduc- tion degrades predictive performance. The Gaussian Process (GP) is initialized with a zero mean function and an RBF kernel.
The p-DKGP is trained for 500 epochs with a learning rate of 0.01 using the Adam optimizer (Kingma & Ba 2014) (with weight decay of 0.01) and dropout rate of 0.2 for regularization. Upon completion, we save the weights (W,, p. b,) and the GP hyperparameters (variance and length- scale) for inference on new test subjects and for transfer learning in the subject-specific model (ss- DKGP). For the subject-specific model, we initialize the ss-DKGP with the saved weights (W, p. by) and the hyperparameters of the population GP. Then, we train the ss-DKGP for 100 epochs with a learning rate of 0.01, during which the deep kernel is frozen; only the spbject-specific GP hyperpa- rameters are updated. The Adam optimizer with a weight decay of 0.05 is used in this stage.
B.2 SPARE MODELS
For each SPARE biomarker, SPARE-AD and SPARE-BA, we build a separate deep kernel regression model with adaptive shrinkage estimation. The input features include the same 145 volumetric ROIs, along with the following covariates: Age at Baseline, Sex, Diagnosis at Baseline, APOE4 Alleles, Education, SPARE-BA, and SPARE-AD at baseline, in addition to Time.
As in the ROI Volume models, the transformation function & is a multilayer perceptron that projects the 153-dimensional input to a 64-dimensional feature space. We employ GP with zero mean function and an RBF kernel. The p-DKGP is trained for 500 epochs with a learning rate of 0.01, using the Adam optimizer with a weight decay of 0.01 and dropout rate of 0.2. The learned weights (Wp. by) and GP hyperparameters (variance and lengthscale) are then saved for subsequent inference and for initializing transfer learning in the subject-specifts model. Transfer learning is performed by initializing the ss-DKGP with the saved weights (W,, p. bp) and the population GP hyperparameters. The ss-DKGP is then trained for 100 epochs with a learning rate of 0.01, during which the deep kernel is detached from the optimization process and only the subject-specific GP hyperparameters are updated using the Adam optimizer with a weight decay of 0.05.
B.3 DETAILS ON THE COMPETING BASELINES
We compare our method against various baselines, including Linear Mixed Effects (LMM) mod- els, Generalized Additive Models (GAMs), Deep Regression, and the Deep Mixed Effects (DME) Chung et al.) 2019). Each baseline model is trained on the cohort of 1800 subjects, since for the development of the baselines we do not need to reserve validation-set subjects as we do for the de- velopment of the Adaptive Shrinkage Estimator. The test set is the 440 subjects. For every subject M,, where X; denotes the 145 ROI Volume measurements acquired
i, we define U, at the first visit, M, comprises the clinical covariates (age at first visit, sex, diagnosis at first visit, education years, and APOE4 alleles), and T, represents the time elapsed since the first visit. Specif- ically, for LMM, we use the 145 ROI Volume measurements at first visit, clinical covariates (age at first visit, sex, diagnosis at first visit, education years, APOE4 alleles) and Time as fixed ef- fects. The Subject ID served a slope. For GAMs, personalization involved fitting a GAM to population data of 1800 subjects, supplemented with each test subject's partially observed trajectory. The second non-linear baseline is the Deep Regression. At first, we train the Deep Regression on the population dataset of 1800 subjects. Then on the personalization, we freeze the first layers of the deep network and we fine tune only the last layer with the subject data. The architecture of the Deep Network is an MLP that consists of an input layer. three hidden layers, and an output layer. The first hidden layer contains 100 neurons, the second hidden layer has 50 neurons, and the third hidden layer again contains 100 neurons. Each hidden layer uses the Rectified Linear Unit (ReL.U) activation function, which in- troduces non-linearity into the model and helps it learn complex data patterns. The MLP is trained using the Stochastic Gradient Descent (SGD) optimization algorithm to minimize the Mean Squared Error (MSE) loss function. For the Deep Mixed Effects (Chung et al., 2019). we used the publicly available code in order to apply the DME method to our data. As a warping mean function, we use MLP. Additionally, we experimented with Transformer model (Vaswani et al. 2017) utilizing positional encoding along the temporal dimension and implemented LSTM models Hochreiter & Schmidhuber (1997). However, both models faced convergence issues during training and did not
random intercept and the interaction term Time:Subject ID as
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yield satisfactory results sparse temporal dataset. Theoretically, Transformer models rely self-attention mechanisms to capture dependencies across sequences, which assume the availability of comprehensive and densely sampled sequential data. In the context of sparse temporal data, the self-attention mechanism cannot function optimally due to insufficient temporal information, lead- ing suboptimal performance. Similarly, LSTM models require temporally aligned and regularly sampled data to maintain the sequential relationships inherent in time series. Without prior prepro- cessing, such as data imputation handle irregularities and missing values, LSTMs struggle to learn effectively from sparse temporal data. As a result, we omitted these models from the quantitative comparisons in the current work.
C ANALYSIS ON POSTERIOR CORRECTION
Our goal to determine the oracle shrinkage parameter a in Equation equation 11 the predictions from the population model (D-DKGP) and the subject-specific model (ss-DKGP). To achieve this, we propose minimizing the Mean Squared Error (MSH) between the combined prediction We and the ground truth Yr over time points. The objective function is defined as:
which combines
f,
Jax) Σ (yt - (oyy, + +1 a)=))
(11)
In this section, we provide a theoretical justification for this formulation, explaining why the in- dependence assumption between the models' errors does not affect the estimation of using this objective function.
Both the P-DKGP and ss-DKGP models provide predictive means y/, and ys, for the ROI value at each time point t. We aim to find the oracle a that minimizes the MSE between the combined prediction Ye and the ground truth yr. The combined prediction is given by:
ye aBJps + (1 ax)ys..
(12)
To find the optimal a, we take the derivative of J(a with respect to a and set zero:
dJ =
in
(y+ - (oyp, + (1 - a))..)) (Um. - ys.) = O.
(13)
da
==
Simplifying, we get:
fn
(yt - (ayp. + (1 - a)ya.)) (Up. - Us.) = O.
(14)
t=
Solving for a, we find:
Elzozs Us (ppr Uoe)
a'
(15)
Elz0(yn (2m y2
This expression shows that the optimal a depends on the covariance between y1 Us, and Vp. Us and the variance of yp. Ye.-
To gain further insight into the dependence of the optimal a² statistical properties of the data, relate Equation [5 to the concepts of covariance and variance. Let us define:
X, Up
Uec
Y, Ut Ve.
(16)
With these definitions, Equation 15 becomes:
0n 1= Y
a
(17)
2t= X?
The numerator and denominator in Equation 17 are related to the sample covariance and variance,
respectively. Specifically, the numerator is proportional to the covariance between Y, and Xt, and
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the denominator is proportional to the variance of Xy:
Cov(Y, 1 fn
(Y, Y) - X),
(18)
11
1=0
1
fn
Var(X)
2 t=
(X, x));
(19)
where Y and X are the sample means of Y, and X,, respectively, and n 4n + is the number of time points.
Assuming that Y, and X, are centered (i.c., Y = and X = 0), which is valid we consider
deviations from their means, Equation 17 simplifies to:
Cov(Y, Cov(Y, n Var(X)
(20)
Var(X)
This expression shows that the optimal a the coefficient that minimizes the residual sum of squares in simple linear regression of Y2 on X, without an intercept. In other words, a' is the scaling factor that best relates the difference between the population and subject-specific predictions (X,) the residuals of the subject-specifie model (Y₁).
If Cov(Y, X)) large and positive, it indicates that when the subject-specific model underpredicts overpredicts (Y, deviates from zero), the difference between the population and subject-specific predictions (X₂) tends to be in the same direction. In this case, a larger (giving more weight to the population model) helps reduce the overall error.
If Cov(Y, X) is small or negative, it suggests that the population model does not provide useful information to correct the subject-speciff model's errors, and a smaller a (giving more weight to the subject-specific model) is preferable.
This analysis confirms that the optimal a depends on the covariance between y/: - Ye, and Vp. Vsa and the variance of Up. Vse- Understanding this dependence provides valuable insight into how the differences between the models" predictions relate the residuals and how to optimally combine them to minimize the prediction error.
C.1 INDEPENDENCE ASSUMPTION AND ITS IMPACT
The combined predictive mean Ye a deterministic function of yp., ye., and a, as given in Equation
It does not involve the errors variances associated with the predictions. As result, the independence or correlation between the models' errors does not influence the calculation of ye- While the independence assumption does affect the estimation of the calculation of ye- does impact the calculation of the combined predictive variance Ve. The variance of the combined prediction is given by:
12
Uc a²rp. + + - +1-a) + 2a - Cov((U Vo.).
(21)
If the errors of the two models are assumed to be independent, the covariance term Cov Vs.) is zero, simplifying ve to:
V.- a
a2)
(22)
Empirical analysis indicates that the errors of the two models are midly correlated, with correlation to range between 0.136 to 0.394. Therefore, the inclusion the covariance term in the calculation of ve to accurately quantify the uncertainty of the combined prediction.
Overall, the theoretical justification demonstrates that the MSE objective function is appropriate for estimating the shrinkage parameter a in our context. avoids the need for the independence as- sumption during estimation and simplifies the optimization process. However, when calculating the predictive variance Ve, it is essential to account for the covariance between the models' predic- tions to accurately quantify uncertainty.
To address this issue, we do:
Estimating Covariance: Empirically estimate Cov(yp, Us.) using validation data.
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Table 3: Correlation between the errors of p-DKGP and ss-DKGP models for different ROI Volume biomarkers and Observations
ROI
3 Observations
Observations
5 OOservatioons
Observations
Hippocampus
0.237
0.337
0.374
0.318
Thalamus Proper R
0.136
0.348
0.302
0.344
Lateral Ventricle R
0.201
0.247
0,319
0.354
Hippocampus
0.341
0.300
0,348
0.208
Amygdala
0,262
0.325
0.355
0.372
Amygdala
0.292
0.356
0.331
0.394
Non-linear functions for Adaptive Shrinkage Estimator
Amyodala
Amygdala
Mean Absolute Error
Hippocampus
R
Hippocampus
Thalamus Proper
Lateral Ventricla
Parahippocampal Gyrus
MAE
0.200
ac
0.175
0.125
0.100
XGBoost
GBM
RF
DNN
XGBoost
GBM
RF
DNN
Figure 5: We present MAE and R2 from 5-fold cross-validation using the 200 held-out subjects from ADNI and BLSA subjects for the Adaprive Shrinkage estimator using XGBoost, GBM, RF and DNN as non-linear functions
Adjusting Variance Calculations: Include the covariance term in the calculation of as per Equation 2T
Reassessing Prediction Intervals: Recompute prediction intervals using the adjusted ve to ensure improved coverage.
C.2 ALTERNATIVES NON-LINEAR FUNCTIONS FOR ADAPTIVE SHRINKAGE ESTIMATOR
We experiment with several non-lineat functions to determine which one learns best the adaptive shrinkage mapping, namely the mapping between and 3pp Uss Vp. Vs, Tobs- We conduct 5-fold cross-validation using XGBoost Regression (XGBoost), Random Forest (RF), Gradient Boosting Machine (GBM), and Deep Neural Network (DNN). The DNN architecture includes linear layer (5x16), ReLU activation, linear layer (16x8), ReLU activation, and final linear layer (8x1). It is trained with MSE loss and optimized using Adam with a learning rate 0.01. Results, presented in Figure | iindicate that XGBoost Regression and Random Forest achieve the best performance in terms of mean absolute error and 22 score on the test set, with both models achieving an average 2 score greater than 0.75 across the majority of the ROI Volumes.
D EXPERIMENTS
D.1 STRATIFIED PERFORMANCE ANALYSIS BY COVARIATES
To thoroughly evaluate our method, we perform stratification of prediction enrors across key demo- graphic and clinical factors: sex, APOE4 Allele status, and education level. This stratification allows examine the model's 's ability varying subpopulations. We report the Mean Absolute Error and corresponding 95% confidence intervals (CIs) for pers-DKGP. alongside with the competing baselines.
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Table 4: XGBoost performance on predicting the adaptive shrinkage a for ROI Volume biomarkers
ROI Volume
MAE
R
Amygdala
0.099
0.830
Amygdala L
0.113
0.796
Hippocampus R
0.113
0.810
Hippocampus
0.118
0.774
Lateral Ventricle R
0.132
0.675
Thalamus Proper R
0.135
0.759
PHG R
0.111
0.783
Stratification by Sex. Our results indicate that pers-DKGP consistently achieves the lowest Mcan AE for both male and female groups. In males, pers-DKGP attains a Mean AE of 0.135 (95% Cl: 120, 0.1 0,1, significantly outperforming LMM, which yields a Mean AE of 0.187 (CI: [0.160, 0.214]). Similarly, for females, pers-DKGP reports Mean AE of 0.145 (CI: [0.130, 0.160), compared to GAM's Mean AE of 0. (CI: [0.165, 0.231J). Although prediction errors are slightly higher in females females-likely due increased biomarker variability-the consistently narrower Cls of Pers-DKGP underscore its enhanced reliability across sexes.
Stratification by APOE4 Alleles Status. Considering the crucial role of the APOE4 Allele in Alzheimer's ' Discase progression, we examine model performance for Non-Carriers, Heterozygous and Homozygous separately. For APOE4 homozygotes, pers-DKGP achieves Mean AE of 0.142 (CI: [0.128, 0. 0.156)). markedly lower than DME's Mean AE of 0.210 (CI: [0.176, 0.244)). For non- carriers, pers pers- DKGG obtains Mean AE of 0.130 (CI: [0.118, 0.142)), outperforming DeepRegr, which records a Mean AE of O. 192 (CI: [0.162,0.222] 0.222)).
Stratification by Education Education level, serving as a proxy for cognitive reserve, introduces additional variability in disease progression predictions. In the subgroup with education levels be- low 16 years, pers-DKGP achieves a mean AE of 0.155 (CI: [0.140, 0.170J, outperforming LMM, which exhibits a mean AE of 0.225 (CI: [0.195, 0.255]). Among subjects with 16 more years of education, pers-DK maintains its advantage, recording mean AE of O. 1.1 (CI: [0.110, 0.130)), whereas GAM shows mean AE of 0.175 (CI: (0.145, 0.205)).
Overall, the stratification of AE demonstrates that Pers outperforms baseline methods in all subpopulations. Its lower dictive accuracy but also greater reliability, even in challenging subgroups such as APOE4 carriers, and individuals with lower education levels.
AE and narrower confidence intervals indicate not only higher pre-
D.2 PERFORMANCE WITH NUMBER OF OBSERVATIONS
Error with Number Observations for SPARE-AD Score. Table 5 presents the mean abso- lute етror and 95% confidence interval for the SPARE-AD biomarker across different numbers of observations (history). A history of I corresponds to using the population model prediction, which employ when only a single acquisition of the subject is available; in this case, we have 1. As we increase the number of observations, we apply posterior correction with adaptive shrinkage inferred by the adaptive shrinkage estimator, allowing ject's individual history. Notably, the mean AE decreases as more observations are included. This demonstrates the benefit of applying Adaptive Shrinkage with increased subject history improve the accuracy of the SPARE-AD biomarker prediction.
adjust the model based the sub-
Table 5: Mean Absolute Error and 95% Confidence Interval for the SPARE-AD biomarker with increasing number observations
Observations
Mean AE
95% CI
1 (a = 1)
0.227
0.003
0.233
0,008
0.219
0.008
0.153
0.010
0.148
0.010
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pers
DespRogr
ROI Volume Biomarkers
3.4
by
Moee AE by APOER Nleles
1.2
1.4
1.2
12
1.0
3.8
Ras
5.4
0.2
0.4
02
2.2
c.
2.8
Mean AE by Sea
0.8
Meon Erucation
0.5
Meae AE By APOE4 Aleles
0.7
7.7
0.6
2.0
203
SPARE
200
Mon
022
Fammat
New
Haraegaas
Sex
Education Years
AFOE4 Alletes
Figure 6: We stratify MAE by key covariates Sex, APOE4 Alleles, and Education Years rig- orously assess model performance across different subpopulations. Error bars denote the 95% con- fidence intervals of the MAE. The top row aggregates metries for seven ROI Volume biomarkers, while the bottom row summarizes the MAE for both SPARE-AD and SPARE-BA.
Error Analysis. Figure ? illustrates the distribution of absolute errors across history levels (1 to 6) using boxplots. The median error indicated by the central line within each box, with the interquartile range (IQR) defining the edges and whiskers extending to 1.5 times the IQR. Outliers are depicted as individual points beyond the whiskers. red line represents the mean absolute error, providing overview of the central tendency.
The results demonstrate a marked reduction in mean absolute error with increasing history, par- ticularly during the carlier transitions: 21.96% decrease from history to 2 and further 15.92% decrease from history 2 to This underscores the significance of incorporating additional longitudi- nal observations. However, the improvements plateau at higher history levels, reflecting diminishing returns. It is important to note that the error will never practically reach zero, owing to the inherent noise and variability of neuroimaging biomarkers. Nevertheless, the results highlight the neces- sity of subject-specific personalization, as individual trajectories often deviate from population- SPARE-AD estimates. With additional follow-up observations, these deviations are better captured, resulting in more accurate and individualized SPARE-AD trajectories. This emphasizes the critical role of model adaptation clinical practice, as refined SPARE-AD estimates can provide valuable insights for predicting disease progression, including transitions to dementia more specifically, progression from MCI Alzheimer's Disease.
D.3 QUALITATIVE EXAMPLES OF ROI VOLUME AND SPARE BIOMARKERS
In this section we provide additional qualitative results on test subjects. We present results for the ROI volume biomarkers as well as the SPARE AD biomarker. The ROI progression models use as input the imaging scan (145 Volumetric ROIs), demographics and clinical variables. The SPARE- AD progression model uses the 145 Volumetric ROIs, demographics and clinical variables well as the SPARE-AD score at baseline.
Empirical Evidence of Predicted SPARE-AD Trajectories for MCI Progressor. figure g we present example of subject that starts as Cognitive Normal at the Age of 74 years old. We use our model (pers-DKGP) in order to predict the longitudinal SPARE-AD changes from the 145 Volumetric ROIs as well as the demographics (Age. Sex, Education Years) and clinical variables such as the Clinical Diagnosis and the APOE4 Alleles. At the first visit the subject, we extrapolate SPARE-AD trajectory that indicates years of observations the MCI the predicted trajectory of the SPARE-AD biomarker indicates no significant longitudinal change in the SPARE-AD trajectory. In the 42 months of observations,
changes related to progression. Within the 2 and half
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Absolute Number Observations
Mean Absolute Error
0:00 00
0.8
0OD
0b00000
Absolute
0.2
0.0
Number of Observations
Figure 7: Boxplots show the distribution of absolute errors across history levels (1 to 6), with the central line indicating the median, the box edges representing the interquartile range (IQR), and whiskers extending to 1.5 times the IQR. Outliers are shown as points beyond the whiskers. The red line connects the mean absolute error for each level.
the predicted SPARE-AD trajectory indicates increasing trend in the SPARE-AD values that indicates increased AD releated patterns the brain. Increased AD-like patterns indicate higher risk of conversion to MCI or Dementia (AD). In almost 5 years of observation, the predicted trajectory indicates a steeper increase in the future SPARE-AD values indicating againg high risk of MCI or AD. The subject finaly is clinically diagnosed with MCI after 80 months of observation. Our method is able to predicted changes of biomarker values that are indicative of Progression and this highlights also the clinical usage of our method
stong predictive tool for progression prediction either for use in the clinical practice or the design of clinical trials. For example, this subject with increasing trend of SPARE-AD trajectory would be ideal subject for recruitment in clinical trial as it converts to demonstrates inclining biomarker trajectory making it subject that is highly likely be part of clinical trial.
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Personalized Predicted Trajectories for SPARE-AD Biomarker for MCI Progressor
Noky Estirrates
Observed
Future
Ooservation
Progressor fest
MCI Progressorng in montre observatio period
MCI P Progressor in months observation pertod
06
2
Time to ribaadddlv ho ronda)
Trme For bas Enmoniis
frmbaad mordal
MCI Progressor in m8 absenation pertod
MCI Progressor mo oeservaiio pariod
0.
Time Pom werebaiee (nmmntis)
Figure 8: We present predicted SPARE-AD trajectories for a Cognitive Normal subject at the base- line age of 74 years old. After 7 years the subject is diagnosed with Mild Cognitive Impairment. The predicted SPARE-AD trajectories predict the increasing attrophy-like patterns 3 years prior the clinical diagnosis of conversion to MCI. This highlights the potential clinical application of tool for progression prediction and clinical trial design.
Empirical Evidence from a Healthy Control and and MCI Progressor. Figure we present qualitative comparison of predicted trajectories for two subjects who begin the study at similar ages-74 (left) and 71 (right), respectively- volumetric loss in three brain regions: the amygdala, hippocampus, and lateral ventricle. The vol- umetric loss is modeled as function of MRI scans alongside clinical and demographic covariates, including age, sex, diagmosis, APOE4 allele status, and years of education.
are cognitively normal at baseline. We analyze the
At the initial visit, both subjects exhibit minimal hippocampal atrophy. However, over successive follow-up observations, the subject on the left (9p) demonstrates markedly steeper decline in hip- pocampal volume compared to the subject on the right, who maintains a more stable hippocampal trajectory. The predicted accelerated decline hippocampal volume for the subject on the left sug- gests elevated risk of progressing to mild cognitive impairment (MCI) or dementia, potentially due to underlying pathology such as Alzheimer's disease (AD) accelerated brain aging. con- trast, the subject the right (91), who remains healthy control throughout the observation period, exhibits only minimal hippocampal volume loss.
This example illustrates the practical application of our method in predicting disease progression, which has significant implications for clinical practice, clinical trial design, and treatment effect estimation. Specifically, in the context of clinical trial design, identifying subjects with steep hip- pocampal atrophy trajectories can inform the recruitment individuals who are more likely to exhibit disease progression, thereby enhancing the efficiency and efficacy of the study.
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MCI Progressor
Healthy Control
Observation
4 Observations
Observations
Observations
Observations
Amygahia R. (standardized)
2.5
0.5
0.5
Héppocampus (standardizer (anddrdiizd)
1.5
0.5
Lateral Ventricle R (standardlized)
0.5
-0.5
20
10
20
30
50
60
70
80
Time from baseline (in months)
Time from baseline (in months)
Figure 9: We present predicted Amygdala and Hippocampal Volume trajectories and Ventricular Enlargement for a Healthy Control and and MCI Progressor. MCI Progressor exhibits steeper vol- ume loss in Amygdala and Hippocampus in comparison with the Healthy Control. MCI Progressor exhibits either accelarated brain aging or in the onset of AD which justifies its faster volume loss.
Empirical Evidence the Personalization Test Subjects. To further validate the efficacy of our method, provide empirical evidence through qualitative analysis in scenarios where individual trajectories either diverge from or align with the true underlying trend. In Figure I0 we present cohort of test subjects (panels (a) (h)) exhibiting variability in progression status, alongside the cor- responding adaptive shrinkage parameter depicetd in the second row-utilized at each person- alization step. Consistently across all examples, we observe that the adaptive shrinkage parameter progressively decreases as the number of observations increases. In several cases, the adjustments remain more conservative, with staying closer to I, which aligns with the foundational intuition of method. This pattern suggests that adequate accumulation of evidence regarding subject's trajectory necessary to shift the adaptive shrinkage parameter toward zero, thereby placing greater trust in the ss- DKDD predictions. This rationale is well- well-founded, substantial evidence crucial for the ss-DKGP to generate meaningful trajectories and mitigate the noise variations inherent in neuroimaging data acquisitions. Additional examples visualized in Figure [1]
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Qualitative Examples of Personalized Volume ROI Trajectries and Adaptive Shrinkage
Observation
Observations
Observations
Observations
-7
7 Observations
Healthy Control
b
Healthy Control
C
MCI Stable
Hopoornnua (annbadoo
Timee from baneliee monthe
Three month))
Narrber
Number Cossraatre
Healthy Control
AD Progressor
MCI Stable
Thre tor buseling morthal
from base (lrmonths)
Time beteline
Allaeii Shrwagg
Adbek
Deseraatioss
Narber Gbseraatons
g
MCI Stable
h
AD Progressor
montr))
Adeer Stambage
Alighe Shewaage
Figure 10: We present qualitative examples where population trajectories deviate from the subject's observed trajectory throughout the observation period (in years). Evidence is provided from eight distinct test subjects.
the first row of each panel (a)-(h), we present the adapted trajectories. The second row of each subfigure visualizes the corresponding adaptive shrinkage for posterior correction for each observation, ranging from 7 observations.
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Qualitative Examples Personalized Volume Trajectries and Adaptive Shrinkage
Doservation
Oiservations
Observations
Observation
Observations
Progressor
MCI Stable
Stable
Amiostaa Deunddent 2011
0.5
Time from buteline 1th meniol
from
Terie from baseline morhho
Adpube Streoog
Numbee Otservationa
Mambee of Otservatons
Number Obssrvatize
MCI Stable
Healthy Control
Healthy Control
Theamu Standed
from bencline munthe!
Nutther Desorviiirs
Namber of Classtattine
Figure 11: We present qualitative examples where population trajectories deviate from the subject's observed trajectory throughout the observation period (in years). Evidence provided from six distinct test subjects. In the first row of each panel (f)-(f), we present the adapted trajectories. The second row each subfigure visualizes the corresponding adaptive shrinkage for posterior correction for each observation, ranging from 4 to 7 observations.
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D.4 ANALYSIS OF ADAPTIVE SHRINKAGE ESTIMATOR
D.4.1 ABLATION ON SHRINKAGE PARAMETER a
Determining the shrinkage for each ROI Volume is non-trivial, particularly for predicting long-term trajectories. This is a difficult task because either subject's trajectory would deviate from popula- tion trends, or subject would have limited acquisitions, making it difficult for the subject-specific model to extrapolate its ROI Volume trajectory. Volume loss in the brain slow process, espe- cially for subject who is young or has not yet developed any pathology. Thus, in the case of limited acquisitions for subject, which are also close in time to the baseline, the additional observations are rather noisy copies of the baselines and do not contain any "signal" of the trajectory of developing atrophy. In that case, the ss-DKGP model would not have enough evidence to extrapolate future ROI Volume. As result, we should find the ideal shrinkage combine the two predictors and eventually leverage both the population's 's ability to make reliable long teng-termmm predictions and the model's is ability to learn short-term predictions. We show that adaptive shrinkage provides the best results compared to any other weighting scheme, as we also present table 6
Table 6: Ablation study on the shrinkage parameter We report the Mean AE along with its 95% percentile CI, Mean Coverage, and Mean Interval Width
ROI
Mean AE (CT) Mean Coverage
Best Constant
Hippocampus
0.257 (0,209)
0.808
0.843
Lateral Ventricle R
0.143 (0.182)
0,853
0.507
Thalamus Proper R
0,241 (0.214)
0,934
1.127
Amygdala
0.349 (0.317)
0.742
0.918
Hippocampus
0.274 (0.245)
0,805
0.850
PHG R
0.423 (0.360)
0.582
0.844
Deterministic
Hippocampus
0,308 (0.275)
0,480
0.459
Lateral Ventricle R
0.156 (92)
0.620
0.310
Thalamus Proper R
0.308 (0.287)
0.512
0.492
Amygdala R
0.418 (0.400)
0.503
0.650
Hippocampus L
0.314 (0.290)
0.487
0.478
PHG R
0.487 (0.457)
0.459
0.681
Adaptive Shrinkage
Hippocampus R
0.243 (0.1
0.795
0.902
Lateral Ventricle R
0.131 (0.186)
0.855
0.626
Thalamus Proper R
0.219 (0.216)
0.849
0.911
Amygdala
0.312 (0.283)
0.762
0.964
Hippocampus
0.258 (0.241)
0.790
0.901
PHG R 0.389 (0.344)
0.745
0.908
D.4.2 INTERPRETATION OF ADAPTIVE SHRINKAGE ESTIMATOR
As we increase the number of observations, we see that, no matter the biomarker, the alpha tends to zero. This aligns with the domain expectation that the longer the time from the baseline of the last observation Tobs, the more likely we are to have observed trajectory trend from the subject's data.
In Figure 12 we visualize the distribution of adaptive shrinkage in the test set as well as in the three external clinical studies. This demonstrates that adaptive shrinkage has learned to assign greater trust to the subject-specifet model as the number of follow-ups increases for a subject. This aligns perfectly with domain expectations and the explainability analysis implemented for the Adaptive Shrinkage Estimator. This property makes the Adaptive Shrinkage Estimator a transparent method for performing posterior correction in the two predictive distributions, P-DKGP and sS-D.GP
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a
ROI Volume
b
SPARE
C OASIS
ROI Volume on External Studies
AIBL
Prevent AD
1000
Otsemddiars
Observatoo
Chnervation
Stsandarrs
$000
Otsersations
Obosteal
Stocmalaars
1000
Conervations
100
Steamadarra
53.33
0.35
3.59
0.75
3.00
10.00
0:25
2.50
1200
0.00
0.25
0.5
0.75 411
1.00
KSS Ap11
Figure 12: We visualize the distribution of adaptive shrinkage a for a) the ROI Volumes, the b) SPARE-BA and SPARE-AD biomarkes and e) the 7 ROI Volumes in the external neuroimaging studies: OASIS, AIBL and PreventAD
SHAP Summary Plot of Adaptive Shrinkage
Ahrinive Estimator
Tobs
Y_PP
y_ss
V_pp
V
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
mean([SHAP valuel) (average impact on model output magnitude)
Figure 13: We calculate SHAP values for the Adaptive Shrinkage Estimator for the SPARE-AD biomarker. As expected, the time of observation Tobs emerges as the most infiuential feature of the Adaptive Shrinkage stimator.
Table 7: Correlation Analysis between Deviation (5,) and Predicted and between Tobs and Pre- dicted a for Large Deviation
Biomarker
Correlation between Tass and Predicted for Large oy
SPARE-BA
-0.640
SPARE-AR
-0.529
Lateral Ventricle
-0.484
Hippocampus L
-0.401
Hippocampus R
-0,381
Thalamus Proper R
-0,555
PHG R
-0.479
Amygdala R
-0.439
28

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Published as a conference paper at ICLR 2025
D.5 COMPARISON ON ALTERNATIVE GP PERSONALIZATION APPROACHES
In this section, we conduct a comparative analysis with other personalized GPs that align with our formulation. Specifically, within the ss-DKGP framework, we do an ablation study transformation, learned from the population model, affects the subject-specific process. To achieve this, we train the ss-DKGP for each subject on the test set without initializing the deep kernel (ss- DKGP no init). We also train a standard subject-specific Gaussian Process (ss-GP) with a zero mean and RBF kernel. This comparison demonstrates the effectiveness of transferring the from the p- DKGP when training the ss-DKGP. Additionally, we explore an alternative personalization approach where the population dataset D, augmented with the subject's is observed trajectory D.. In this setting, again employ the & transformation learned from the p-DKGP. This approach is referred to as the ft-DKGP (fine-tuned DKGP). We perform transfer learning by initializing the weights of the deep kernel with (Wp. bp). The ft-DKGP is trained for 500 epochs using the same learning rate as the P-DKGP. During this process, the deep kernel is detached from the optimization procedure, and only the hyperparameters of the subject-specific GP are updated. The Adam optimizer with weight decay of 0.05 is utilized.
how the &
Mean AE
Mean Interval
Mean Coverage
Interval
1.5
Mean Coverage
3..8
Mean A6
R.6
2.0
9.4
0.2
0.5
DKcP DKGP
S6DKGGP
1s no mnt
59-0P
R.DKSP
pars DRSP
50 WKDKOP
31-KKP no wit
55
R.DKGP
DKGP
S-OKG) no Pit
350 GP
Figure 14: Comparison of predictive performance and uncertainty quantification across various GP models, averaged over three regions of interest (ROIs): Hippocampus, Lateral Ventricle, and Tha- lamus Proper, indicating that the personalized DKGP models achieve the best prediction accuracy and highest coverage. The bar plot displays the Mean performance metrics (AE, interval length and coverage) across these ROIs, while the line represents the standard deviation.
Among the ss-DKGP, ss- sK-D init, and ss-GP models, we observe that ss- ss-DKGP achieves the lowest Absolute Error (AE) with a significant margin compared to the other two settings. This indicates that leveraging the population transformation & is crucial for the effective training of DKGP. This finding supports our hypothesis that the transformation & successfully captures the most predictive features for ROI progression, which are beneficial for ss-DKGP training. We observe that ft-DKGP model achieves performance that is close to the ss-DKGP model. However, ft-DKGP fails to personalize on unseen times, since the predicted trajectory falls back to the population trend. This is not the optimal way to personalize since the trajectory does not adapt to the subject specific trend. Additionally, it is not computationally efficient to retrain the model with the entire population data every time we need to personalize a subject.
Furthermore, the pers-DKGP model achieves the lowest AE, which an additional indication in favor of our approach. It highlights the strength of including the P-DKGP model in the final per- sonalized prediction. Knowing solely the observed trajectory of subject is not enough in case of limited and noisy observations. In that case should trust the p-DKGP model more, which trans- lates to an a parameter close to 1. Interestingly, this intuition aligns with the predicted a that we got during the personalization from the XGBoost regression. To verify that, we gathered the predicted from the personalization process from the 7 ROIs. We plotted the distribution of a with the num- ber of observations ranging from 4 till 7. The plot is shown in Figure 12 It clearly depicts that, as the number of observations increases, the distributions tend to show more noticeable skewness the right, with higher densities in the lower a ranges and decreasing densities towards higher a values. This trend suggests that more observations are taken into account in personalization, the shrinkage parameter tends to be smaller. That translates to more trust the ss-DKGP prediction. This is highly intuitive because as observation time Tabs increases, more acquisitions we obtain for a subject and thus the more information the ss- DS-DKGP captures about the progression of a ROI over time.
29


Reviews:
[{"review_id": "RFbJu6lp7O", "reviewer": "Reviewer_7rm7", "summary": "This paper introduces a new personalized deep kernel regression framework to predict brain volume changes over time. The framework combines a population model, which captures population-level trends in brain volume changes, with a subject-specific model that tailors predictions to each individual subject’s trajectories. The optimal combination coefficient is determined using a shrinkage estimator to achieve the best personalized brain trajectory predictions. The framework is validated through metrics of predictive accuracy, uncertainty quantification, and comparisons with existing statistical and deep learning models, demonstrating superior performance. Additionally, the authors validated the framework's generalizability on brain datasets not seen during training.", "questions": "This reviewer is happy to adjust the score if the questions and concerns are well addressed. \n\n- If an individual ROI differs significantly from the population mean, would it be possible that the optimization of Eq. (17) become biased towards minimizing the error between $y_t$ and $(1-\\alpha) y_{st} $, as $y_{st}$ is learned to be fitted to a personalized brain trajectory. In this scenario, the value of $\\alpha$ would likely be low if the changes in a particular ROI are subtle or difficult for the network to detect, and vice versa. It would be good if the authors can provide either empirical evidence or theoretical justification on how the model balances between population and individual trends in such cases.\n\n- In Eq. (11), the expectation term appears to be conditioned on the learned latent representation $z_p$ from the population level. However, this condition seems to be missing in the final formulation. Please clarify if this is a typo. \n\n- Could the authors clarify how the six brain ROIs were selected and extracted from the original images? Additionally, please specify what information about the ROIs is input into the model (e.g., volume, size, or full ROI data).\n\n- It is challenging to well justify the results in Fig. 3. The differences between the population and personalized trajectories may depend heavily on the distribution of training data and examine whether it is biased toward cognitively normal individuals / mild cognitive impairment (MCI) / AD. Each experiment should ideally account for and justify this factor.\n\n- To enhance clarity, it would be helpful if the authors could summarize the other personalized trajectories for each of the six different ROIs in Fig. 3.\n\n- It would be helpful to increase the font size of the legend in Fig. 3.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper is overall well written and organized; hence it is easy for readers to follow.\n\n- The motivation to develop a more precise predictive tool for diagnosing Alzheimer’s disease (AD) has strong clinical value.\n\n- The proposed method is new in its approach to modeling both population-level and subject-specific brain trajectories. By combining these models to predict personalized brain volume changes over time through adaptive shrinkage estimation in a deep kernel regression framework, the author show a superior performance of the model compared with existing statistical and deep learning approaches.", "weaknesses": "- While the approach of modeling population-level and individual-level trajectories for brain changes is promising, the current design, which trains these models separately, may be suboptimal. There is substantial research in computational anatomy [e.g., 1-3] demonstrating that individual trajectories should ideally be modeled as a hierarchical structure built upon the population-level mean. This approach naturally captures the hierarchical relationship between fixed group-level information and individual variability.\n\n   [1] Hierarchical linear modeling (HLM) of longitudinal brain structural and cognitive changes in alcohol-dependent individuals during sobriety, 2007.\n\n   [2] A hierarchical geodesic model for diffeomorphic longitudinal shape analysis, 2013.\n\n   [3] Hierarchical multi-geodesic model for longitudinal analysis of temporal trajectories of anatomical shape and covariates, 2019.\n\n- The connection between population-level trends and personalized trajectories in the proposed model is not clearly established.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pN5beohfca", "reviewer": "Reviewer_Q2Nf", "summary": "The authors proposed a method to model brain-changing trajectories. The method can capture brain volume trajectories from a large and diverse cohort. They introduced a personalization step that generates subject-specific models for individual trajectories. The paper was well written and easy to follow, although the method and the results were interesting. However, several points are unclear in the manuscript.", "questions": "1. The subject-specific model ss-DKGP also used parameters of the population-level p-DKGP, how to guarantee the correctness of the predicted subject-specific results? If the answer is YES, why do we need to combine the prediction resultes of p-DKGP and ss-DKGP?\n2. Expanding on the limitations of existing techniques would strengthen the positioning of your contribution.\n3. The description of how external clinical studies were handled in the validation process could be clearer. Were there any preprocessing steps specific to the external datasets, and if so, how might these affect the generalization results?\n4. The results demonstrate strong performance, but additional ablation studies would further strengthen the claims of the paper. For instance, how does the model perform when fewer ROIs are considered, or when different biomarkers are used? Additionally, could the method handle non-monotonic biomarkers, or is it limited to monotonic progressions?\n5. The evaluation of external datasets is a strong point of the paper. However, it would be useful to see how the model handles missing data in longitudinal studies. What imputation strategies were employed, if any, and how did they impact the results?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The performance of the proposed method was evaluated on different neuroimaging data, which confirmed the generalizability and applicability of the method across different clinical contexts.", "weaknesses": "1. The paper lacks of providing a comprehensive discussion of the wider literature on disease progression modeling. The interpretation of the results from the experiments is inadequate, raising questions about the model’s practical implications. \n2. Adding a clear comparison of the simulation data set will help readers better understand the advantages of the approach. \n3. Interpretation and Presentation of Results: The authors should provide a thorough interpretation of these results and a clearer discussion of the model’s real-world implications and its utility. In particular, the authors should clarify the implications of the findings from the experiment and how the modeling informs participant-level monitoring of disease progression.\n4. Context and Literature Review: The authors should expand the context and literature review to properly situate their work within the broader field of disease progression modeling. The discussion should include a comparative analysis with other relevant works.\n5. The references in this paper are relatively old. Please add the literature review of the past two years.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3wyfeTqXQk", "reviewer": "Reviewer_yf4S", "summary": "This paper proposes a framework for personalized prediction to forecast longitudinal regional brain volumetric changes. Specifically, the authors provide a pipeline for integrating the population Deep Kernel Gaussian Process (DKGP) and the personalized DKGP by applying the Adaptive Posterior Shrinkage Estimation. The central innovation of this paper is the adaptive shrinkage mechanism, which could dynamically balance predictions from the population (p-DKGP) with the specific models (ss-DKGP) of the subject. Specifically, the authors propose an approach to learn the shrinkage parameter $\\alpha$ to weigh the two models (as a linear combination). By combining these two models, this framework could make accurate predictions of brain volumes and could be stably generalized to diverse clinical populations. To sum up, the paper provides a flexible approach to modeling brain volume trajectories, which could offer a lot of potential applications in clinical settings.", "questions": "1. Could you elaborate on the rationale for freezing the latent transformation (MLP) parameters in the subject-specific model?\n2. Did you explore the latent space and do some related analysis, such as subpopulation analysis or clustering of subjects with similar disease/ brain-trajectory patterns? \n3. What is the rationale for the choice of XGBoost to learn the shrinkage parameter $\\alpha$? Did you try with other models?\n4. How sensitive is the model's performance with different $\\alpha$?  Can you provide some exploration of the impact of $\\alpha$ on the prediction accuracy? \n4. The proposed model combines the variance of the GPs under the assumption of independence. How do you justify this assumption?", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The logic of the paper is very clear and easy to follow. The paper's structure is well organized. \n2. The combination of population and subject-specific models and its application on data generalization for brain-related disease (AD, brain aging patterns) is a valuable contribution to the existing literature in this field.", "weaknesses": "1. The contributions only have limited novelty. For example, a similar idea has already been proposed by Eleftheriadis et al., 2017. This paper already has a detailed demonstration of how the Gaussian Process can incorporate domain or individual-specific trends in a multi-task setup, which weakens the novelty of the proposed approach. Also, applying the machine learning techniques (like XGBoost) for weight estimation is also a common practice. The learning of the shrinkage parameters $\\alpha$ might be a good application in the engineering field but is not a very novel way in ML research. A Bayesian Neural Network or probabilistic models that could provide more principled approaches for weight determination might be a better approach, such as *Weight Uncertainty in Neural Networks* by Blundell et al., 2015. \n\n2. The usage of latent space is very limited. In the current setup, the latent space serves primarily as a way to reduce the dimensionality of the input data and capture basic non-linear interactions through the MLP. However, it doesn’t appear that the authors further analyze or interpret the latent space itself. It would be very interesting if the authors could explore the temporal dynamics and conduct trajectory analysis in the latent space. \n\n3. In the experiment parts, the authors only compared some very traditional models; it would be much better if the authors could compare their approach with some more advanced deep learning based techniques. \n\nReference:\nEleftheriadis, S., Rudovic, O., Deisenroth, M. P., & Pantic, M. (2017). Gaussian Process Domain Experts for Modeling of Facial Affect. IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 26(10), 4697–4711. https://doi.org/10.1109/TIP.2017.2721114\n\nBlundell, C., Cornebise, J., Kavukcuoglu, K. &amp; Wierstra, D.. (2015). Weight Uncertainty in Neural Network. <i>Proceedings of the 32nd International Conference on Machine Learning</i>, in <i>Proceedings of Machine Learning Research</i> 37:1613-1622 Available from https://proceedings.mlr.press/v37/blundell15.html.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "f1jicHYfYw", "reviewer": "Reviewer_enPj", "summary": "The authors have developed a deep kernel regression model to predict longitudinal brain volume changes. The model is made for personalised predictions and has two parts, one population model for global prediction and one subject-specific model for personalised predictions. The model is compared to several commonly used models on two open data sets. The resulting models are further evaluated on three external test sets. The results are convincing, and the method appears to work well.", "questions": "Comments/questions/required changes:\n - How much data from individuals do you need for the personalised GP to be useful/to help in the predictions?\n - The step on Line 12 of Algorithm 1 is not atomic. Split this up or explain clearly somewhere what is being done.\n - It says the values in parentheses in Table 1 are standard deviations. Present the standard errors (standard deviations of the means) instead, so that the results can actually be compared properly.\n\nMinor comments:\n - Do use \\citet rather than \\cite or \\citep when you mention a reference in the body of the text, so that the names are not in parentheses in those cases.\n - Some more LaTeX-related issues: Remove blank lines after equations to avoid indented sentences after. If an equation ends a sentence, put a full stop after it (or, as in Equation 7, put the comma after the equation, not at the beginning of the next sentence). In general, make equations parts of sentences. There is a lot of space between Equations 6 and 7 and between Equations 11 and 12, probably because of an empty line between them (but do make these equations part of the sentence instead!).\n - Equation 2: The \\Phi has not been defined.\n - Lines 82-83: These should rather be tuples, and not sets, since the order matters.\n - Line 163: A \\cdot is missing as a wildcard for the first argument.\n - Line 417: A space missing after the colon.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is well-written and in general easy to follow. The method is straight-forward, clear, and appears to work well, and is applied with convincing results to real-world datasets. I really like that you evaluated on multiple external data; this strengthens the confidence in and validity of the results.", "weaknesses": "It is not entirely clear where the methodological novelty lies. Personalised GPs have been proposed before (as mentioned in the introduction) and weighting schemes have also been proposed before (also correctly mentioned in the introduction). The adaptive shrinkage is interesting, and appears to work well, but must also be considered a small contribution.\n\nIt is great that the method is evaluated on no less than three external test sets, but it would have been good/more convincing if it was also trained on more than just two training sets.\n\nThere should be en ablation for the personalisation (or, equivalently, presenting results with \\alpha=1), to see if it actually improves the results over just having the population level model.\n\nThere are not confidence intervals in the bar plots, which means it is not possible to compare those results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)