Title: Idiographic Personality Gaussian Process for Psychological Assessment
Venue: NeurIPS 2024
Accepted: yes
Keywords: Applications -- Cognitive science

Gaussian process

Latent variable model
Abstract: We develop a novel measurement framework based on Gaussian process coregionalization model to address a long-lasting debate in psychometrics: whether psychological features like personality share a common structure across the population or vary uniquely for individuals. We propose idiographic personality Gaussian process (IPGP), an intermediate model that accommodates both shared trait structure across individuals and "idiographic" deviations. IPGP leverages the Gaussian process coregionalization model to conceptualize responses of grouped survey batteries but adjusted to non-Gaussian ordinal data, and exploits stochastic variational inference for latent factor estimation. Using both synthetic data and a novel survey, we show that IPGP improves both prediction of actual responses and estimation of intrapersonal response patterns compared to existing benchmarks. In the survey study, IPGP also identifies unique clusters of personality taxonomies, displaying great potential in advancing individualized approaches to psychological diagnosis.

Idiographic Personality Gaussian Process for Psychological Assessment

Abstract

We develop a novel measurement framework based on a Gaussian process coregionalization model to address a long-lasting debate in psychometrics: whether psychological features like personality share a common structure across the population, vary uniquely for individuals, or some combination. We propose the idiographic personality Gaussian process (IPGP) framework, an intermediate model that accommodates both shared trait structure across a population and 'idiographic' deviations for individuals. IPGP leverages the Gaussian process coregionalization model to handle the grouped nature of battery responses, but adjusted to non-Gaussian ordinal data. We further exploit stochastic variational inference for efficient latent factor estimation required for idiographic modeling at scale. Using synthetic and real data, we show that IPGP improves both prediction of actual responses and estimation of individualized factor structures relative to existing benchmarks. In a third study, we show that IPGP also identifies unique clusters of personality taxonomies in real-world data, displaying great potential to advance individualized approaches to psychological diagnosis and treatment.

1 Introduction

Building models for the evaluation of latent traits from observed responses is crucial to understand long-term behaviors through repeated quantitative assessments. These are used, for example, to study emotional stability after medical treatment or the development of academic ability during secondary education [1-3]. However, existing frameworks face several interrelated limitations. First, there are strong reasons to believe that standard taxonomies may over-generalize, failing to distinguish between related psychological phenomena that often differ in etiology, symptoms, and biological processes between individuals [e.g., 1, 4], and this may lead to inaccuracy when making predictions [5]. A related issue is that measurement models are rarely individualized, instead assuming that (1) the correlation between latent traits of interest and survey responses are invariant across individuals and (2) the relationship between the latent traits themselves are the same for everyone. Lastly, current models are almost always developed for cross-sectional data that are collected only once from each respondent, which overlooks any potential dynamics of psychological processes.

To address these limitations, previous research has adopted three different approaches, each inadequate in its own way. First, recent work has proposed an idiographic approach that builds a completely distinct taxonomy for each individual [6-8]. However, complete personalization can sacrifice generalizability and interpretability for clinicians, as any possible population commonality is ignored. A second line of research focuses on building dynamic psychometric models of time-series data via some variant of item response theory [9, 10, 3], longitudinal structural equation modeling [11-13], vectorized autoregression [14, 15] and/or Gaussian process (GP) latent trajectories [16-18]. However, all these models adopt the nomothetic approach, assuming that the responses of all individuals

share an identical latent structure. Finally, there is a smaller body of work that adopts intermediate approaches to create individualization while maintaining group commonality [e.g., 19]. However, prior research models quantitative responses directly, ignoring the latent structures that are often the actual focus of domain researchers.

In this work, we propose an idiographic personality Gaussian process (IPGP) framework for assessing dynamic psychological taxonomies from time-series survey data. This framework combines nomothetic and idiographic approaches by employing a common structure to explain typical circumstances, while allowing individual structures to accommodate deviations into distinct forms. We leverage the Gaussian process coregionalization model based on multi-task kernels to conceptualize responses of grouped survey batteries, adjusted to non-Gaussian ordinal data, and utilize IPGP for hypothesis testing of domain theories. Methodologically, our approach combines Gaussian process latent variable models (GPLVM) [20], Gaussian process dynamic systems (GPDM) [17, 18] and GP ordinal regression for Likert-type survey data [21, 22]. Despite involving latent variables and GPs, IPGP differs from GPLVM in two ways: (1) it optimizes the factor loading matrix while marginalizing the latent variables, and (2) it accommodates categorical data using a non-Gaussian ordered probit likelihood. Computationally, our framework exploits stochastic variational inference for latent factor estimation, contrasting with other GP measurement models relying on Gibbs sampling that may not scale efficiently to intensive longitudinal setups [18, 23].

To our knowledge, our work presents the first multi-task Gaussian process latent variable model for dynamic idiographic assessment. While multi-task GPs have found recent applications in areas such as causal inference [24-26], environmental science [27-29], and biomedical research [30], their potential remains largely unexplored in psychology. Current psychometric models typically focus on cross-sectional settings without dynamics [6, 31, 8] or single task settings that ignore inter-battery correlations [32, 33]. We conducted an extensive simulation study comparing IPGP against benchmark methods and analyzed an existing cross-sectional personality dataset. Our results demonstrate that IPGP simultaneously enhances the estimation of idiographic taxonomies and improves the prediction of responses. Additionally, we collected a novel IRB-approved longitudinal dataset. When applied to this data, IPGP not only shows superior performance in response prediction, but also suggests unique personality taxonomies. These findings highlight IPGP's significant potential for advancing individualized approaches to psychological diagnosis and treatment.

2 Background

We start by laying out the ordinal factor model for building standard taxonomy from survey data [34, 35]. We then briefly discuss several existing idiographic longitudinal models in psychological assessment and review the Gaussian process model.

Ordinal factor analysis. Consider the scenario where some set of units, i ∈ { 1 , . . . , N } , repeatedly answer the same set of j ∈ { i, . . . , J } survey items over t ∈ { 1 , . . . , T } periods with ordinal observations y ijt ∈ { 1 , . . . , C } up to C levels. For example, the responses could be Likert-typed, ranging from 'strongly disagree' to 'strongly agree'. The latent factor model posits that the j th underlying latent variable f ( i ) j ( t ) for unit i at time t can be recovered as w T j x i ( t ) , where x i ( t ) ∈ R D are unit-level latent factors and w j ∈ R D are factor loadings. The f ( i ) j ( t ) s are then mapped to ordinal responses via an ordered probit model: p ( y ijt = c | f ( i ) j ( t ) = f ) = Φ( b c -f ) -Φ( b c -1 -f ) with threshold parameters b 0 < · · · < b C . Usually b 0 and b C are fixed to -∞ and + ∞ so that the resulting categorical probability vector sums to 1 , while b 1 , . . . , b C -1 are allowed to move freely. Stacking x i ( t ) s, w j s and y ijt 's into matrices x , w and tensor y , the joint likelihood can be written as L ( y | x , w ) = ∏ i ∏ j ∏ t p ( y ijt | x i ( t ) , w j ) . Identification is guaranteed with standard orthogonality and normalization constraints [36]. This factor model is also known as an item response model [37, 38], and can be estimated via maximum likelihood, weighted least squares, or an EM algorithm [39-41].

Idiographic longitudinal assessment. In psychological assessment, the idiographic approach emphasizes intrapersonal variation by requiring distinct loadings, while the nomothetic approach identifies general interpersonal variation assuming shared factor loadings [42]. In terms of data collection, the idiographic approach usually surveys each individual multiple times ( N = 1 and large

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Gaussian process. A Gaussian process (GP) can be used to define a distribution over f such that the values of f at arbitrary finite subset of X have a joint multivariate Gaussian [43]. A GP ( µ, K ) is specified with a mean function µ : X → R and a positive-definite kernel function K : X × X → R ; evaluating these functions pointwise determines the mean and covariance of these finite-dimensional distributions. The most common kernel is the squared exponential (RBF) kernel K ( x 1 , x 2 ) = exp( -1 2 x 1 T Px 2 ) with precision matrix P = diag (1 /ℓ 2 1 , . . . , 1 /ℓ 2 d ) and d = card ( X ) . The posterior of a GP can be derived analytically for a Gaussian likelihood but must be approximated in modeling latent variables with categorical indicators. We discuss the variational approximation method we use for inference in Sec. (3).

3 Methodology

We propose an idiographic personality Gaussian process (IPGP) framework for assessing individualized dynamic psychological taxonomies from time-series survey data. Instead of joint estimation of latent factors and their loadings that cannot guarantee rotational and scaling invariance, we marginalize out the latent variables and focus on learning taxonomies of loadings. The overall architecture of IPGP is illustrated in Figure (1), where the ordinal input observations across the indicators are modeled as ordinal transformations of latent dynamic GP with individualized RBF kernels and loading matrices.

3.1 Multi-task learning

Typically in psychological assessment, survey questions are meticulously grouped such that each group gauges a particular latent trait (e.g. dimension of personality). Hence, we conceptualize the assessment of psychological traits as a multi-task learning problem, where each question represents a distinct task but can be correlated with other tasks. A multi-task GP is an extension of the single-task GP but for vector-valued functions [31]. To motivate the multi-task framework, first consider the

two-task scenario with two T × 1 vector f 1 ( i ) and f 2 ( i ) denoting the latent temporal processes of unit i for question j = 1 , 2 . To fix the scale of latent factors, a time-level Gaussian process prior is placed on x i ( t ) ∼ GP ( 0 , K ( i ) time ) . By exploiting affine property of Gaussians, the induced joint distribution of vectorized [ f 1 ( i ) , f 2 ( i ) ] T can be written as:

whose covariance of shape 2 T × 2 T contains four block matrices K ( i ) time scaled by different w T j w j ′ ( j, j ′ ∈ { 1 , 2 } ). Specifically, w T 1 w 2 controls the inter-task covariance between these two tasks and w T j w j s ( j ∈ { 1 , 2 } ) control their intra-task variance. This multi-task structure is also known as the linear model of coregionalization (LMC) [44], which models output functions as linear combinations of several independent latent processes. In our case, each dimension in x i ( t ) represents one latent process, which jointly defines the functions as f j ( i ) = w T j x i ( t ) . To extend this, let f ( i ) = [ f 1 ( i ) , . . . , f J ( i ) ] T represents the flattened JT × 1 vector consisting of all J tasks. We write f ( i ) in a formal multi-task GP notation using Kronecker product ⊗ :

where K ( i ) task denotes the unit-specific task covariance matrix, consisting of the self inner products of D × J shared interpersonal loading W pop and D ∗ × J unit-specific low-rank Z ( i ) ( D ∗ < D ) for intrapersonal deviations that serves to be the additional idiographic component, independent of W pop. In our experiments, we found degraded performance for D ∗ = 2 but extra computational costs so we focused on D ∗ = 1 . The Kronecker product ⊗ then multiplies each entry in the J × J task covariance with K ( i ) time , and returns the stacked JT × JT covariance for f ( i ) . Through the use of time kernel K ( i ) time , properties of the latent trait trends such as periodicity or autocorrelation could be incorporated. Here we use the common RBF kernel K ( i ) time ( t, t ′ ) = exp ( -1 2 ( t -t ′ ) 2 /ℓ 2 i ) to account for dynamic changes in the latent attributes, whose bandwidth is determined by the unit-specific length scale ℓ i , but any other kernel can substitute RBF as practitioners see fit. Finally, the latent variables f ( i ) s are further projected to response space by the ordered probit model.

3.2 Variational inference

Due to the non-Gaussian ordinal likelihood, we adopt the variational inference technique (VI) with inducing points introduced in [33]. Dropping superscript for demonstration, VI utilizes a variational distribution q ( u ) = N ( µ u , Σ u ) on inducing variables u to approximate p ( f | y ) using the conditional p ( f | u ) . Hence, the conditional log-likelihood log p ( y | u ) can be lower bounded by the expected log-likelihood w.r.t. p ( f | u ) , after exploiting the non-negativity of Kullback-Leibler (KL) divergence between p ( f | u ) and p ( f | y ) :

Furthermore, a lower bound on model evidence (ELBO) can be obtained by combining Eq. (4) and an inequality derived by another KL divergence KL [ q ( u ) ∥ p ( u | y )] ≥ 0 (see Appendix B for details):

where the KL divergence KL [ q ( u ) ∥ p ( u )] between the variational q ( u ) and prior p ( u ) can be computed in closed form as both distributions are Gaussians. The expectation of log-likelihood log p ( y | f ) under the marginal distribution q ( f ) = ∫ p ( f | u ) q ( u ) d u is intractable but can be numerically approximated using Gauss-Hermite quadrature method. The variational parameters µ u and Σ u , individualized loadings w i and diag ( v ) as well as likelihood parameters { b c } s are then optimized to maximize this lower bound. Finally, the predictive likelihood of new p ( y ∗ ) = ∫ p ( y ∗ | f ∗ ) p ( f ∗ | u ) q ∗ ( u ) d u is obtained by marginalizing out the optimized q ∗ ( u ) . Throughout our experiments, we adopt stochastic inference for computational scalability.

3.3 Theory testing

Our IPGP framework also naturally facilitates downstream tasks such as domain theory testing between models with and without shared or idiographic components. We adopt posterior odds ratio test, using posterior p ( M i | y ) = p ( y |M i ) p ( M i ) ∑ i p ( y |M i ) p ( M i ) over a pool of models {M i } conditioning on observations y with prior weights p ( M i ) , as the hypothesis test to determine whether the latent structures for each individual are indeed distinct or are simply explainable by interpersonal commonality. Specifically, we refer the multi-task model in Eq. (3) as the idiographic model, and compare it with an nomothetic model without unit-specific components: K pop task = W pop W T pop .

Note that compared to this baseline nomothetic model, our proposed idiographic model in Eq. (3) introduces additional unit-level JN loading parameters that enlarge the hyperparameter optimization space. Hence, we propose to first learn the interpersonal loading matrix W pop using the standard cross-sectional data from a nomothetic model that focuses on learning of population taxonomy, and then use the estimated W pop as informative prior in the full model. We will show empirically in Sec. (4) that with this stronger prior IPGP achieves a more precise estimation of individual taxonomies.

4 Experiments

We now evaluate IPGP in learning idiographic latent taxonomies and predicting actual responses against baseline methods from both psychometrics and Gaussian process literature in three experiments: a simulation study, a re-analysis of a large cross-sectional dataset, and a pilot study of repeated measures of the Big Five [45] personality traits.

4.1 Simulation and ablation

Setup. Our simulation considers longitudinal data of N = 10 units over T = 30 periods. We assume that the latent traits of each unit i have dimension D = 5 , and each latent vector is generated independently from a GP x ( d ) i ( t ) ∼ GP ( 0 , K ( i ) time ) with a unit-specific length scale uniformly randomly picked from ℓ ( i ) time ∈ [10 , 20 , 30] . We split J = 20 items into D subsets of size J/D = 4 , such that each subset dominates one dimensional in the latent traits. Specifically, we set high value of 3 in the population factor loading matrix W pop for entries corresponding to the k th subset for dimension k , and low values drawn from Unif [ -1 , 1] otherwise. We also set each unit-specific loading w i from Unif [ -1 , 1] . To introduce sparsity and reverse coding, we randomly set half of the loadings to zero and invert the signs of the remaining half. Finally, we generate the y ijt s according to the ordered probit model with C = 5 levels, and apply 80% / 20% splitting for training and testing.

Metrics and baselines. We consider two sets of metrics for evaluation: (1) the in-sample and out-of-sample predictive accuracy (ACC) and log-likelihood (LL) of the actual responses, (2) the correlation matrix distance (CMD) between the estimated factor loading matrix and the true ones, which is defined for two covariance matrices R 1 , R 2 as d ( R 1 , R 2 ) = 1 -tr ( R 1 R 2 ) ∥ R 1 ∥ f ∥ R 2 ∥ f [46] where f is the Frobenius norm. Note that CMD becomes zero if R 1 , R 2 are equal up to a scaling factor, and one if they are orthogonal after flattening. We compare IPGP to (1) various latent variable models for ordinal responses, including the graded response model (GRM) [37], the generalized partial credit model (GPCM) [47] and the sequential response model (SRM) [48], (2) Gaussian process dynamic model (GPDM) [17, 18] where the continuous predictions are rounded to the nearest ordinal level, (3) latent structural model (LSM) [13, 49] with trait-dependent latent variables and (4) time-varying vector autoregression (TVAR) with regularized kernel smoothing [15]. We also compare IPGP with several ablated models: (1) IPGP-NOM without the idiographic kernel, (2) IPGP-IND without the population kernel, (3) IPGP-LOW with lower-rank factors of 2 than actual rank of 5 in the synthetic setup and (4) IPGP-NP where the population kernel is learned from scratch rather than fixed to the informative prior. Note that W pop in the full IPGP model is fixed as learned from IPGP-NOM.

Results. We use 100 inducing points and the ADAM optimizer with learning rate 0 . 05 to optimize ELBO for 10 epoches with batch size of 256 . We repeat our simulation with 25 different random seeds using 300 cores on Intel Xeon 2680 CPUs. Table 1 shows the comparison of the average predictive accuracy, log-likelihood, and correlation matrix distance between IPGP and baselines and ablated models. Our IPGP model (indicated in bold) significantly outperforms all ablated models and baseline methods in estimated correlation matrix, predictive accuracy, and log-likelihood of both training and testing sets in pairedt tests. We found that IPGP is able to predict the correct labels due to its idiographic components and proper likelihood, since IPGP-NOM and IPGP-GL are two of the worst ablations for all prediction metrics. In addition, IPGP-IND and IPGP-NP have the worst correlation matrix estimation, implying that a well-informed population kernel is crucial in recovering the underlying factor structures.

Robustness of IPGP. To assess IPGP's robustness to rank misspecification, we conducted additional exploratory factor analysis using our simulated data. We tested model performance with ranks of 2, 5, and 8, where 5 represents the true rank of the data. As shown in Table 2, both the true-rank (5) and higher-rank (8) models significantly outperform the low-rank (2) model. However, increasing the rank beyond the true rank of 5 yields no additional performance benefits: the high-rank model is not significantly better than the true-rank model in a paired t-test. These results demonstrate two key points: first, low-rank approximations inherently lack the capacity to fully capture the underlying structure, and second, increasing the rank beyond the true rank provides no additional benefit. This underscores the importance of careful exploratory analysis in practical applications to determine the appropriate rank.

4.2 Cross-sectional factor analysis

We next validate the popular Big Five personality theory using standard cross-sectional data via a factor analysis, where a range of factors are tested and then compared according to model evidence. This serves to show that the model works appropriately to detect known latent traits even in nondynamic settings, and to validate the informative prior for the W pop matrix in our next experiment. We utilize an existing dataset called life outcomes of personality replication (LOOPR) [50], which is collected from 5,347 unique participants on the Big Five Inventory [51] consisting of 60 questions. Our validation considers a range of latent trait dimension counts from D = 1 , . . . , 5 . For each dimension count, we first apply principal component analysis (PCA) directly on the correlation matrix

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of the cross-sectional observations to learn a vanilla population factor loading matrix. We then initialize W pop in our model with this vanilla loading matrix, and optimize the loading matrix jointly with the variational parameters. Note that T = 1 in LOOPR, so we drop the idiographic components.

Validation of Big Five. Table 3 presents the predictive accuracy and averaged log-likelihood for our method and various baselines (excluding TVAR due to its lack of low-rank assumption) across different values of D in LOOPR. Bold numbers indicate the best model for each D , while italic numbers highlight the best model across all D values. Although IPGP with D = 5 shows slightly lower in-sample predictive accuracy compared to the D = 3 model, it demonstrates significantly stronger model evidence than all alternatives. Posterior odds ratio test reveals that the second-best model is exp( -79) ≈ 5 × 10 -35 times less likely than the five-factor model.

We further evaluated IPGP's performance through exploratory analysis, testing model ranks from 1 to 10. Results shown in Table 4 provide strong support for a rank-5 model, which achieves both higher model evidence and lower BIC, strengthening the evidence for the Big Five theory. The BIC follows a V-shaped pattern, decreasing as the rank approaches 5 and increasing thereafter, indicating that rank-5 represents an optimal balance point: ranks below 5 provide insufficient model capacity, while higher ranks lead to overfitting. These findings demonstrate that when analyzing psychological measurements from standard cross-sectional data, IPGP successfully identifies the correct underlying factor structure, making it valuable for downstream applications.

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0.198

0.037

0.039

0.015

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0.068

-0.069

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0.000

0.072

0.042

0.039

0.068

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0.034

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-0.002

0.066

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0.050

0.100

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0.050

0.000

0.050

-0.144

-0.002

0.100

0.050

0.000

-0.065

0.066

-0.358

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Figure 3: Four residual correlations as identified by our k-mean clustering. Each heatmap displays the trait-level residual correlation averaged across corresponding batteries for one cluster, with darker red and blue indicating larger positive and negative deviations. For instance, agreeableness (A) is more correlated to extraversion (E) than the population profile in the first profile, but less correlated to openness (O) in the second profile. Moreover, these two directions of deviations are even exacerbated in the third and fourth profiles.

Estimated interpersonal variation. Figure 2 compares the raw correlation matrix with our estimated Big Five correlation matrix. Both matrices exhibit a distinctive block pattern, characterized by strong correlations between questions within each Big Five factor and weak correlations across different factors. We also observe that questions related to negative emotionality demonstrate slight negative correlations with questions measuring extraversion and conscientiousness, suggesting the presence of meaningful trait-by-trait interaction effects.

4.3 Longnitudinal Pilot Study

To further demonstrate IPGP in longitudinal setting for learning idiographic psychological taxonomies, we collected intensive longitudinal data using experience sampling measures (ESM). We highlight the predictive ability of IPGP through a prediction and a leave-one-trait-out cross-validation task. We also illustrate how IPGP identifies unique personality taxonomies that might advance individualized approaches to psychological diagnosis and inspire new theory.

Data collection. We employed an experience sampling method (ESM) design where participants completed personality assessments six times daily over a three-week period, allowing for a maximum of 126 assessments per person. Our study included 93 valid student participants, yielding 8,770 total assessments with an average of 94 assessments per participant. We based our assessment on the BFI-2 [52], which provides comprehensive coverage of the trait space and ensures proper identification of latent factors. While the original BFI-2 contains 60 items (four items for each of the three sub-factors within each Big-Five domain), we modified it for our ESM design by removing items unsuitable for contextualized assessment. To reduce participant fatigue and minimize learning effects from repeated measures, we implemented a planned missing design: participants randomly responded to two out of three items for each sub-factor, resulting in a streamlined 30-item assessment. Note that our data is collected from a student sub-population as non-representative samples, and future studies may explore the model's applicability across diverse populations.

Comparison between nomothetic and idiographic models. We run the full IPGP model with idiographic component and unit-specific time kernel on the collected longitudinal data. Again we set the ranks of the population and individual loading matrices to 5 and 1 respectively, and incorporate prior knowledge of the cross-sectional data by fixing the population loadings as the Big Five loadings estimated in Sec. (4.2) and optimizing the individual loadings. We contrast our proposed idiographic model (IPGP) and baselines in Table 5, which shows the in-sample prediction, averaged log-likelihood and posterior odds ratio. We found that IPGP outperforms IPGP-NOM with higher predictive accuracy and log-likelihood, and is decisively favored by a posterior odds ratio of exp(1 . 06 × 10 4 ) .

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0.15

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-1.8

-1.6

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-1.9

-1.8

-1.7

-1.6

IPGP-NOM

Predictive performance of IPGP. We also evaluate the out-of-sample performance of the idiographic and nomothetic models using two prediction tasks: forecasting future responses and leave-one-trait-out cross validation. For the forecasting task, we train both models with data from the first 40 days and predict future responses for the last 5 days. For the cross-validation task, we predict responses of each trait by training on data belonging to the other four traits, where 20% of responses for one trait was held out (randomly choosing the trait and items to remove). Figure (4) shows the predictive accuracy and log-likelihood of IPGP and IPGP-NOM for the forecasting task over varying horizons and for the leave-one-trait-out cross-validation task. IPGP has consistently better performance than IPGP-NOM in both tasks, except for being slightly less accurate in predicting extraversion. Overall, IPGP is favored than IPGP-NOM by posterior odds ratios of exp(43) and exp(716) in these two tasks.

Discovery of taxonomies. Despite our small cohort size ( 93 respondents), we also explore how the profiles of personality that substantially differ from the interpersonal commonality cluster into informative groups. Specifically, we first perform a k -mean clustering using all 93 estimated individual correlation matrix with CMD as the distance metric, and then compute the residual correlation between each estimated clustering centroid and the population correlation. Figure (3) illustrates four residual correlations identified by our k -mean clustering. Each heatmap displays the residual correlation at the trait level averaged between the corresponding batteries for one cluster, with darker red and blue indicating larger positive and negative deviations. For instance, agreeableness (A) is more correlated to extraversion (E) than population profile in the first profile, but less correlated to openness (O) in the second profile. Moreover, these two directions of deviations are even exacerbated in the third and fourth profiles.

The discovery of unique personality taxonomies suggests a potential resolution to the longstanding idiographic versus nomothetic debate in personality and psychometric sciences. Our findings indicate that the optimal approach lies between these two extremes, rather than fully embracing either perspective. The four distinct profiles that we identified, while derived from the Big Five framework, demonstrate how individuals can meaningfully deviate from a common taxonomy. These deviations may provide valuable insights into individuals' motivations, behavioral patterns, and self-concepts. For example, individuals matching Profile 4 show a strong correlation between Extraversion and Agreeableness, possibly reflecting a tendency toward warm and socially engaging behavior (such as someone who naturally connects with and shows kindness to everyone at social gatherings). Furthermore, these distinct profiles enhance the predictive power of individual-level ( N = 1 ) models by allowing them to learn from people with similar characteristic patterns.

5 Related work

Idiographic assessment captures critical individual characteristics that are often lost in simplified taxonomies of psychological behaviors [53]. Evidence from multiple psychometric fields demonstrates that nomothetic models, which focus solely on interpersonal variation, often lack generalizability

[1]. Researchers have proposed various solutions to address this limitation. Song and Ferrer [54] enhanced dynamic factor models with random effects to analyze psychological processes. Jongerling et al. [55] developed a multilevel first-order autoregressive model incorporating random intercepts to track daily positive effects across weeks. Beltz et al. [19] bridged nomothetic and idiographic approaches by introducing individual components to the group iterative multiple model (GIMME) for clinical data analysis. However, these methods share a common limitation: they model in response space rather than latent space when handling ordinal survey data.

Gaussian process latent variable model (GPLVM) is a dimensional reduction method for Gaussian data, where the latent variables are optimized after integrating out the function mappings [20, 56]. Our proposed framework differs from GPLVM as we optimize the factor loading matrix while marginalizing the latent variables. In addition, our model contrasts GPLVM and (variational) Gaussian process dynamical model (GPDM) [16, 17] in our non-Gaussian ordered logistic observation model and the use of multi-task kernel for latent structure. Finally, our longitudinal framework with stochastic variational inference learning differs from the static GP item response theory (GPIRT) [23] with more computationally demanding Gibbs sampling.

Longitudinal measurement models have gained prominence as researchers increasingly incorporate temporal dynamics into psychological theories through longitudinal survey designs [57, 58]. This development has spawned various methodological approaches. One family of methods includes longitudinal structural equation models (SEM), such as multiple-group longitudinal SEM and longitudinal growth curve models, designed for repeated measurement studies [11]. The M plus software later extended these capabilities by implementing dynamic SEM with Bayesian Gibbs sampling [13, 49]. Another stream of research produced dynamic item response models [9, 10, 3] and time-varying vector autoregressive models [14, 15] to track latent trait trajectories. While multi-task Gaussian process time series have been successfully applied to Gaussian observations in behavioral research [18] and found applications in causal inference [24, 26], environmental science [28], and biomedical research [30], they remain unexplored for survey batteries with non-Gaussian likelihood where exact inference is not possible.

6 Conclusion

We introduce the idiographic personality Gaussian process (IPGP) model, a novel approach for personalized psychological assessment that learns intrapersonal taxonomies from longitudinal ordinal survey data, a configuration that remains underexplored in Gaussian process dynamic systems literature. Our model leverages Gaussian process coregionalization to capture the between-battery structure and employs stochastic variational inference to ensure scalable computation. Looking ahead, we envision extending IPGP to other psychological domains, such as emotion research, and enhancing it by incorporating contextual information about behaviors and activities.

Our proposed IPGP framework also provides insight for domain theory testing, contributing to the substantive debate in psychometrics surrounding the shared versus unique structures of psychological features. Our experimental results show that IPGP is decisively favored over the nomothetic baseline, and substantive deviations from the common trend persist in for many individuals.

Acknowledgments and Disclosure of Funding

This work was supported by the 2023 Seed Grant of Transdisciplinary Institute in Applied Data Sciences at Washington University in St Louis. YC and RG were supported by the National Science Foundation (NSF) under award number IIS-1845434.

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A Notations of IPGP

Throughout our notation, superscript ( i ) indicates unit and underscript j indicates task.

y ( i ) = [ y 1 ( i ) , . . . , y J ( i ) ] T represents the flattened JT × 1 response vector consisting of all J tasks and T periods for unit i .

f ( i ) = [ f 1 ( i ) , . . . , f J ( i ) ] T represents the flattened JT × 1 latent vector consisting of all J tasks and T periods for unit i , mapped to response space via ordered-probit likelihood.

W pop is the D × J ( D latent dimensions, J tasks) shared interpersonal loading matrix.

Z ( i ) is the D ∗ by J unit-specific low-rank loading matrix that serves to be the additional idiographic component, independent of W pop.

K ( i ) task is the unit-specific task covariance matrix with shared component W T pop W pop and a low-rank approximation Z ( i ) T Z ( i ) of D ∗ < D for unit-specific deviations.

K ( i ) time is the time covariance for dynamic changes in the latent attributes. We used an RBF kernel K ( i ) time ( t, t ′ ) = exp ( -( t -t ′ ) 2 /ℓ 2 i ) with unit-specific bandwidth ℓ i s.

W pop is a global parameter estimated for the entire population while Z ( i ) is a unit-level parameter. These are combined in Eq.(3) to create a unique kernel for each unit that combines both components.

B Mathematical Details of Evidence Lower Bound

We provide the full mathematical details of the evidence lower bound defined in Eq. (6). As KL divergence is always non-negative, we first consider the KL divergence between p ( f | u ) and p ( f | y ) :

Moving E p ( f | u ) log p ( y | f ) to the R.H.S of the above inequality will lead to Eq. (4). We then exploit the inequality given by KL [ q ( u ) ∥ p ( u | y )] ≥ 0 :

Rearranging the above inequality, applying Eq. (4) and exploiting notation q ( f ) = ∫ p ( f | u ) q ( u ) d u leads to the ELBO:

C Data Collection and Demographics

The LOOPR dataset from Soto [50] in our first case study used Qualtrics and quota sampling to ensure representative samples of the U.S. population, and hence was very diverse:

age: 11% ages 18-24, 18% ages 25-34, 17% ages 35-44, 19% ages 45-54, 17% ages 55-64, 18% ages 65 and older

sex: 52% female, 48% male

race/ethnicity: 74% non-Hispanic white/Caucasian, 11% black/African American, 10% Hispanic/Latino, 3% Asian/Asian American, 2% American Indian/Native American

educational attainment: 10% did not complete high school, 33% high school graduate, 28% some college, 19% college graduate, 10% graduate or professional degree

annual household income ($): 14% <20,000, 12% 20,000-29,999, 11% 30,000-39,999, 15% 40,000-49,999, 26% 50,000-79,999, 22% 80,000+

As social and personality psychology often faces challenges with model building due to reliance on non-representative samples, past researches rely heavily on student populations. We follow such practice but aim to broaden our sample within budget and available diversity. Participants in our longitudinal study (the third experiment) are mostly college students with an average age of 20.23 (SD = 1.94); 70% are female, 26% are male and 4% are self-identified as other; ethnicities self-reported as 42% Caucasian, 39% Asian, 12% African American, and 7% Other. We used the BFI-2 [52], a widely used personality measure across cultures and ages. We acknowledge the typical biases of convenient sampling in higher education, including socio-economic and ethnic diversity limitations.

D Runtime in Simulation Study

Table 6 shows the average runtime of IPGP and its competitors in the simulation study. IPGP does require more time to train due to the enlarged model space.

E Estimated Correlations of Selective Individuals

Figure (5) shows the estimated correlations of selective individuals for the identified four profiles in the longitudinal study.

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Energy.2

Compass.1

Compass.2

Compass.4

Respect.1

Respect.4

Respect.3

Trust.2

Trust.3

Organiz.2

Organiz.1

Organiz.3

Product.2

Product.3

Product.1

Respons.3

Respons.2

Respons.4

Anxiety.3

Anxiety.4

Anxiety.1

Depres.4

Depres.2

Depres.3

Violat.2

Violat.3

Violat.1

Curious.3

Curious.4

Curious.1

Aesth.3

Aesth.1

Aesth.2

Creativ.2

Creativ.3

Creativ.1

Sociab.1

Sociab.4

Sociab.3

Assert.1

Assert.2

Assert.3

Energy.3

Energy.1

Energy.2

Compass.1

Compass.2

Compass.4

Respect.1

Respect.4

Respect.3

Trust.2

Trust.3

Organiz.2

Organiz.1

Organiz.3

Product.2

Product.3

Product.1

Respons.3

Respons.2

Respons.4

Anxiety.3

Anxiety.4

Anxiety.1

Depres.4

Depres.2

Depres.3

Violat.2

Violat.3

Violat.1

Curious.3

Curious.4

Curious.1

Aesth.3

Aesth.1

Aesth.2

Creativ.2

Creativ.3

Creativ.1

Extraversion

Agreeableness

Conscientiousness

Negative Emotionality

Open-Mindedness

Extraversion

︷

︸︸

︷

Agreeableness

︷

︸︸

︷

Conscientiousness

︷

︸︸

︷

Negative Emotionality

︷

︸︸

︷

Open-Mindedness

︷

︸︸

︷

-

1

.

00

-

0

.

75

-

0

.

50

-

0

.

25

0

.

00

0

.

25

0

.

50

0

.

75

1

.

00

Sociab.1

Sociab.4

Sociab.3

Assert.1

Assert.2

Assert.3

Energy.3

Energy.1

Energy.2

Compass.1

Compass.2

Compass.4

Respect.1

Respect.4

Respect.3

Trust.2

Trust.3

Organiz.2

Organiz.1

Organiz.3

Product.2

Product.3

Product.1

Respons.3

Respons.2

Respons.4

Anxiety.3

Anxiety.4

Anxiety.1

Depres.4

Depres.2

Depres.3

Violat.2

Violat.3

Violat.1

Curious.3

Curious.4

Curious.1

Aesth.3

Aesth.1

Aesth.2

Creativ.2

Creativ.3

Creativ.1

Sociab.1

Sociab.4

Sociab.3

Assert.1

Assert.2

Assert.3

Energy.3

Energy.1

Energy.2

Compass.1

Compass.2

Compass.4

Respect.1

Respect.4

Respect.3

Trust.2

Trust.3

Organiz.2

Organiz.1

Organiz.3

Product.2

Product.3

Product.1

Respons.3

Respons.2

Respons.4

Anxiety.3

Anxiety.4

Anxiety.1

Depres.4

Depres.2

Depres.3

Violat.2

Violat.3

Violat.1

Curious.3

Curious.4

Curious.1

Aesth.3

Aesth.1

Aesth.2

Creativ.2

Creativ.3

Creativ.1

Extraversion

Agreeableness

Conscientiousness

Negative Emotionality

Open-Mindedness

Extraversion

︷

︸︸

︷

Agreeableness

︷

︸︸

︷

Conscientiousness

︷

︸︸

︷

Negative Emotionality

︷

︸︸

︷

Open-Mindedness

︷

︸︸

︷

-

1

.

00

-

0

.

75

-

0

.

50

-

0

.

25

0

.

00

0

.

25

0

.

50

0

.

75

1

.

00

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

︸

︷︷

︸

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a multi-output Gaussian process model for classification in the context of psychological studies. It's based on a linear model of co-regionalisation leveraging unit-level latent factors and RBF kernels to jointly model population and individual personality traits to tackle an old debate in psychometrics regarding their original nature. The authors leverage a variational formulation to derive a lower bound for the model evidence and optimise the individual-specific, population, and GP-related parameters. An extensive simulation and real-data study is proposed, presenting longitudinal and multi-subject survey analyses, personality correlations and predictions.\n\nSTRENGTHS:\nThis paper is remarkably well-written and presented. The flow of derivations is easy to follow, and the illustrations are helpful and of excellent quality. The treated problem is far from trivial, with data that present several degrees of correlations (time, individual, population) and technical constraints (missing and categorical data). The experiments are impressive, with extensive comparison against many competitors and excellent results overall. The method shows great promise in providing practical insight into the domain, and even though I'm not a specialist in psychology, the ability to reconcile idiographic and nomothetic approaches seems particularly valuable.\n\nWEAKNESSES:\nOne could argue that the methodological novelty is limited as the presented multi-output GP model is fairly well-known in the GP community. However, I think its similarities and differences with GPLVM and GPDM are well presented, and the overall application is far from trivial regarding the nature of data (categorical outputs, latent variables, missing data, ...). This relative weakness is more than compensated by the strength of results and the potential such a methodology brings in a field like psychology, where the signal-on-noise ratio is generally low, and the nature of measurements is often challenging.\n\nQUESTIONS:\nI only have a few questions regarding computation times, scaling and robustness of the inter-output covariance matrices, which are well-known limitations of co-regionalisation GP models in practice. \n\n- In this sense, could you provide numerical evidence and comparison regarding training and prediction times for IPGP and competitors?  \n- Even though the number of tasks/topics remains probably limited in psychological applications, could you discuss a bit more, in Section 4.1 **Result**, the decrease in performance between full IPGP and the low-rank version. What could we expect for higher dimensions? Would the low-rank approximation remain robust for a more significant gap with the actual rank? \n- In my experience, several multi-output GP models (or maybe their implementations) are somewhat unstable when it comes to estimating $\\textbf{K}_{task}$, and pathological cases can often arise. Have you experienced such problems? And if so, how did you manage to tackle this issue? \n\n**Typos:**\n\nPage 5: The indices of the CMD definition have a slight error. All terms are written as $R_1$, and the norm is 'f' instead of '$l_2$'  \nPage 6: The title of 4.2 \"Cross-secitonal\" ==> Cross-sectional  \nPage 7: \"Both correlation matrices **displace** a block pattern\" ==> display?  \nPage 7: \"questions corresponding negative emotionality\" ==> corresponding to\n\nLIMITATIONS:\nIn my opinion, the limitations are adequately discussed. The application spectrum and position in the related literature are well documented.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-08"}

comments: SUMMARY:
Gives a multi-task/output GP formulation for multiple time-series (or intrinsic co-regionalisation model) for pyschological assessments. Design of the factor loadings informs the task-correlations and reflects the knowledge of individual's correlation between his responses and inter-person correlations. More of an application paper rather than a paper advancing machine learning; but a good application nonetheless.

STRENGTHS:
1. The application is a natural fit for the model chosen, and results are good and convincing. 
2. The use of Bayes factor for model testing make the paper better as an reference for this application.
3. Sections 4.2 and 4.3 evaluates the model from different aspects.

WEAKNESSES:
1. The paper is weak from the perspective of advancing state-of-the-art machine learning algorithms.
2. The paper is totally unrelated to GPLVM and tasks that GPLVM are designed for. Mentions of GPLVM only confuses the reader.
3. Equation 3 needs fixing.
4. In section 3.2, the method is variational inference, and not stochastic variational inference.
5. Description of the setup in section 4.1 needs to be clearer to explain also in terms of $K_{task}$.
6. Line 192 mentions "informative prior". What is this "informative prior"?

*Minor*
7. Line 319: "addressing" is too strong. Suggest to change to "contributing".
8. Line 321: "than" -> "over"

QUESTIONS:
I do not have any critical questions.

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-10

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n1. This paper introduces an innovative measurement framework utilizing the Gaussian process coregionalization model to resolve the question of whether psychological attributes such as personality exhibit a universal structure among the populace or are uniquely individualized. \n2. An Idiographic Personality Gaussian Process (IPGP), a hybrid model that accounts for both the commonality of traits across people and individual-specific \"idiographic\" variations. \n3. Gaussian process coregionalization model to interpret the responses from grouped survey batteries, adapted for non-Gaussian ordinal data, and employs stochastic variational inference for estimating latent factors. \n4. The application of IPGP on both synthetic data and an original survey demonstrates its performance.\n\nSTRENGTHS:\n1. The paper’s main innovation lies in its unique combination of multitask Gaussian process, which results in a conceptualization of the subject matter.\n2. The innovative use of multitask Gaussian process in this paper offers a promising solution to a long-standing challenge of psychological assessment.\n3. The quantitative and qualitative methods in this paper provide a potential in advancing psychological diagnosis and treatment.\n4. The interdisciplinary nature of the research is interesting. \n5. The authors provide a holistic view of the issue at hand.\n\nWEAKNESSES:\n1. there are many MTGPs, the author ignored comparison with them.\n2. incorrect description, such as: \"multi-task structure is also known as the linear model of coregionalization (LMC)\".\n3. This represents a particular implementation of MTGP, but it doesn't introduce any novel methodological advancements.\n4. The paper presents a valuable contribution, but its innovative aspects are limited, as it largely builds upon existing theories without introducing significant new insights.\n5. The literature review appears somewhat limited. Expanding it to include more recent or relevant MTGPs could provide a more comprehensive context for the research.\n6. The statistical analysis used in the study seems inadequate given the complexity of the data.\n7. The research seems to be more of an incremental advancement rather than a novel contribution, which may limit its impact on the field.\n8. The experimental evidence provided in the paper is not as comprehensive as it should be to support the claims made, suggesting a need for more extensive testing.\n9. Only one experiment is related to psychological assessment.\n10. No idiographic personality is discovered and presented in the paper.\n\nQUESTIONS:\nPlease see the weakness above\n\nLIMITATIONS:\nPlease see the weakness above", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nUPDATE: I am updating my scores in light of the excellent authors' response.\n\nThis paper considers psychometric data composed of ordinal responses $y_{ijt}$, each of which represent how unit _i_ answered survey item _j_ during time period _t_. The key characteristic of this item-response data is that the same $N$ units are longitudinally surveyed on the same $J$ items over $T$ periods. \n\nThe paper takes an ordered logit factor modeling approach to analyzing such data, wherein $f_j^{(i)}(t) = \\mathbf{w}_j^\\top \\mathbf{x}_i(t)$ is the latent ideal point of unit _i_ on on item _i_ at time _t_, and $\\mathbf{w}_j$ and $ \\mathbf{x}_i(t)$ are the $K$-dimensional loadings and factor vectors, respectively.\n\nThe paper moreover advocates an \"idiographic approach [which] emphasizes _intrapersonal_ variation by requiring distinct loadings $\\mathbf{w}_j^{(i)}$\" which are different for each unit $i$. It is able to effectively achieve this by exploiting the repeated measurements of each unit-item $(i,j)$ pair over different time periods $t$.\n\nThe proposed model is an instance of a multi-task Gaussian process (MTGP). Intrapersonal variation is modeled using a unit-specific kernel $K_{time}^{i}$. The full covariance is then $JT \\times JT$, resulting from a Kronecker product of $\\mathbf{K}_{\\textrm{time}}^{(i)}$ (applied to input $T$ periods) with a low-rank matrix $J \\times J$ matrix that represents covariance between survey items (or \"tasks\"). This all extends the previously-presented linear model of coregionalization (LMC), which was originally developed for the simpler case where observations of $(i,j)$ pairs are not repeated over time.\n\nThe paper derives a variational inference inference algorithm for the model, which follows closely from previous work.\n\nThe paper then reports three sets of experiments: 1) a synthetic study that reports excellent results on parameter recovery (when the true parameters are known), 2) a re-analysis study that purports to show that a non-dynamic (i.e., non-idiographic) version of proposed model is able to identify the correct factor structure (K=5) from survey data designed to assess the \"Big Five\" psychometric traits, and 3) an illustrative case study involving a novel longitudinal data set.\n\nSTRENGTHS:\nThe paper is very well-written. It brings an interesting application to life and convinces the reader that the proposed modeling approach is well-tailored to the problem at hand. The paper gives a clear review of the prerequisite concepts and presents its modeling approach clearly. \n\nThe paper covers related work in psychometrics well is convincing that the modeling approach is novel within that applied community. Experiments bear out this claim, as the proposed model performs much better than models which are currently used in psychometrics.\n\nThe paper presents a novel longitudinal survey dataset composed of $N=93$ subjects who were given personality assessment surveys over the course of three weeks. If I am reading the paper correctly, each subject was asked to complete a survey _six times per day_. This data sounds highly non-trivial to collect, and should be considered a main contribution in itself.\n\nWEAKNESSES:\nThe paper is vague about its technical contributions. It makes the following hedged novelty statement: \"the first multi-task GP latent variables model _for dynamic idiographic assessment_\". I took this to mean that the proposed approach is not very new technically, but it has never yet been applied to dynamic idiographic assessments. However, the paper also makes the general claim that it \"advances the literatures on Gaussian process latent variable models...\". I am of the opinion that tailoring existing modeling frameworks to new applications does provide a technical contribution, as it contributes a new view and set of interpretations/metaphors that can help to better understand the abstract model. I think the paper does contribute in this way. But I am unsure if the paper further contributes more substantially to the area of Gaussian process latent variable models, as the paper is vague about that.\n\nThere is sloppiness in some of the main equations. For instance, equation 2 confuses a distribution with a random variable stating $p(\\mathbf{f}^{(i}) \\sim \\cdots$, and I believe equation 3 confuses an inner with an outer product (shouldn't it be $w_i w_i^\\top$?). There's also some confusing overloading of symbols between the background and the model sections. The background sets up the idea that an ideographic approach has distinct loadings $w_j^{(i)}$ different across $i$. But the proposed model itself does not seem to sport this as each $w_j$ is global for each task. I believe effectively the model does achieve an idiographic interpretation through the unit-specific kernel, but the connection between the background and model section is not clear.\n\nI think the second experiment has some major flaws. The paper claims this experiment validates the \"Big Five\" because model performance peaks at $K=5$. However, the model is only fit for $K=1...5$. We can clearly see from the log likelihood numbers that model performance for IPGP is monotonically increasing in $K$ for the values considered, suggesting that it will likely be even better at larger values of $K$; if that were true, it would not validate the \"Big Five\" theory. As this is an applied paper, I would expect a much higher level of rigor on one of the two applied case studies.\n\nQUESTIONS:\nAs this is an emergency review, after the reviewer discussion period, I am not sure if the authors will be able to reply. But if they are able, I would just ask that they respond to the various points made in previous sub-sections.\n\nLIMITATIONS:\nThis paper did involve a fairly intense human subjects experiment for data collection, but the paper reports it was IRB-approved. Psychometrics of this kind are fairly common; I do not think the paper introduces any new potential negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-08-15"}