Table 1
Glossary.
| SYMBOL | INTERPRETATION |
|---|---|
| s(t) | A vector [s1(t); s2 (t); s3(t)] indicating time-variant symptom scores that are bounded between 0 and 1: s1(t)- psychosis, s2 (t)– depression, s3(t) – mania. The transformation from y to s is parameterized by u. |
| y(t) | A vector [y1(t); y2 (t); y3(t)] indicating the time-variant latent psychopathology. The mapping from x to y is parameterized by B. |
| x(t) | A vector [x1(t); x2 (t); x3(t)] indicating the latent time-variant pathophysiology state. Values at each time point are determined by the Lorenz system of differential equations (Figure 1). Parameterized by A and τ. |
| v(t) | A 1-by-T vector. Exogeneous inputs. Output of a DCT. |
| T | A time-invariant scalar. The number of assessment points. Determined by the number of available data points in the empirical or simulated data. |
| A | A time-invariant vector [a1; a2; a3]. Lorenz attractor parameters. In the current paper, a1= 10, a3 = 1. Only a2, namely the Rayleigh parameter, is a free parameter. |
| B | A time-invariant 3-by-3 matrix. The weights of the linear combination, mapping from x to y. In the current paper, b1,3, b2,2, b3,1 were fixed at 0. |
| C | A 1-by-8 vector. Coefficients of the DCT parameterizing exogenous inputs. |
| D | The design matrix of the cosine discrete function of time. |
| x | A time-invariant vector [x1(0); x2 (0); x3(0)], indicating the initial values of the pathophysiology states. |
| u | A time-invariant vector [u1; u2; u3]. Thresholding the symptom scores from psychopathology. In the current paper, they are fixed at 1 (i.e., u1 = u2 = u3 = 1). |
| τ | A time-invariant scalar. Time constant of dynamics. |
Figure 1
The graphical model and definition of dynamics. Orange empty circles (dashed) denote time-invariant parameters; deep blue empty circles (solid) denote time-variant variables whereas the deep blue filled circle denotes a time-variant observable/measurable variable; letters in light blue denote parameters whose values are allowed to vary in the current study. Level 3 models exogenous inputs, such as stressors, v(t) at time t. This is defined as a general linear model with a design matrix D (fixed) encoding a discrete cosine transform (DCT) basis set, with associated parameters C. Here we use the notation Dt to indicate row t of matrix D. Level 2 models the dynamics of pathophysiology x(t) as a Lorenz attractor (a set of 3 differential equations) with parameters A. A linear transform yields psychopathology y(t) parameterized by A. Finally, level 1 transforms psychopathology to observed symptoms s(t) in the range [0 1] which was the unit of the actual measurements.
Figure 2
Model fitting results. A. Estimated and actual symptom scores. The estimated symptoms (dashed) overlaid on the empirical symptoms (solid) for three subjects. Each row corresponds to a different kind of symptom score (psychosis, depression and mania). B. Estimated latent variables. For the same three subjects, the estimated exogeneous inputs, the pathophysiology trajectory (the Lorenz attractor) and the psychopathology are plotted from bottom to top. For the pathophysiology state, the blue dot marks the initial state of the trajectory in the state space defined by (x1,x2,x3).
Figure 3
Subject-level parameter estimation and recovery. A. Estimated parameters. These bar graphs report the estimated parameters from the model fitting using nine subjects’ empirical data. Blue bars are posterior expected values and error bars are 90% credible intervals (the interval within which the parameter falls with 0.9 probability). B. Face validity of the parameters. Parameter values used to generate simulated symptom data were sampled from the posterior densities shown in panel A. This plot shows the estimated parameters from all 108 virtual subjects plotted against the parameters used to generate their data. Each dot is a virtual subject, the dashed line is the diagonal, and the red solid line is the line of best fit. R values denotes Pearson’s correlation coefficients.
Figure 4
Assessment of model and group identifiability. A-B. Group membership recovery results. Assessment of model and group identifiability. A-B. Group membership recovery results. Panel A depicts the 108 subjects × 9 groups design matrix (blue = 0, yellow = 1), used in the hierarchical linear regression model (PEB) to test for group membership. Panel B depicts the posterior probability of the estimated group membership ranging from 0 (blue) to 1 (yellow). Each row of this panel is the result of a model comparison, where for each subject, nine PEB models were compared in which the subject was assigned to group 1…9. Coloured lines on the off-diagonal elements indicate mis-classification of subjects. C. Illustration of leave-one-out cross-validation, applied to predicting whether subjects belonged to group 1 or group 5. The true group allocation parameter is on the horizontal axis and the estimated group membership parameter is on the vertical axis. D. Leave-one-out cross-validation results for all 36 group pairs. The correlation coefficients between the group allocation parameter and the estimated parameter are displayed (black text for values > 0.4).