feat(psychometric): recover Driver later-start later-occasion variance of predetermined T0VAR

CHANGELOG.md

@@ -4,6 +4,7 @@ All notable changes to TEPP are documented here. The format follows Keep a Chang
 
 ## [Unreleased]
 
+- `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 3–5, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T11:05Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar later-start later-occasion variance of §4.3 predetermined `T0VAR`. Section 4.3 treats the first time point as predetermined when no assumptions are made about the process prior to the initial time point. Free `T0VAR` `p_0` is then estimated. Section 4.3 notes that the process gradually transitions from the initial variances toward the stationary variances, and that the initial time point need not reflect the first measurement occasion (`startoffset`). Equation 3 writes `η(t) = exp(A Δt) η(t0) + … +` the stochastic integral. Equation 4 writes that the integral exhibits covariance `Q_Δt`. After a later start `u` the within-subject state variance is `e^{2 a u} p_0 + Q_u`. Evolving that later start over `s` is `e^{2 a s}(e^{2 a u} p_0 + Q_u) + Q_s`. Chapman–Kolmogorov writes `Q_{u+s} = e^{2 a s} Q_u + Q_s`, so the later-occasion map over `u + s` is the same composition. Trait variance and `addedTIPREDVAR` are time-invariant between-subject and do not enter `Q_s`. The later-start later-occasion composition is `trait + e^{2 a s}(e^{2 a u} p_0 + Q_u) + Q_s + (B / a)² v`. Form the later-start within-subject variance first, then evolve that state, then include the trait, then include the TI extra variance, then add. Setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map. Stationary later-occasion variance uses `−q / (2 a)` in place of `p_0` and is not this map when `p_0` is free. Later-occasion variance at `u` omits `Q_s` and is not this map when `s > 0`. Later-start lagged covariance is `e^{a s}` of the later state and omits `Q_s`; it is not this map. Evolving the later total as if it were all state (`e^{2 a s}` of `trait + e^{2 a u} p_0 + Q_u + (B / a)² v` plus `Q_s`) is not this map. Later-occasion variance over the lag interval alone ignores `startoffset` and omits `e^{2 a s} Q_u`; it is not this map when `u > 0`. As `u → 0+` the composition approaches later-occasion variance over `s`. As `s → 0+` the composition approaches later-occasion variance at `u`. As `s → ∞` with stable `a < 0` the carried later state vanishes and `Q_s` approaches `−q / (2 a)`, so the composition approaches contemporaneous stationary `T0VAR`. A zero trait, a zero initial variance, a zero diffusion, and a zero TI contribution is exactly zero. A zero initial variance, a zero diffusion, and a zero TI contribution is exactly the trait. Trait-only variance does not require a stable drift. `a ≥ 0` cannot hold a finite TI extra variance when that contribution is nonzero and fails closed. Nonzero diffusion with `a ≥ 0` is a growing process and is kept. Both intervals must be event time and strictly positive. Equation 5 of that later-start later-occasion variance is `λ²(trait + e^{2 a s}(e^{2 a u} p_0 + Q_u) + Q_s + (B / a)² v) + θ + ψ`. `MANIFESTVAR` is not that later-start later-occasion observed variance. The later-start later-occasion latent variance is not the later-start later-occasion observed variance. Predetermined later observed variance at `u` omits `Q_s` and is not that observed variance when `s > 0`. Later-start lagged observed covariance omits `Q_s` and `θ` and is not that observed variance. Stationary later-occasion observed variance is not that observed variance when `p_0` is free. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T11:05Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T11:05Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 3–5, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T10:27Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar later-start lagged covariance of §4.3 predetermined `T0VAR`. Section 4.3 treats the first time point as predetermined when no assumptions are made about the process prior to the initial time point. Free `T0VAR` `p_0` is then estimated. Section 4.3 notes that the process gradually transitions from the initial variances toward the stationary variances, and that the initial time point need not reflect the first measurement occasion (`startoffset`). Equation 3 writes `η(t) = exp(A Δt) η(t0) + … +` the stochastic integral. Equation 4 writes `cov(η_t, η_{t-1}) = A_Δt cov(η_{t-1})`. After a later start `u` the within-subject state variance is `e^{2 a u} p_0 + Q_u`. Lagging that later start over `s` is `e^{a s}(e^{2 a u} p_0 + Q_u)`. Trait variance and `addedTIPREDVAR` are time-invariant between-subject and do not decay with `e^{a s}`. The lagged composition is `trait + e^{a s}(e^{2 a u} p_0 + Q_u) + (B / a)² v`. Form the later-start within-subject variance first, then lag that state, then include the trait, then include the TI extra variance, then add. Setting `p_0 = −q / (2 a)` recovers the stationary lagged map. Stationary lagged covariance uses `−q / (2 a)` in place of `p_0` and is not this map when `p_0` is free. First-occasion lagged covariance `trait + e^{a s} p_0 + (B / a)² v` omits `e^{a s} Q_u` and is not this map when `u > 0`. Later-occasion variance includes `Q_u` without lagging that later state and is not this map. Evolving the later total as if it were all state (`e^{a s}` of `trait + e^{2 a u} p_0 + Q_u + (B / a)² v`) is not this map. As `u → 0+` the composition approaches first-occasion lagged covariance. As `s → 0+` the composition approaches later-occasion variance at `u`. As `s → ∞` with stable `a < 0` the state term vanishes. A zero trait, a zero initial variance, a zero diffusion, and a zero TI contribution is exactly zero. A zero initial variance, a zero diffusion, and a zero TI contribution is exactly the trait. Trait-only variance does not require a stable drift. `a ≥ 0` cannot hold a finite TI extra variance when that contribution is nonzero and fails closed. Nonzero diffusion with `a ≥ 0` is a growing process and is kept. Both intervals must be event time and strictly positive. Equation 5 of that later-start lagged covariance is `λ²(trait + e^{a s}(e^{2 a u} p_0 + Q_u) + (B / a)² v) + ψ`. Independent `ε_t` does not enter. `MANIFESTVAR` is not that later-start lagged observed covariance. The later-start lagged latent covariance is not the later-start lagged observed covariance. First-occasion lagged observed covariance omits `e^{a s} Q_u` and is not that observed covariance when `u > 0`. Predetermined later observed variance includes `Q_u` and `θ` and is not that later-start lagged observed covariance. Stationary lagged observed covariance is not that observed covariance when `p_0` is free. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T10:27Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T10:27Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 5, p. 5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T10:03Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar first-occasion variance of §4.3 predetermined `T0VAR`. Section 4.3 treats the first time point as predetermined when no assumptions are made about the process prior to the initial time point. Free `T0VAR` `p_0` is then estimated. Between-subject `TRAITVAR` and `addedTIPREDVAR` are inherently stationary. The first-occasion composition is `trait + p_0 + (B / a)² v`. Form the free first-occasion state variance first, then include the trait, then include the TI extra variance, then add. Setting `p_0 = −q / (2 a)` recovers the stationary first-occasion map. Stationary first-occasion variance uses `−q / (2 a)` in place of `p_0` and is not this map when `p_0` is free. Free `T0VAR` `p_0` is not this map. The lagged map `trait + e^{a Δt} p_0 + (B / a)² v` decays the state and is not this map. The later-occasion map `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` includes `Q_Δt` and is not this map. As `Δt → 0+` those maps approach this composition. A zero trait, a zero initial variance, and a zero TI contribution is exactly zero. A zero initial variance and a zero TI contribution is exactly the trait. Trait-only variance does not require a stable drift. `a ≥ 0` cannot hold a finite TI extra variance when that contribution is nonzero and fails closed. Equation 5 of that first-occasion variance is `λ²(trait + p_0 + (B / a)² v) + θ + ψ`. `MANIFESTVAR` is not that first-occasion observed variance. The predetermined first-occasion latent variance is not the predetermined first-occasion observed variance. Stationary first-occasion observed variance is not that observed variance when `p_0` is free. Predetermined later observed variance includes `Q_Δt` and is not that first-occasion observed variance. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T10:03Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T10:03Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).
 - `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 3–5, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T09:04Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar lagged covariance of §4.3 predetermined `T0VAR`. Section 4.3 treats the first time point as predetermined when no assumptions are made about the process prior to the initial time point. Free `T0VAR` `p_0` is then estimated. Equation 3 writes `η(t) = exp(A Δt) η(t0) + …`. Equation 4 writes `cov(η_t, η_{t-1}) = A_Δt cov(η_{t-1})`. Trait variance and `addedTIPREDVAR` are time-invariant between-subject and do not decay with `e^{a Δt}`. The lagged composition is `trait + e^{a Δt} p_0 + (B / a)² v`. Form the lagged free first-occasion covariance first, then include the trait, then include the TI extra variance, then add. Setting `p_0 = −q / (2 a)` recovers the stationary lagged map. Stationary lagged covariance uses `−q / (2 a)` in place of `p_0` and is not this map when `p_0` is free. Evolving `trait + p_0 + (B / a)² v` as if it were all state (`e^{a Δt}` of that total) is not this map. Free `T0VAR` `p_0` is not this map. The later-occasion map `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` includes `Q_Δt` and is not this map. As `Δt → ∞` with stable `a < 0` the state term vanishes. As `Δt → 0+` the composition approaches `trait + p_0 + (B / a)² v`. A zero-diffusion carry with `a ≥ 0` is `e^{a Δt} p_0` and is kept. Trait-only variance does not require a stable drift. The interval must be event time and strictly positive. Equation 5 of that lagged covariance is `λ²(trait + e^{a Δt} p_0 + (B / a)² v) + ψ`. Independent `ε_t` does not enter. `MANIFESTVAR` is not that lagged observed covariance. The predetermined lagged latent covariance is not the predetermined lagged observed covariance. Predetermined later observed variance includes `Q_Δt` and `θ` and is not that lagged observed covariance. Stationary lagged observed covariance is not that observed covariance when `p_0` is free. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T09:04Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T09:04Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*).