§ 01 Framework · one lens, three operations
§ 01
The unifying lens

A matrix · calculus · dynamics framework

The lab's methods are unified by a single idea: a research problem becomes tractable once it is written as linear algebra, optimized with matrix calculus, and where behaviour unfolds over time, closed with a differential equation. Read across the equations below and the family resemblance is the point.

01 · LINEAR ALGEBRA

Structure as decomposition

Persons × items, students × schools, studies × moderators are all matrices. Latent structure is recovered by factoring the covariance the model implies.

\[ \boldsymbol{\Sigma}(\boldsymbol{\theta})=\boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^{\!\top}+\boldsymbol{\Theta} \]
SEMCFAIRTPCA
02 · MATRIX CALCULUS

Estimation as a gradient

Every estimator is the stationary point of an objective. Fisher information, score equations, and back-propagation are one differential idea applied across scales.

\[ \hat{\boldsymbol{\theta}}=\arg\min_{\boldsymbol{\theta}}\;\mathcal{L}(\boldsymbol{\theta}),\quad \mathbf{H}=\frac{\partial^2\mathcal{L}}{\partial\boldsymbol{\theta}\,\partial\boldsymbol{\theta}^{\!\top}} \]
MLFIMLHMCCAT
03 · DIFFERENTIAL EQUATIONS

Change as a flow

Motivation, anxiety, and engagement are not snapshots but trajectories. Coupled ODEs describe how affective states co-evolve and settle into attractors.

\[ \dot{\boldsymbol{\xi}}=\mathbf{A}\boldsymbol{\xi}+\mathbf{g}(\boldsymbol{\xi}),\quad \boldsymbol{\xi}(0)=\boldsymbol{\xi}_0 \]
LCSDSEMPhaseODE

How the three pillars connect

A unified pipeline: data becomes matrices, matrices yield gradients, gradients trace dynamics.

Raw Data n × p observations LINEAR ALGEBRA Σ = ΛΦΛ′ + Θ decompose structure MATRIX CALCULUS ∇L(θ) = 0 estimate parameters DIFF. EQUATIONS dξ/dt = f(ξ) model dynamics feedback: model fit → refine structure

Methods × pillars

Each method draws on one or more pillars. The table shows which mathematical operation dominates.

Method Linear Algebra Matrix Calculus Diff. Equations
Multilevel SEM covariance decomposition ML / FIML estimation —
HLM / Mixed Models design matrices REML / Bayes —
IRT / CAT item parameters Fisher information —
Meta-Analysis effect aggregation weighted estimation —
Text Mining / NLP embeddings, SVD gradient descent —
Dynamical Systems state-space form parameter estimation ODE / phase flow
Latent Change Score latent structure SEM estimation discrete dynamics