§ 00 — Multidimensional Quantitative Methods · Applied Linguistics

Mining the latent manifold of social-science data.

We treat every dataset as a high-dimensional object: a matrix to be decomposed, an operator to be differentiated, a system to be integrated over time. From multilevel structural equation models to computerized adaptive testing, the lab develops methods that surface structure the literature has not yet perceived.

Explore the 3D figures → The unifying framework
The object of study
\[ \mathbf{X}\in\mathbb{R}^{\,n\times p}\;\xrightarrow{\;\Phi\;}\;\boldsymbol{\Xi}\in\mathbb{R}^{\,n\times k},\quad k\ll p \]
\[ \frac{d\boldsymbol{\xi}}{dt}=\mathbf{f}\!\big(\boldsymbol{\xi};\boldsymbol{\theta}\big),\qquad \nabla_{\boldsymbol{\theta}}\,\mathcal{L}=\mathbf{0} \]

observed data  →  latent dimensions  →  dynamics in time

§ 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.

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.

\[ \boldsymbol{\Sigma}(\boldsymbol{\theta})=\boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^{\!\top}+\boldsymbol{\Theta} \]
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.

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

\[ \dot{\boldsymbol{\xi}}=\mathbf{A}\boldsymbol{\xi}+\mathbf{g}(\boldsymbol{\xi}),\quad \boldsymbol{\xi}(0)=\boldsymbol{\xi}_0 \]
§ 02 Methods · computation · toolchain
§ 02
Toolchain

Methods & computation

Reproducible, bilingual (English · 繁體中文) pipelines built primarily in R and Stan, with interactive deployment through Shiny.

R · tidyverselavaan / OpenMxlme4 · brms Stan (HMC)mirt · catRmetafor jiebaR · text2vectorch / kerasR Shiny Bayesian inferencedeSolve (ODEs)Quarto