Figures · L-DAHS

§ 03 Figures · three 3D scenes

§ 03

Interactive 3D illustrations

Seeing in three dimensions

Three interactive figures, each built from a simulated 3D dataset that mirrors the structures the lab studies. Rotate, zoom, and hover to read the geometry.

Figure 1

The latent trait space

Three hundred simulated learners, each placed by three correlated latent factors: Aptitude, Motivation, and Anxiety, drawn from a structured covariance matrix with three latent profiles. Point colour encodes a fourth derived dimension (achievement), so the figure shows a 4D structure projected into a rotatable cube.

\[ \boldsymbol{\xi}_i=\boldsymbol{\mu}_{g(i)}+\mathbf{L}\mathbf{z}_i,\quad \mathbf{z}_i\sim\mathcal{N}(\mathbf{0},\mathbf{I}),\ \ \mathbf{L}\mathbf{L}^{\!\top}=\boldsymbol{\Sigma} \]

x — Aptitude (ξ₁) y — Motivation (ξ₂) z — Anxiety (ξ₃) colour — Achievement (η)

Figure 2

A two-dimensional response surface

The probability of item mastery as a smooth surface over two latent abilities, in the spirit of a multidimensional item-response model. The ridge running diagonally is the compensatory region where strength on one trait offsets weakness on the other: the kind of interaction a flat table of coefficients hides but a surface makes obvious.

\[ P(\theta_1,\theta_2)=\dfrac{1}{1+\exp\!\big[-(a_1\theta_1+a_2\theta_2-d)\big]} \]

x — Ability θ₁ y — Ability θ₂ height — P(mastery)

Figure 3

Affect dynamics in phase space

Several simulated learners released from nearly-identical starting states, each trajectory integrating a coupled nonlinear system of motivation, engagement, and anxiety. Tiny differences in initial affect produce divergent paths around a shared attractor: a visual argument for why static models miss regime shifts that dynamical ones catch.

\[ \dot{M}=\sigma(E-M),\quad \dot{E}=M(\rho-A)-E,\quad \dot{A}=ME-\beta A \]

x — Motivation y — Engagement z — Anxiety colour — time t

§ 04 Gallery · eight more figures

§ 04

Additional figures · one or two per research focus

A gallery of figures.

Eight publication-style figures spanning the lab's research foci. Each is tagged by the methodology it serves and whether it is 2D or 3D.

Fig A

Test information surface

CAT · IRT3-D

Where a multidimensional test measures most precisely. Each item contributes a ridge of Fisher information; their sum is the landscape an adaptive algorithm climbs when choosing the next item.

\[ \mathcal{I}(\boldsymbol{\theta})=\sum_i \big(a_{1i}^2+a_{2i}^2\big)\,P_i(\boldsymbol{\theta})\big[1-P_i(\boldsymbol{\theta})\big] \]

Fig B

CAT convergence

CAT

A simulated examinee. The ability estimate and its shrinking 95% band close in on the true value as items are administered.

\[ \mathrm{SE}=\mathcal{I}^{-1/2} \]

Fig C

Forest plot — meta-analytic synthesis

Meta-analysis2-D

Twelve simulated studies with marker size scaled to precision, and the random-effects pooled estimate as a diamond at the foot. The canonical synthesis figure.

\[ \hat{\delta}_{\text{RE}}=\frac{\sum_i w_i\,d_i}{\sum_i w_i},\qquad w_i=\frac{1}{\,s_i^2+\hat{\tau}^2\,} \]

Fig D

Funnel plot

Bias

Effect size against its standard error. Asymmetry at the base is the visual signature of small-study or publication bias.

\[ \hat{\delta}\pm 1.96\,s \]

Fig E

Random-slope spaghetti plot

HLM · multilevel2-D

Twenty classrooms, each its own regression line coloured by intercept, scattered around the fixed-effect line (white). Exactly the picture random intercepts and slopes are meant to capture.

\[ y_{ij}=(\beta_0+u_{0j})+(\beta_1+u_{1j})\,x_{ij}+\varepsilon_{ij} \]

Fig F

Correlation matrix heatmap

SEM2-D

The block structure that a measurement model assumes, made visible: high within-construct correlations on the diagonal blocks, low cross-construct correlations off them.

\[ \mathbf{R}=\mathbf{D}^{-1/2}\,\boldsymbol{\Sigma}\,\mathbf{D}^{-1/2} \]

Fig G

Gradient descent on a loss surface

Matrix calculus3-D

Estimation as a walk downhill. The path zig-zags down an elongated quadratic valley toward the minimum, the geometric reason ill-conditioned problems are slow to converge.

\[ \mathbf{w}_{t+1}=\mathbf{w}_t-\eta\,\nabla\mathcal{L}(\mathbf{w}_t) \]

Fig H

Phase portrait with vector field

Dynamical systems2-D

A two-state affect system: the arrows are the flow, the dashed lines the nullclines, and the bright curves are trajectories spiralling into a stable equilibrium, a planar companion to the 3D attractor.

\[ \dot{M}=-\gamma M+\omega A,\qquad \dot{A}=-\omega M-\gamma A \]

§ 05 Text mining · NLP · two interactive figures

§ 05

From the text mining course

Text mining & NLP

Two interactive figures from the lab's text mining work on learner language: a co-occurrence network you can pull apart and a sentiment map you can read point by point.

Figure I

N-gram co-occurrence network

A bigram co-occurrence network built from a simulated corpus of English-learning text. Each node is a word sized by frequency, each link a pair that co-occurs, and the colours mark three themes the lab tracks: skills, affect, and instruction. Drag a node to pull the web apart, hover to light up its strongest collocates, or search for a word to find it.

\[ \mathrm{PMI}(w_1,w_2)=\log\dfrac{P(w_1,w_2)}{P(w_1)\,P(w_2)} \]

Skills Affect Instruction link — co-occurrence

Figure J

NRC emotion radar of learner feedback

The NRC Emotion Lexicon scores text on eight emotions rather than a single positive or negative axis. Each radar here is the emotion profile of a simulated learner, from a confident voice rich in joy and trust to an anxious one driven by fear. Use the dropdown to switch between learner profiles and the whole corpus, and hover a vertex to read each emotion score.

\[ \mathbf{e}(d)=\frac{1}{|d|}\sum_{w\in d}\operatorname{NRC}(w),\quad \mathbf{e}\in[0,1]^{8} \]

joy · trust · anticipation fear · sadness anger · disgust radius — emotion intensity

Figure K

Topic model of the learner corpus

A latent Dirichlet allocation over simulated learner writing. The left heatmap shows how strongly each word loads on each of five topics, and the right bars show how three sample documents split across those topics. Hover a cell to read its topic, word, and weight.

\[ \theta_d\sim\mathrm{Dir}(\alpha),\quad z_{dn}\sim\mathrm{Mult}(\theta_d),\quad w_{dn}\sim\mathrm{Mult}(\beta_{z_{dn}}) \]

topic-word weight β rows — topics · cols — words

Figure L

Affect and topics over a term

Four indices tracked across a sixteen-week term: confidence and joy climbing, anxiety easing, and the speaking topic rising and falling around assessment. The dashed marker is the midterm week. Click a legend entry to toggle a line, or hover to read a week.

\[ \bar{x}_t=\frac{1}{k}\sum_{i=0}^{k-1}x_{t-i} \]

Confidence Joy Anxiety Speaking topic

Figure M

Speech spectrogram with pitch contour

A simulated spectrogram of a short spoken phrase: time runs left to right, frequency bottom to top, and colour is energy. The bright horizontal bands are vowel formants, the brief broadband flecks are consonants, and the white curve traces the pitch (F0) as intonation rises and falls. Hover to read frequency and energy.

\[ X(t,f)=\Big|\sum_n x[n]\,w[n-t]\,e^{-j2\pi f n}\Big|^2 \]

energy (dB) pitch F0 x — time · y — frequency