learningcurves

MATLAB implementations of the Prerau Lab’s state-space learning-curve estimators. Given a time series of trial-by-trial performance — binary (correct/incorrect), continuous (reaction time), or both recorded simultaneously — these estimators recover the underlying cognitive state (an unobservable latent variable representing the subject’s understanding of the task) as it evolves across trials, with proper uncertainty quantification.

Methodologically, this is a state-space framework:

• State equation. The cognitive state x_k evolves as a Gaussian random walk (with optional AR(1) drift) across trials. You can’t observe x_k directly.

• Observation equation. What you can observe are performance measures on each trial. Binary (correct/incorrect) observations follow a Bernoulli whose probability is a logistic function of x_k. Continuous observations (e.g., log reaction time) follow a Gaussian with mean linear in x_k.

• Inference. Expectation-maximization (or particle-filter Monte Carlo) produces a smoothed estimate E[x_k | data] with per-trial confidence bands.

The key conceptual contribution: fusing binary and continuous measures into one estimate gives a more credible and accurate cognitive-state trajectory than either measure analyzed separately.

Methods in this toolbox

Subdirectory	Observation model	Backend	When to use
binsmoother/	Binary only	EM + forward/backward filter (Smith–Brown style)	You only have correct/incorrect per trial
mixedsmoother/	Binary + continuous simultaneously	EM + forward/backward filter	You have paired binary + continuous (e.g., correct-or-not + reaction time) per trial
particle_filter/	Binary, continuous, or mixed	Sequential Monte Carlo	Missing data in either channel; non-Gaussian state noise; or you want posterior particles rather than Gaussian summaries

All three recover smoothed trajectories of the same underlying cognitive state — the difference is which observations they can consume and which approximations they make.

Install

addpath(genpath('/path/to/learningcurves'));

Requirements

• MATLAB R2020a or later

• Statistics and Machine Learning Toolbox — for random sampling in the particle filter and some diagnostics

• No other toolboxes required

binsmoother

Binary-only EM smoother. Fits a Gaussian-random-walk state model with a Bernoulli-logistic observation model. Canonical use: k out of N correct trials per session, recover a smooth p(correct | trial) curve with 5/50/95-percentile bands.

Entry point:

[p05, p95, pmid, pmode, pmatrix, xnew, signewsq] = ...
    binsmoother(Responses, SigE, BackgroundProb, NumberSteps)

Argument	Meaning
Responses	1×K number-correct-per-trial (supports “k out of N” when max(Responses) > 1)
SigE	initial guess for random-walk standard deviation of the latent state
BackgroundProb	chance-level correct probability
NumberSteps	max EM iterations (converges when state-variance change < 1e-8)

Returns p05/p95/pmid/pmode curves on p(correct), a per-trial pmatrix giving the posterior probability that performance exceeded chance, and the latent state xnew with variance signewsq.

Based on Smith, Brown et al., J. Neurophysiol. 2004. Our implementation was used as the binary-only special case in the mixed-filter work.

mixedsmoother

Simultaneous binary + continuous EM smoother. This is the method introduced in Prerau et al., Biological Cybernetics 2008 and extended with the EM estimator in Prerau et al., J. Neurophysiol. 2009.

Model:

State:       x_k = ρ · x_{k-1} + v_k,       v_k ~ N(0, σ²_v)
Binary obs:  P(n_k correct) = 1 / (1 + exp(-(μ_1 + γ·x_k)))
Continuous:  z_k = α + β · x_k + e_k,        e_k ~ N(0, σ²_e)

All parameters (α, β, γ, ρ, σ²_e, σ²_v) are estimated by EM — no tuning required beyond initial guesses. γ is typically fixed to zero in the default implementation (binary probability is modulated by μ_1 alone) but can be freed for joint-coupling models.

Entry point:

[alph, beta, gamma, rho, sig2e, sig2v, xnew, signewsq, muone, a] = ...
    mixedlearningcurve(N, Z, background_prob, rhog, alphag, betag, sig2eg, sig2vg)

Argument	Meaning
N	2×K — row 1: number correct per trial; row 2: max possible correct per trial
Z	1×K continuous observations (typically log(reaction_time))
background_prob	chance-level probability of correct response
rhog, alphag, betag, sig2eg, sig2vg	initial guesses for the EM

Returns the EM-estimated model parameters, the smoothed latent state trajectory xnew with variance signewsq, the logit-transformed chance-bias muone, and the backward-smoother gain a.

Because the continuous channel tightens the otherwise-uncertain binary estimate, the mixed smoother typically produces substantially narrower confidence bands and more accurate recovery of the true state in simulation than the binary-only or continuous-only smoothers (Prerau et al. 2008, 2009).

particle_filter

Sequential Monte Carlo estimator for cases where the EM smoothers’ assumptions (Gaussian state noise, linear continuous observation model, complete data) don’t hold. Produces a full posterior distribution as a set of weighted particles at each trial, rather than a Gaussian mean+variance.

Entry point:

[param_ests, particles] = learningcurve_pfilter(times, data, ...
    num_particles, smoother, prog_bar, plot_on)

Argument	Meaning
times	1×T observation times in seconds
data	2×T — row 1: binary observations (0/1), row 2: continuous; NaN marks missing data in either channel
num_particles	particle-population size (default: 5000)
smoother	true: forward + backward smoothing; false: forward-only filter (faster, higher posterior variance)
prog_bar, plot_on	UI options

Returns param_ests (estimated parameters and posterior summaries) and particles (the raw particle matrix across time — useful for arbitrary posterior queries).

Model mirrors the mixedsmoother’s (Gaussian-random-walk state, Bernoulli binary obs, Gaussian continuous obs) but the particle filter representation handles:

• Missing data — either channel can be NaN on any trial without special treatment

• Non-Gaussian state noise — just change the state-transition sampler

• Non-linear continuous observation models — just change the likelihood function

Specialized variants in this folder:

• binary_learningcurve_pfilter_ptile.m — binary-only particle filter with percentile extraction

• learningcurve_pfilter_ptile.m — mixed with percentile extraction on the posterior

• learningcurve_pfilter_ptile_bino_only.m — binary-only percentile variant

• sleeponset_pfilterEEG_3states.m — 3-state variant for EEG-based sleep-onset detection

Citation

If you use this toolbox, please cite the appropriate methods paper(s):

Primary: EM smoother for binary + continuous

Prerau MJ, Smith AC, Eden UT, Kubota Y, Yanike M, Suzuki W, Graybiel AM, Brown EN. “Characterizing learning by simultaneous analysis of continuous and binary measures of performance.” Journal of Neurophysiology 102(5):3060–3072, 2009. doi: 10.1152/jn.91251.2008

Applied to monkey within-session rapid association-learning data (Wirth/Suzuki) and mouse multi-day T-maze learning (Graybiel). Introduces formal definitions of the learning curve, reaction-time curve, ideal-observer curve, learning trial, and between-trial performance comparisons.

Foundational: mixed-filter algorithm

Prerau MJ, Smith AC, Eden UT, Yanike M, Suzuki WA, Brown EN. “A mixed filter algorithm for cognitive state estimation from simultaneously recorded continuous and binary measures of performance.” Biological Cybernetics 99(1):1–14, 2008. doi: 10.1007/s00422-008-0227-z

Introduces the recursive mixed filter. Kalman filter (for continuous-only data) and the Smith–Brown binary recursive filter are shown as special cases.

A machine-readable citation is in CITATION.cff — GitHub’s “Cite this repository” button uses it.

Related concepts

The 2009 paper formalizes several trial-based concepts used downstream:

• Learning curve — P(correct | trial) over trials, with confidence bands.

• Reaction-time curve — posterior E[reaction_time | trial] over trials, also with bands.

• Ideal-observer curve — posterior probability that a trial is above chance given all data observed up to and including that trial (accounts for retrospective knowledge).

• Learning trial — the first trial at which the posterior probability of above-chance performance exceeds a user-specified criterion (e.g., 0.95).

• Between-trial comparisons — posterior probability that performance at trial j exceeds performance at trial i < j.

All are derived directly from the xnew / signewsq outputs of the smoothers in this toolbox.

Documentation

Full API reference: https://preraulab.github.io/learningcurves/

Each function’s full docstring is visible both in help <function> at the MATLAB prompt and on the hosted site.

License

BSD 3-Clause. See LICENSE.

Reference

• API Reference
  • binsmoother
  • mixedsmoother
  • particle_filter