ssMousetrack-package {ssMousetrack}

Bayesian State-Space Modeling of Mouse-Tracking Experiments Via Stan

Description

The ssMousetrack package allows analysing mouse-tracking experiments via Bayesian state-space modeling. The package estimates the model using Markov Chain Monte Carlo, variational approximations to the posterior distribution, or optimization, as implemented in the rstan package. The user can use the customary R modeling syntax to define equations of the model and Stan syntax to specify priors over the model parameters.

The sections below provide an overview of the state-space model implemented by the ssMousetrack package.

Details

(i) Mouse-tracking data

The raw data of a mouse-tracking experiment for I individuals and J stimuli consist of a collection of arrays (x,y)_{ij} = (x_0,...,x_{N_{ij}}; y_0,...,y_{N_{ij}}) which contain ordered N_{ij} x 1 sequences of x-y Cartesian coordinates as mapped to the computer-mouse pointer. The x-y coordinates are pre-processed according to the following steps:

1. Realigning: the arrays (x,y)_{ij} are re-aligned on a common sampling scale, so that N indicates the cumulative amount of progressive time from 0% to N = 100%, with N being the same over i=1,...,I and j=1,...,J

2. Normalization: the aligned arrays (x,y)_{ij} are normalized so that (x_0,y_0)_{ij}=(0,0) and (x_N,y_N)_{ij}=(1,1) for each i=1,...,I and j=1,...,J

3. Translation: the normalized arrays (x,y)_{ij} are translated into the quadrant [-1,1]x[0,1]

4. atan2 projection: the final arrays (x,y)_{ij} are projected onto a lower-subspace via the atan2 function by getting the ordered collection of radians (y)_{ij} = (y_0,...,y_N) in the subset of reals (0,\pi]^N, for each i=1,...,I and j=1,...,J.

The final I x J x N array of data Y contains the mouse-tracking trajectories expressed in terms of angles. These trajectories lie on the arc defined by the union of two disjoint sets, namely the sets \{y_0,...,y_N: y_n \geq \pi/2 \} (target's hemispace) and \{y_0,...,y_N: y_n < (3\pi)/4 \} (distractor's hemispace), with \pi/2 and (3\pi)/4 being the location points for target and distractor, respectively.

Note that, the current version of ssMousetrack package requires the number of stimuli J to be the same over the subjects i=1,...,I.

The pre-processed mouse-tracking trajectories are analysed using the state-space modeling described below.

(ii) Model representation

The array Y contains the observed data expressed in angles. The measurement equation of the model is:

y_{ij}^{(n)} \sim vonMises\big(\mu_{ij}^{(n)},\kappa_{ij}^{(n)}\big)

where \mu_{ij}^{(n)} and \kappa_{ij}^{(n)} are the location and the concentration parameters for the vonMises probability law. The moving mean on the arc \mu_{ij}^{(n)} is defined as:

\mu_{ij}^{(n)} := G(\beta,x_{i}^{(n)})

with \beta being a J x 1 array of real parameters representing the contribution of the j-th stimulus on the observed trajectory y_{ij} = (y^{(0)},...,y^{(N)}) whereas G is a non-linear function mapping reals to the subset (0,\pi] of the form: (i) \big[ (1 + \exp(\beta - x_{i}^{(n)})) \big]\pi^{-1} (logistic), (ii) \big[ \exp(-\beta \exp(-x_{i}^{(n)})) \big]\pi (gompertz). In the G equation, x_i^{(n)} is a real random quantity obeying to the law:

x_i^{(n)} \sim Normal\big( x_i^{(n-1)},\sigma^2_i \big)

which represents a random walk process with time-fixed variance \sigma^2_i. The terms x_{i} = (x_{i}^{(0)},...,x_{i}^{(N)}) are the individual latent dynamics unaffected by the stimuli (i.e., how individual differ in executing the task) whereas \beta contains the experimental effects regardless to the individual dynamic (i.e., how experimental variables act on the individual dynamics to produce the observed responses).

The terms \beta = (\beta_1,...,\beta_J) are defined according to the following linear combination:

\beta_j := \sum_{k=1}^K z_{jk}\gamma_k

where z_{jk} is an element of the J x K dummy matrix Z representing main and high-order effects of the experimental design.

The terms \kappa_{ij} = (\kappa_{ij}^{(0)},...,\kappa_{ij}^{(N)}) are computed as follows:

\kappa_{ij}^{(n)} := \exp^{o}\big(\delta_{ij}^{(n)}\big)

where \delta^{(n)}_{ij} = |y_{ij}^{(n)}-(3\pi)/4| (if y_{ij}^{(n)} < \pi/2) or \delta^{(n)}_{ij} = |y_{ij}^{(n)}-\pi/4| (if y_{ij}^{(n)} \geq \pi/2). The function \exp^o is the exponential function scaled in the natural range of the parameters \kappa_{ij} (positive real numbers).

(iii) Bayesian formulation

The state-space model in the ssMousetrack package requires estimating the array of latent trajectories X and the K x 1 parameters \gamma. Let \Theta representing both the unknown quantities, the posterior density after factorization is:

f(\Theta|Y) \propto f(\gamma) \prod_{i=1}^I \prod_{j=1}^J f(\gamma|y_{ij}) \prod_{i=1}^I \prod_{j=1}^J f(x_i|y_{ij})

Sampling from f(\Theta|Y) is solved via marginal MCMC where the term f(x_i|y_{ij}) is approximated by means of Kalman filtering/smoothing. The marginal Likelihood of the model used for the rejection criterion of the MCMC sampler is approximated with the Normal distribution using the Kalman filter theory.

References

Calcagnì, A., Lombardi, L., & D'Alessandro, M. (2018). A state space approach to dynamic modeling of mouse-tracking data. Frontiers in Psychology: Quantitative Psychology and Measurement, 10, 2716

Calcagnì, A., Lombardi, L., & D'Alessandro, M. (2018). A state space approach to dynamic modeling of mouse-tracking data. Under review

Calcagnì, A., Lombardi, L., & D'Alessandro (2018, August). Probabilistic modeling of mouse-tracking data: A statespace approach. Paper presented at the 2018 European Mathematical Psychology Group Meeting (EMPG 2018), Genova, Italy

Calcagnì, A., Lombardi, L., D'Alessandro, M., & Sulpizio S. (2018, March). A subject oriented state-space approach to model mouse-tracking data. Paper presented at the 60th Conference of Experimental Psychologists (TeaP 2018), Marburg, Germany

Freeman, J. B. (2018). Doing psychological science by hand. Current Directions in Psychological Science, In press, 1-9

Särkkä, S. (2013). Bayesian Filtering and Smoothing. Cambridge University Press

Durbin, J., & Koopman, S. J. (2012). Time series analysis by state space methods (Vol. 38). Oxford University Press

Andrieu, C., Doucet, A., & Holenstein, R. (2010). Particle markov chain monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3), 269s-342

Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2004). Bayesian Data Analysis (Second edition). Chapman & Hall/CRC.

See Also

https://mc-stan.org/ for more information on the Stan C++ language used by ssMousetrack package

Jokkala, J. (2016). Github repository: kalman-stan-randomwalk, https://github.com/juhokokkala/kalman-stan-randomwalk