Appendix A. Factor Model Parameter Priors and Updates
Parameters are updated using a Markov Chain Monte Carlo (MCMC) scheme. Denote the full conditional distribution of a parameter as .
Observation Means (): For each pollutant , assume independent prior distributions, .
Let be a vector of pollutant for all locations and time points, be a matrix of factor loadings, be a block diagonal matrix with each of the diagonal blocks equal to , and be a vector of all factors across all time. Then, we can rewrite the conditional distribution as .
The resulting full conditional will then be, .
Observation Variances (): For each pollutant , assume independent prior distributions, .
Referring to the notation defined above, the resulting full conditional will be
Factor Variances (): For each factor , assume independent prior distributions, .
Let be a block diagonal matrix with each of the diagonal blocks equal to , be a vector of factor for , and be a vector of factor for .
The resulting full conditional will be, .
Spatially Varying Factor Loadings (): For each factor-pollutant pair, assume a Gaussian process prior for the factor loadings, diag(. Let be the block diagonal matrix of the covariance matrices associated with the factor loadings for pollutant .
Let a matrix of the spatial factors at time , a matrix of the stacked on top of each other for , and a vector of the factor loadings at all locations for pollutant . Then, we can rewrite the conditional distribution as
Using standard multivariate Normal theory, the resulting full conditional will be, where and .
Spatially Constant Factor Loadings (): For each pollutant assume independent prior distributions,
Let be the matrix of factors across time.
Then, the resulting full conditional will be, where and .
Factor Evolution Coefficients (): To ensure stationarity in time, the factor evolution coefficients are restricted to the interval , so assume a Truncated Gaussian process prior, .
Let be a vector of evolution coefficients, be the matrix stacking diagonal matrices of for .
Then, the resulting full conditional will be , where and
Factors (): Assume is a realization from a Gaussian process for each factor . Let and be an block diagonal matrix corresponding to the covariance matrix for the collection of all factors at time 0, . Let and be the diagonal covariance matrices associated with and , respectively.
The factors are updated through a Forward Filtering Backwards Sampling (FFBS) algorithm [21
Forward Filtering: For , compute and , where , , , , and . Then, sample .
Backwards Sampling: For sample where , , and .
When data are unobserved at a particular time,
, we assume it is missing at random. Following the reasoning of [23
], there is no additional information incorporated into the posterior and the factors are updated setting
in the FFBS algorithm.
Spatial Parameters (): There are parameters , parameters and parameters describing the variance in the spatial processes. Denote these as with priors . Similar for the range parameters with priors .