two model classes (SoFR and FoSR) • There are a lot of other ways to encounter the need for variable selection in functional regression • Scalar-on-image regression • Function-on-function regression
Issue isn’t large number of functional predictors – it’s the size of the single predictor • Scientific assumptions encourage sparse solutions • Together, both support the use of variable selection approaches
that regression coefficient is zero if and only if binary variable is zero • Promote smoothness and clustering through MRFs • Clustering via an Ising prior: p ( ) = ( a, b ) exp " a · + X l ( X l02 l blI ( l = l0 ) )# • Smoothness via a conditional autoregressive prior for coefficients [ l | l = 1, l, l] ⇠ N[¯ l , 2 b /dl] • Full disclosure – I’m not really a fan of this method anymore.
Comes up more frequently than scalar-on-image • Many longitudinal studies can be thought of in this context • Ease of data collection can make this a variable selection problem • E.g. ambulatory blood pressure monitoring Yi(t) = 0(t) + p X k=1 Xik(t) k(t) + i(t)
of the functional linear concurrent model have the following form: X i X⇤T i X⇤ i ! 1 X i X⇤T i yi ! • Hard to directly apply group variable selection tools
(1 k)v1)I] k ⇠ Bern(⇡) • Estimate using Variational Bayes • Reasonably computationally efficient • Can jointly model an FPCA expansion of residual curves as well • Main tuning parameter is v0 • Controls width of spike prior • Too narrow and all groups are omitted; too wide and all groups are included • Not really variable selection, but that has some helpful properties
Scalar-on-Image Regression via Spatial Bayesian Variable Selection. JCGS. • Wang and Zhu (2016). Generalized Scalar-on-Image Regression Models via Total Variation. JASA. • Kang, Reich and Staicu (Under review). Scalar-on-image regression via soft-thresholded Gaussian processes. • Goldsmith and Schwartz (2017). Variable Selection in the Functional Linear Concurrent Model. Statistics in Medicine.