seudo inverse for a GAM model is defined as below, where it can
hat this format is almost identical with the LSE estimate for an
del,
ൌሺ܁࢚܁ሻି܁࢚ܡ
(4.31)
Bayesian linear regression algorithm
esian learning provides a more robust approach to model
r estimation. The Bayesian linear regression algorithm (BLR) is
example [Buerkner, 2017]. The posterior probability of a BLR
n be calculated based on the following definition,
ሺܟ|ܡ, ܆ሻൌሺܡ|܆, ܟ, Σሻൈሺܟ|αሻ
ሺܡ|܆ሻ
(4.32)
and Σ stand for the regression coefficients and the covariance
stands for the hyper-parameter of the a priori structure for w.
ihood function is shown below, where ઽൌ܆௧ܟെܡ,
ሺܡ|܆, ܟ, Σሻൌ
1
ሺ2ߨሻே/ଶ√Σ
exp ൬െ1
2 ઽ௧Σିଵઽ൰
(4.33)
ussian a priori structure is defined as below, where d stands for
er of independent variables,
ሺܟ|ߙሻൌቀߙ
2ߨቁ
ௗ/ଶ
exp ቀെߙ
2 ܟ௧ܟቁ
(4.34)
a priori structures can also be used in BLR algorithm [Buerkner,
he R package brms can be used for constructing a model for a
using BLR. The R function for BLR in this package is named as
e call of this R function is shown below,
brm.model=brm(formula,data,family)