Page 170 - Artificial Intelligence for Computational Modeling of the Heart

142 Chapter 4 Data-driven reduction of cardiac models

duration (APD). The first model then reads f 1 : θ → [V peak ,V rest ,
APD], and can be estimated for instance using the projection pur-
suit regression (PPR) method [367]. The central idea of PPR is to
define auxiliary features by computing linear combinations of the
model inputs in an optimal direction; and then model the output
as a nonlinear function of the auxiliary features. Let f 1 represent
any of the outputs of f 1 ;using PPR,the modelhas theform:

s

f 1 (θ) = g j (θω j ), (4.4)
j=1

p
where ω j is a unit column vector in R , and function g j is esti-
mated as part of the model creation. The model can in fact be
built in an iterative manner: given a projection direction ω k , apply
a univariate smoother to obtain an estimation of g k by minimiz-
ing the discrepancy between model and data; given a function g k ,
use Gauss–Newton search to find the optimal updated weights ω k
minimizing the fitting error. The number of ridge terms s is deter-
mined by a backward deletion procedure.
The second step of the statistical learning procedure is to in-
clude predicted phenomenological features of the action poten-
tial as model inputs together with the CRN model parameters,
for estimating the embedding coordinates v dr .Thisstepestimates
amodel f 2 :[θ,f 1 (θ)] → v dr , which can be built for instance us-
ing multivariate adaptive regression spline (MARS [365]), or the
PPR method as in the first step. MARS is a non-parametric regres-
sion method extending linear regression by fitting linear or cubic
splines to capture data non-linearity. Let f 2 represent any of the
outputs of f 2 ; using MARS, the model has the form:

J

f 2 (θ,f 1 (θ)) = β 0 + β j h j (θ,f 1 (θ)), (4.5)
j=1

where h j (x) is a function of the form max(x −c,0) or max(c−x,0) or
their products, with c being a constant. The model is estimated by
a greedy approach fitting the splines to the training data, and then
a backward deletion pass is performed on the splines to minimize
over-fitting.
This multi-step regression approach can be further extended
by iteratively using the estimated components of v dr as additional
inputs. More precisely, v dr (1) can be estimated using [θ,f 1 (θ)] as
inputs; v dr (2) can be estimated using [θ,f 1 (θ),v dr (1)],and simi-
larly for the following components of v dr .