Page 169 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 4 Data-driven reduction of cardiac models 141

PCA projects the high-dimensional space Ω AP onto the re-
pca
duced space Ω spanned by principal components z l
AP
(l ∈{1,··· ,q}) that are orthogonal (uncorrelated) and maximize
the observed covariance. The principal components are com-
puted as Z = Y zs W where Y zs is the observation matrix converted
in z-scores, Z =[z 1 ,··· ,z q ]∈ R n×q , W =[w 1 ,··· ,w q ]∈ R m×q .The
w l ’s are the eigenvectors of the covariance matrix Y zs Y T zs in de-
creasing order of corresponding eigenvalues. Given an observa-
tion by the PCA coefficient row vector z pca , the reconstruction
pca T
from space Ω to space Ω AP is by v = ¯ y + z pca W σ Y .Here
AP
¯ y and σ Y are the mean and standard deviation of Y,and repre-
sents an element-wise product.
LLE seeks a projection of data onto a low-dimensional space
Ω lle by a neighborhood-preserving mapping. The standard LLE
AP
algorithm is performed in three steps. First, k nearest neighbors
i
are identified for each data point v , based on a given distance
i
metric; then, the barycentric coordinates w ij of v relative to its
k neighbors are computed; finally, the embedding coordinates
z i ∈ Ω lle are found by minimizing the global embedding cost
lle AP
i j 2
function Φ(z lle ) = |z − w ij z | , with given weights w ij .To
i lle j lle
reconstruct an observation, mapping z lle from space Ω lle to space
AP
Ω AP ,firstthe k nearest neighbors in space Ω lle are identified, and
AP
then the observation is reconstructed as the linear combination of
its nearest neighbor, using the same weights.
4.2.1.3 Statistical learning
We define a data-driven model of action potential dynamics,
p
m
generating an AP profile v ∈ R given a sample θ ∈ R of the CRN
q
p
model parameters. The model writes v dr = f(θ), f : R → R ,and
it is designed to predict the embedding coordinates described in
section 4.2.1.2, which are then used to reconstruct the AP profile v.
Depending on the chosen embedding, v dr = z pca or v dr = z lle .In
the following, the parameter values θ are standardized using z-
scores when used as input for the regression model. To simplify
notation, we will not use a new symbol for the standardized pa-
rameter values.
The regression problem is split into two parts. In the first step,
the relationship between CRN model parameters and handcrafted
features of the action potential profile are estimated. These phe-
nomenological features are then used as additional input for the
second regression step, estimating the embedding coordinates of
the AP, to increase the accuracy of the overall prediction.
The additional features used in the regression are peak voltage
(V peak ), resting membrane potential (V rest ), and action potential