Single-Trial Analysis of Bioelectromagnetic Signals: The Quest for Hidden Information - Modeling in Computational Biology and Biomedicine - page 233

trials, but with a variable latency between stimulation and response. The noisy 1-D
manifold, which can be revealed by PCA, can then be parameterized by the latency
of the response. In order to capture this “degree of freedom” manifold-learning
techniques can be applied: by providing low-dimensional representation of the data,
they offer an efficient way of exploiting the structure present in a dataset.
The rest of this section is divided in two parts: first, Sect. 7.2.1 explores the low-
dimensional representations afforded through the PCA. Section 7.2.2 next shows
how Laplacian Eigenmaps can be used to reorder a dataset according to its principal
modes of variation. This section relates research results originally presented in [ 10 ].

7.2.1 Principal Components Analysis of a Multitrial Dataset

Let, ,

which are drawn with a probability distribution p which has support on a low-
dimensional smooth sub-manifold

( x i ) i =1 ,...,K be K trials, considered as elements of a metric space ( X ,d X )

.
The goal of PCA is to represent the data in a new referential, via a rotation that
diagonalizes the empirical covariance matrix. Representing the data in the leading
PCA directions is a valuable tool in exploratory analysis as it makes visible the
structure present in the data. Chapters 3 and 5 of this topic apply PCA to other
biomedical data.
To illustrate this, a dataset containing 1,000 time series was simulated, each with
512 time samples. The time series mainly differ by their latency. Figure 7.3 presents
this dataset: in (a), nine out of 1,000 time series are plotted; (b) displays a raster plot,
i.e., an image whose 1,000 lines consist of the time series, in color-scale; (c) and (d)
show 2D and 3D PCA projections of the dataset , in which each dot represents one
of the time series. Fig. 7.3 d clearly displays how the dataset is organized along a 1D
structure.
It is possible to exploit this 1D structure in view of reordering the time-series
according to their latency. This reordering is equivalent to finding a parameterization
of the curve in Fig. 7.3 d. This challenge is addressed in the following sections by a
nonlinear embedding method called the Graph Laplacian.

M of X

7.2.2 Nonlinear Embedding via the Graph Laplacian

Given a set of points ( x i ) i , each representing a time series, the aim is to recover

the structure of M via an embedding function f that maps the ( x i ) i into a

n . The embedding function f provides a
low-dimensional representation of the dataset and also a parameterization of the
manifold. When

low-dimensional Euclidian space R

has a 1D structure, the first coordinate of f can be used to
reorder the points (i.e., in our context, the time series), provided that f satisfies a
“regularity” constraint, sometimes referred to as a minimal distortion property: if

M