Single-Trial Analysis of Bioelectromagnetic Signals: The Quest for Hidden Information - Modeling in Computational Biology and Biomedicine - page 234

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Fig. 7.3 PCA analysis of a set of 1,000 jittered synthetic time series of 512 time samples. ( a )Nine
time series out of 1,000. ( b ) Original raster plot. ( c ) 2D PCA projection. ( d ) 3D PCA projection

two points x and z are close according to the manifold M ,then f ( x ) and f ( z ) must

n . To express this regularity constraint in mathematical terms,

also be close in R

notice that for n =1 , a Taylor expansion provides the following inequality [ 2 ]

| f ( z ) − f ( x ) |≤ d M ( x, z ) ||∇ f ( x ) || + o ( d M ( x, z )) ,

where ∇ f stands for the gradient of f and d M is the geodesic distance on the

manifold between points x and z . The notation g ( z )= o ( d M ( x, z )) means that

)
d M ( x,z )

g ( z

as z tends towards x .
In order to obtain an embedding that satisfies the “regularity” constraint,
Laplacian-based methods control the smoothness of f globally by minimizing

0

tends to

M ||∇

f ( x ) || 2 p ( x ) dx , provided that

M ||

) || 2 p f ( x ( x ) dx =1 .

(7.3)

The latter condition removes the scaling indeterminacy for the function f .