Proper Orthogonal Decomposition

Principle

The package contains a processing algorithm for decomposing an input using a limited number of predefined signals from a bigger set.

Decomposition is performed by iterating projections onto the straight lines generated by the reference signals.

The proposed algorithm takes a vector space approach. A signal, or more precisely its sequence of N temporal samples, is mapped to a point P in a vector space of dimension N. A value taken by a signal P at sample time t_i becomes the coordinate of P along the axis t_i.

The set of reference signals represents a library the algorithm micks from in order to synthetize or approximate any input. The reference points form a cloud in the space described above. A linear combination of appropriately selected reference points will approximate the target signal S.

Use case: asset replication

In order to illustrate how the algorithm works, let's assume we replicate a fund using two assets stock 1 and stock 2, with only 3 data points for past performances.

The chart below shows the first and final projections in the case of a 3D space: 3 daily performances for a 2 stocks-universe, plus a target fund to replicate.

The first projection is performed on the second stock (yellow), since it better matches the fund performances than those of the first stock.

The interactive version of the chart above is available at math3d.org.

From a time perspective:

Typical sizes for asset replication: 1000 dimensions (~5 years of daily data), 200 candidate assets for use in replication, target portfolio with 5 assets.

Documentation

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Publication

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