Constructing local discriminative features for signal classification

Transcription

1 Constructing local discriminative features for signal classification

2 Local features for signal classification Outline Motivations Problem formulation Lifting scheme Local features Conclusions Toy example

3 Local features for signal classification 3 Motivations I Measurement (sampling) Data preprocesing eg. (feature extraction) Classifier

4 Local features for signal classification i Widely used feature extractors PCA ICA Wavelets Many others Motivations II Main problem is that features extracted by these methods are not always proper for classification.

5 Local features for signal classification 5 Problem formulation Find features that have following properties Locality - each single feature depends on a part of the original signal Discriminative - feature creation is supervised Linearity (?) - discriminative biorthogonal bases

6 Local features for signal classification The Lifting Scheme Method to construct wavelets proposed by Wim Sweldens. Very easy and natural way of constructing wavelets. No Fourier transform is needed for understanding, all is done in spatial (time) domain. Very simple to implement. Trivially invertible. No problem with bounded domains such as intervals and signals of length not being a power of two. Efficient (natural parallelism).

7 Local features for signal classification 7 The Lifting Scheme - update-first variation x split x odd x even update c predict d

8 Local features for signal classification 8 The Lifting Scheme - operators Let x R T for T = t for some t Z SPLIT - splits signal x into even and odd subsignals k = 1,,...,T/ x e (k) = x(k) x o (k) = x(k 1) UPDATE - creates coarse approximation c of the signal x c(k) = x e(k) + x o (k), k = 1,,...,T/ PREDICT - creates wavelets coefficients d d(k) = PREDICT k (x e (k), c), k = 1,,...,T/

9 Local features for signal classification 9 Linear predictors I w k x(k) d(k) c(k) w k 1 w k c(k+1) Coefficients d(k) are given by the following formulas (just example) d(k) = x e (k) (w 1 c(k) + w c(k + 1)) not regularised (1) d(k) = w x e (k) (w 1 c(k) + w c(k + 1)) regularised ()

10 Local features for signal classification 1 Linear predictors II The problem is to find coefficients d(k) (given by (1) or ()) such that they will have different values for examples from different classes. We propose two approaches Not regularised Find such vector w that coefficients d(k) given by (1) discriminate two classes well and such that d(k) is zero for signals being locally polynomial up to some degree p. Regularised Find vector ( w o w by () discriminate two classes well. ) such that coefficients d(k) given To find such vectors we used modified Support Vector Machines.

11 Local features for signal classification 11 Conclusions and future research Efficient Natural parallelism. Locality Each predictor depends only on small sample of the whole signal. This can be used for finding interesting parts of analysed signals. Nonlinearity If there is no need for linearity we can use nonlinear SVM for calculating features by replacing inner products by kernel functions. Multidimensional signals Method can be quite easily extended for multidimensional signals such as images. Multiple Kernel Learning Method of optimising all coefficients at once by using convex combination of local kernels.

12 Local features for signal classification 1 Toy example - well known Waveform data Figure 1: Training data (two classes out of three)

13 Local features for signal classification Figure : Coarse approximations and classification error for three decomposition levels(error for original dataset equals.1)

14 Local features for signal classification 1 Analysis Synthesis Base vector Time (samples) Base Vector Time (samples) Figure 3: Supports of analysis and synthesis discriminative basis (regularised)

15 Local features for signal classification Figure : Best synthesis (left) and analysis (right) base vectors for each decomposition level