67 lines
1.7 KiB
Plaintext
67 lines
1.7 KiB
Plaintext
three methods for approaching ICA with consideration for time series:
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1) Temporal ICA with smoothing
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Consider spatial mixing model X = A*S
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X = N x T
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A = N x N
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S = N x T
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the distribution of each x is not necessarily independent of the distribution of any other other x
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the distribution of each s is independent of the distribution of each other s
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spatial dimensions correspond to multivariate data
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goal is to unmix multivariate data into new independent distributions
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there is some temporal structure in the data
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smooth the data X over time
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2) ICA where each observation is a T-dimensional data point
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X = A * S
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X = T x N
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A = T x T
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S = T x N
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dot product of two data points does not produce a zero
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if we can find some mapping of the data points onto an independent basis
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we are unaware of how many independent components truly underly the data
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given N observations, it is possible to recover no more than N components
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we often care to recover less components
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the process we actually consider should be some mixing process with noise
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assume N observations
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each observation is some superposition of temporal patterns
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these temporal patterns occur in each observation independently
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suppose we have recovered N' principle components that account for most of the variance of our data
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thus, we have e set
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We can then consider rotations of these principal components
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PCA provides a transformation on the loadings of the components that maximizes the amount of variance accounted for by each component
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ICA provides a transformation on the loadings of the components that maximizes the statistical independence in the distributions of the loadings of the components
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1st dimension varies over basis functions
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