Dimensionality Reduction

The curse of dimensionality.

Dimensionality Reduction is reducing the “dimension” of the data set. In time series analysis, the dimension is the number of data points.

PAA

Reduce the number of points by taking the average value for each segment. Given $X$ points, we want $N$ segments → Reduced PAA → Reconstructed PAA.

Length of each segment = $\frac{X}{N}$.

Example 1:

Limitations

APCA

An adaptive PAA; it is called adapative, because it allows segments to be of variable lengths.

Haar Discrete Wavelet Transform (DWT)

• Find DWT Coefficients
• Truncate DWT Coefficients
• Reconstruct DWT
• Truncate if padded
• Replace approximates with actual averages

If mismatch in desired segments and achieved segments:

• will have to merge segments
  • we merge adjacent segments with least deviation in value
  • merging is averaging
  • repeat this process until desired segments = achieved segments
    Example: $C=\{4.5,0.5,1,-1,1,1,-1,1\}$

$4.5+0.5=5$
$4.5-0.5=4$

$5+1=6$
$5-1=4$

$4+-1=3$
$4--1=5$

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$6+1=7$
$6-1=5$

$f$