Time Series and Partitions

My project took a new direction after my presentation. I had the idea to implement partitions along the time series points instead of using uniform partitions. The reason behind this new approach is that the finite representation given by it would be true to the time series. This approach would also improve the quality of the finite representation, bringing it closer to an outer approximation. Since every time series point is on a partition, every interval and their image gets picked up by the finite representation.

However, partitions along time series points presents three major problems.

1. partitions that are within two points of closet value along the domain or range that have more than one partition within their values (this will lose edges)

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Solution: Allow only one partition between two points of closest value.

 

2. the next problem occurs when a horizontal partition is between the maximum of a function in an interval and the maximum range-value, time series point in the same interval.

3. lastly, another problem occurs when a horizontal partition is between the minimum of a function in an interval and the minimum range-value, time series point in the same interval.

Solution for 2 and 3: create an envelope for an interval and use the max. and min. on the envelope to determine which partitions to remove.

Once these solutions are implemented, a finite representation of a one-dimensional time series should match exactly to its underlying outer approximation. This would then allow me the apply Conley Index Theory without error from incomplete information, which would occur if uniform partitions were used.