Correlation dimension

[qubit0]

Is there a way to compute the correlation dimension for a given embedding dimension in this framework?

Let's say that if I were to take x time series of the Lorenz system and compute the correlation dimension for each embedding dimension then I would get a plot where the correlation dimension would increase with the embedding dimension for a while but after a while correlation dimension would be independent of embedding dimension.

If I were to do the same thing for the noise then the correlation dimension would linearly increase with the embedding dimension for all along.

[ulupo]

Hi! I'm not too familiar with this definition. Currently, we select optimal Takens parameters as described here: https://giotto-ai.github.io/gtda-docs/0.5.1/modules/generated/time_series/embedding/gtda.time_series.takens_embedding_optimal_parameters.html#gtda.time_series.takens_embedding_optimal_parameters, corresponding to the code in

giotto-tda/gtda/time_series/embedding.py

Lines 98 to 114 in 7b3e47d

	mutual_information_list = Parallel(n_jobs=n_jobs)(
	delayed(_mutual_information)(X, time_delay, n_bins=100)
	for time_delay in range(1, max_time_delay + 1))
	time_delay = \
	mutual_information_list.index(min(mutual_information_list)) + 1
	n_false_nbhrs_list = Parallel(n_jobs=n_jobs)(
	delayed(_false_nearest_neighbors)(X, time_delay, dim, stride=stride)
	for dim in range(1, max_dimension + 3))
	variation_list = [np.abs(n_false_nbhrs_list[dim - 1]
	- 2 * n_false_nbhrs_list[dim] +
	n_false_nbhrs_list[dim + 1])
	/ (n_false_nbhrs_list[dim] + 1) / dim
	for dim in range(2, max_dimension + 1)]
	dimension = variation_list.index(min(variation_list)) + 2
	return time_delay, dimension

[qubit0]

One such example is here: https://github.com/notsebastiano/GP_algorithm

[ulupo]

Thank you! That is interesting. Currently we don't have the bandwidth to study this and implement it in detail, but we'd be happy if you'd like to contribute this feature. We can keep discussing here how this would go about.