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Merge pull request #57 from JuliaDynamics/PR-Hegger_Cao_dimension
hegger dimension and automated Cao's method
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| export optimal_traditional_de | ||
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|
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| """ | ||
| optimal_traditional_de(s, method = "afnn", dmethod = "mi_min"; kwargs...) → 𝒟, τ, E | ||
| Produce an optimal delay embedding `𝒟` of the given timeseries `s` by | ||
| using the traditional approach of first finding an optimal (and constant) delay | ||
| time using [`estimate_delay`](@ref) with the given `dmethod`, and then an optimal | ||
| embedding dimension, by calculating an appropriate statistic for each dimension `d ∈ 1:dmax`. | ||
| Return the embedding `𝒟`, the optimal delay time `τ` | ||
| (the optimal embedding dimension `d` is just `size(𝒟, 2)`) and the actual | ||
| statistic `E` used to estimate optimal `d`. | ||
| Notice that `E` is a function of the embedding dimension, which ranges from 1 to `dmax`. | ||
| For calculating `E` to estimate the dimension we use the given `method` which can be: | ||
| * `"afnn"` (default) is Cao's "Averaged False Nearest Neighbors" method[^Cao1997], | ||
| which gives a ratio of distances between nearest neighbors. | ||
| * `"ifnn"` is the "Improved False Nearest Neighbors" from Hegger & Kantz[^Hegger1999], | ||
| which gives the fraction of false nearest neighbors. | ||
| * `"fnn"` is Kennel's "False Nearest Neighbors" method[^Kennel1992], which gives | ||
| the number of points that cease to be "nearest neighbors" when the dimension | ||
| increases. | ||
| * `"f1nn"` is Krakovská's "False First Nearest Neighbors" method[^Krakovská2015], | ||
| which gives the ratio of pairs of points that cease to be "nearest neighbors" | ||
| when the dimension increases. | ||
| For more details, see individual methods: [`delay_afnn`](@ref), [`delay_ifnn`](@ref), | ||
| [`delay_fnn`](@ref), [`delay_f1nn`](@ref). | ||
| The special keywords `` denote for which possible embedding | ||
| dimensions should the statistics be computed for. | ||
| !!! warn "Careful in automated methods" | ||
| While this method is automated if you want to be **really sure** of the results, | ||
| you should directly calculate the statistic and plot its values versus the | ||
| dimensions. | ||
| ## Keywords | ||
| The keywords | ||
| ``` | ||
| τs = 1:100, dmax = 10 | ||
| ``` | ||
| denote which delay times and embedding dimensions `ds ∈ 1:dmax` to consider when calculating | ||
| optimal embedding. All remaining keywords are propagated to the low level functions: | ||
| ``` | ||
| fnn_thres::Real = 0.05, slope_thres::Real= 0.2, w::Int=1, | ||
| rtol=10.0, atol=2.0, τs = 1:100, metric = Euclidean(), r::Real=2.0 | ||
| ``` | ||
| ## Description | ||
| We estimate the optimal embedding dimension based on the given delay time gained | ||
| from `dmethod` as follows: For Cao's method the optimal dimension is reached, | ||
| when the slope of the `E₁`-statistic (output from `"afnn"`) falls below the | ||
| threshold `slope_thres` (Default is .05) and the according stochastic test turns | ||
| out to be false, i.e. when the `E₂`-statistic is not "equal" to 1 for all en- | ||
| countered dimensions. We treat `E₂`-values as equal to 1, when `1-E₂ ≤ fnn_thres`. | ||
| For all the other methods we return the optimal embedding dimension | ||
| when the corresponding FNN-statistic (output from `"ifnn"`, `"fnn"` or `"f1nn"`) | ||
| falls below the fnn-threshold `fnn_thres` (Default is .05) AND the slope of the | ||
| statistic falls below the threshold `slope_thres`. Note that with noise | ||
| contaminated time series, one might need to adjust `fnn_thres` according to the | ||
| noise level. | ||
| See also the file `test/compare_different_dimension_estimations.jl` for a comparison. | ||
| [^Cao1997]: Liangyue Cao, [Physica D, pp. 43-50 (1997)](https://www.sciencedirect.com/science/article/pii/S0167278997001188?via%3Dihub) | ||
| [^Kennel1992]: M. Kennel *et al.*, [Phys. Review A **45**(6), (1992)](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.45.3403). | ||
| [^Krakovská2015]: Anna Krakovská *et al.*, [J. Complex Sys. 932750 (2015)](https://doi.org/10.1155/2015/932750) | ||
| [^Hegger1999]: Hegger & Kantz, [Improved false nearest neighbor method to detect determinism in time series data. Physical Review E 60, 4970](https://doi.org/10.1103/PhysRevE.60.4970). | ||
| """ | ||
| function optimal_traditional_de(s::AbstractVector, dimensionmethod::String = "afnn", | ||
| delaymethod::String= "mi_min"; | ||
| fnn_thres::Real = 0.05, slope_thres::Real = .05, dmax::Int = 10, w::Int=1, | ||
| rtol=10.0, atol=2.0, τs = 1:100, metric = Euclidean(), r::Real=2.0, | ||
| ) | ||
|
|
||
| # TODO: This function needs to be reworked to use dimensions 2:dmax. | ||
| # starting with dimension 1:dmax is pointless, because 1 can **never** be | ||
| # a proper embedding dimension!!! | ||
|
|
||
| @assert dimensionmethod ∈ ("afnn", "fnn", "ifnn", "f1nn") | ||
| τ = estimate_delay(s, delaymethod, τs) | ||
| ds = 1:dmax | ||
| γs = ds .- 1 # TODO: This must be updated to dimension in 2.0 | ||
|
|
||
| if dimensionmethod=="afnn" | ||
| dimension_statistic = delay_afnn(s, τ, ds, metric) | ||
| Y, τ = find_cao_optimal(s, τ, dimension_statistic, slope_thres) | ||
| E2 = stochastic_indicator(s, τ, ds) | ||
| flag = is_stochastic(E2, fnn_thres) | ||
| flag && println("Stochastic signal, valid embedding NOT achieved ⨉.") | ||
| elseif dimensionmethod=="fnn" | ||
| dimension_statistic = delay_fnn(s, τ, ds; rtol, atol) | ||
| Y, τ = find_fnn_optimal(s, τ, dimension_statistic, fnn_thres, slope_thres) | ||
| elseif dimensionmethod=="ifnn" | ||
| dimension_statistic = delay_ifnn(s, τ, ds; r, w, metric) | ||
| Y, τ = find_fnn_optimal(s, τ, dimension_statistic, fnn_thres, slope_thres) | ||
| elseif dimensionmethod=="f1nn" | ||
| dimension_statistic = delay_f1nn(s, τ, ds, metric) | ||
| Y, τ = find_fnn_optimal(s, τ, dimension_statistic, fnn_thres, slope_thres) | ||
| end | ||
| return Y, τ, dimension_statistic | ||
| end | ||
|
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||
|
|
||
| """ | ||
| Helper function for selecting the appropriate embedding dimension from the | ||
| statistic when using Kennel's, Hegger's or Krakovskas's method. | ||
| """ | ||
| function find_fnn_optimal(s::Vector{T}, τ::Int, rat::Vector, fnn_thres::Real, | ||
| slope_thres::Real) where {T} | ||
| @assert length(rat) > 1 | ||
| y = abs.(diff(rat)) | ||
| flag = false | ||
| m = 0 | ||
| for i = 2:length(rat) | ||
| if rat[i] ≤ fnn_thres && y[i-1] ≤ slope_thres | ||
| m = i | ||
| break | ||
| elseif rat[i] > rat[i-1] | ||
| flag = true | ||
| m = i-1 | ||
| break | ||
| end | ||
| end | ||
| if m == length(rat) || m == 0 | ||
| println("Sufficiently small FNNs NOT reached. "* | ||
| "Valid embedding NOT achieved ⨉.") | ||
| elseif flag | ||
| println("Algorithm stopped due to increasing FNNs. "* | ||
| "Double-check the FNN-statistic.") | ||
| else | ||
| println("Algorithm stopped due to sufficiently small FNNs. "* | ||
| "Valid embedding achieved ✓.") | ||
| end | ||
| Y = embed(s, max(1, m), τ) # you can embed in 1 dimension in latest version | ||
| return Y, τ | ||
| end | ||
|
|
||
|
|
||
| """ | ||
| Helper function for selecting the appropriate embedding dimension from the | ||
| statistic when using Cao's method. | ||
| """ | ||
| function find_cao_optimal(s::Vector{T}, τ::Int, rat::Vector, thres::Real) where {T} | ||
| m = 0 | ||
| y = abs.(diff(rat)) | ||
| for i = 1:length(rat)-1 | ||
| if y[i] ≤ thres && rat[i] > 0.5 | ||
| m = i | ||
| break | ||
| end | ||
| end | ||
| if m == 0 | ||
| println("NO convergence of E₁-statistic. "* | ||
| "Valid embedding NOT achieved ⨉.") | ||
| else | ||
| Y = m > 1 ? embed(s, m, τ) : s | ||
| println("Algorithm stopped due to convergence of E₁-statistic. "* | ||
| "Valid embedding achieved ✓.") | ||
| end | ||
| Y = embed(s, max(1, m), τ) # you can embed in 1 dimension in latest version | ||
| return Y, τ | ||
| end | ||
|
|
||
| """ | ||
| Helper function for Cao's method, whether to decide if the input signal is | ||
| stochastic or not. | ||
| """ | ||
| function is_stochastic(rat::Vector{T}, thres::Real) where {T} | ||
| cnt = 0 | ||
| for i = 1:length(rat) | ||
| if abs(1-rat[i]) ≥ thres | ||
| cnt += 1 | ||
| end | ||
| end | ||
| if cnt == 0 | ||
| flag = true | ||
| else | ||
| flag = false | ||
| end | ||
| return flag | ||
| end |
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