Convergent Cross Mapping

[MKLau]

Based on Taken's Theorem, a series of lagged vectors taken from a time series can be used to reconstruct the manifold that created the time series.

This can be seen in the following plots using the Lorenz "Butterfly" attractor.

Top figure = Lorenz manifold Bottom figure = reconstructed (aka. shadow) manifold using time lags (tau = 2) using three time lags

[MKLau]

Random vector

[amellison17]

Matt - this is really cool. Am I right that we can apply it to the simulated pitcher plant data and to Amanda's O2-time-series data?

Aaron

From: MKLau [mailto:notifications@github.com] Sent: Monday, September 01, 2014 4:25 PM To: HFpostdoc/unstable_states Cc: Ellison, Aaron Subject: Re: [unstable_states] Convergent Cross Mapping - Lorenz Attractor (#55)

Assigned #55https://github.com/HFpostdoc/unstable_states/issues/55 to @amellison17https://github.com/amellison17.

— Reply to this email directly or view it on GitHubhttps://github.com/HFpostdoc/unstable_states/issues/55#event-159758750.

[MKLau]

Great!

Yes, any time series with an underlying manifold.

I still need to program a nearest neighbors algorithm to do the cross mapping, but currently we can use the time lag function I wrote to generate "shadow manifold" plots.

These are essentially phase plots but can be interpreted as having properties of the true manifold based on Taken's Theorem.

Matt

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On Mon, Sep 01, 2014 at 7:59 PM, Aaron Ellison notifications@github.com<mailto:notifications@github.com> wrote:

Matt - this is really cool. Am I right that we can apply it to the simulated pitcher plant data and to Amanda's O2-time-series data?

Aaron

From: MKLau [mailto:notifications@github.com] Sent: Monday, September 01, 2014 4:25 PM To: HFpostdoc/unstable_states Cc: Ellison, Aaron Subject: Re: [unstable_states] Convergent Cross Mapping - Lorenz Attractor (#55)

Assigned #55https://github.com/HFpostdoc/unstable_states/issues/55 to @amellison17https://github.com/amellison17.

— Reply to this email directly or view it on GitHubhttps://github.com/HFpostdoc/unstable_states/issues/55#event-159758750.

— Reply to this email directly or view it on GitHubhttps://github.com/HFpostdoc/unstable_states/issues/55#issuecomment-54097862.

[amellison17]

wicked cool! Aaron

From: MKLau [mailto:notifications@github.com] Sent: Monday, September 01, 2014 8:11 PM To: HFpostdoc/unstable_states Cc: Ellison, Aaron Subject: Re: [unstable_states] Convergent Cross Mapping (#55)

Great!

Yes, any time series with an underlying manifold.

I still need to program a nearest neighbors algorithm to do the cross mapping, but currently we can use the time lag function I wrote to generate "shadow manifold" plots.

These are essentially phase plots but can be interpreted as having properties of the true manifold based on Taken's Theorem.

Matt

Sent using CloudMagichttps://cloudmagic.com/k/d/mailapp?ct=pa&cv=5.1.3&pv=4.1.2

On Mon, Sep 01, 2014 at 7:59 PM, Aaron Ellison notifications@github.com<mailto:notifications@github.com> wrote:

Matt - this is really cool. Am I right that we can apply it to the simulated pitcher plant data and to Amanda's O2-time-series data?

Aaron

From: MKLau [mailto:notifications@github.com] Sent: Monday, September 01, 2014 4:25 PM To: HFpostdoc/unstable_states Cc: Ellison, Aaron Subject: Re: [unstable_states] Convergent Cross Mapping - Lorenz Attractor (#55)

Assigned #55https://github.com/HFpostdoc/unstable_states/issues/55 to @amellison17https://github.com/amellison17.

— Reply to this email directly or view it on GitHubhttps://github.com/HFpostdoc/unstable_states/issues/55#event-159758750.

— Reply to this email directly or view it on GitHubhttps://github.com/HFpostdoc/unstable_states/issues/55#issuecomment-54097862.

— Reply to this email directly or view it on GitHubhttps://github.com/HFpostdoc/unstable_states/issues/55#issuecomment-54098213.

[MKLau]

Here is the method applied to the Sensitivity O2 data:

E = 3 and tau = 25

Run 1

Run 51

Run 91

[MKLau]

Here is time lags 1 and 2 for Run 1 colored by time:

Here is time lags 1 and 2 for Run 51 colored by time:

Here is time lags 1 and 2 for Run 91 colored by time: