Predicting system trajectories toward critical transitions

• US 10,460,008 B2
• Filed: 03/13/2014
• Issued: 10/29/2019
• Est. Priority Date: 03/14/2013
• Status: Active Grant

• Current Assignee: HRL Laboratories LLC (The Boeing Co.)
• Original Assignee: HRL Laboratories LLC (The Boeing Co.)
• Inventors: Ni, Kang-Yu, Lu, Tsai-Ching
• Primary Examiner(s): Ishizuka, Yoshihisa
• Application Number:
  US14/209,314
• Publication Number:
  US 20170308505A1
• Time in Patent Office: 2,056 Days
• Field of Search: 702183
• CPC Class Codes: G06F 17/10 Complex mathematical operations function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544
• CPC Class Codes: G06F 17/14 Fourier, Walsh or analogous domain transformations , e.g. Laplace, Hilbert, Karhunen-Loeve, transforms for correlation function computation G06F17/156; spectrum analysers G01R23/16

Abstract

Described is a system for predicting system trajectories toward critical transitions. The system transforms a set of multivariate time series of observables of a complex system into a set of symbolic multivariate time series. Then pair-wise time series of a transfer entropy (TE) measure are determined, wherein the TE measure quantifies the amount of information transfer from a source to a destination in the complex system. An associative transfer entropy (ATE) measure is determined which decomposes the pair-wise time series of TE to associative states of asymmetric, directional information flows, wherein the ATE measure is comprised of an ATE+ influence class and a ATE− influence class. The system estimates ATE+, TE, and ATE− trajectories over time, and at least one of the ATE+, TE, and ATE− trajectories is used to predict a critical transition in the complex system.

Claims

1. A system for predicting critical transitions, the system comprising:

an electronic circuit having a plurality of components, each component having a time series of voltage measurements; and

one or more processors and a non-transitory computer-readable medium having executable instructions encoded thereon such that when executed, the one or more processors perform operations of;

transforming the time series of voltage measurements from the plurality of components of the electronic circuit into a set of symbolic multivariate time series;

determining a transfer entropy (TE) measure between two time series in the set of symbolic multivariate time series over a sliding time window;

determining an associative transfer entropy (ATE) measure, wherein the ATE measure is comprised of an ATE+ positive influence class and an ATE−

negative influence class, and wherein the ATE measure captures information transfer for a state association D_K between a variable x_j and a variable x_i, for i≠

j according to the following;

2. The system as set forth in claim 1, wherein the one or more processors further perform an operation of estimating the TE trajectory by the natural logarithm (ln) with an unknown coefficient a and a constant c according to the following:

g(t)=a ln(t)+c,

where g(t) represents the estimated TE trajectory, and where t represents time.

3. The system as set forth in claim 2, wherein the one or more processors further perform an operation of estimating the unknown coefficient a with various discrete time steps by taking the derivative of g(t)=a ln(t)+c to obtain

4. The system as set forth in claim 3, wherein the one or more processors further perform an operation of determining the unknown coefficient a according to the following:

for a fixed timestep Δ

t in a window [T_start, T_end], determining the following matrix equation;

5. The system as set forth in claim 4, wherein the one or more processors further perform an operation of determining a predicted value for a future time T_end+t_d according to the following:

F<sub>Δ

t</sub>(t_d)=a<sub>Δ

t</sub> ln(T_end+t_d)+c<sub>λ

t</sub>,

where F<sub>Δ

t</sub> is a predicted value for a future time, and where t_d is a desired time length for prediction.

6. The system as set forth in claim 1, wherein the electronic circuit is a non-foster circuit.

7. A computer-implemented method for predicting critical transitions in an electronic circuit having a plurality of components, comprising an act of:

causing one or more processors to execute instructions stored on a non-transitory memory such that upon execution, the one or more processors perform operations of;

transforming time series of voltage measurements from the plurality of components of the electronic circuit into a set of symbolic multivariate time series;

determining a transfer entropy (TE) measure between two time series in the set of symbolic multivariate time series over a sliding time window;

determining an associative transfer entropy (ATE) measure, wherein the ATE measure is comprised of an ATE+ positive influence class and an ATE−

negative influence class, and wherein the ATE measure captures information transfer for a state association D_K between a variable x_j and a variable x_i, for i≠

j according to the following;

8. The method as set forth in claim 7, wherein the one or more processors further perform an operation of estimating the TE trajectory by the natural logarithm (ln) with an unknown coefficient a and a constant c according to the following:

g(t)=a ln(t)+c,

where g(t) represents the estimated TE trajectory, and where t represents time.

9. The method as set forth in claim 8, wherein the data processor further performs an operation of estimating the unknown coefficient a with various discrete time steps by taking the derivative of g(t)=a ln(t)+c to obtain

10. The method as set forth in claim 9, wherein the data processor further performs an operation of determining the unknown coefficient a according to the following:

for a fixed timestep Δ

t in a window [T_start, T_end], determining the following matrix equation;

11. The method as set forth in claim 10, wherein the data processor further performs an operation of determining a predicted value for a future time T_end+t_d according to the following:

F<sub>Δ

t</sub>(t_d)=a<sub>Δ

t</sub> ln(T_end+t_d)+c<sub>Δ

t</sub>,

where F<sub>Δ

t</sub> is a predicted value for a future time, and where t_d is a desired time length for prediction.

12. A computer program product for predicting critical transitions in an electronic circuit having a plurality of components, the computer program product comprising computer-readable instructions stored on a non-transitory computer-readable medium that are executable by a computer having one or more processors for causing the one or more processors to perform operations of:

transforming time series of voltage measurements from the plurality of components of the electronic circuit into a set of symbolic multivariate time series;

determining a transfer entropy (TE) measure between two time series in the set of symbolic multivariate time series over a sliding time window;

determining an associative transfer entropy (ATE) measure, wherein the ATE measure is comprised of an ATE+ positive influence class and an ATE−

negative influence class, and wherein the ATE measure captures information transfer for a state association D_K between a variable x_j and a variable x_i, for i≠

j according to the following;

13. The computer program product as set forth in claim 12, further comprising instructions for causing the processor to perform an operation of estimating the TE trajectory by the natural logarithm (ln) with an unknown coefficient a and a constant c according to the following:

g(t)=a ln(t)+c,

where g(t) represents the estimated TE trajectory, and where t represents time.

14. The computer program product as set forth in claim 13, further comprising instructions for causing the processor to perform an operation of estimating the unknown coefficient a with various discrete time steps by taking the derivative of g(t)=a ln(t)+c to obtain

15. The computer program product as set forth in claim 14, further comprising instructions for causing the processor to perform an operation of determining the unknown coefficient a according to the following:

for a fixed timestep Δ

t in a window [T_start, T_end], determining the following matrix equation;

16. The computer program product as set forth in claim 15, further comprising instructions for causing the processor to perform an operation of determining a predicted value for a future time T_end+t_d according to the following:

F<sub>Δ

t</sub>(t_d)=a<sub>Δ

t</sub> ln(T_end+t_d)+c<sub>Δ

t</sub>,

where F<sub>Δ

t</sub> is a predicted value for a future time, and where t_d is a desired time length for prediction.