PREDICTING SYSTEM TRAJECTORIES TOWARD CRITICAL TRANSITIONS

• US 20170308505A1
• Filed: 03/13/2014
• Published: 10/26/2017
• 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
• Granted Patent: US 10,460,008 B2
• Time in Patent Office: Days
• 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 system trajectories toward critical transitions, the system comprising:

one or more processors installed as an embedded decision support module in a vehicle entity 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 a set of multivariate time series of observables of a complex system into a set of symbolic multivariate time series, wherein the complex system is a heterogeneously networked dynamical system of vehicle entities;

determining a transfer entropy (TE) measure between two time series, wherein the TE measure quantifies an amount of information transfer from a source vehicle entity to a destination vehicle entity in the complex system;

determining an associative transfer entropy (ATE) measure by decomposing the TE measure to associative states of asymmetric, directional information flows, the associative states being an ATE+ positive influence class and an ATE−

negative influence class,

wherein an influence from the source vehicle entity on the destination vehicle entity is positively correlated in the ATE+ positive influence class such that if a value of the source vehicle entity is increasing, a value of the destination vehicle entity is increasing, and an influence from the source vehicle entity on the destination vehicle entity is negatively correlated in the ATE−

negative influence class such that if a value of the source vehicle entity is increasing, a value of the destination vehicle entity is decreasing;

estimating ATE+, TE, and ATE−

trajectories over time; and

predicting a critical transition in the complex system using at least one of the ATE+, TE, and ATE−

trajectories for analysis of directional information influences among vehicle entities of the complex system to avoid a catastrophic failure of the complex system.

2. The system as set forth in claim 1, wherein vehicle electrical systems 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>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.

6. A computer-implemented method for predicting system trajectories toward critical transitions, comprising an act of:

causing one or more processors installed as an embedded decision support module in a vehicle entity to execute instructions stored on a non-transitory memory such that upon execution, the one or more processors perform operations of;

transforming a set of multivariate time series of observables of a complex system into a set of symbolic multivariate time series, wherein the complex system is a heterogeneously networked dynamical system of vehicle entities;

determining a transfer entropy (TE) measure between two time series, wherein the TE measure quantifies an amount of information transfer from a source vehicle entity to a destination vehicle entity in the complex system;

determining an associative transfer entropy (ATE) measure by decomposing the TE measure to associative states of asymmetric, directional information flows, the associative states being an ATE+ positive influence class and an ATE−

negative influence class,

wherein an influence from the source vehicle entity on the destination vehicle entity is positively correlated in the ATE+ positive influence class such that if a value of the source vehicle entity is increasing, a value of the destination vehicle entity is increasing, and an influence from the source vehicle entity on the destination vehicle entity is negatively correlated in the ATE−

negative influence class such that if a value of the source vehicle entity is increasing, a value of the destination vehicle entity is decreasing;

estimating ATE+, TE, and ATE−

trajectories over time; and

predicting a critical transition in the complex system using at least one of the ATE+, TE, and ATE−

trajectories for analysis of directional information influences among vehicle entities of the complex system to avoid a catastrophic failure of the complex system.

7. The method as set forth in claim 6, 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.

8. The method as set forth in claim 7, 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

9. The method as set forth in claim 8, 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;

10. The method as set forth in claim 9, 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.

11. A computer program product for predicting system trajectories toward critical transitions, 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 installed as an embedded support module in a vehicle entity for causing the one or more processors to perform operations of:

transforming a set of multivariate time series of observables of a complex system into a set of symbolic multivariate time series, wherein the complex system is a heterogeneously networked dynamical system of vehicle entities;

determining a transfer entropy (TE) measure between two time series, wherein the TE measure quantifies an amount of information transfer from a source vehicle entity to a destination vehicle entity in the complex system;

determining an associative transfer entropy (ATE) measure by decomposing the TE measure to associative states of asymmetric, directional information flows, the associative states being an ATE+ positive influence class and an ATE−

negative influence class,

wherein an influence from the source vehicle entity on the destination vehicle entity is positively correlated in the ATE+ positive influence class such that if a value of the source vehicle entity is increasing, a value of the destination vehicle entity is increasing, and an influence from the source vehicle entity on the destination vehicle entity is negatively correlated in the ATE−

negative influence class such that if a value of the source vehicle entity is increasing, a value of the destination vehicle entity is decreasing;

estimating ATE+, TE, and ATE−

trajectories over time; and

predicting a critical transition in the complex system using at least one of the ATE+, TE, and ATE−

trajectories for analysis of directional information influences among vehicle entities of the complex system to avoid a catastrophic failure of the complex system.

12. The computer program product as set forth in claim 11, 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.

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 unknown coefficient a with various discrete time steps by taking the derivative of g(t)=a ln(t)+c to obtain

14. The computer program product as set forth in claim 13, 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;

15. The computer program product as set forth in claim 14, 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.