Predicting system trajectories toward critical transitions
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.
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:
T
x
j
→
x
i
D
k
=
∑
(
x
i
,
t
+
1
,
x
i
,
t
,
x
j
,
t
)
∈
D
k
p
(
x
i
,
t
+
1
,
x
i
,
t
,
x
j
,
t
)
log
p
(
x
i
,
t
+
1
|
x
i
,
t
,
x
j
,
t
)
p
(
x
i
,
t
+
1
|
x
i
,
t
)
,
where Σ represents a summation, t represents a unit of time, p represents a joint probability of the variables, and | represents a conditional probability, wherein the variables x j and x i represent states of at least one component of the electronic circuit;
estimating ATE+, TE, and ATE− trajectories over time; and
based on the estimated ATE+, TE, and ATE− trajectories, predicting instability in the electronic circuit.
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
ℊ
′
(
t
)
=
a
t
,
then obtaining a discretized version of the unknown coefficient a according to the following:
Δℊ
Δ
t
=
a
1
t
.
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 Δt in a window [T start , T end ], determining the following matrix equation:
[
1
t
1
+
Δ
t
/
2
1
t
2
+
Δ
t
/
2
⋮
1
t
k
+
Δ
t
/
2
]
a
=
[
ℊ
(
t
1
+
Δ
t
)
-
ℊ
(
t
1
)
Δ
t
ℊ
(
t
2
+
Δ
t
)
-
ℊ
(
t
2
)
Δ
t
⋮
ℊ
(
t
k
+
Δ
t
)
-
ℊ
(
t
k
)
Δ
t
]
,
where t 1 , . . . , t k +Δt∈[T start , T end ]; and
determining an approximation of a Δt for the fixed timestep Δt according to the following:
a
Δ
t
=
arg
a
min
[
1
t
1
+
Δ
t
/
2
1
t
2
+
Δ
t
/
2
⋮
1
t
k
+
Δ
t
/
2
]
a
-
[
ℊ
(
t
1
+
Δ
t
)
-
ℊ
(
t
1
)
Δ
t
ℊ
(
t
2
+
Δ
t
)
-
ℊ
(
t
2
)
Δ
t
⋮
ℊ
(
t
k
+
Δ
t
)
-
ℊ
(
t
k
)
Δ
t
]
2
,
where argmin a represents a minimization, t k represents a k th timestep, and wherein the constant c corresponding to this timestep is
c Δt =g ( T end )− a Δt ln( T end ).
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 Δt ( t d )= a Δt ln( T end +t d )+ c λt ,
where F Δt 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:
T
x
j
→
x
i
D
k
=
∑
(
x
i
,
t
+
1
,
x
i
,
t
,
x
j
,
t
)
∈
D
k
p
(
x
i
,
t
+
1
,
x
i
,
t
,
x
j
,
t
)
log
p
(
x
i
,
t
+
1
|
x
i
,
t
,
x
j
,
t
)
p
(
x
i
,
t
+
1
|
x
i
,
t
)
,
where Σ represents a summation, t represents a unit of time, p represents a joint probability of the variables, and | represents a conditional probability, wherein the variables x j and x i represent states of at least one component of the electronic circuit;
estimating ATE+, TE, and ATE− trajectories over time; and
based on the estimated ATE+, TE, and ATE− trajectories, predicting instability in the electronic circuit.
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
ℊ
′
(
t
)
=
a
t
,
then obtaining a discretized version of the unknown coefficient a according to the following:
Δℊ
Δ
t
=
a
1
t
.
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 Δt in a window [T start , T end ], determining the following matrix equation:
[
1
t
1
+
Δ
t
/
2
1
t
2
+
Δ
t
/
2
⋮
1
t
k
+
Δ
t
/
2
]
a
=
[
ℊ
(
t
1
+
Δ
t
)
-
ℊ
(
t
1
)
Δ
t
ℊ
(
t
2
+
Δ
t
)
-
ℊ
(
t
2
)
Δ
t
⋮
ℊ
(
t
k
+
Δ
t
)
-
ℊ
(
t
k
)
Δ
t
]
,
where t 1 , . . . , t k +Δt∈[T start , T end ]; and
determining an approximation of a Δt for the fixed timestep Δt according to the following:
a
Δ
t
=
arg
a
min
[
1
t
1
+
Δ
t
/
2
1
t
2
+
Δ
t
/
2
⋮
1
t
k
+
Δ
t
/
2
]
a
-
[
ℊ
(
t
1
+
Δ
t
)
-
ℊ
(
t
1
)
Δ
t
ℊ
(
t
2
+
Δ
t
)
-
ℊ
(
t
2
)
Δ
t
⋮
ℊ
(
t
k
+
Δ
t
)
-
ℊ
(
t
k
)
Δ
t
]
2
,
where argmin a represents a minimization, t k represents a k th timestep, and wherein the constant c corresponding to this timestep is
c Δt =g ( T end )− a Δt ln( T end ).
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 Δt ( t d )= a Δt ln( T end +t d )+ c Δt ,
where F Δt 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:
T
x
j
→
x
i
D
k
=
∑
(
x
i
,
t
+
1
,
x
i
,
t
,
x
j
,
t
)
∈
D
k
p
(
x
i
,
t
+
1
,
x
i
,
t
,
x
j
,
t
)
log
p
(
x
i
,
t
+
1
|
x
i
,
t
,
x
j
,
t
)
p
(
x
i
,
t
+
1
|
x
i
,
t
)
,
where Σ represents a summation, t represents a unit of time, p represents a joint probability of the variables, and | represents a conditional probability, wherein the variables x j and x i represent states of at least one component of the electronic circuit;
estimating ATE+, TE, and ATE− trajectories over time; and
based on the estimated ATE+, TE, and ATE− trajectories, predicting instability in the electronic circuit.
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
g
′
(
t
)
=
a
t
,
then obtaining a discretized version of the unknown coefficient a according to the following:
Δ
g
Δ
t
=
a
1
t
.
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 Δt in a window [T start , T end ], determining the following matrix equation:
[
1
t
1
+
Δ
t
/
2
1
t
2
+
Δ
t
/
2
⋮
1
t
k
+
Δ
t
/
2
]
a
=
[
g
(
t
1
+
Δ
t
)
-
g
(
t
1
)
Δ
t
g
(
t
2
+
Δ
t
)
-
g
(
t
2
)
Δ
t
⋮
g
(
t
k
+
Δ
t
)
-
g
(
t
k
)
Δ
t
]
,
where t 1 , . . . , t k +Δt∈[T start , T end ]; and
determining an approximation of a Δt for the fixed timestep Δt according to the following:
a
Δ
t
=
arg
a
min
[
1
t
1
+
Δ
t
/
2
1
t
2
+
Δ
t
/
2
⋮
1
t
k
+
Δ
t
/
2
]
a
-
[
g
(
t
1
+
Δ
t
)
-
g
(
t
1
)
Δ
t
g
(
t
2
+
Δ
t
)
-
g
(
t
2
)
Δ
t
⋮
g
(
t
k
+
Δ
t
)
-
g
(
t
k
)
Δ
t
]
2
,
where argmin a represents a minimization, t k represents a k th time point, and wherein the constant c corresponding to this timestep is
c Δt =g ( T end )− a Δt ln( T end ).
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 Δt ( t d )= a Δt ln( T end +t d )+ c Δt ,
where F Δt is a predicted value for a future time, and where t d is a desired time length for prediction.