System and method for efficient estimation of cumulative distribution function
A computer-implemented system and method for estimating a Cumulative Distribution Function (CDF) are provided. The method includes: receive input data representing a volume V of a target space indicating a future target event; compute, using the trained neural network, an estimation of a first flux through a boundary of the volume V; compute, using the trained neural network, an estimation of a second flux through a boundary of a volume W of a base space based on the estimation of the first flux through the boundary of the volume V; generate, using the trained neural network, an estimation of a CDF for the volume V based on the second flux through the boundary of the volume W; compute a probability for the future target event based on the estimated CDF for the volume V; and generate a control command based on the probability for the future target event.
1 . A computer-implemented system for estimating a Cumulative Distribution Function (CDF), the system comprising:
at least one processor; and
memory in communication with said at least one processor storing an CDF Estimator comprising a trained neural network and machine-readable instructions, wherein the instructions, when executed at said at least one processor, cause said system to:
receive input data representing a volume V of a target space indicating a future target event;
compute, using the trained neural network, an estimation of a first flux through a boundary of the volume V;
compute, using the trained neural network, an estimation of a second flux through a boundary of a volume W of a base space based on the estimation of the first flux through the boundary of the volume V;
generate, using the trained neural network, an estimation of a CDF for the volume V based on the second flux through the boundary of the volume W;
compute a probability for the future target event based on the estimated CDF for the volume V; and
generate a control command based on the probability for the future target event.
2 . The system of claim 1 , wherein, during training of the neural network, the instructions when executed at said at least one processor, cause said system to: receive, as training input, historical data representative of past target events and wherein the boundary of the volume V of the target space is defined based in part on the historical data.
3 . The system of claim 2 , wherein, during training of the neural network, the instructions when executed at said at least one processor, cause said system to: use a model fitting process to generate a Probability Distribution Function (PDF) based on the historical data.
4 . The system of claim 3 , wherein, during the training of the neural network, one or more weights of the neural network are iteratively updated to improve the estimation of the CDF by selecting points on the boundary of the volume V.
5 . The system of claim 1 , wherein the estimation of the CDF for the volume V is based on a flux through the transformed boundary of the volume V in the base space.
6 . The system of claim 1 , wherein the estimation the CDF for the volume V is based on a flux through the untransformed boundary of the volume V in the target space.
7 . The system of claim 1 , wherein the trained neural network is configured to compute:
P ( x∈V )=∫ V p X ( x ) dx,
wherein P(x∈V) is a cumulative density over the volume V, and x represents a random variable with the complex target distribution p X (x).
8 . The system of claim 7 , wherein:
p
X
(
x
)
=
p
Y
(
y
)
abs
(
❘
"\[LeftBracketingBar]"
d
g
dx
❘
"\[RightBracketingBar]"
)
,
wherein: |·| is a determinant, abs(·) denotes absolute value, g is inverse of f,
d
g
dx
is (d×d)—Jacobian J of g, x represents the random variable with the complex target distribution p X (x) using a bijective mapping f: d → d , y is a random variable with a base distribution p Y (y).
9 . The system of claim 8 , wherein p Y (y) comprises a Gaussian distribution.
10 . The system of claim 8 , wherein p Y (y) comprises a uniform distribution.
11 . A computer-implemented method for estimating a Cumulative Distribution Function (CDF), the method comprising:
receiving input data representing a volume V of a target space indicating a future target event;
computing, using a trained neural network, an estimation of a first flux through a boundary of the volume V;
computing, using the trained neural network, an estimation of a second flux through a boundary of a volume W of a base space based on the estimation of the first flux through the boundary of the volume V;
generating, using the trained neural network, an estimation of a CDF for the volume V based on the second flux through the boundary of the volume W;
computing a probability for the future target event based on the estimated CDF for the volume V; and
generating a control command based on the probability for the future target event.
12 . The method of claim 11 , wherein, during training of the neural network, historical data representative of past target events are used as training input, and wherein the boundary of the volume V of the target space is defined based in part on the historical data.
13 . The method of claim 12 , wherein method comprises: using a model fitting process to generate a Probability Distribution Function (PDF) based on the historical data.
14 . The method of claim 13 , wherein during the training of the neural network, one or more weights of the neural network are iteratively updated to improve the estimation of the CDF by selecting points on the boundary of the volume V.
15 . The method of claim 11 , wherein the estimation of the CDF for the volume V is based on a flux through the transformed boundary of the volume V in the base space.
16 . The method of claim 11 , wherein the estimation the CDF for the volume V is based on a flux through the untransformed boundary of the volume V in the target space.
17 . The method of claim 11 , wherein:
P ( x∈V )=∫ p X ( x ) dx,
wherein P(x∈V) is a cumulative density over the volume V, and x represents a random variable with the complex target distribution p X (x).
18 . The method of claim 17 , wherein:
p
X
(
x
)
=
p
Y
(
y
)
abs
(
❘
"\[LeftBracketingBar]"
d
g
dx
❘
"\[RightBracketingBar]"
)
,
wherein: |·| is a determinant, abs(·) denotes absolute value, g is inverse of f,
d
g
dx
is (d×d)—Jacobian J of g, x represents the random variable with the complex target distribution p X (x) using a bijective mapping f: d → d , y is a random variable with a base distribution p Y (y).
19 . The method of claim 18 , wherein p Y (y) comprises a Gaussian distribution.
20 . The method of claim 18 , wherein p Y (y) comprises a uniform distribution.