Systems and methods for data fusion mapping estimation
Systems and methods are disclosed for generating a probability density to estimate the probability that an event will occur in a region of interest. The methods input spatial event data comprising one or more events occurring in the region of interest along with auxiliary data related to the region of interest. The auxiliary data comprises non-event data having spatial resolution such that the probability density estimate for the region of interest is calculated based on a function of the auxiliary data and the event data. In particular, the auxiliary data is used to generate a penalty functional used in the calculation of the probability density estimate.
1. A system for generating a probability density to estimate the probability that an event will occur in a region of interest, comprising:
a processor;
programming executable on said processor for:
inputting spatial event data comprising one or more events occurring in the region of interest;
inputting auxiliary data related to the region of interest;
wherein the auxiliary data comprises non-event data having spatial resolution;
wherein the auxiliary data comprises spatial data defining a valid region where the one or more events are to occur and an invalid region where events are not to occur;
calculating a probability density estimate for the region of interest based on a function of the auxiliary data and the event data;
wherein the probability density estimate is calculated using a Maximum Penalized Likelihood Estimation (MPLE) model;
wherein said MPLE model comprises a penalty functional that encodes the auxiliary data in populating a valid region of said probability density estimate;
wherein the penalty functional is configured to generate a probability density map within the region of interest that smoothes data in the valid region and minimizes non-zero density estimates in the invalid region;
wherein the penalty functional comprises a H 1 Sobolev functional.
2. A system for generating a probability density to estimate the probability that an event will occur in a region of interest, comprising:
a processor;
programming executable on said processor for:
inputting spatial event data comprising one or more events occurring in the region of interest;
inputting auxiliary data related to the region of interest;
wherein the auxiliary data comprises non-event data having spatial resolution;
wherein the auxiliary data comprises spatial data defining a valid region where the one or more events are to occur and an invalid region where events are not to occur;
calculating a probability density estimate for the region of interest based on a function of the auxiliary data and the event data;
wherein the probability density estimate is calculated using a Maximum Penalized Likelihood Estimation (MPLE) model;
wherein said MPLE model comprises a penalty functional that encodes the auxiliary data in populating a valid region of said probability density estimate;
wherein the penalty functional is configured to generate a probability density map within the region of interest that smoothes data in the valid region and minimizes non-zero density estimates in the invalid region;
wherein the penalty functional comprises a total variation (TV) functional;
and
wherein the probability density estimate is calculated according to the equation:
u
^
(
x
)
=
arg
min
∫
Ω
u
ⅆ
x
=
1
,
0
≤
u
{
∫
Ω
∇
u
ⅆ
x
+
λ
∫
Ω
u
∇
·
θ
ⅆ
x
-
μ
∑
i
=
1
n
log
(
u
(
x
i
)
)
}
;
wherein u(x) is the desired probability density for x ε R 2 ,
wherein the known location of events occur at x 1 , x 2 , . . . , x n ;
wherein μ corresponds to weighting of maximum likelihood compared to the penalty functional;
wherein
θ
=
∇
(
1
D
)
∇
(
1
D
)
ɛ
;
and
wherein ( 1 D ) is a characteristic function of the valid region.
3. A system as recited in claim 1 :
wherein the probability density estimate is calculated according to the equation:
u
^
(
x
)
=
arg
min
∫
Ω
u
ⅆ
x
=
1
,
0
≤
u
{
1
2
∫
Ω
z
ɛ
2
∇
u
2
ⅆ
x
-
μ
∑
i
=
1
n
log
(
u
(
x
i
)
)
}
;
wherein u(x) is the desired probability density for x ε R 2 ;
wherein the known location of events occur at x 1 , x 2 , . . . , x n ;
wherein μ corresponds to weighting of maximum likelihood compared to the penalty functional; and
wherein z ε →(1−δ(∂D)).
4. A system as recited in claim 1 or 2 , wherein the auxiliary data comprises geographical data.
5. A system as recited in claim 4 , wherein the auxiliary data comprises geographical aerial image of the region of interest.
6. A system as recited in claim 1 or 2 , wherein the auxiliary data comprises census data relating to the region of interest.
7. A system for generating a probability density map of a region of interest, comprising:
a processor;
programming executable on said processor for:
inputting spatial event data comprising one or more events occurring in the region of interest;
inputting auxiliary data related to the region of interest;
wherein the auxiliary data comprises non-event data having spatial resolution defining a valid region where the one or more events are to occur and an invalid region where events are not to occur;
calculating a probability density estimate for the region of interest based on a function of the auxiliary data and the event data;
wherein the probability density estimate is calculated using a Maximum Penalized Likelihood Estimation (MPLE) model;
wherein said MPLE model comprises a penalty functional that encodes the auxiliary data in populating a valid region of said probability density estimate;
generating a probability density map of the region of interest corresponding to said probability density estimate;
wherein the penalty functional is configured to smooth data in the valid region and minimize non-zero density estimates in the invalid region of the probability density map; and
wherein the penalty functional comprises a H 1 Sobolev functional.
8. A system for generating a probability density map of a region of interest, comprising:
a processor;
programming executable on said processor for:
inputting spatial event data comprising one or more events occurring in the region of interest;
inputting auxiliary data related to the region of interest;
wherein the auxiliary data comprises non-event data having spatial resolution defining a valid region where the one or more events are to occur and an invalid region where events are not to occur;
calculating a probability density estimate for the region of interest based on a function of the auxiliary data and the event data;
wherein the probability density estimate is calculated using a Maximum Penalized Likelihood Estimation (MPLE) model;
wherein said MPLE model comprises a penalty functional that encodes the auxiliary data in populating a valid region of said probability density estimate;
generating a probability density map of the region of interest corresponding to said probability density estimate;
wherein the penalty functional is configured to smooth data in the valid region and minimize non-zero density estimates in the invalid region of the probability density map;
wherein the penalty functional comprises a total variation (TV) functional;
and
wherein the probability density estimate is calculated according to the equation:
u
^
(
x
)
=
arg
min
∫
Ω
u
ⅆ
x
=
1
,
0
≤
u
{
∫
Ω
∇
u
ⅆ
x
+
λ
∫
Ω
u
∇
·
θ
ⅆ
x
-
μ
∑
i
=
1
n
log
(
u
(
x
i
)
)
}
;
wherein u(x) is the desired probability density for x ε R 2 ;
wherein the known location of events occur at x 1 , x 2 , . . . , x n ;
wherein μ corresponds to weighting of maximum likelihood compared to the penalty functional;
wherein
θ
=
∇
(
1
D
)
∇
(
1
D
)
ɛ
;
and
wherein ( 1 D ) is a characteristic function of the valid region.
9. A system as recited in claim 7 :
wherein the probability density estimate is calculated according to the equation:
u
^
(
x
)
=
arg
min
∫
Ω
u
ⅆ
x
=
1
,
0
≤
u
{
1
2
∫
Ω
z
ɛ
2
∇
u
2
ⅆ
x
-
μ
∑
i
=
1
n
log
(
u
(
x
i
)
)
}
;
wherein u(x) is the desired probability density for x ε R 2 ;
wherein the known location of events occur at x 1 , x 2 , . . . , x n ;
wherein μ corresponds to weighting of maximum likelihood compared to the penalty functional; and
wherein z ε →(1−δ(∂D)).
10. A system as recited in claim 7 or 8 , wherein the auxiliary data comprises geographical data.
11. A system as recited in claim 10 , wherein the auxiliary data comprises an aerial image of the region of interest.
12. A system as recited in claim 7 or 8 , wherein the auxiliary data comprises census data relating to the region of interest.
13. A system for generating a probability density to estimate the probability that an event will occur in a region of interest, comprising:
a processor;
programming executable on said processor for:
inputting spatial event data comprising one or more events occurring in the region of interest;
inputting auxiliary data related to the region of interest;
wherein the auxiliary data comprising non-event data having spatial resolution; and
calculating a probability density estimate for the region of interest based on a function of the auxiliary data and the event data;
wherein the auxiliary data is used to generate a penalty functional to calculate the probability density estimate;
wherein the auxiliary data comprises spatial data defining a valid region where the one or more events are to occur and an invalid region where events are not to occur;
wherein the penalty functional is configured to generate a probability density map within the region of interest that restricts population of non-zero density estimates in the invalid region;
wherein the penalty functional comprises a total variation (TV) functional;
wherein the a probability density estimate is calculated according to the equation:
u
^
(
x
)
=
argmin
∫
Ω
udx
=
1
,
0
≤
u
{
∫
Ω
∇
u
ⅆ
x
+
λ
∫
Ω
u
∇
·
θ
ⅆ
x
-
μ
∑
i
=
1
n
log
(
u
(
x
i
)
)
}
;
wherein u(x) is the desired probability density for x ε R 2 ,
wherein the known location of events occur at x 1 , x 2 , . . . , x n ;
wherein μ corresponds to weighting of maximum likelihood compared to the penalty functional;
wherein
θ
=
∇
(
1
D
)
∇
(
1
D
)
ɛ
;
and
wherein ( 1 D ) is a characteristic function of the valid region.
14. A system for generating a probability density to estimate the probability that an event will occur in a region of interest, comprising:
a processor;
programming executable on said processor for:
inputting spatial event data comprising one or more events occurring in the region of interest;
inputting auxiliary data related to the region of interest;
wherein the auxiliary data comprising non-event data having spatial resolution; and
calculating a probability density estimate for the region of interest based on a function of the auxiliary data and the event data;
wherein the auxiliary data is used to generate a penalty functional to calculate the probability density estimate;
wherein the auxiliary data comprises spatial data defining a valid region where the one or more events are to occur and an invalid region where events are not to occur;
wherein the penalty functional is configured to generate a probability density map within the region of interest that restricts population of non-zero density estimates in the invalid region;
wherein the penalty functional comprises a H 1 Sobolev functional;
wherein the a probability density estimate is calculated according to the equation:
u
^
(
x
)
=
argmin
∫
Ω
udx
=
1
,
0
≤
u
{
1
2
∫
Ω
z
ɛ
2
∇
u
2
ⅆ
x
-
μ
∑
i
=
1
n
log
(
u
(
x
i
)
)
}
;
wherein u(x) is the desired probability density for x ε R 2 ;
wherein the known location of events occur at x 1 , x 2 , . . . , x n ;
wherein μ corresponds to weighting of maximum likelihood compared to the penalty functional; and
wherein z ε →(1δ(∂D)).