System and Method for Adjusting Distributions of Data Using Mixed Integer Programming
Exemplary embodiments of the present disclosure are related to systems, methods, and computer-readable medium to facilitate modifying a distribution of data elements to more closely resemble a reference distribution. In exemplary embodiments a modification constraint can be assigned to limit a modification of data elements in a subject distribution and a reference distribution can be identified. Data elements in the subject distribution can be programmatically modified to generate a modified distribution based on a reference distribution, wherein a modification of the data elements can be constrained in response to the modification constraint.
1 . A computer-implemented method of adjusting a distribution of data elements, the method comprising:
assigning a modification constraint to limit a modification of data elements in a subject distribution;
identifying a reference distribution; and
executing code to modify at least one of the data elements in the subject distribution to generate a modified distribution based on a reference distribution, a modification of the at least one of the data elements being constrained in response to the modification constraint.
2 . The computer-implemented method of claim 1 , wherein the modification constraint is a maximum offset that can be applied to the data elements.
3 . The computer-implemented method of claim 1 , wherein the modification constraint is a maximum dissimilarity between the modified distribution and the reference distribution.
4 . The computer-implemented method of claim 1 , wherein executing code to modify at least one of the data elements comprises solving a mixed-integer linear program to minimize an offset applied to the at least one data element and minimize a dissimilarity between the subject distribution and the reference distribution.
5 . The computer-implemented method of claim 1 , wherein the modified distribution is a histogram having bins to which the data elements are assigned.
6 . The computer-implemented method of claim 5 , wherein the modification constraint prohibits assigning the data elements to more than one of the bins subsequent to modification of the data elements.
7 . The computer-implemented method of claim 6 , wherein modifying at least one of the data elements comprises applying an offset to the at least one of the data elements to modify a data value of the at least one of the data elements to be a center value of one of the bins
8 . The computer-implemented method of claim 7 , wherein the offset is applied to modify the data value of the at least one of the data elements so that the data element remains in an originally assigned bin.
9 . The computer-implemented method of claim 7 , wherein the offset is applied to modify the data value of the at least one of the data elements so that the data value corresponds to the center value of a different bin than an original bin to which the data element was assigned.
10 . The computer-implemented method of claim 5 , wherein modifying at least one of the data elements comprises applying an offset to the at least one of the data elements, wherein the offset is a convex combination of two consecutive bin edges.
11 . The computer-implemented method of claim 5 , wherein the modification constraint is a dissimilarity measure between the modified distribution and the reference distribution.
12 . The computer-implemented method of claim 11 , wherein the dissimilarity measure is defined on a bin-by-bin basis by comparing corresponding pairs of bins of the subject distribution and the reference distribution.
13 . The computer-implemented method of claim 11 , wherein the dissimilarity measure is determined utilizing a Minkowski distance giving by:
(
∑
j
p
j
-
q
j
t
)
1
/
t
where j denotes a bin index, p j denotes a population of a bin b j in the reference histogram, q j denotes a quantity of data elements of the subject distribution that fall into the bin b j , and t denotes an order of the Minkowski distance.
14 . The computer-implemented method of claim 11 , wherein the dissimilarity measure is determined utilizing a scaled distance measure given by:
(
∑
j
w
j
(
p
j
-
q
j
)
t
)
1
/
t
where j denotes a bin index, denotes a population of a bin b j in the reference histogram, q j denotes a quantity of data elements of the subject distribution that fall into the bin b j , t denotes an order of the scaled distance measure, and w denotes a weighting factor.
15 . The computer-implemented method of claim 11 , wherein the dissimilarity measure is determined utilizing a Kullback-Leibler Divergence dissimilarity measure given by:
∑
j
=
1
m
p
j
log
p
j
q
j
where j denotes a bin index, denotes a population of a bin b j in the reference histogram, q j denotes a quantity of data elements of the subject distribution that fall into the bin b j .
16 . A non-transitory computer-readable medium storing instruction executable by a processing device, wherein execution of the instructions by the processing device implements a computer-implemented method of adjusting a distribution of data elements comprising:
assigning a modification constraint to limit a modification of data elements in a subject distribution;
identifying a reference distribution; and
executing code to modify at least one of the data elements in the subject distribution to generate a modified distribution based on a reference distribution, a modification of the at least one of the data elements being constrained in response to the modification constraint.
17 . The computer-readable medium of claim 16 , wherein the modification constraint is a maximum offset that can be applied to the data elements.
18 . The computer-readable medium of claim 16 , wherein the modification constraint is a maximum dissimilarity between the modified distribution and the reference distribution.
19 . The computer-readable medium of claim 16 , wherein the modified distribution is a histogram having bins to which the data elements are assigned.
20 . The computer-readable medium of claim 19 , wherein the modification constraint prohibits assigning the data elements to more than one of the bins subsequent to modification of the data elements.
21 . The computer-readable medium of claim 20 , wherein modifying at least one of the data elements comprises applying an offset to the at least one of the data elements to modify a data value of the at least one of the data elements to be a center value of one of the bins
22 . The computer-readable medium of claim 19 , wherein the modification constraint is a dissimilarity measure between the modified distribution and the reference distribution.
23 . The computer-readable medium of claim 11 , wherein the dissimilarity measure is defined on a bin-by-bin basis by comparing corresponding pairs of bins of the subject distribution and the reference distribution.
24 . A system for adjusting a distribution of data elements comprising:
a non-transitory computer-readable medium storing executable code for implementing an adjustment of a distribution; and
a processing device programmed to execute the code to:
assign a modification constraint to limit a modification of data elements in a subject distribution;
identify a reference distribution; and
modify at least one of the data elements in the subject distribution to generate a modified distribution based on a reference distribution, a modification of the at least one of the data elements being constrained in response to the modification constraint.
25 . The system of claim 24 , wherein the modification constraint is a maximum offset that can be applied to the data elements.
26 . The system of claim 24 , wherein the modification constraint is a maximum dissimilarity between the modified distribution and the reference distribution.
27 . The system of claim 24 , wherein the modified distribution is a histogram having bins to which the data elements are assigned.
28 . The system of claim 27 , wherein the modification constraint prohibits assigning the data elements to more than one of the bins subsequent to modification of the data elements.
29 . The system of claim 28 , wherein modifying at least one of the data elements comprises applying an offset to the at least one of the data elements to modify a data value of the at least one of the data elements to be a center value of one of the bins
30 . The system of claim 27 , wherein the modification constraint is a dissimilarity measure between the modified distribution and the reference distribution.
31 . The system of claim 30 , wherein the dissimilarity measure is defined on a bin-by-bin basis by comparing corresponding pairs of bins of the subject distribution and the reference distribution.