Multivariate responses using classification and regression trees systems and methods
View Patent ↗The present invention is a method of allowing inclusion of more than one variable in a Classification and Regression Tree (CART) analysis. The method includes predicting y using p exploratory variables, where y is a multivariate, continuous response vector, describing a probability density function at “parent” and “child” nodes using a multivariate normal distribution, which is a function of y, and defining a split function where “child” node distributions are individualized, compared to the parent node. In one embodiment a system is configured to implement the multivariate CART analysis for predicting behavior in a non-performing loan portfolio.
1. A method of allowing inclusion of more than one variable in a classification and regression tree analysis for valuing a portfolio of non-performing loans and predicting future performance of the portfolio of non-performing loans using p explanatory variables, said method comprising the steps of:
defining a first “parent” node representing the portfolio of non-performing loans;
defining a split function to determine whether to create “child” nodes by generating a probability density function of the p explanatory variables at a corresponding parent node using a multivariate normal distribution;
creating “child” nodes when a split function value for the corresponding parent node and child nodes indicates that the parent node is statistically non-homogeneous with respect to at least one of the p variables, wherein statistical non-homogeneity is determined by comparing the split function value for the corresponding parent and child nodes, and wherein statistical non-homogeneity indicates a greater predictive value included within at least one of the created child nodes as compared to the corresponding parent node;
repeating said steps of defining a split function and creating “child” nodes until the parent node is statistically homogeneous;
calculating y based on the p explanatory variables and the defined split functions, wherein y is a multivariate response vector representing a predicted recovery amount and a predicted timing value, the predicted recovery amount including at least one amount predicted to be recovered for each non-performing loan included within the portfolio of non-performing loans, the predicted timing value including at least one value predicting when each predicted recovery amount will be recovered, wherein the calculation is performed by a computer; and
determining a value of the portfolio of non-performing loans based on the calculated y.
2. A method according to claim 1 wherein said step of defining a split function further comprises the step of defining p(y)=ƒ(y|μ,Σ), where p(y) represents a statistical distribution generated at each “parent” and “child” node.
3. A method according to claim 2 wherein said step of defining a split function further comprises the step of defining the split function as
φ
(
s
,
P
)
=
KL
(
p
L
p
R
,
p
P
)
=
∫
log
(
p
L
p
R
p
P
)
p
L
p
R
ⅆ
y
=
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
where
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
signifies the expected value, taken over the joint distribution arising from the child nodes.
4. A method according to claim 2 further comprising the step of defining the probability density function as
p
(
y
)
=
f
(
y
μ
,
Σ
)
=
(
2
π
)
-
nr
2
Σ
-
1
2
exp
{
-
1
2
tr
(
y
-
μ
)
Σ
-
1
(
y
-
μ
)
′
}
,
where p(y) represents a statistical distribution generated at each “parent” and “child” node which is also a multivariate distribution function, and n=sample size (number of observations), r=number of response variables, y=n×r matrix of response values, μ=n×r matrix of mean response values, where each row is the same r vector mean, and Σ=r×r matrix of covariance values for the responses.
5. A method according to claim 4 wherein said step of defining a split function further comprises the step of defining the split function as
φ
(
s
,
P
)
=
KL
(
p
L
p
R
,
p
P
)
=
∫
log
(
p
L
p
R
p
P
)
p
L
p
R
ⅆ
y
=
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
where
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
signifies the expected value, taken over the joint distribution arising from the child nodes.
6. A method according to claim 5 , wherein for a probability density function for p(y), the split function is calculated using matrix calculus as:
φ
(
s
,
P
)
=
-
n
L
2
log
(
Σ
L
Σ
)
-
n
R
2
log
(
Σ
R
Σ
)
-
n
×
r
2
+
n
L
2
tr
(
Σ
-
1
Σ
L
)
+
n
R
2
tr
(
Σ
-
1
Σ
R
)
+
1
2
tr
(
Σ
-
1
(
μ
L
-
μ
)
′
(
μ
L
-
μ
)
)
+
1
2
tr
(
Σ
-
1
(
μ
R
-
μ
)
′
(
μ
R
-
μ
)
)
.
7. A method according to claim 6 wherein said step of defining a split function further comprises the step of using maximum likelihood estimation (MLE) for parameter estimation of μ and Σ.
8. A method according to claim 6 further comprising the step of choosing a split such that the split function φ is maximized.
9. A system for performing multivariate classification and regression tree analysis for valuing a portfolio of non-performing loans using p explanatory variables, said system comprising:
at least one computer having a user interface for enabling a user to input information relating to the portfolio of non-performing loans;
a server in communication with the at least one computer, said server configured to read input information relating to the portfolio of non-performing loans, said server further configured to perform the steps of:
defining a first “parent” node representing the portfolio of non-performing loans,
defining a split function to determine whether to create “child” nodes by generating a probability density function of the p explanatory variables at a corresponding parent node using a multivariate normal distribution,
creating “child” nodes when a split function value for the corresponding parent node and child nodes indicates that the parent node is statistically non-homogeneous with respect to at least one of the p variables, wherein statistical non-homogeneity is determined by comparing the split function value for the corresponding parent and child nodes, and wherein statistical non-homogeneity indicates a greater predictive value included within at least one of the created child nodes as compared to the corresponding parent node,
repeating said steps of defining a split function and creating “child” nodes until the parent node is statistically homogeneous,
calculating y based on the p explanatory variables and the defined split functions, wherein y is a multivariate response vector representing a predicted recovery amount and a predicted timing value, the predicted recovery amount including at least one amount predicted to be recovered for each non-performing loan included within the portfolio of non-performing loans, the predicted timing value including at least one value predicting when each predicted recovery amount will be recovered, and
determining a value of the portfolio of non-performing loans based on the calculated y; and
a network connecting said computer to said server.
10. A system according to claim 9 wherein said server is configured to describe a probability density function further according to p(y)=ƒ(y|μ,Σ), where p(y) represents a statistical distribution generated at each “parent” and “child” node.
11. A system according to claim 10 wherein said server configured to define a split function as
φ
(
s
,
P
)
=
KL
(
p
L
p
R
,
p
P
)
=
∫
log
(
p
L
p
R
p
P
)
p
L
p
R
ⅆ
y
=
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
where
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
signifies the expected value, taken over the joint distribution arising from the child nodes.
12. A system according to claim 10 wherein said server is configured to describe a probability density function according to
p
(
y
)
=
f
(
y
μ
,
Σ
)
=
(
2
π
)
-
nr
2
Σ
-
1
2
exp
{
-
1
2
tr
(
y
-
μ
)
Σ
-
1
(
y
-
μ
)
′
}
,
where p(y) is a multivariate distribution function, and n=sample size (number of observations), r=number of response variables, y=n×r matrix of response values, μ=n×r matrix of mean response values, where each row is the same r vector mean, and Σ=r×r matrix of covariance values for the responses.
13. A system according to claim 12 wherein said server configured to define a split function as
φ
(
s
,
P
)
=
KL
(
p
L
p
R
,
p
P
)
=
∫
log
(
p
L
p
R
p
P
)
p
L
p
R
ⅆ
y
=
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
where
E
L
,
R
[
log
(
p
L
p
R
p
P
)
]
signifies the expected value, taken over the joint distribution arising from the child nodes.
14. A system according to claim 13 , wherein for a probability density function for p(y), said server is configured to calculate the split function using matrix calculus as:
φ
(
s
,
P
)
=
-
n
L
2
log
(
Σ
L
Σ
)
-
n
R
2
log
(
Σ
R
Σ
)
-
n
×
r
2
+
n
L
2
tr
(
Σ
-
1
Σ
L
)
+
n
R
2
tr
(
Σ
-
1
Σ
R
)
+
1
2
tr
(
Σ
-
1
(
μ
L
-
μ
)
′
(
μ
L
-
μ
)
)
+
1
2
tr
(
Σ
-
1
(
μ
R
-
μ
)
′
(
μ
R
-
μ
)
)
.
15. A system according to claim 10 wherein said server is configured to define a split function using maximum likelihood estimation (MLE) for parameter estimation of μ and Σ.
16. A system according to claim 10 wherein said server is configured to choose a split such that the split function φ is maximized.
17. A system according to claim 9 wherein said server is configured to allow a user to submit information relating to non-performing loan portfolios via the Internet.
18. A system according to claim 9 wherein said server is configured to allow a user to submit information relating to non-performing loan portfolios via an Intranet.
19. A system according to claim 9 wherein said network is one of a wide area network and a local area network.