U.S. patent application number 14/486659 was filed with the patent office on 2015-01-01 for interaction detection for generalized linear models.
The applicant listed for this patent is International Business Machines Corporation. Invention is credited to Yea J. Chu, Sier Han, Jing-Yun Shyr.
Application Number | 20150006605 14/486659 |
Document ID | / |
Family ID | 51489233 |
Filed Date | 2015-01-01 |
United States Patent
Application |
20150006605 |
Kind Code |
A1 |
Chu; Yea J. ; et
al. |
January 1, 2015 |
INTERACTION DETECTION FOR GENERALIZED LINEAR MODELS
Abstract
Provided are techniques for interaction detection for
generalized linear models. Basic statistics are calculated for a
pair of categorical predictor variables and a target variable from
a dataset during a single pass over the dataset. It is determined
whether there is a significant interaction effect for the pair of
categorical predictor variables on the target variable by:
calculating a log-likelihood value for a full generalized linear
model without estimating model parameters; calculating the model
parameters for a reduced generalized linear model with a recursive
marginal mean accumulation technique using the basic statistics;
calculating a log-likelihood value for the reduced generalized
linear model; calculating a likelihood ratio test statistic using
the log-likelihood value for the full generalized linear model and
the log-likelihood value for the reduced generalized linear model;
calculating a p-value of the likelihood ratio test statistic; and
comparing the p-value to a significance level.
Inventors: |
Chu; Yea J.; (Chicago,
IL) ; Han; Sier; (Xi'an, CN) ; Shyr;
Jing-Yun; (Naperville, IL) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
International Business Machines Corporation |
Armonk |
NY |
US |
|
|
Family ID: |
51489233 |
Appl. No.: |
14/486659 |
Filed: |
September 15, 2014 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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13793724 |
Mar 11, 2013 |
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14486659 |
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Current U.S.
Class: |
708/446 |
Current CPC
Class: |
G06Q 10/067 20130101;
G06Q 10/06 20130101; G06F 17/18 20130101 |
Class at
Publication: |
708/446 |
International
Class: |
G06F 7/60 20060101
G06F007/60 |
Claims
1. A method, comprising: calculating, using a computer, basic
statistics for a pair of categorical predictor variables and a
target variable from a dataset during a single pass over the
dataset; and determining, using the computer, whether there is a
significant interaction effect for the pair of categorical
predictor variables on the target variable by: calculating, using
the computer, a log-likelihood value for a full generalized linear
model without estimating model parameters; calculating, using the
computer, the model parameters for a reduced generalized linear
model with a recursive marginal mean accumulation technique using
the basic statistics; calculating, using the computer, a
log-likelihood value for the reduced generalized linear model;
calculating, using the computer, a likelihood ratio test statistic
using the log-likelihood value for the full generalized linear
model and the log-likelihood value for the reduced generalized
linear model; calculating, using the computer, a p-value of the
likelihood ratio test statistic; and comparing, using the computer,
the p-value to a significance level.
2. The method of claim 1, wherein the full generalized linear model
is of the form g(.mu.)=Xi.beta.i+Xj.beta.j+(Xi.times.Xj).beta.ij,
wherein g(.mu.) is a link function of .mu. and .mu. is an expected
value vector of the target variable Y, wherein Xi and Xj are the
categorical predictor variables, and wherein .beta.i, .beta.j, and
.beta.ij are the model parameters.
3. The method of claim 1, wherein the reduced generalized linear
model is of the form g(.mu.)=Xi.beta.i+Xj.beta.j, wherein g(.mu.)
is a link function of .mu. and .mu. is an expected value vector of
the target variable Y, wherein Xi and Xj are the categorical
predictor variables, and wherein .beta.i and .beta.j are the model
parameters.
4. The method of claim 1, wherein the recursive marginal mean
accumulation technique calculates search directions for the model
parameters calculation by an iterative process based on
accumulating weighted marginal means.
5. The method of claim 1, further comprising: performing, using the
computer, m-way interaction detection among m categorical
predicator variables, where m>2.
6. The method of claim 1, further comprising: performing, using the
computer, m-way interaction detection among multiple possible
combinations of m categorical predictor variables, where m>1,
wherein the dataset contains predictor variables, and the basic
statistics for each of the possible combinations are calculated
during a single pass over the dataset.
7. The method of claim 1, wherein a Software as a Service (SaaS) is
provided to perform the method.
8-21. (canceled)
Description
FIELD
[0001] Embodiments of the invention relate to interaction detection
for generalized linear models.
BACKGROUND
[0002] Business analysts like to know which factors (e.g.,
categorical predictors) impact a target variable of interest and by
how much the factors impact the target variable. A target variable
may be described as a field that is predicted or influenced by one
or more of the factors in a model. A categorical predictor may be
described as a field that has a finite number of nominal or ordinal
categories as values.
[0003] A linear regression model may be used to answer such
questions from business analysts. Furthermore, in many business
scenarios, the interaction between factors may be relevant.
[0004] An Analysis of Variance (ANOVA) technique works in linear
regression models that assume the target variable follows a normal
distribution and the linear relationship exists between the target
variable and factors, but the ANOVA technique is not applicable in
more general models.
[0005] As an example, a software company wants to determine which
characteristics of customers will affect their decision to buy or
not to buy a product. For this example, a logistic regression model
is more appropriate because the target variable (buy or not to buy
a product) is binary, a Bernoulli distribution is used, and the
mean of the target variable is to be between 0 and 1 (so a function
of the target variable mean is assumed to be linearly related to
factors, which is called a "logit link function").
[0006] As another example, if or when a car insurance company wants
to analyze which factors contribute the most to customer's claim
size, then a seasoned analyst knows to fit a gamma regression to
damage claims for cars because it is more appropriate to the
analysis of positive range data by using a gamma distribution and
an inverse link function to relate the mean of the target variable
to a linear combination of the factors.
[0007] In a further example, a shipping company is concerned about
damage to cargo ships caused by waves and wants to determine which
factors (such as ship types, years of construction, etc.) are more
prone to damage, then the incident counts are modeled as occurring
at a Poisson rate and a log-linear model (with a Poisson
distribution and a log link function) is used.
[0008] Many such general models belong to so called "generalized
linear models". The generalized linear model expands the linear
regression model so that the target variable is linearly related to
the predictors via a specified link function. Moreover, the
generalized linear model allows for the target variable to have a
non-normal distribution.
[0009] Because the ANOVA technique is not applicable in generalized
linear models, a likelihood ratio test may be used to detect
interaction. The likelihood ratio test compares log-likelihood
values between the full and reduced generalized linear models. For
a two-way interaction, the full model includes two factors (also
called "main effects") and an interaction effect, while the reduced
model includes two main effects (without an interaction effect).
Computation of log-likelihood value in the reduced model is an
iterative process and requires many data passes.
SUMMARY
[0010] Provided is a method for interaction detection for
generalized linear models. Certain embodiments provide a two-way
interaction detection in which basic statistics are calculated for
a pair of categorical predictor variables and a target variable
from a dataset during a single pass over the dataset. It is
determined whether there is a significant interaction effect for
the pair of categorical predictor variables on the target variable
by: calculating a log-likelihood value for a full generalized
linear model without estimating model parameters; calculating the
model parameters for a reduced generalized linear model with a
recursive marginal mean accumulation technique using the basic
statistics; calculating a log-likelihood value for the reduced
generalized linear model; calculating a likelihood ratio test
statistic using the log-likelihood value for the full generalized
linear model and the log-likelihood value for the reduced
generalized linear model; calculating a p-value of the likelihood
ratio test statistic; and comparing the p-value to a significance
level.
[0011] In certain embodiments, m-way interaction detection is
performed among multiple possible combinations of m categorical
predictor variables, where m.gtoreq.2, wherein the dataset contains
predictor variables, and the basic statistics for each of the
possible combinations are calculated during a single pass over the
dataset.
[0012] Also provided is a computer program product for interaction
detection for generalized linear models. The computer program
product comprises a computer readable storage medium having program
code embodied therewith, the program code executable by at least
one processor to calculate, by the at least one processor, basic
statistics for a pair of categorical predictor variables and a
target variable from a dataset during a single pass over the
dataset; and determine, by the at least one processor, whether
there is a significant interaction effect for the pair of
categorical predictor variables on the target variable by:
calculating, by the at least one processor, a log-likelihood value
for a full generalized linear model without estimating model
parameters, calculating, by the at least one processor, the model
parameters for a reduced generalized linear model with a recursive
marginal mean accumulation technique using the basic statistics,
calculating, by the at least one processor, a log-likelihood value
for the reduced generalized linear model, calculating, by the at
least one processor, a likelihood ratio test statistic using the
log-likelihood value for the full generalized linear model and the
log-likelihood value for the reduced generalized linear model,
calculating, by the at least one processor, a p-value of the
likelihood ratio test statistic, and comparing, by the at least one
processor, the p-value to a significance level.
[0013] Moreover, provided is a computer system for interaction
detection for generalized linear models. The computer system
includes one or more processors, one or more computer-readable
memories and one or more computer-readable, tangible storage
devices, and program instructions, stored on at least one of the
one or more storage devices for execution by at least one of the
one or more processors via at least one of the one or more
memories, to: calculate basic statistics for a pair of categorical
predictor variables and a target variable from a dataset during a
single pass over the dataset; and determine whether there is a
significant interaction effect for the pair of categorical
predictor variables on the target variable by: calculating a
log-likelihood value for a full generalized linear model without
estimating model parameters, calculating the model parameters for a
reduced generalized linear model with a recursive marginal mean
accumulation technique using the basic statistics, calculating a
log-likelihood value for the reduced generalized linear model,
calculating a likelihood ratio test statistic using the
log-likelihood value for the full generalized linear model and the
log-likelihood value for the reduced generalized linear model,
calculating a p-value of the likelihood ratio test statistic, and
comparing the p-value to a significance level.
BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS
[0014] Referring now to the drawings in which like reference
numbers represent corresponding parts throughout:
[0015] FIG. 1 illustrates a computing architecture in accordance
with certain embodiments.
[0016] FIG. 2 illustrates, in a flow diagram, a two-way interaction
detection process in accordance with certain embodiments.
[0017] FIG. 3 illustrates, in Table 1, example target variable
distributions and link functions for the distributions in
accordance with certain embodiments.
[0018] FIG. 4 illustrates Table 2, which shows, for a pair of
categorical predictors, X.sub.1 and X.sub.2, a list of statistics
to be collected and computed in accordance with certain
embodiments.
[0019] FIG. 5 illustrates Table 3, which shows that the formulae
for computing the values of the log-likelihood functions in
accordance with certain embodiments.
[0020] FIG. 6 illustrates, in Table 4, distributions, variance
functions, and first derivatives in accordance with certain
embodiments.
[0021] FIG. 7 illustrates, in Table 5, some commonly used link
functions, the inverse forms, and the first and second derivatives
in accordance with certain embodiments.
[0022] FIG. 8 illustrates, in a flow diagram, a recursive marginal
mean accumulation technique, which is a doubly iterative process,
in accordance with certain embodiments.
[0023] FIG. 9 illustrates a cloud computing node in accordance with
certain embodiments.
[0024] FIG. 10 illustrates a cloud computing environment in
accordance with certain embodiments.
[0025] FIG. 11 illustrates abstraction model layers in accordance
with certain embodiments.
DETAILED DESCRIPTION
[0026] The descriptions of the various embodiments of the present
invention have been presented for purposes of illustration, but are
not intended to be exhaustive or limited to the embodiments
disclosed. Many modifications and variations will be apparent to
those of ordinary skill in the art without departing from the scope
and spirit of the described embodiments. The terminology used
herein was chosen to best explain the principles of the
embodiments, the practical application or technical improvement
over technologies found in the marketplace, or to enable others of
ordinary skill in the art to understand the embodiments disclosed
herein.
[0027] Embodiments provide a technique to detect possible
multiplicative interaction effects for generalized linear
predictive models based on basic statistics computed in a single
data pass. For large and distributed data sources, a map-reduce
approach may be used.
[0028] FIG. 1 illustrates a computing architecture in accordance
with certain embodiments. Computing device 100 includes Interaction
Detection for Generalized Linear Models (IDGLM) system 110 that
outputs one or more significant interaction effects 120. A
significant interaction effect (also referred to as a statistically
significant interaction effect) may be interpreted as an
interaction effect has an impact on a target based on a pattern
(rather than by chance).
[0029] Computing device 100 is coupled to data store 150. Data
store 150 includes one or more data sets 160, information for a
generalized linear model 170, basic statistics 172, full model 174,
and reduced model 176. Each data set 160 includes data for one or
more target variables 162 and/or data for one or more categorical
predictors 164.
[0030] FIG. 2 illustrates, in a flow diagram, a two-way interaction
detection process in accordance with certain embodiments. IDGLM
system 110 receives data set 200 (that includes the data for a
target variable and the data for a set of categorical predictors)
and information for generalized linear model 202 (including a
distribution and a link function) to perform a two-way interaction
detection process.
[0031] In block 204, IDGLM system 110 computes and/or collects
basic statistics. In certain embodiments, IDGLM system 110 computes
and/or collects basic statistics for the target variable and each
pair of two categorical predictors for possible predictors in a
single data pass.
[0032] In blocks 206-214, IDGLM system 110 conducts a likelihood
ratio test. In particular, for each pair of categorical predictors,
IDGLM system 110 conducts a likelihood ratio test between the full
model (two main effects and an interaction effect) and the reduced
model (excluding the interaction effect under the null hypothesis
of no interaction effect).
[0033] In block 206, IDGLM system 110 computes the log-likelihood
value for the full model. In certain embodiments, IDGLM system 110
computes the log-likelihood value for the full model based on the
basic statistics (computed in block 204) without estimating
parameters.
[0034] In block 208, IDGLM system 110 computes the log-likelihood
value for the reduced model using a recursive marginal mean
accumulation technique. In certain embodiments, IDGLM system 110
estimates parameters for the reduced model with the recursive
marginal mean accumulation technique based on the basic statistics
(computed in block 204), and, during the iterative process of
parameter estimation, the log-likelihood value for the reduced
model is also computed.
[0035] In block 210, IDGLM system 110 computes likelihood ratio
test statistics. In certain embodiment, IDGLM system 110 computes
likelihood ratio test statistics as two times the difference
between two log-likelihood values.
[0036] In block 212, IDGLM system 110 computes a p-value. In
certain embodiments, IDGLM system 110 computes the p-value for the
likelihood ratio test statistic based on a chi-squared
distribution.
[0037] In block 214, IDGLM system 110 determines significant
interaction. In certain embodiments, IDGLM system 110 determines
whether an interaction effect for the pair of categorical
predictors is significant.
[0038] IDGLM system 110 outputs significant interaction effects 216
(e.g., in a list), which may be used for subsequent analyses.
[0039] Although traditionally, parameter estimation for the reduced
model in block 208 needs many data passes, IDGLM system 110 avoids
many data passes with a "recursive marginal mean accumulation"
technique to estimate parameters, which use basic statistics
computed in a single data pass. Therefore, IDGLM system 110
performs a single data pass to detect possible multiplicative
interaction effects for generalized linear models.
[0040] Furthermore, IDGLM system 110 is able to extend the two-way
to m-way interaction detection, where m>2. In principle, the
basic statistics for m categorical predictors with the target
variable may be collected and computed in a single data pass. Then
likelihood ratio test statistics are computed based only on these
basic statistics.
[0041] In certain embodiments, the interaction detection focuses on
the generalized linear model. Thus, merely to enhance
understanding, a brief introduction of the generalized linear model
is provided.
Generalized Linear Model
[0042] A generalized linear model of a target variable y with a set
of categorical predictors X has the form shown in Equation (1):
.eta.=g(E(y))=g(.mu.)=X.beta., y.about.F, (1)
where .eta. is the linear predictor; g(.cndot.) is the monotonic
differentiable link function which states how the expected value or
mean of y, E(y)=.mu., is related to the linear predictor .eta.; F
is the target variable's probability distribution.
[0043] Choosing different combinations of a proper probability
distribution and a link function may result in different models.
FIG. 3 illustrates, in Table 1 300, example target variable
distributions and link functions for the distributions in
accordance with certain embodiments. Embodiments are not limited to
these examples. Embodiments may be applied to any target variable
distribution that belongs to the exponential family of
distributions and any link function that is monotonic
differentiable.
[0044] For each pair of categorical predictors, say X.sub.1 and
1.sub.2, IDGLM system 110 tests whether the multiplicative
interaction effect X.sub.1.times.X.sub.2 is significant in the
following full model shown in Equation (2):
.eta.=g(.mu.)=X.sub.1.beta..sub.1+X.sub.2.beta..sub.2+(X.sub.1.times.X.s-
ub.2).beta..sub.3 (2)
[0045] In certain embodiments, if the null hypothesis H.sub.0:
.beta..sub.3=0 is not rejected, then the interaction effect
X.sub.1.times.X.sub.2 is not considered in the subsequent analyses
and it becomes the following reduced model shown in Equation
(3):
.eta.=g(.mu.)=X.sub.1.beta..sub.1+X.sub.2.beta..sub.2 (3)
[0046] In certain embodiments, if some categorical predictors are
continuous, then they are transformed to categorical by a
technique, such as the equal width technique, the equal frequency
technique or other techniques. This transformation may need a data
pass.
[0047] In certain embodiments, the interaction detection process
consists of the following two processes: [0048] 1. Collect and
compute basic statistics among the target variable and each pair of
categorical predictors for all possible categorical predictors in a
single data pass. For large and distributed data sources, a
map-reduce approach may be used. FIG. 4 illustrates Table 2 400,
which shows, for a pair of X.sub.1 and X.sub.2, a list of
statistics to be collected and computed in accordance with certain
embodiments. [0049] 2. For each pair of categorical predictors,
conduct a likelihood ratio test with the null hypothesis
H.sub.0:.beta..sub.3=0 in the following 5 sub-processes: [0050] 2.1
Compute the log-likelihood value for the full model based on
statistics in process(1) without estimating parameters. Denote it
as .theta..sub.full. [0051] 2.2 Estimate parameters for the reduced
model with the recursive marginal mean accumulation technique based
on statistics in process (1). During the iterative process of
parameter estimation, the log-likelihood value for the reduced
model is also computed. Denote it as .theta..sub.reduced. [0052]
2.3 Compute the likelihood ratio test statistic:
.chi..sup.2=2(.theta..sub.full-.theta..sub.reduced). [0053] 2.4
Compute the p-value:
p=1-P.sub.2(.chi..sub.df.sup.2.ltoreq..chi..sup.2), where
.chi..sub.df.sup.2 is a random variable that follows a chi-squared
distribution with df degrees of freedom, and df is the difference
of the number of parameters between the full model and the reduced
model. [0054] 2.5 If p<.alpha., where .alpha. is a significant
level (the default is 0.05) then the interaction effect
X.sub.1.times.X.sub.2 is significant and will be included in the
subsequent analyses.
Log-Likelihood Computation
[0055] The log-likelihood functions may be different for different
distributions. FIG. 5 illustrates Table 3 500, which shows that the
formulae for computing the values of the condensed log-likelihood
functions in accordance with certain embodiments. With the
condensed log-likelihood function, some terms are omitted from the
whole log-likelihood function because these terms are the same for
the full and reduced models thus do not affect the likelihood ratio
test statistics.
[0056] In certain embodiments, for the full model, the expectation
of y in each cell, .mu..sub.ij or .mu..sub.ijk is replaced by
.gamma..sub.ij or .gamma..sub.ijk, respectively.
[0057] In certain embodiments, for the reduced model, .mu..sub.ij
or .mu..sub.ijk is computed together with parameters estimation
using the recursive marginal mean accumulation technique.
Recursive Marginal Mean Accumulation Technique
[0058] To compute the log-likelihood value for the reduced model,
Equation (3), IDGLM system 110 estimates parameters in the reduced
model. In certain embodiments, since there is no closed form
solution, unless it is a linear model (distribution is normal and
link function is identity), the parameters can be estimated by
using the maximum likelihood technique. The common techniques used
are the Newton type technique, in which the first derivative
(gradient) and/or second derivative (Hessian) are needed to update
the parameters, or Iteratively Reweighted Least Squares (IRLS)
technique, in which the "pseudo" target variable and weights are
generated to do a weighted least square regression. Typically, both
techniques are iterative processes in which each iteration performs
one pass through the data. In certain embodiments, IDGLM system 110
provides a new "recursive marginal mean accumulation" technique,
which is a doubly iterative process where: 1) updating parameters
is iterative and 2) computing search direction is iterative.
However, then recursive marginal mean accumulation does not need a
data pass.
[0059] Equation (3) for many distributions (except nominal
multinomial, which will be discussed below) may be simplified, for
the combination of and as shown in Equation (4):
.eta..sub.ij=g(.mu..sub.ij)=.alpha..sub.i+.gamma..sub.j (4)
where .alpha..sub.i and .gamma..sub.j are the parameters for
X.sub.1=i and X.sub.2=j respectively, and can be called "row
parameter" and "column parameter", because their increments are
computed by the row marginal mean and column marginal mean of a
two-way table.
[0060] The doubly iterative process is described with the follow
operations: [0061] (a) Set the initial values of .alpha..sub.i and
.gamma..sub.j to be 0, for i=1, . . . , R and j=1, . . . , S, and
compute initial value of
.mu..sub.ij=g(.alpha..sub.i+.gamma..sub.j).sup.-1, see Table 5 in
FIG. 7 for the corresponding inverse forms, g(.cndot.).sup.-1.
[0062] (b) Compute the initial log-likelihood value by plugging
initial values of .mu..sub.ij into formulae in Table 3 (FIG. 5).
[0063] (c) A R.times.S two-way table is created with the elements
w.sub.ij and s.sub.ij in each cell, where, as shown in Equation (5)
and Equation (6):
[0063] w ij = N ij V ( .mu. ij ) ( g ' ( .mu. ij ) ) 2 + N ij ( y _
ij - .mu. ij ) .times. V ( .mu. ij ) g '' ( .mu. ij ) + V ' ( .mu.
ij ) g ' ( .mu. ij ) ( V ( .mu. ij ) ) 2 ( g ' ( .mu. ij ) ) 3 and
( 5 ) s ij = 1 w ij .times. N ij ( y _ ij - .mu. ij ) V ( .mu. ij )
g ' ( .mu. ij ) ( 6 ) ##EQU00001## [0064] and V(.mu..sub.ij) is the
variance function, V'(.mu..sub.ij) is the first derivative of
V(.mu..sub.ij), and g'(.mu..sub.ij) and g''(.mu..sub.ij) are the
first and second derivatives of the link function, g(.mu..sub.ij),
respectively.
[0065] FIG. 6 illustrates, in Table 4 600, distributions, variance
functions, and first derivatives in accordance with certain
embodiments. In particular, Table 4 lists the variance functions
and the corresponding first derivatives for distributions, except
nominal multinomial. FIG. 7 illustrates, in Table 5 700, some
commonly used link functions, the inverse forms, and the first and
second derivatives in accordance with certain embodiments. [0066]
(d) Compute the search directions, d.alpha..sub.i and
d.gamma..sub.j, for row and column parameters, iteratively, by the
following sub-operations: [0067] (d-1) Set the initial values of
d.alpha..sub.i and d.gamma..sub.j to be 0. [0068] (d-2) Update the
search direction for row parameter by adding the marginal mean of
the corresponding row as shown in Equation (7):
[0068] d.alpha..sub.i=d.alpha..sub.i+s.sub.i.cndot., (7) [0069]
where s.sub.i.cndot., is the weighted marginal mean of s.sub.ij for
row i,i=1, . . . , R, as shown in Equation (8):
[0069] s i .cndot. = j = 1 S w ij .times. s ij j = 1 S w ij . ( 8 )
##EQU00002## [0070] (d-3) Update the two-way table by subtracting
row marginal mean for each row as shown in Equation (9):
[0070] s.sub.ij=s.sub.ij-s.sub.i.cndot., (9) [0071] (d-4) Update
the search direction for column parameter by adding the marginal
mean of the corresponding column as shown in Equation (10):
[0071] d.gamma..sub.j=d.gamma..sub.j+s.sub..cndot.j (10) [0072]
where s.sub..cndot.1 is the weighted marginal mean of s.sub.ij for
column j, j=1, . . . , S, as shown in Equation (11):
[0072] s .cndot. j = i = 1 R w ij .times. s ij i = 1 R w ij . ( 11
) ##EQU00003## [0073] (d-5) Update the two-way table by subtracting
column marginal mean for each column as shown in Equation (12):
[0073] s.sub.ij=s.sub.ij-s.sub..cndot.j. (12) [0074] (d-6) Check
whether the search directions converge by the following
criterion
[0074] max(|s.sub.i.cndot.|, |s.sub..cndot.j|)<.epsilon..sub.i,
[0075] where .epsilon..sub.1 is a specified tolerance level. [0076]
If the criterion is not met, go back to (d-2), otherwise go to (e).
[0077] (e) Update the row and column parameters as shown in
Equation (13):
[0077] .alpha..sub.i=.alpha..sub.i+.epsilon..times.d.alpha..sub.i,
and
.gamma..sub.j=.gamma..sub.j+.epsilon..times.d.gamma..sub.j, (13)
[0078] where is a step length in a line search technique. [0079]
(f) Compute the log-likelihood value with the updated target
variable mean which is computed with the updated parameter
estimates. [0080] (g) Check whether the parameters converge: the
absolute difference of log-likelihood values in two successive
iterations is less than a specified tolerance level, say
.epsilon..sub.2, which can be different from .epsilon..sub.1.
[0081] (h) If the criterion is not met, go back to (c), otherwise
stop and output the final log-likelihood value.
[0082] Note that for nominal multinomial, computation is more
complex in the following operations: [0083] (a) The estimated
expectations for each category of the target variable as shown in
Equation (14):
[0083] .pi. ij , k = { exp ( .alpha. ik + .gamma. jk ) 1 + k = 1 K
- 1 exp ( .alpha. ik + .gamma. jk ) , k = 1 , , K - 1 , 1 1 + k = 1
K - 1 exp ( .alpha. ik + .gamma. jk ) , k = K ( 14 ) ##EQU00004##
[0084] (b) The log-likelihood value as shown in Equation (15):
[0084] i = 1 R j = 1 S k = 1 K N ij , k .times. ln ( .pi. ij , k )
( 15 ) ##EQU00005## [0085] (c) In the R.times.S two-way table,
W.sub.ij is extended from a scalar to a matrix and .theta..sub.ij
to a vector as shown in Equation (16) and Equation (17):
[0085]
w.sub.ij=N.sub.ij(diag(.pi..sub.ij)-.pi..sub.ij.times..pi..sub.ij-
.sup.T) and (16)
s.sub.ij=N.sub.ijw.sub.ij.sup.-1( y.sub.ij-.pi..sub.ij), (17)
[0086] where .pi..sub.ij.sup.T=(.pi..sub.ij,1, . . . ,
.pi..sub.ij,k-1) and y.sub.ij.sup.T=( y.sub.ij,1, . . . ,
y.sub.ij,K-1) [0087] (d) The search directions, d.alpha..sub.i and
d.gamma..sub.j, are extended to vectors, d.alpha..sub.i and
d.gamma..sub.j. [0088] The weighted marginal means of s.sub.ij for
row i, i=1, . . . , R, and for column j, j=1, . . . S, are extended
to vectors as shown in Equation (18) and Equation (19):
[0088] s i .cndot. = ( j = 1 S w ij ) - 1 .times. ( j = 1 S w ij
.times. s ij ) and ( 18 ) s .cndot. j = ( i = 1 R w ij ) - 1
.times. ( i = 1 R w ij .times. s ij ) . ( 19 ) ##EQU00006## [0089]
(e) The parameters, .alpha..sub.i and .gamma..sub.j, are extended
to vectors, .alpha..sub.i and .gamma..sub.j.
[0090] FIG. 8 illustrates, in a flow diagram, a recursive marginal
mean accumulation technique, which is a doubly iterative process,
in accordance with certain embodiments. IDGLM system 110 receives
basic statistics 800 and initial values for parameters 802
(operation (a) of the doubly iterative process). In block 804,
IDGLM system 110 computes an initial log likelihood (operation (b)
of the doubly iterative process). In block 806, IDGLM system 110
creates a two-way table (operation (c) of the doubly iterative
process).
[0091] In blocks 808-818, IDGLM system 110 performs the recursive
marginal mean accumulation (operation (d) of the doubly iterative
process). In particular, in block 808, IDGLM system 110 sets the
initial search directions for row and column parameters to zeros
(operation (d-1) of the doubly iterative process). In block 810,
IDGLM system 110 updates the search directions for row parameters
by adding row marginal means to the current search directions for
row parameters (operation (d-2) of the doubly iterative process).
In block 812, IDGLM system 110 updates the two-way table by
extracting the row marginal mean from each row (operation (d-3) of
the doubly iterative process). In block 814, IDGLM system 110
updates the search direction for column parameters by adding column
marginal means (operation (d-4) of the doubly iterative process).
In block 816, IDGLM system 110 updates the two-way table by
extracting column marginal mean from each column (operation (d-5)
of the doubly iterative process). In block 818, IDGLM system 110
determines whether there is search direction convergence (operation
(d-6) of the doubly iterative process). If so, processing continues
to block 820, otherwise, processing loops back to block 810.
[0092] In block 820, IDGLM system 110 updates the row and column
parameters (operation (e) of the doubly iterative process). In
block 822, IDGLM system 110 computes the log-likelihood based on
the updated parameters (operation (f) of the doubly iterative
process). In block 824, IDGLM system 110 determines whether there
is parameters convergence (operation (g) of the doubly iterative
process). If so, processing continues to output final
log-likelihood value 826, otherwise, processing loops to block 806
(operation (h) of the doubly iterative process).
Extension to m-Way Interaction Detection
[0093] The two-way interaction detection technique (see paragraphs
48-55) may be extended to m-way interaction detection. The full
model then contains all the main effects, two-way interaction
effects, . . . , and m-way interaction effects. The reduced model
is the model that the m-way interaction effect excludes from the
full model. The likelihood ratio test is used to test whether an
m-way interaction effect is significant. Similar to the situation
of two-way interaction detection, the basic statistics between the
target variable and m categorical predictors are collected first.
Then the log-likelihood value of the full model is computed based
on these basic statistics. For the reduced model, the extended
recursive marginal mean accumulation technique for computation of
the log-likelihood is described as follows: [0094] (a) Set initial
parameters corresponding to all main effects, two-way interaction
effects, . . . , and (m-1)-way interaction effects to be 0. [0095]
(b) Compute the initial log-likelihood value based on initial
parameters. [0096] (c) Similar to the two-way interaction
detection, create an m-way table. [0097] (d) Compute the search
directions iteratively by the following iterative operations:
[0098] (d-1) Set the initial search directions of one-way main
effects, two-way interaction effects, . . . , (m-1)-way interaction
effects to be 0. [0099] (d-2) Select one dimension in the m-way
table, then update corresponding search direction of one-way main
effect by adding the marginal mean of this dimension, and update
the m-way table by subtracting the marginal mean of this dimension.
Such process is repeated for each of other main effects. [0100]
(d-3) Select two dimensions in the m-way table, then update
corresponding search direction of two-way interaction effect by
adding the marginal mean of the two-dimensional table, and update
the m-way table by subtracting the marginal mean of the
two-dimensional table. Such process is repeated for each of other
two-way interaction effects. [0101] (d-4) Similar to (d-2) or
(d-3), update the search directions from three-way to (m-1)-way
interaction effects. [0102] (d-5) Check whether the search
directions converge: if the maximum absolute marginal mean of all
marginal means from one-way main effects to (m-1)-way interaction
effects is less than a tolerance level. If the criterion is met,
then go to the operation (e), otherwise go back to operation (d-2).
[0103] (e) Similar to the operation (e) in the two-way interaction
detection, update the parameters from one-way main effects to
(m-1)-way interaction effects. [0104] (f) Compute the
log-likelihood value with the updated target variable mean which is
computed with the updated parameter estimates. [0105] (g) Check
whether the parameters converge: the absolute difference of
log-likelihood values in two successive iterations is less than a
specified tolerance level. [0106] (h) If the criterion is not met,
go back to (c), otherwise stop and output the final log-likelihood
value.
Cloud Computing
[0107] It is understood in advance that although this disclosure
includes a detailed description on cloud computing, implementation
of the teachings recited herein are not limited to a cloud
computing environment. Rather, embodiments of the present invention
are capable of being implemented in conjunction with any other type
of computing environment now known or later developed.
[0108] Cloud computing is a model of service delivery for enabling
convenient, on-demand network access to a shared pool of
configurable computing resources (e.g. networks, network bandwidth,
servers, processing, memory, storage, applications, virtual
machines, and services) that can be rapidly provisioned and
released with minimal management effort or interaction with a
provider of the service. This cloud model may include at least five
characteristics, at least three service models, and at least four
deployment models.
[0109] Characteristics are as follows:
[0110] On-demand self-service: a cloud consumer can unilaterally
provision computing capabilities, such as server time and network
storage, as needed automatically without requiring human
interaction with the service's provider.
[0111] Broad network access: capabilities are available over a
network and accessed through standard mechanisms that promote use
by heterogeneous thin or thick client platforms (e.g., mobile
phones, laptops, and PDAs).
[0112] Resource pooling: the provider's computing resources are
pooled to serve multiple consumers using a multi-tenant model, with
different physical and virtual resources dynamically assigned and
reassigned according to demand. There is a sense of location
independence in that the consumer generally has no control or
knowledge over the exact location of the provided resources but may
be able to specify location at a higher level of abstraction (e.g.,
country, state, or datacenter).
[0113] Rapid elasticity: capabilities can be rapidly and
elastically provisioned, in some cases automatically, to quickly
scale out and rapidly released to quickly scale in. To the
consumer, the capabilities available for provisioning often appear
to be unlimited and can be purchased in any quantity at any
time.
[0114] Measured service: cloud systems automatically control and
optimize resource use by leveraging a metering capability at some
level of abstraction appropriate to the type of service (e.g.,
storage, processing, bandwidth, and active user accounts). Resource
usage can be monitored, controlled, and reported providing
transparency for both the provider and consumer of the utilized
service.
[0115] Service Models are as follows:
[0116] Software as a Service (SaaS): the capability provided to the
consumer is to use the provider's applications running on a cloud
infrastructure. The applications are accessible from various client
devices through a thin client interface such as a web browser
(e.g., web-based email). The consumer does not manage or control
the underlying cloud infrastructure including network, servers,
operating systems, storage, or even individual application
capabilities, with the possible exception of limited user-specific
application configuration settings.
[0117] Platform as a Service (PaaS): the capability provided to the
consumer is to deploy onto the cloud infrastructure
consumer-created or acquired applications created using programming
languages and tools supported by the provider. The consumer does
not manage or control the underlying cloud infrastructure including
networks, servers, operating systems, or storage, but has control
over the deployed applications and possibly application hosting
environment configurations.
[0118] Infrastructure as a Service (IaaS): the capability provided
to the consumer is to provision processing, storage, networks, and
other fundamental computing resources where the consumer is able to
deploy and run arbitrary software, which can include operating
systems and applications. The consumer does not manage or control
the underlying cloud infrastructure but has control over operating
systems, storage, deployed applications, and possibly limited
control of select networking components (e.g., host firewalls).
[0119] Deployment Models are as follows:
[0120] Private cloud: the cloud infrastructure is operated solely
for an organization. It may be managed by the organization or a
third party and may exist on-premises or off-premises.
[0121] Community cloud: the cloud infrastructure is shared by
several organizations and supports a specific community that has
shared concerns (e.g., mission, security requirements, policy, and
compliance considerations). It may be managed by the organizations
or a third party and may exist on-premises or off-premises.
[0122] Public cloud: the cloud infrastructure is made available to
the general public or a large industry group and is owned by an
organization selling cloud services.
[0123] Hybrid cloud: the cloud infrastructure is a composition of
two or more clouds (private, community, or public) that remain
unique entities but are bound together by standardized or
proprietary technology that enables data and application
portability (e.g., cloud bursting for load balancing between
clouds).
[0124] A cloud computing environment is service oriented with a
focus on statelessness, low coupling, modularity, and semantic
interoperability. At the heart of cloud computing is an
infrastructure comprising a network of interconnected nodes.
[0125] Referring now to FIG. 9, a schematic of an example of a
cloud computing node is shown. Cloud computing node 910 is only one
example of a suitable cloud computing node and is not intended to
suggest any limitation as to the scope of use or functionality of
embodiments of the invention described herein. Regardless, cloud
computing node 910 is capable of being implemented and/or
performing any of the functionality set forth hereinabove.
[0126] In cloud computing node 910 there is a computer
system/server 912, which is operational with numerous other general
purpose or special purpose computing system environments or
configurations. Examples of well-known computing systems,
environments, and/or configurations that may be suitable for use
with computer system/server 912 include, but are not limited to,
personal computer systems, server computer systems, thin clients,
thick clients, handheld or laptop devices, multiprocessor systems,
microprocessor-based systems, set top boxes, programmable consumer
electronics, network PCs, minicomputer systems, mainframe computer
systems, and distributed cloud computing environments that include
any of the above systems or devices, and the like.
[0127] Computer system/server 912 may be described in the general
context of computer system executable instructions, such as program
modules, being executed by a computer system. Generally, program
modules may include routines, programs, objects, components, logic,
data structures, and so on that perform particular tasks or
implement particular abstract data types. Computer system/server
912 may be practiced in distributed cloud computing environments
where tasks are performed by remote processing devices that are
linked through a communications network. In a distributed cloud
computing environment, program modules may be located in both local
and remote computer system storage media including memory storage
devices.
[0128] As shown in FIG. 9, computer system/server 912 in cloud
computing node 910 is shown in the form of a general-purpose
computing device. The components of computer system/server 912 may
include, but are not limited to, one or more processors or
processing units 916, a system memory 928, and a bus 918 that
couples various system components including system memory 928 to a
processor or processing unit 916.
[0129] Bus 918 represents one or more of any of several types of
bus structures, including a memory bus or memory controller, a
peripheral bus, an accelerated graphics port, and a processor or
local bus using any of a variety of bus architectures. By way of
example, and not limitation, such architectures include Industry
Standard Architecture (ISA) bus, Micro Channel Architecture (MCA)
bus, Enhanced ISA (EISA) bus, Video Electronics Standards
Association (VESA) local bus, and Peripheral Component
Interconnects (PCI) bus.
[0130] Computer system/server 912 typically includes a variety of
computer system readable media. Such media may be any available
media that is accessible by computer system/server 912, and it
includes both volatile and non-volatile media, removable and
non-removable media.
[0131] System memory 928 can include computer system readable media
in the form of volatile memory, such as random access memory (RAM)
930 and/or cache memory 932. Computer system/server 912 may further
include other removable/non-removable, volatile/non-volatile
computer system storage media. By way of example only, storage
system 934 can be provided for reading from and writing to a
non-removable, non-volatile magnetic media (not shown and typically
called a "hard drive"). Although not shown, a magnetic disk drive
for reading from and writing to a removable, non-volatile magnetic
disk (e.g., a "floppy disk"), and an optical disk drive for reading
from or writing to a removable, non-volatile optical disk such as a
CD-ROM, DVD-ROM or other optical media can be provided. In such
instances, each can be connected to bus 918 by one or more data
media interfaces. As will be further depicted and described below,
memory 928 may include at least one program product having a set
(e.g., at least one) of program modules that are configured to
carry out the functions of embodiments of the invention.
[0132] Program/utility 940, having a set (at least one) of program
modules 942, may be stored in memory 928 by way of example, and not
limitation, as well as an operating system, one or more application
programs, other program modules, and program data. Each of the
operating system, one or more application programs, other program
modules, and program data or some combination thereof, may include
an implementation of a networking environment. Program modules 942
generally carry out the functions and/or methodologies of
embodiments of the invention as described herein.
[0133] Computer system/server 912 may also communicate with one or
more external devices 914 such as a keyboard, a pointing device, a
display 924, etc.; one or more devices that enable a user to
interact with computer system/server 912; and/or any devices (e.g.,
network card, modem, etc.) that enable computer system/server 912
to communicate with one or more other computing devices. Such
communication can occur via Input/Output (I/O) interfaces 922.
Still yet, computer system/server 912 can communicate with one or
more networks such as a local area network (LAN), a general wide
area network (WAN), and/or a public network (e.g., the Internet)
via network adapter 920. As depicted, network adapter 920
communicates with the other components of computer system/server
912 via bus 918. It should be understood that although not shown,
other hardware and/or software components could be used in
conjunction with computer system/server 912. Examples, include, but
are not limited to: microcode, device drivers, redundant processing
units, external disk drive arrays, RAID systems, tape drives, and
data archival storage systems, etc.
[0134] Referring now to FIG. 10, illustrative cloud computing
environment 1050 is depicted. As shown, cloud computing environment
1050 comprises one or more cloud computing nodes 910 with which
local computing devices used by cloud consumers, such as, for
example, personal digital assistant (PDA) or cellular telephone
1054A, desktop computer 1054B, laptop computer 1054C, and/or
automobile computer system 1054N may communicate. Nodes 910 may
communicate with one another. They may be grouped (not shown)
physically or virtually, in one or more networks, such as Private,
Community, Public, or Hybrid clouds as described hereinabove, or a
combination thereof. This allows cloud computing environment 1050
to offer infrastructure, platforms and/or software as services for
which a cloud consumer does not need to maintain resources on a
local computing device. It is understood that the types of
computing devices 1054A-N shown in FIG. 10 are intended to be
illustrative only and that computing nodes 910 and cloud computing
environment 1050 can communicate with any type of computerized
device over any type of network and/or network addressable
connection (e.g., using a web browser).
[0135] Referring now to FIG. 11, a set of functional abstraction
layers provided by cloud computing environment 1050 (FIG. 10) is
shown. It should be understood in advance that the components,
layers, and functions shown in FIG. 11 are intended to be
illustrative only and embodiments of the invention are not limited
thereto. As depicted, the following layers and corresponding
functions are provided:
[0136] Hardware and software layer 1160 includes hardware and
software components. Examples of hardware components include
mainframes, in one example IBM.RTM. zSeries.RTM. systems; RISC
(Reduced Instruction Set Computer) architecture based servers, in
one example IBM pSeries.RTM. systems; IBM xSeries.RTM. systems; IBM
BladeCenter.RTM. systems; storage devices; networks and networking
components. Examples of software components include network
application server software, in one example IBM WebSphere.RTM.
application server software; and database software, in one example
IBM DB2.RTM. database software. (IBM, zSeries, pSeries, xSeries,
BladeCenter, WebSphere, and DB2 are trademarks of International
Business Machines Corporation registered in many jurisdictions
worldwide).
[0137] Virtualization layer 1162 provides an abstraction layer from
which the following examples of virtual entities may be provided:
virtual servers; virtual storage; virtual networks, including
virtual private networks; virtual applications and operating
systems; and virtual clients.
[0138] In one example, management layer 1164 may provide the
functions described below. Resource provisioning provides dynamic
procurement of computing resources and other resources that are
utilized to perform tasks within the cloud computing environment.
Metering and Pricing provide cost tracking as resources are
utilized within the cloud computing environment, and billing or
invoicing for consumption of these resources. In one example, these
resources may comprise application software licenses. Security
provides identity verification for cloud consumers and tasks, as
well as protection for data and other resources. User portal
provides access to the cloud computing environment for consumers
and system administrators. Service level management provides cloud
computing resource allocation and management such that required
service levels are met. Service Level Agreement (SLA) planning and
fulfillment provide pre-arrangement for, and procurement of, cloud
computing resources for which a future requirement is anticipated
in accordance with an SLA.
[0139] Workloads layer 1166 provides examples of functionality for
which the cloud computing environment may be utilized. Examples of
workloads and functions which may be provided from this layer
include: mapping and navigation; software development and lifecycle
management; virtual classroom education delivery; data analytics
processing; transaction processing; and interaction detection for
generalized linear models.
[0140] Thus, in certain embodiments, software or a program,
implementing interaction detection for generalized linear models in
accordance with embodiments described herein, is provided as a
service in a cloud environment.
[0141] In certain embodiments, computing device 100 has the
architecture of computing node 910. In certain embodiments,
computing device 100 is part of a cloud environment. In certain
alternative embodiments, computing device 100 is not part of a
cloud environment.
Additional Embodiment Details
[0142] As will be appreciated by one skilled in the art, aspects of
the present invention may be embodied as a system, method or
computer program product. Accordingly, aspects of the present
invention may take the form of an entirely hardware embodiment, an
entirely software embodiment (including firmware, resident
software, micro-code, etc.) or an embodiment combining software and
hardware aspects that may all generally be referred to herein as a
"circuit," "module" or "system." Furthermore, aspects of the
present invention may take the form of a computer program product
embodied in one or more computer readable medium(s) having computer
readable program code embodied thereon.
[0143] Any combination of one or more computer readable medium(s)
may be utilized. The computer readable medium may be a computer
readable signal medium or a computer readable storage medium. A
computer readable storage medium may be, for example, but not
limited to, an electronic, magnetic, optical, electromagnetic,
infrared, or semiconductor system, apparatus, or device, or any
suitable combination of the foregoing. More specific examples (a
non-exhaustive list) of the computer readable storage medium would
include the following: an electrical connection having one or more
wires, a portable computer diskette, a hard disk, a random access
memory (RAM), a read-only memory (ROM), an erasable programmable
read-only memory (EPROM or Flash memory), an optical fiber, a
portable compact disc read-only memory (CD-ROM), an optical storage
device, a magnetic storage device, or any suitable combination of
the foregoing. In the context of this document, a computer readable
storage medium may be any tangible medium that can contain, or
store a program for use by or in connection with an instruction
execution system, apparatus, or device.
[0144] A computer readable signal medium may include a propagated
data signal with computer readable program code embodied therein,
for example, in baseband or as part of a carrier wave. Such a
propagated signal may take any of a variety of forms, including,
but not limited to, electro-magnetic, optical, or any suitable
combination thereof. A computer readable signal medium may be any
computer readable medium that is not a computer readable storage
medium and that can communicate, propagate, or transport a program
for use by or in connection with an instruction execution system,
apparatus, or device.
[0145] Program code embodied on a computer readable medium may be
transmitted using any appropriate medium, including but not limited
to wireless, wireline, optical fiber cable, RF, etc., or any
suitable combination of the foregoing.
[0146] Computer program code for carrying out operations for
aspects of the present invention may be written in any combination
of one or more programming languages, including an object oriented
programming language such as Java, Smalltalk, C++ or the like and
conventional procedural programming languages, such as the "C"
programming language or similar programming languages. The program
code may execute entirely on the user's computer, partly on the
user's computer, as a stand-alone software package, partly on the
user's computer and partly on a remote computer or entirely on the
remote computer or server. In the latter scenario, the remote
computer may be connected to the user's computer through any type
of network, including a local area network (LAN) or a wide area
network (WAN), or the connection may be made to an external
computer (for example, through the Internet using an Internet
Service Provider).
[0147] Aspects of the present invention are described below with
reference to flow diagram (e.g., flowchart) illustrations and/or
block diagrams of methods, apparatus (systems) and computer program
products according to embodiments of the invention. It will be
understood that each block of the flowchart illustrations and/or
block diagrams, and combinations of blocks in the flowchart
illustrations and/or block diagrams, can be implemented by computer
program instructions. These computer program instructions may be
provided to a processor of a general purpose computer, special
purpose computer, or other programmable data processing apparatus
to produce a machine, such that the instructions, which execute via
the processor of the computer or other programmable data processing
apparatus, create means for implementing the functions/acts
specified in the flowchart and/or block diagram block or
blocks.
[0148] These computer program instructions may also be stored in a
computer readable medium that can direct a computer, other
programmable data processing apparatus, or other devices to
function in a particular manner, such that the instructions stored
in the computer readable medium produce an article of manufacture
including instructions which implement the function/act specified
in the flowchart and/or block diagram block or blocks.
[0149] The computer program instructions may also be loaded onto a
computer, other programmable data processing apparatus, or other
devices to cause a series of operational steps to be performed on
the computer, other programmable apparatus or other devices to
produce a computer implemented process such that the instructions
which execute on the computer or other programmable apparatus
provide processes for implementing the functions/acts specified in
the flowchart and/or block diagram block or blocks.
[0150] The flowcharts and block diagrams in the Figures illustrate
the architecture, functionality, and operation of possible
implementations of systems, methods and computer program products
according to various embodiments of the present invention. In this
regard, each block in the flowcharts or block diagrams may
represent a module, segment, or portion of code, which comprises
one or more executable instructions for implementing the specified
logical function(s). It should also be noted that, in some
alternative implementations, the functions noted in the block may
occur out of the order noted in the figures. For example, two
blocks shown in succession may, in fact, be executed substantially
concurrently, or the blocks may sometimes be executed in the
reverse order, depending upon the functionality involved. It will
also be noted that each block of the block diagrams and/or
flowcharts illustration, and combinations of blocks in the block
diagrams and/or flowcharts illustration, can be implemented by
special purpose hardware-based systems that perform the specified
functions or acts, or combinations of special purpose hardware and
computer instructions.
[0151] In addition, the illustrated operations of the flow diagrams
and block diagrams show certain events occurring in a certain
order. In alternative embodiments, certain operations may be
performed in a different order, modified or removed. Moreover,
operations may be added to the above described logic and still
conform to the described embodiments. Further, operations described
herein may occur sequentially or certain operations may be
processed in parallel. Yet further, operations may be performed by
a single processing unit or by distributed processing units.
[0152] The code implementing the described operations may further
be implemented in hardware logic or circuitry (e.g., an integrated
circuit chip, Programmable Gate Array (PGA), Application Specific
Integrated Circuit (ASIC), etc. The hardware logic may be coupled
to a processor to perform operations.
[0153] Devices that are in communication with each other need not
be in continuous communication with each other, unless expressly
specified otherwise. In addition, devices that are in communication
with each other may communicate directly or indirectly through one
or more intermediaries.
[0154] A description of an embodiment with several components in
communication with each other does not imply that all such
components are required. On the contrary a variety of optional
components are described to illustrate the wide variety of possible
embodiments of the present invention.
[0155] Further, although process steps, method steps, algorithms or
the like may be described in a sequential order, such processes,
methods and algorithms may be configured to work in alternate
orders. In other words, any sequence or order of steps that may be
described does not necessarily indicate a requirement that the
steps be performed in that order. The steps of processes described
herein may be performed in any order practical. Further, some steps
may be performed simultaneously.
[0156] When a single device or article is described herein, it will
be readily apparent that more than one device/article (whether or
not they cooperate) may be used in place of a single
device/article. Similarly, where more than one device or article is
described herein (whether or not they cooperate), it will be
readily apparent that a single device/article may be used in place
of the more than one device or article or a different number of
devices/articles may be used instead of the shown number of devices
or programs. The functionality and/or the features of a device may
be alternatively embodied by one or more other devices which are
not explicitly described as having such functionality/features.
Thus, other embodiments of the present invention need not include
the device itself.
[0157] The terminology used herein is for the purpose of describing
particular embodiments only and is not intended to be limiting of
the invention. As used herein, the singular forms "a", "an" and
"the" are intended to include the plural forms as well, unless the
context clearly indicates otherwise. It will be further understood
that the terms "comprises" and/or "comprising," when used in this
specification, specify the presence of stated features, integers,
steps, operations, elements, and/or components, but do not preclude
the presence or addition of one or more other features, integers,
steps, operations, elements, components, and/or groups thereof.
[0158] The terms "an embodiment", "embodiment", "embodiments", "the
embodiment", "the embodiments", "one or more embodiments", "some
embodiments", and "one embodiment" mean "one or more (but not all)
embodiments of the present invention(s)" unless expressly specified
otherwise.
[0159] The terms "including", "comprising", "having" and variations
thereof mean "including but not limited to", unless expressly
specified otherwise.
[0160] The enumerated listing of items does not imply that any or
all of the items are mutually exclusive, unless expressly specified
otherwise.
[0161] The corresponding structures, materials, acts, and
equivalents of all means or step plus function elements in the
claims below are intended to include any structure, material, or
act for performing the function in combination with other claimed
elements as specifically claimed. The description of embodiments of
the present invention has been presented for purposes of
illustration and description, but is not intended to be exhaustive
or limited to the invention in the form disclosed. Many
modifications and variations will be apparent to those of ordinary
skill in the art without departing from the scope and spirit of the
invention. The embodiments were chosen and described in order to
best explain the principles of the invention and the practical
application, and to enable others of ordinary skill in the art to
understand the invention for various embodiments with various
modifications as are suited to the particular use contemplated.
[0162] The foregoing description of embodiments of the invention
has been presented for the purposes of illustration and
description. It is not intended to be exhaustive or to limit the
embodiments to the precise form disclosed. Many modifications and
variations are possible in light of the above teaching It is
intended that the scope of the embodiments be limited not by this
detailed description, but rather by the claims appended hereto. The
above specification, examples and data provide a complete
description of the manufacture and use of the composition of the
embodiments. Since many embodiments may be made without departing
from the spirit and scope of the invention, the embodiments reside
in the claims hereinafter appended or any subsequently-filed
claims, and their equivalents.
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