Transfer of Global Measures of Dependence into Cumulative Local ()
1. Introduction
The ideas of transferring integral measure of dependence into local measures are probably not new. We got the clue from the works of [1] -[3] . Cooke raised the idea of the use of the indicator random variables (r.v.’s) in regression models to consider the regression coefficients as measures of local dependence. Genest [4] gave some more motivations on the need of transfer of overall (integrated) measures to the cumulative local functions. Dimitrov [2] revived some old measures of dependence between random events proposed by Obreshkov [5] , and noticed their convenience in the use of the studies of local dependence between r.v.s. [6] is an attempt looking at the cumulative dependence from the point of view of the so called Sibuya function, which is also used in studies of dependence.
We propose here a review of the classic statistical analysis approach, where the procedures are more or less routine to be used in the analysis of local dependence. The manuscript is built in the following way: Section 2 introduces the idea of the transfer in detail. Section 3 considers the covariance and correlation matrices between indicator functions for a finite family of random events. At this point it is worth noticing that the expressions for the entries 
of the covariance matrix for identificators of random events 
coincide with the quantity 
introduced as connection between the events 
and 
Similarly, the entries 
of the correlation matrix coincide with the expression called in [5] as correlation coefficient between the events 
and
. Obreshkov approach is using different categories. Following these ideas, [2] exposes a relatively complete analysis of the two-dimensional dependence between random events, and establishes some more detailed properties of the Obreshkov measures of dependence. Similar expressions for the correlation between random events, are used in [4] . They call it correlation coefficient associated with pairs of dichotomous random variables. Long before [7] , p.311 uses correlation coefficient between indicator functions of two events, but does not elaborate any details.
Let 
and 
be two 
-sub-algebras of
. The survey of [3] proposes the measure
(1.1)
to be used as magnitude of dependence between 
and
. This measure is used for studying properties of a sequence 
of r.v.’s, when choosing 
and 
generated by two remote members—
and 
of the sequence. The 
is called mixing. The similarity with the Obreshkov’s connection measure between events is obvious. However, this measure of dependence 
is a global measure limited within the interval
, and does not have clear meaning of magnitude. Some of its properties are summarized in the statements of Section 3. Section 4 extends these results as integrated local measures of dependence between finite sequence of r.v.’s. Explicit presentations in terms of joint and marginal cumulative distribution functions (c.d.f.’s) are obtained. Some invariant properties are established in Section 5; Examples (explicit analytic and graphic) with popular copula distributions, and a bivariate Poisson distribution with dependent components illustrate these properties. 3-d surfaces with level curves consisting of points with constant cumulative magnitude of local dependence (covariance or correlation magnitude) in the sample space are drawn (Sections 5, 6, and 7).
2. The Idea for Measure Transfer
Let 
be random variables (r.v.’s) defined on a probability space 
with joint cumulative distribution function (c.d.f.)
and marginals
Most of the conventional measures of dependence between r.v.’s used in the practice are integral (global). Usually they have the form of expected values of some specifically selected function
. Sometimes 
can be a real number, a vector, or a matrix. The dependence measure “of type
” is expressed by the equation
Let 
be a finite set of random events in the sample space 
of an experiment, where the r.v.’s 
are defined. Introduce the indicator r.v.’s
Then the domain of the r.v.’s 
is a subset of the sample space 
of the r.v.’s 
Definition 1. The quantity
is called measure of dependence of the type 
between the random events
.
Assume, the measure of dependence of type 
is introduced, and its properties are well established and understood. For any choice of the point 
define the random events
(1.2)
Obviously, 
define the marginal distribution functions of the r.v.’s
. Put these events in Definition 1, and get naturally the next:
Definition 2. The quantities
are called cumulative local measure of dependence of type 
between the r.v.’s 
at the point
.
As an illustration of the above idea we consider the frequently used and well known integral measures within a finite sequence of r.v.’s, the covariance and correlation matrices.
3. Covariance and Correlation between Random Events
Introduce vector-matrices 
and
, and their transposed 
and
. The covariance matrix between r.v.’s 
is
(1.3)
By implementing the idea of the previous section, we introduce Covariance matrix between random events
.
First, observe that 
and
Introduce vectors
and
Formally, the covariance matrix 
between the random events 
is defined by
Its entries are
(1.4)
On the main diagonal of the covariance matrix 
we recognize the entries
(1.5)
where 
is the complement event to the event
. By making use of the standard approach we introduce the correlation matrix between random events
whose entries are the correlation coefficients between the indicators of respective events. Based on (1.4) and (1.5) for
, they have the form
(1.6)
We call these entries correlation coefficients between random events 
and
. As one should expect, the entries on main diagonal are equal to 1. Each entry of the correlation matrix for
, satisfies the inequalities
(1.7)
Let us focus on some basic properties of the entries of covariance and correlation matrices. They reveal also the information contained in terms of these matrices. Details can be seen in [2] .
Theorem 1. a) The covariance 
and the correlation coefficient 
between events 
and 
equal 0 if and only if 
and 
are independent. The case includes the situation when one of the events 
or 
is the sure, or the impossible event (then the two events also are independent);
b) 
if and only if events 
and 
coincide;
c) 
if and only if events 
and 
coincide (or equivalently, if and only if the events 
and 
coincide);
d) The covariances between events 
and 
satisfy the inequalities
(1.8)
The equalities in the middle of (0.8) hold only if the two events 
and 
obey the hypotheses stated in parts a), b) and c);
e) The covariance and the correlation coefficients between the events 
and 
and their complements 
and 
are related by the equalities
and
The advantage of the considered measures is that they are completely determined by the individual (marginal) probabilities 
of the random events, and by their joint probabilities 
An interesting fact is that we do not need the joint probability to calculate conditional probabilities between the events. If either of the measures of dependence 
or 
is known then the the conditional probability 
can be found when marginal probabilities 
are known. It holds Theorem 2. Given the measures 
or 
and marginal probabilities 
, the conditional probabilities of one event, given the other, can be calculated by the equations
(1.9)
Remark 1. If the covariance, or the correlation between two events 
and 
is positive, the conditional probability of one event, given the other one, increases, compare to its unconditional probability, i.e. 
and the exact quantity of increase equals to
(1.10)
Conversely, if the correlation coefficient is negative, the information that one of the events has occurred, the chances that the other one occurs decrease. The net amount of decrease in probability is also given by (1.10).
4. Covariance and Correlation Functions Describe Cumulative Local Dependence for Random Variables
Here we continue developing the idea of Section 2 to illustrate how the cumulative local measures of dependence between r.v.’s are obtained and expressed in terms of the conventional probability distribution functions. Let 
be arbitrary finite sequence of r.v.’s. For any choice of the real numbers
, let 
be the random events as defined in (1) 
The covariance and correlation matrices for random events 
immediately turn into respective integrated measures of local dependence between the r.v.’s 
at the selected point
. We discuss these matrices here, and focus on their specific forms in terms of the respective distribution functions. The covariance matrix function
whose entries are defined for any pair 
by the equations
(1.11)
represents the cumulative local covariance structure between r.v.’s 
at the point
. Analogously, we obtain the correlation matrix function
whose entries according to Equation (1.6), are
(1.12)
Obviously, these matrices are functions of the joint c.d.f.’s 
and of the marginals 
of the participating r.v.’s. The properties of these entries are described by Theorem 1, and Theorem 2. We notice here the following particular and important facts.
Theorem 3. a) The equality
holds if and only if the two events
, and 
are independent at the point
;
b) The equality
holds if and only if the two events
, and 
coincide at the point
;
c) The equality
holds if and only if the two events
, and the complement 
coincide at the point
. Then also the events
, and 
coincide.
d) Given the dependence function measures 
or 
and marginal c.d.f.’s 
, the conditional probabilities 
and 
can be evaluated by the expressions
and
Proof. The statement is just an illustration of the transfer of the results of Theorems 1 and 2 from random events to the case of events depending on parameters. We omit details.
Theorem 3 d) provides the opportunity to make predictions for either of the events
, or
, given that the other event occurs, or does not occur. The absolute values of the increase (or decrease) of the chances of the r.v. 
to get values less than a number 
under conditions of known values of the variable 
compare to these chances without conditions, can be evaluated as shown in Remark 1. The evaluation is expressed in terms of the local measures of dependence and the marginal c.d.f.’d of participating r.v.’s.
Remark 4. [8] introduces positive quadrant dependence between two r.v.’s by the requirement: for all 
to be fulfilled 
This is equivalent to the validity of 
everywhere. The reverse inequality, when it remains true for all 
Nelsen calls as negative quadrant dependence. Also there ([8] p. 189) the author gives some discussion about Local Positive (and Local Negative) Quadrant Dependencies between r.v.’s 
and 
at a point
, and does not go any further in its possible use. We are confident, that having local measure of dependence at a point, and especially, the measure of its magnitude given by the correlations functions 
offers the opportunity to use more. Namely, it is possible to classify the points in the plane according to their magnitude of dependence. For instance, there might be points of marginal independence, where the covariance equals zero. There can be areas of positive and negative dependence, depending on the sign of the covariance. There can be curves of constant cumulative correlations, e.g. curves given by the equations, like 
or
, etc. Our examples below provide effective proofs of such facts.
5. Monotone Transformations of the Random Variables and Related Dependence Functions. Copula
Here we study how the dependence functions behave under increasing or decreasing transformations. Before we move on, we introduce the following notations in order to make exposition clearer. We will use the notations 
and 
for the covariance and correlation matrices of the random variables 
where 
are arbitrary functions defined on the range of the arguments
. Also we introduce notations 
and 
to mark the variables for which the respective dependence functions are pertained.
Now we present the results with the following statement:
Theorem 4. Given arbitrary continuous functions 
on the supports of the continuous r.v.’s 
and 
respectively, such that 
and 
do exist, we have:
If 
and 
are all increasing functions, or if 
are all decreasing functions, then the values of the respective covariance and correlation functions at the respective points are related by the equations
and
Also the relations
and
are valid.
Proof. The statement follows from the fact that the following equivalence between the random events at any point 
are valid:
If 
and 
are both increasing functions, then the following random events are equivalent
and equivalent are also the events
for all
.
If 
and 
are both decreasing functions, then the following random events are equivalent
and also equivalent are
for all
.
Next, we use the facts that the covariance and correlation matrix functions introduced in (1.11) and (1.12) for pairs of random events 
and 
are the same for any pairs of equivalent events
, as well as for the pair of their complements
, according to Lemma 2 part b) are equal. The reason is that the respective probabilities used in the definition of the matrix entries of equivalent events also are equal.
Assume now, that 
are continuous r.v.’s, and 
is their continuous distribution function, as well as its all marginals are. According to [8] , there exists unique n-dimensional copula 
(a n-dimensional distribution on the n-dimensional hyper-cube 
with uniform 1-dimensional marginals and any 2- and higher-dimensional marginals also being copulas) associated with
, such that it is true
(1.13)
Then it is true also
(1.14)
Denote by 
and by 
the covariance and the correlation matrix functions corresponding to the uniformly distributed on 
r.v.’s 
whose joint distribution function is the copula
. We introduce the notations 
and 
for the covariance and correlation matrix functions corresponding to the original r.v.’s
.
The following statement holds:
Theorem 5. If 
is a continuous 
-dimensional distribution function and 
is its associated copula, then the following relationships are true
and also
Proof. The proof of these relationships is a simple consequence from Theorem 4, when we notice that the copula is just the distribution of a set of random variables, obtained after a monotonically increasing transformation 
or 
is applied to each component of the original random vector
, or to the vector 
Both theorems explain how after transformation the local cumulative dependence structure for the new variables is transfered to the points related with the same transformations at each coordinate of the other set of random variables. The fact of curiosity is that the relationships in Theorem 5 completely repeats the relationships (1.13) and (1.14) which define the copula.
The next examples illustrate the covariance and correlation functions for two popular multivariate copula, which we borrow from [8] .
Examples
1) Farlie-Gumbel-Morgenstern (FGM) n-copula
The FGM n-copula is given by the equation
where the coefficients 
and 
The entries of the covariance and the correlation matrices are respectively
and
We observe here, that the covariance and correlation functions keep constantly the sign of the parameter
.
Moreover, the correlation functions 
do not exceed the numbers 
for all
.
2) Clayton Multivariate Archimedean copula
The copula is given for any
, by the equation
where 
and the parameter 
For negative values of 
the maximum of the expression in the brackets and 0 must be taken, and 
must be fulfilled. For instance, if 
we should have
.
The entries of the covariance and the correlation matrices in case of positive
, are respectively
and
6. Maps for Dependence Structures on the Sample Space
A thorough analysis of the results in previous sections leads to an interesting opportunity to divide the sample space for any pair of r.v.’s 
into areas of positive, or negative dependence between the events 
and
. For this purpose, one can use the following curves:
Curve of Independence, 
the set of all points 
in 
where the correlation function
. In other words,
Here the events 
and 
are independent for all points 
.
Curve of Positive Complete Dependence, 
the set of all points
, where the correlation function
. In other words,
Here the events 
and 
coincide for all points 
.
Curve of Negative Complete Dependence, 
the set of all points
, where the correlation function
. In other words,
Here either of the events 
and 
is equivalent to the complement of the other one for all points
.
Additional mapping can be drown by making use of the Correlation Level Curves at level 
At each point of such a curve
, the correlation function 
between the events 
and 
equals to
. and
.
When 
is positive, the random variables 
are positively associated at the points
. When 
is negative, the random variables 
are negatively associated ate the points
.
In particular applications such curves may have important impact on understanding the dependence structure between the random variables
. By the way, some of these curves may happen to be empty sets.
One can ilustrate these curves in the next examples, using 3-d plots of the surfaces 
and 
with the level curves. In our plots we have used the program system Maple, and graph just the correlation function.
Examples
Here we illustrate the information one may visually get from the mapping of the 3-d graphs for the considered above copula. We plot their correlation function 
with
. Since the pictures are the same for any pair of indices
, we use notations without any indices.
1) The correlation functions for FGM copula (continued)
The two-dimensional marginal distributions of the multivariate FGM copula are given by one and the same equation, and we drop the subscripts:
where 
The correlation function is respectively
and satisfies the inequalities 
On Figure 1 we see the 3-dimensional graph of the surface
, for the cases 
and
.
2) Clayton Archimedean copula (continued)
The two-dimensional marginal distributions of the multivariate Clayton copula also are given by the same equations, and we drop the subscripts:
The copula is given by the equation
and in the multivariate case it is required that 
However the Clayton copula in the two-dimensional case is defined for all 
and 
The respective correlation function is
and this relation is valid for all
.
To have a base for comparison (with the pictures in [8] p. 120, Figure 4.2), we consider the case
, and present several level curves on the 3-d graph of 
Actually, the level curves 
are visible on the surface from which we see that 
is between 0 and 1.0. Results are pictured of Figure 2.
The 3-d graphs for this particular measure shown on Figure 2(b) of dependence between the two components of the copula closely imitate the situations given in the scatter-plots in the reference book [8] , but are more impressive and information giving.
Together with the surface plots, the correlation function and its level curves are quite more informative than the simulated scatter-plots used in copula.
(a) (b)
Figure 1. Correlation function for the Farlie-Gumbel-Morgenstern copula with level curves, θ = ±1. (a) FGM-copula correlation function surface, θ = −1; (b) FGM copula correlation function surface, θ = 1.
(a) (b)
Figure 2. Correlation function for the Clayton copula, θ = 4. (a) Clayton copula correlation function surface-a look from (1,1) corner; (b) Clayton copula correlation function surface-a look from (9,0) corner.
7. On the Use of the New Measures
At this point we believe that the introduced local measures of dependence can be used in similar ways as the covariance and correlation matrices are used in the multivariate analysis. However, because we think that the ideas developed here are so natural and simple, there will be no wonder if what we propose here is already in use. But, with the best of our knowledge, we have never seen anything similar, and we dare to say it under the risk we are not the first who offers such approach.
The results of Section 3 for random events may suit analysis of non-numerical, categorical, and finite discrete cases of random data similar to the ways as the numeric data are treated.
The results of Section 4 show that we are getting an opportunity to treat mixed discrete and continuous variables equally, using the cumulative distribution functions (marginal, and the two-dimensional).
If one treats the induces 
of the events 
as time variables (time is
), the above approach may offer interesting applications in the areas of Time Series Analysis, as well as in the studies of sequences of random variables.
Obtained results may have impact on various side effect questions, and we leave it for other’s comments. We insist on main practical orientation of our results-same and similar as everything else build on the use of the covariance and correlation matrices in the multivariate analysis.
8. Further Expansion of the Idea of Transforming Measures
In addition, the new measures may be used in studying the local structures of dependence within rectangular areas of the sample spaces when the sets of random events are defined as type of Cartesian products
and by making use of the main idea of the transfer from the Introduction.
Another extension that may allow the involvement of the probability density functions (p.d.f.) could be obtained as in Section 4, by considering the elementary events
whose probabilities are
Here 
and 
are the marginal and the joint probability densities of the r.v.’s 
and of the pair 
Since the densities do not have meaning of probabilities, for now we cannot use them in similar constructions. However, the picture changes in case of discrete variables.
As an illustration of this approach in the study of local dependence structure we consider the discrete case. The r.v.’s 
and 
have joint distribution 
and marginals 
and 
Then the covariance function 
is defined by the equalities
The respective correlation function will be defined by
Graphing these functions, one can see the local dependence structure between the two variables at any point 
and to evaluate its magnitude, and in addition will see the entire picture of this mutual dependence in the field-the range of the pair 
As an illustration of this suggestion we consider the simplest Bivariate Poisson distribution constructed in the following way, borrowed from [9] . There are three Poisson distributed r.v.’s 
of parameters 
Then it is supposed 
The dependence between 
and 
comes from the involvement of 
in both components. Their marginal distributions are Poisson with parameters 
and 
correspondingly. The joint distribution is given by
After substitution of this form of the joint distribution and the marginal Poisson distributions in the above expressions, we get the covariance and correlation functions and the study of the local dependence can start. We omit detailed analytical expressions for these functions which is not a problem to get but are too cumbersome to write here.
On Figure 3 we give an illustration of these functions by their 3-d graphs in the case when 
and 
The magnitude of dependence is not strong (the local correlation coefficient is too small). However, we observe that on the square 
the association is positive, when on the rectangle 
this dependence is negative and also is a bit stronger. Outside these areas the two variables are almost independent.
More exotic spots of dependences can be obtained when in Definition 2 the events 
are chosen as
where 
are any Borel sets on the real line, with 
In our opinion, there is a number of follow up questions in the multivariate dependence analysis, and more illustrations on real statistical data will trace the utility of the offered approach.
9. Conclusions
In this paper we discuss the possibility to transfer conventional global measures of dependence onto measures of
(a) (b)
Figure 3. Surfaces explain local dependence structure in the Bivariate Poisson distribution. (a) Dependent PoissonCovariance function; (b) Dependent Poisson-Correlation function.
dependence between random events, and then using another transfer to turn them into cumulative local measures of dependence between random variables.
We illustrate the method on the example of covariance and correlation matrices between random variables. Important results can be found in the discussion on some specific properties of the new measures, and in the qualitative information that may be derived from the meaning of these measures.
The behavior of these measures under monotonic transformations is established. It shows full coincidence between the pictures in the sample space of the original random variables and their respective copulas.
We found that locally the random variables may have areas of positive dependence and areas of negative dependence, as well as areas of local independence. This seems something interesting from informational point of view compared to the global measures of dependence.
Our suggestion is that the local measures of dependence offer important topological pictures to study. Our examples illustrate the new opportunities in this field.
Acknowledgements
The authors acknowledge FAPESP (Grant 06/60952-1, and Grant 03/10105-2), which gathered us for several days, where we had the opportunity to start the work on this exciting topic. We are thankful to Professor Roger Cook for his then unexpected spontaneous support during the Third Brazilian Conference on Statistical Modeling in Insurance and Finance, Maresias, Brazil.