Nonlinear Principal and Canonical Directions from Continuous Extensions of Multidimensional Scaling ()
1. Introduction
Let 
be a random variable on a probability space 
with range
, absolutely continuous cdf 
and density 
w.r.t. the Lebesgue measure. Our main purpose is to expand (a function of) 
as
(1)
where 
is a sequence of uncorrelated random variables, which can be seen as a decomposition of the so-called geometric variability 
defined below, a dispersion measure of 
in relation with a suitable distance function 
Here orthogonality is synonymous with a lack of correlation.
Some goodness-of-fit statistics, which can be expressed as integrals of the empirical processes of a sample, have expansions of this kind [1-3]. Expansion (1) is obtained following a similar procedure, except that we have a sequence of uncorrelated rather than independent random variables. Finite orthogonal expansions appear in analysis of variance and in factor analysis. Orthogonal expansions and series also appear in the theory of stochastic processes, in martingales in the wide sense ([4], Chap. 4; [5], Chap. 10), in non-parametric statistics [6], in goodness-of-fit tests [7,8], in testing independence [9] and in characterizing distributions [10].
The existence of an orthogonal expansion and some classical expansions is presented in Section 2. A continuous extension of matrix formulations in multidimensional scaling (MDS), which provides a wide class of expansions, is presented in Section 3. Some interesting expansions are obtained in Section 4 for a particular distance as well as additional results, such as an optimal property of the first dimension. Section 5 contains an inequality concerning the variance of a function. Section 6 is devoted to diagonal expansions from the continuous scaling point of view. Sections 7 and 8 are devoted to canonical correlation analysis, including a continuous generalization. This paper extends the previous results on continuous scaling [11-14], and other related topics [15,16].
2. Existence and Classical Expansions
There are many ways of obtaining expansion (1). Our aim is to obtain some explicit expansions with good properties from a multivariate analysis point of view. However, before doing this, let us prove that a such expansion exists and present some classical expansions.
Theorem 1. Let 
be an absolutely continuous r.v. with density 
and support 
and 
a measurable function such that 
and 
Then there exists a sequence 
of uncorrelated r.v.’s such that
(2)
with 
where the series 
converges in the mean square as well as almost surely.
Proof. Consider the Lebesgue spaces 
and 
of measurable functions 
on 
such that 
and 
respectively. Obviously, 
and 
are separable Hilbert spaces with quadratic-norms 
and
, respectively. Moreover 
given by 
is a linear isometry of 
onto 
i.e., 
Let 
be an orthonormal basis for
. Accordingly, 
given by
is an orthonormal basis for
. The assumption
, together with
, is equivalent to
. In fact, 
Hence 
where 
are the Fourier coefficients and 
Letting 
we deduce that
, where the series converges in 
and 
where 
is Kronecker’s delta. Replacing 
by
, and defining
, we obtain 
This series converges almost surely, since the series 
converges in 
We may suppose 
Next, for
, we have cov 
as asserted. Finally, note that 
In particular, the series in (2) converges also in the mean square. W Some classical expansions for 
or a function of 
are next given.
2.1. Legendre Expansions
Let 
be the cdf of 
An all-purpose expansion can be obtained from the shifted Legendre polynomials 
on 
where 
The first three are
Note that 
Thus we may consider the orthogonal expansion
where 
and 
are the Fourier coefficients. This expansion is quite useful due to the simplicity of these polynomials, is optimal for the logistic distribution, but may be improved for other distributions, as it is shown below.
2.2. Univariate Expansions
A class of orthogonal expansions arises from
(3)
where both 
are probability density functions. Then 
is a complete orthonormal set w.r.t. 
2.3. Diagonal Expansions
Lancaster [17] studied the orthogonal expansions
(4)
where 
is a bivariate probability density with marginal densities 
Then 
are complete orthonormal sets w.r.t 
and 
respectively. Moreover 
and 
where 
is the n-th canonical correlation between 
and 
Expansion (4) can be viewed as a particular extension of Theorem 3, proved below, when the distance is the so-called chi-square distance. This is proved in [18]. See Section 6.
3. Continuous Scaling Expansions
In this section we propose a distance-based approach for obtaining orthogonal expansions for a r.v.
, which contains the Karhunen-Loève expansion of a Bernoulli process related to 
as a particular case. We will prove that we can obtain suitable expansions using continuous multidimensional scaling on a Euclidean distance.
Let 
be a dissimilarity function, i.e., 
and 
for all 
Definition 1. We say that 
is a Euclidean distance function if there exists an embedding 
where 
is a real separable Hilbert space with quadratic norm
, such that 
We may always view the Hilbert space 
as a closed linear subspace of 
In this case, we may identify 
with 
where for 
is a vector in 
Accordingly, for 
(5)
Definition 2. 
is called a Euclidean configuration to represent 
The geometric variability of 
w.r.t. 
is defined by
The proximity function of 
to 
is defined by
(6)
The double-centered inner product related to 
is the symmetric function
(7)
These definitions can easily be extended to random vectors. For example, if 
is 
is an observation of 
and 
is the Euclidean distance in
, then 
and 
is the Mahalanobis distance from 
to 
The function 
is symmetric, semi-definite positive and satisfies
(8)
It can be proved [19], that there is an embedding 
of 
into 
such that
G is the continuous counterpart of the centered inner product matrix computed from a 
distance matrix and used in metric multidimensional scaling [20,21]. The Euclidean embedding or method of finding the Euclidean coordinates from the Euclidean distances was first given by Schoenberg [22]. The concepts of geometric variability and proximity function, were originated in [23] and are used in discriminant analysis [19], and in constructing probability densities from distances [24]. In fact, 
is a variant of Rao’s quadratic entropy [25]. See also [14,26].
In order to obtain expansions, our interest is on 
and 
i.e., we substitute 
by the r.v. 
For convenience and analogy with the finite classic scaling, let us use a generalized product matrix notation (i.e., a “multivariate analysis notation”), following [18]. We write 
as 
where 
is a Euclidean centered configuration to represent 
according to (5), (8), i.e., we substitute 
by 
in 
and suppose that each 
has mean 0. The covariance matrix of 
is 
where for 
and may be expressed as
where 
stands for the continuous diagonal matrix 
and the row × column multiplication, denoted by 
is evaluated at 
and then follows an integration w.r.t. 
In the theorems below, 
and 
stands for 
and 
Theorem 2. Suppose that for a Euclidean distance 
the geometric variability 
is finite. Define 
Then:
1. 
2. 
is a Hilbert-Schmidt kernel, i.e.,
Proof. Let 
be i.i.d. and write 
From (7) 
and hence using (8),
This proves 1). Next, 
is p.s.d., so for 
the 
matrix with entries 
is p.s.d. In particular, for 
with 
the determinant
is non-negative. Thus
Integrating this inequality over 
gives
and 2) is also proved. W Theorem 3. Suppose that 
for a Euclidean distance 
is finite. Then the kernel 
admits the expansion
(9)
where 
is a complete orthonormal set of eigenfunctions in
. Let
(10)
and 
Then 
and 
is a sequence of centered and uncorrelated r.v.’s, which are principal components of 
where 
and
Proof. The eigendecomposition (9) exists because 
is a Hilbert-Schmidt kernel and 
by Theorem 2. Next, (9) and (10) can be written as
(11)
i.e., 
and 
where 
Thus, for all 
Next, for 
we have 
In particular, since
we have 
Moreover, from (10) we also have
where 
is Kronecker’s delta, showing that the variables 
are centered and uncorrelated.
Recall the product matrix notation. The principal components of 
such that 
are 
where
(12)
is the spectral decomposition of 
Premultiplying (12) by 
and postmultiplying by 
we obtain 
and therefore
Thus the columns of 
are eigenfunctions of 
This shows that 
see
(11), and 
contains the principal components of 
The rows in 
may be accordingly called the principal coordinates of distance
. This name can be justified as follows.
Let us write the coordinates 
and suppose that 
is another Euclidean configurations giving the same distance
. The linear transformation 
is orthogonal and 
with 
Then the r.v.’s 
are uncorrelated and
Thus 
is the first principal coordinate in the sense that 
is maximum. The second coordinate 
is orthogonal to 
and has maximum variance, and so on with the others coordinates. W The following expansions hold as an immediate consequence of this theorem:
(13)
(14)
(15)
where 
and 
, with 
are sequences of centered and uncorrelated random variables, which are principal components of
. We next obtain some concrete expansions.
4. A Particular Expansion
If 
is a continuous r.v. with finite mean and variance, 
say, and 
is the ordinary Euclidean distance
, then it is easy to prove that 
Then from (14) and taking 
we obtain 
which provides the trivial expansion 
A much more interesting expansion can be obtained by taking the square root of
.
4.1. The Square Root Distance
Let us consider the distance function
(16)
The double-centered inner product 
is next given.
Definition 3. Let 
be defined as 
where 
is defined in (10).
We immediately have the following result.
Proposition 1. The function 
satisfies:
1. 
2. 
3. 
Proposition 2. Assuming 
i.i.d., if
, then
(17)
Proof. From 
and combining (7) and (6), we obtain 
where 
which satisfies 
Hence 
and (17) holds. W Proposition 3. The following expansion holds
(18)
Proof. Using (17) and expanding 
and setting 
we get (18). W Replacing 
by 
we have the expansion
(19)
and, as a consequence [12]:
where 
is distributed as 
and 
This expansion also follows from (15).
If 
from (19) we can also obtain the expansions
(20)
as well as
(21)
where the convergence is in the mean square sense [27].
4.2. Principal Components
Related to the r.v. 
with cdf 
let us define the stochastic process 
where 
is the indicator of 
and follows the Bernoulli 
distribution with
. For the distance (16), the relation between
, the Bernoulli process 
and 
is
(22)
where 
and 
is a realization of X. Thus X is a continuous configuration for 
Note that, if 
is finite, then
The covariance kernel 
of X is given by 
Let us consider the integral operator 
with kernel 
where 
is an integrable function on 
Let 
be the countable orthonormal set of eigenfunctions of 
i.e., 
We may suppose that 
constitutes a basis of 
and the eigenvalues 
are arranged in descending order. As a consequence of Mercer’s theorem, the covariance kernel 
can be expanded as
(23)
Theorem 4. The functions 
(see definition 3), satisfy:
1. 
2. 
is a countable set of uncorrelated r.v.’s such that 
3. 
are principal coordinates for the distance 
i.e.,
Proof. To prove 1), let us use the multiplication “*” and write 
as 
where
i.e., 
with 
where 
if 
and 0 otherwise. Then 
is a centered continuous configuration for 
and clearly 
Arguing as in Theorem 3, the centered principal coordinates are 
i.e.,
2) is a consequence of Theorem 3. An alternative proof follows by taking 
in the formula for the covariance [28]:
(24)
where 
See Theorem 6.
To prove 3), let 
and 
Then 
and 
see (22), is
This proves that 
are principal coordinates for the distance 
W The above results have been obtained via continuous scaling. For this particular distance, we get the same results by using the Karhunen-Loève (also called Kac-Siegert) expansion of X, namely,
(25)
where 
i.e., 
Thus each 
is a principal component of X and the sequence 
constitutes a countable set of uncorrelated random variables.
4.3. The Differential Equation
Let 
where 
It can be proved [12] that the means 
eigenvalues 
and functions 
satisfy the second order differential equation
(26)
The solution of this equation is well-known when 
is 
uniform.
Examples of eigenfunctions 
principal components 
and the corresponding variances 
are [12,27,29]:
1. 
is 
uniform: 
2. 
is exponential with unit mean. If 
where 
is the n-th positive root of 
and 
are the Bessel functions of the first order.
3. 
is standard logistic 
If 
(27)
where 
are the shifted Legendre polynomials on 
4. 
is Pareto with 
where
and 
Note that the change of variable 
transforms 
in 
and (26) in
Hence 
and 
providing solutions for the variable 
For instance, we immediately can obtain the principal dimensions of the Pareto distribution with cdf 
4.4. A Comparison
The results obtained in the previous sections can be compared and summarized in Table 1, where 
is a random variable with absolutely continuous cdf 
density 
and support 
The continuous scaling expansion is found w.r.t. the distance (16). Note that we reach the same orthogonal expansions (we only include two), but this continuous scaling approach is more general, since by changing the distance we may find other
Table 1 . Principal components and principal directions of a random variable.
principal directions and expansions. This distance-based approach may be an alternative to the problem of finding nonlinear principal dimensions [30].
4.5. Some Properties of the Eigenfunctions
In this section we study some properties of the eigenfunctions 
and their integrals 
Proposition 4. The first eigenfunction 
is strictly positive and satisfies
Proof. 
is positive, so 
is also positive (Perron-Frobenius theorem). On the other hand 
which satisfies 
W Clearly, if 
is positive, 
is increasing and positive. Moreover, any 
satisfies the following bound.
Proposition 5. If 
is finite then 
is also finite and
(28)
Proof. 
is an eigenfunction and from (24)
Hence 
W Proposition 6. The principal components 
of 
constitutes a complete orthogonal system of 
Proof. The orthogonality has been proved as a consequence of (24). Let 
be a continuous function such that 
Suppose that 
exists. As 
is a complete system 
and integrating, we have 
But 
which shows that 
must be constant. W
4.6. The First Principal Component
In this section we prove two interesting properties of 
and the first principal component 
see [10].
Proposition 7. The increasing function 
characterizes the distribution of 
Proof. Write 
Then 
satisfies the differential equation (see (26))
(29)
where 
When the function 
is given, 
and 
may be obtained by solving the equations
and the density of 
is 
W Proposition 8. For a fixed 
let 
denote the squared correlation between 
and a function 
The average of 
weighted by 
is maximum for 
i.e.,
Proof. Let us write (see Proposition 6) 
Then 
and we can suppose 
From (25)
As 
we have
Thus the supreme is attained at 
W
5. An Inequality
The following inequality holds for 
with the normal 
distribution [31,32]:
where 
is an absolutely continuous function and 
has finite variance. This inequality has been extended to other distributions by Klaassen [33]. Let us prove a related inequality concerning the function of a random variable and its derivative.
If 
is the first principal dimension, then 
and 
We can define the probability density 
with support 
given by
Theorem 5. Let 
be a r.v. with pdf 
If 
is an absolutely continuous function and 
has finite variance, the following inequality holds
(30)
with equality if 
is 
Proof. From Proposition 6, we can write 
where 
Then 
and
, so
From Parseval’s identity 
Thus we have
proving (30). Moreover, if 
we have 
and 
W Inequality (30) is equivalent to
(31)
Some examples are next given.
5.1. Uniform Distribution
Suppose 
uniform 
Then 
and 
We obtain
5.2. Exponential Distribution
Suppose 
exponential with mean 1. Then 
and
where 
satisfies 
Inequality (31) is
where 
5.3. Pareto Distribution
Suppose 
with density 
for 
Then 
and
where 
satisfies 
Inequality (31) is
where 
5.4. Logistic Distribution
Suppose that 
follows the standard logistic distribution. The cdf is
and the density is 
This distribution has especial interest, as the two first principal components are 
i.e., proportional to the cdf and the density, respectively. Note that 
can be obtained directly, as if we write 
then 
and (29) gives
so 
Similarly, we can obtain 
Besides
i.e., the expectation of 
w.r.t. 
with density 
is maximum for 
Now 
and 
The density 
is just
, therefore inequality (30) for the logistic distribution reduces to
In general, if 
is logistic with variance 
then 
with 
Noting that the functions 
are orthogonal to 
and using (24), we obtain
As 
the Cauchy-Schwarz inequality proves that
6. Diagonal Expansions
Correspondence analysis is a variant of multidimensional scaling, used for representing the rows and columns of a contingency table, as points in a space of low-dimension separated by the chi-square distance. See [15,34]. This method employs a singular value decomposition (SVD) of a transformed matrix. A continuous scaling expansion, viewed as a generalization of correspondence analysis, can be obtained from (3) and (4).
6.1. Univariate Case
Let 
be two densities with the same support 
Define the squared distance
The double-centered inner product is given by
and the geometric variability is
which is a Pearson measure of divergence between 
and 
If 
is an orthonormal basis for 
we may consider the expansion
and defining 
then
and
However, 
are not the principal coordinates related to the above distance. In fact, the continuous scaling dimension is 1 for this distance and it can be found in a straightforward way.
6.2. Bivariate Case
Let us write (4) as
(32)
where 
are the canonical functions and 
is the sequence of canonical correlations. Note that 
and 
Suppose that 
is absolutely continuous w.r.t. 
and let us consider the Radom-Nikodym derivative
The so-called chi-square distance between 
is given by
Let 
the Pearson contingency coefficient is defined by
The geometric variability of the chi-square distance is 
In fact
The proximity function is
and the double-centered inner product is
We can express (32) as
This SVD exists provided that 
Multiplying 
by himself and integrating w.r.t. 
we readily obtain
Comparing with (9) we have the principal coordinates 
where 
which satisfy
Then
See [18] for further details.
Finally, the following expansion in terms of cdf’s holds [35]:
(33)
where 
Using a matrix notation, this diagonal expansion can be written as
where 
stands for the diagonal matrix with the canonical correlations, and 
7. The Covariance between Two Functions
Here we generalize the well-known Hoeffding’s formula
which provides the covariance in terms of the bivariate and univariate cdf’s. The proof of the generalization below uses Fubini’s theorem and integration by parts, being different from the proof given in [28].
Let us suppose that the supports of 
are the intervals 
respectively, although the results may also hold for other subsets of 
We then have
Theorem 6. If 
are two functions defined on 
respectively, such that:
1. Both functions are of bounded variation.
2. 
3. The expectations 
are finite.
Then:
(34)
Proof. The covariance exists and is
where 
Integration by parts gives
and similarly 
By Fubini’s theorem for transition probabilities
where 
is the cdf of 
given
, we can write
We first integrate with respect to 
Setting 
to find 
integration by parts gives
Since 
setting 
we find
and setting 
again integration by parts gives
where 
Therefore
A last simplification shows that 
W
8. Canonical Analysis
Given two sets of variables, the purpose of canonical correlation analysis is to find sets of linear combinations with maximal correlation. In Section 6.2 we studied, from a multidimensional scaling perspective, the nonlinear canonical functions of 
and 
with joint pdf 
Here we find the canonical correlations and functions for several copulas.
Let 
be a bivariate random vector with cdf
, where 
and 
are 
uniform. Then 
is a cdf called copula [36]. Let us suppose 
symmetric. Then 
Therefore, in finding the canonical functions, we can suppose 
Let us consider the symmetric kernels
We seek the canonical functions 
and the corresponding canonical correlations, i.e.,
(35)
Definition 4. A generalized eigenfuction of 
w.r.t. 
with eigenvalue 
is a function 
such that
(36)
From the theory of integral equations, we have
(37)
Definition 5. On the set of functions in 
with 
we define the inner products
Clearly, see Theorem 6, if 
is eigenfunction of 
w.r.t. 
with eigenvalue 
we have the covariance 
and the variance 
Therefore the correlation between 
is:
Thus the canonical correlations of 
are the eigenvalues of 
w.r.t. 
Justified by the embedding in a Euclidean or Hilbert space via the chi-square distance, the geometric dimension of a copula 
is defined as the cardinal of the set 
of canonical correlations. The dimension can be finite, infinite countable (cardinality of
) or uncountable (cardinality of the continuum
).
We next illustrate these results with some copulas. Since 
is a copula, the so-called FréchetHoeffding upper bound, we firstly suggest a procedure for constructing copulas and performing canonical analysis. This procedure is based on the expansion (23) for the logistic distribution.
For this distribution, if 
we have [27]:
where 
is given in (27). With the change 
we find:
(38)
with 
and
where 
are Legendre polynomials and 
are shifted Legendre polynomials on
. Thus 
etc.
8.1. FGM Copula
If we consider only 
in (38), we obtain the Farlie-Gumbel-Morgenstern copula:
Then 
and 
if 
Thus 
is an eigenfunction with eigenvalue 
Then 
and 
are the canonical functions with canonical correlation 
The geometric dimension is one.
8.2. Extended FGM Copula
By taking more terms in expansion (38), we may consider the following copula
where 
The canonical correlations are 
The canonical functions are 
respectively. This copula has dimension 3.
Clearly 
reduces to the FGM copula if 
When the dimension is 2, i.e., 
we have a copula with cubic sections [37]. A generalization is given in [38].
8.3. Cuadras-Augé Copula
The Cuadras-Augé family of copulas [39] is defined as the weighted geometric mean of 
and 
For this copula, the canonical correlations constitute a continuous set. If 
for 
and 0 otherwise, it can be proved [40] that the set 
of eigenpairs of 
w.r.t. 
is given by
Thus, the set of canonical functions and correlations for the Cuadras-Augé copula is the uncountable set 
with dimension of the power of the continuum. In particular, the maximum correlation is the parameter 
with canonical function the Heaviside distribution 
The maximum correlation 
was obtained by Cuadras [28].
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