Robust multiple-set linear canonical analysis based on minimum covariance determinant estimator

1 Introduction

Many multivariate statistical methods are based on empirical covariance operators. That is the case for multiple regression, principal components analysis, factor analysis, linear discriminant analysis, linear canonical analysis, multiple-set linear canonical analysis, and so on. However, these empirical covariance operators are known to be extremely sensitive to outliers. That is an undesirable property that makes the preceding methods themselves sensitive to outliers. For overcoming this problem, robust alternatives for these methods have been proposed in the literature, mainly by replacing the aforementioned empirical covariance operators by robust estimators. In this vein, robust versions of multivariate statistical methods have been introduced, especially for multiple regression (

[21]), principal components analysis ([8],[10],[16],[22]), factor analysis ([19]), linear discriminant analysis ([6],[9],[14]), linear canonical analysis ([5],[24]). Multiple-set linear canonical analysis (MSLCA) is an important multivariate statistical method that analyzes the relationship between more than two random vectors, so generalizing linear canonical analysis. It has been introduced for many years (e.g., [12]) and has been studied since then under different aspects (e.g., [15],[23],[25]

). A formulation of MSLCA within the context of Euclidean random variables has been made recently (

[18]) and permitted to obtain an asymptotic theory for this analysis when it is estimated by using empirical covariance operators. To the best of our knowledge, such estimation of MSLCA is the one that have been tackled in the literature, despite the fact that it is known to be nonrobust as it is sensitive to outliers. So, there is a real interest in introducing a robust estimation of MSLCA as it was done for the others multivariate statistical methods. This can be done by using robust estimators of the covariance operators of the involved random vectors instead of the empirical covariance operators. Among such robust estimators, the minimum covariance determinant (MCD) estimator has been extensively studied ([1], [2],[3],[7]), and it is known to have good robustness properties. Also, its asymptotic properties have been obtained ([1],[2],[3]) mainly under elliptical distribution.

In this paper, we propose a robust version of MSLCA based on MCD estimator of the covariance operator. We start by recalling, in Section 2, the notion of MSLCA for Euclidean random variables and we study its robustness properties by deriving the influence functions of the functionals that lead to its estimator from the empirical covariance operators. It is proved that the influence function of the operator that determines MSLCA is not bounded. In Section 3, we introduce a robust estimation of MSLCA (denoted by RMSLCA) by using the MCD estimator of the covariance operator on which this analysis is defined. Then we derive the influence function of the operator that determines RMSLCA, which is proved to be bounded, and that of the canonical coefficients and the canonical directions. Section 4 is devoted to asymptotic properties of RMSLCA. We obtain limiting distributions that are then used in Section 5 where a robust test for mutual non-correlation is introduced. The robustness properties of this test are studied through the derivation of the second order influence function of the test statistic under the null hypothesis. The proofs of all theorems and propositions are postponed in Section 6.

2 Influence in multiple-set canonical analysis

In this section we recall the notion of multiple-set linear canonical analysis (MSLCA) of Euclidean random variables as introduced by Nkiet[18], and also its estimation based on empirical covariance operators. Then, the robustness properties of this analysis are studied through derivation of the influence functions that correspond to the functionals related to it.

2.1 Multiple-set linear canonical analysis

Letting $(Ω, A, P)$

be a probability space, and

K

be an integer such that

K \geq 2

, we consider random variables

X_{1}, \dots, X_{K}

defined on this probability space and with values in Euclidean vector spaces

X_{1}, \dots, X_{K}

respectively. We then consider the space

X := X_{1} \times \dots X_{K}

which is also an Euclidean vector space equipped with the inner product

⟨ \cdot, \cdot ⟩_{X}

defined by:

\forall α \in X, \forall β \in X, ⟨ α, β ⟩_{X} = K \sum k = 1 ⟨ α_{k}, β_{k} ⟩_{k},

where $⟨ \cdot, \cdot ⟩_{k}$ is the inner product of $X_{k}$ and $α = (α_{1}, \dots, α_{K})$ , $β = (β_{1}, \dots, β_{K})$ . From now on, we assume that the following assumption holds :

( $A_{1}$ ): for $k \in {1, \dots, K}$ , we have $E (X_{k}) = 0$ and $E (∥ X_{k} ∥_{k}^{2}) < + \infty$ , where $∥ \cdot ∥_{k}$ denotes the norm induced by $⟨ \cdot, \cdot ⟩_{k}$ .

Then, we consider the random vector $X = (X_{1}, \dots, X_{K})$ with values in $X$ , and we can give the following definition of multiple-set linear canonical analysis (see [18]):

Definition 2.1

The multiple-set linear canonical analysis (MSLCA) of $X$ is the search of a sequence ${(α^{(j)})}_{1 \leq j \leq q}^{(j)}$ of vectors of $X$ , where $q = dim (X)$ , satisfying:

α^{(j)} = arg max α \in C_{j} E (< α, X >_{X}^{2}),

(1)

where $C_{1} = {α \in X / \sum_{k = 1}^{K} v a r (< α_{k}, X_{k} >_{k}) = 1}$ and for $j \geq 2$ :

C_{j} = {α \in C_{1} / K \sum k = 1 c o v (< α_{k}^{(r)}, X_{k} >_{k}, < α_{k}, X_{k} >_{k}) = 0, \forall r \in {1, \dots, j - 1}} .

A solution of the above maximization problem is obtained from spectral analysis of an operator that will know be specified. For $(k, ℓ) \in {1, \dots, K}^{2}$ , let us consider the covariance operators

V_{k ℓ} = E (X_{ℓ} \otimes X_{k}) = V_{ℓ k}^{*} and V_{k} := V_{k k},

where $\otimes$

denotes the tensor product such that

x \otimes y

is the linear map :

h \mapsto< x, h > y

, and

A^{*}

denotes the adjoint of

A

. Letting

τ_{k}

be the canonical projection

τ_{k} : α \in X \mapsto α_{k} \in X_{k}

which adjoint operator $τ_{k}^{*}$ is the map

τ_{k}^{*} : t \in X_{k} \mapsto (0, \dots, 0      k - 1 times, t, 0, \dots, 0) \in X,

we consider the operators defined as

Φ = K \sum k = 1 τ_{k}^{*} V_{k} τ_{k} and Ψ = K \sum k = 1 K \sum ℓ = 1 ℓ \neq k τ_{k}^{*} V_{k ℓ} τ_{ℓ} .

(2)

The covariance operator $V_{k}$ is a self-adjoint and positive operator; we assume throughout this paper that it is invertible. Then, it is easy to check that $Φ$ is also self-adjoint positive and invertible operator, and we consider

T = Φ^{- 1 / 2} Ψ Φ^{- 1 / 2} .

The spectral analysis of this last operator gives a solution of the maximization problem specified in Definition 2.1. Indeed, if ${β^{(1)}, \dots, β^{(q)}}$ is an orthonormal basis of $X$ such that $β^{(j)}$

is an eigenvector of

T

associated with the

j

-th largest eigenvalue

ρ_{j}

, then we obtain a solution of (1) by taking

α^{(j)} = Φ^{- 1 / 2} β^{(j)}

, and we have

ρ_{j} =< β^{(j)}, T β^{(j)} >_{X}

. Finally, the MSLCA of

X

is the family

{(ρ_{j}, α^{(j)})}_{1 \leq j \leq q}^{(j)}

obtained as indicated above. The

ρ_{j}

’s are termed the canonical coefficients and the

α^{(j)}

’s are termed the canonical directions.

Note that $T$ can be expressed as a function of the covariance operator $V = E (X \otimes X)$ of $X$ . Indeed, denoting by $L (X)$ the space of linear maps fom $X$ to itself, and considering the linear maps $f$ and $g$ from $L (X)$ to itself defined as

f (A) = K \sum k = 1 τ_{k}^{*} τ_{k} A τ_{k}^{*} τ_{k} and g (A) = K \sum k = 1 K \sum ℓ = 1 ℓ \neq k τ_{k}^{*} τ_{k} A τ_{ℓ}^{*} τ_{ℓ},

(3)

it is easy to check, by using properties of tensor produts (see [11]), that

V_{k ℓ} = E ((τ_{ℓ} (X)) \otimes (τ_{k} (X))) = τ_{k} E (X \otimes X) τ_{ℓ}^{*} = τ_{k} V τ_{ℓ}^{*}, V_{k} = τ_{k} V τ_{k}^{*}

(4)

and, therefore, from (2), (3) and (4), it follows

T = f {(V)}^{- 1 / 2} g (V) f {(V)}^{- 1 / 2} .

2.2 Estimation based on empirical covariance operator

Now, we recall the classical way for estimating MSLCA by using empirical covariance operators (see, e.g., [18]). For $k = 1, \dots, K$ , let ${X_{k}^{(i)}}_{1 \leq i \leq n}$ be an i.i.d. sample of $X_{k}$ . We then consider the sample means and empirical covariance operators defined for $(k, ℓ) \in {1, \dots, K}^{2}$ as

	${¯ ¯¯¯ ¯ X}_{k \cdot n} = \frac{1}{n} n \sum i = 1 X_{k}^{(i)}, {ˆ V}_{k ℓ \cdot n}$	$=$	$\frac{1}{n} n \sum i = 1 (X_{ℓ}^{(i)} - {¯ ¯¯¯ ¯ X}_{ℓ \cdot n}) \otimes (X_{k}^{(i)} - {¯ ¯¯¯ ¯ X}_{k \cdot n})$
		$=$	$\frac{1}{n} n \sum i = 1 X_{ℓ}^{(i)} \otimes X_{k}^{(i)} - {¯ ¯¯¯ ¯ X}_{ℓ \cdot n} \otimes {¯ ¯¯¯ ¯ X}_{k \cdot n},$

and ${ˆ V}_{k \cdot n} := {ˆ V}_{k k \cdot n}$ . These permit to define random operators, with values in $L (X)$ , as

(5)

and to estimate $T$ by

{ˆ T}_{n} = {ˆ Φ}_{n}^{- 1 / 2} {ˆ Ψ}_{n} {ˆ Φ}_{n}^{- 1 / 2} .

(6)

Considering the eigenvalues ${ˆ ρ}_{1 \cdot n} \geq {ˆ ρ}_{2 \cdot n} \dots \geq {ˆ ρ}_{q \cdot n}$ of ${ˆ T}_{n}$ , and ${{ˆ β}_{n}^{(1)}, \dots, {ˆ β}_{n}^{(q)}}$ an orthonormal basis of $X$ such that ${ˆ β}_{n}^{(j)}$ is an eigenvector of ${ˆ T}_{n}$ associated with ${ˆ ρ}_{j \cdot n}$ , we can estimate $ρ_{j}$ by ${ˆ ρ}_{j \cdot n}$ , $β^{(j)}$ by ${ˆ β}_{n}^{(j)}$ and $α^{(j)}$ by ${ˆ α}_{n}^{(j)} = {ˆ Φ}_{n}^{- 1 / 2} {ˆ β}_{n}^{(j)}$ .

The random operator ${ˆ T}_{n}$ can also be expressed as a function of the empirical covariance operator of the $X^{(i)}$ ’s that are defined as $X^{(i)} = (X_{1}^{(i)}, \dots, X_{K}^{(i)})$ ; this empirical covariance operator is

	${ˆ V}_{n}$	$=$	$\frac{1}{n} n \sum i = 1 (X^{(i)} - {¯ ¯¯¯ ¯ X}_{n}) \otimes (X^{(i)} - {¯ ¯¯¯ ¯ X}_{n})$
		$=$	$\frac{1}{n} n \sum i = 1 X^{(i)} \otimes X^{(i)} - {¯ ¯¯¯ ¯ X}_{n} \otimes {¯ ¯¯¯ ¯ X}_{n} .$

where ${¯ ¯¯¯ ¯ X}_{n} = n^{- 1} \sum_{i = 1}^{n} X^{(i)} .$ Since ${ˆ V}_{k ℓ \cdot n} = τ_{k} {ˆ V}_{n} τ_{ℓ}^{*}$ and ${ˆ V}_{k \cdot n} = τ_{k} {ˆ V}_{n} τ_{k}^{*}$ , we straighforwardly obtain from (3), (5) and (6):

{ˆ T}_{n} = f {({ˆ V}_{n})}_{n}^{- 1 / 2} g ({ˆ V}_{n}) f {({ˆ V}_{n})}_{n}^{- 1 / 2} .

(7)

2.3 Influence functions

For studying the effect of a small amount of contamination at a given point on MSLCA it is important, as usual in robustness litterature (see [13]), to use influence function. More precisely, we have to derive expressions of the influence functions related to the functionals that give $T$ , $ρ_{j}$ and $α^{(j)}$ (for $1 \leq j \leq q$ ) at the distribution $P_{X}$ of $X$ . Recall that the influence function of a functional $S$ at $P$ is defined as

IF (x; S, P) = lim ε ↓ 0 \frac{S ((1 - ε) P + ε δ_{x}) - S (P)}{ε},

where $δ_{x}$ is the Dirac measure putting all its mass in $x$ .

First, we have to specify the functionals related to $T$ , $ρ_{j}$ and $α^{(j)}$ (for $1 \leq j \leq q$ ) and their empirical counterparts. Let us consider the functional $T$ given by

T (P) = f {(V (P))}^{- 1 / 2} g (V (P)) f {(V (P))}^{- 1 / 2} .

where $V$ is the functional defined as

V (P) = \int x \otimes x d P (x) - (\int x d P (x)) \otimes (\int x d P (x)) .

Applying this functional to the distribution $P_{X}$ of $X$ gives $V (P_{X}) = V$ and, therefore, $T (P_{X}) = T$ . For $j \in {1, \dots, q}$ , denoting by $R_{j}$ (resp. $B_{j}$ ; resp. $A_{j}$ ) the functional such that $R_{j} (P)$ is the $j$ -th largest eigenvalue of $T (P)$ (resp. the associated eigenvector; resp. $A_{j} (P) = f {(V (P))}^{- 1 / 2} B_{j} (P)$ ), we have $R_{j} (P_{X}) = ρ_{j}$ , $B_{j} (P_{X}) = β^{(j)}$ and $A_{j} (P_{X}) = α^{(j)}$ .

Furthermore, denoting by $P_{n}$ the empirical measure corresponding to the sample ${X^{(1)}, \dots, X^{(n)}}$ , we have

V (P_{n}) = {ˆ V}_{n}, T (P_{n}) = {ˆ T}_{n}, R_{j} (P_{n}) = {ˆ ρ}_{j \cdot n}, B_{j} (P_{n}) = {ˆ β}_{n}^{(j)} % and A_{j} (P_{n}) = {ˆ α}_{n}^{(j)} .

These functionals are to be taken into account in order to derive the influence functions related to MSLCA. We make the following assumption:

$(A_{2})$ : For all $k \in {1, . . ., K}$ , we have $V_{k} = I_{k}$ , where $I_{k}$ denotes the identity operator of $X_{k}$ .

Then, we have the following theorem that gives the influence function of $T$ .

Theorem 1

We suppose that the assumptions $(A_{1})$ and $(A_{2})$ hold. Then, for any vector $x = (x_{1}, \dots, x_{K}) \in X$ we have:

IF (x; T, P_{X}) = K \sum k = 1 K \sum ℓ = 1 ℓ \neq k - \frac{1}{2} τ_{k}^{*} (x_{k} \otimes x_{k}) V_{k ℓ} τ_{ℓ} - \frac{1}{2} τ_{ℓ}^{*} V_{ℓ k} (x_{k} \otimes x_{k}) τ_{k} + τ_{k}^{*} (x_{ℓ} \otimes x_{k}) τ_{ℓ} .

(8)

As $T$ determines MSLCA, it is important to ask whether its influence function is bounded. If so, we say that MSLCA is robust because it would mean that a contamination at the point $x$ has a limited effect on $T$ . The following proposition shows that $IF (x; T, P_{X})$ is not bounded. We denote by $∥ \cdot ∥_{L (X)}$ the operators norm defined as $∥ A ∥_{L (X)} = \sqrt{tr (A A^{*})}$ .

Proposition 1

We suppose that the assumptions $(A_{1})$ and $(A_{2})$ hold. Then, there exists $x_{0} \in X$ such that:

lim t \to + \infty ∥ IF (t x_{0}; T, P_{X}) ∥_{L (X)} = + \infty .

Now, we give in the following theorem, the influence functions related to the canonical coefficients and the canonical directions.

Theorem 2

We suppose that the assumptions $(A_{1})$ and $(A_{2})$ hold. Then, for any $x \in X$ and any $j \in {1, \dots, q}$ , we have:

(i) IF (x; ρ_{j}, P_{X})

=

K \sum k = 1 K \sum ℓ = 1 ℓ \neq k < β_{k}^{(j)}, x_{k} >_{k} < x_{ℓ} - V_{ℓ k} x_{k}, β_{ℓ}^{(j)} >_{ℓ} .

(ii) We suppose, in addition, that $ρ_{1} > ρ_{2} > \dots > ρ_{q}$ . Then :

$IF (x, α^{(j)}, P_{X})$	$=$	$K \sum k = 1 K \sum ℓ = 1 l \neq k q \sum m = 1 m \neq j \frac{1}{ρ_{j} - ρ_{m}} (< β_{k}^{(m)}, x_{k} >_{k} < x_{ℓ}, β_{ℓ}^{(j)} >_{ℓ}$
		$- \frac{1}{2} < β_{k}^{(m)}, x_{k} >_{k} < x_{k}, V_{k ℓ} β_{ℓ}^{(j)} >_{k}$
		$- \frac{1}{2} < x_{k}, V_{k ℓ} β_{ℓ}^{(m)} >_{ℓ} < x_{k}, β_{k}^{(j)} >_{k}) β^{(m)}$
		$- \frac{1}{2} (K \sum k = 1 [τ_{k}^{*} (x_{k} \otimes x_{k}) τ_{k} + < β_{k}^{(j)}, x_{k} >_{k}^{2} I] - 2 I) β^{(j)},$

where $I$ denotes the identity operator of $X$ .

Remark 1

Romanazzi[20] derived influence functions for the squared canonical coefficients and the canonical directions obtained from linear canonical analysis (LCA) of two random vectors. LCA is in fact a particular case of MSLCA obtained when $K = 2$ (see [18]). With Theorem 2 we recover the results of [20] when whe take $K = 2$ . We will only show it below for the canonical coefficients. For $j \in {1, . . ., q}$ , by applying Theorem 2 with $K = 2$ , we obtain

$IF (X; ρ_{j}, P_{X})$	$=$	$< β_{1}^{(j)}, x_{1} >_{1} < β_{2}^{(j)}, x_{2} - V_{21} x_{1} >_{2}$	(10)
		$+ < β_{2}^{(j)}, x_{2} >_{2} < β_{1}^{(j)}, x_{1} - V_{12} x_{2} >_{1}$
	$=$	$2 < β_{1}^{(j)}, x_{1} >_{1} < x_{2}, β_{2}^{(j)} >_{2} - < β_{1}^{(j)}, x_{1} >_{1} < x_{1}, V_{12} β_{2}^{(j)} >_{1}$
		$- < x_{2}, V_{21} β_{1}^{(j)} >_{2} < x_{2}, β_{2}^{(j)} >_{2} .$

The linear canonical analysis (LCA) of $X_{1}$ and $X_{2}$ is obtained from the spectral analysis of $R = V_{12} V_{21}$ (since $V_{1} = I_{1}$ and $V_{2} = I_{2}$ ). If we denote by $λ_{j}$ , $η_{1}^{(j)}$ , $η_{2}^{(j)}$ the related squared canonical coefficients and canonical vectors, it is known (see Remark 2.2 in [18]) that

λ_{j} = ρ_{j}^{2}, η_{k}^{(j)} = \sqrt{2} β_{k}^{(j)} (k = 1, 2), V_{12} η_{2}^{(j)} = ρ_{j} η_{1}^{(j)}, and V_{21} η_{1}^{(j)} = ρ_{j} η_{2}^{(j)} .

(11)

Then, putting $u_{j} =< x_{1}, η_{1}^{(j)} >_{1}$ and $v_{j} =< x_{2}, η_{2}^{(j)} >_{2}$ , we deduce from (10), (11) and the equality $IF (X; ρ_{j}^{2}, P_{X}) = 2 ρ_{j} IF (X; ρ_{j}, P_{X})$ that:

IF (X; ρ_{j}^{2}, P_{X}) = 2 ρ_{j} u_{j} v_{j} - ρ_{j}^{2} u_{j}^{2} - ρ_{j}^{2} v_{j}^{2},

(12)

what is the result obtained in [20].

3 Robust multiple-set linear canonical analysis (RMSLCA)

It has been seen that the MSLCA based on empirical covariance operator is not robust since $IF (x; T, P_{X})$ is not bounded. There is therefore an interest in proposing a robust version of MSLCA. In this section, we introduce such a version by replacing in (7) the empirical covariance operator by a robust estimator of $V$ . More precisely, we use the minimum covariance determinant (MCD) estimator of $V$ . We consider the following assumption:

( $A_{3}$ ) : the distribution $P_{X}$ of $X$ is an elliptical contoured distribution with density

f_{X} (x) = (det (V))^{- 1 / 2} h (< x, V^{- 1} x >_{X}),

where $h : [0, + \infty [\to [0, + \infty [$ is a function having a strictly negative derivative $h^{'}$ .

We first define the estimator of MSLCA based on MCD estimator of $V$ , then we derive the related influence functions.

3.1 Estimation of MSLCA based on MCD estimator

Letting $γ$ be a fixed real such that $0 < γ < 1$ , we consider a subsample $S \subset {X^{(1)}, . . ., X^{(n)}}$ of size $h_{n} \geq ⌈ n γ ⌉$ , where $X^{(i)} = (X_{1}^{(i)}, \dots, X_{K}^{(i)})$ , and we define the empirical mean and covariance operator based on this subsample by:

{ˆ M}_{n} (S) = \frac{1}{h_{n}} \sum X^{(i)} \in S X^{(i)}

and

{ˆ V}_{n} (S) = \frac{1}{h_{n}} \sum X^{(i)} \in S (X^{(i)} - {ˆ M}_{n} (S)) \otimes (X^{(i)} - {ˆ M}_{n} (S)) .

We denote by ${ˆ S}_{n}$ the subsample of ${X^{(1)}, . . ., X^{(n)}}$ which minimizes the determinant $det ({ˆ V}_{n} (S))$ of ${ˆ V}_{n} (S)$ over all subsamples of size $h_{n}$ . Then, the MCD estimators of the mean and the covariance operator of $X$ are ${ˆ M}_{n} ({ˆ S}_{n})$ and ${ˆ V}_{n} ({ˆ S}_{n})$ , respectively. It is well known that the these estimators are robusts and have high breakdown points (see, e.g., [21]). From them, we can introduce an estimator of MSLCA which is expected to be also robust. Indeed, putting

{˜ V}_{n} := {ˆ V}_{n} ({ˆ S}_{n}),

we consider the random operators with values in $L (X)$ defined as

{˜ Φ}_{n} = K \sum k = 1 τ_{k}^{*} {˜ V}_{k \cdot n} τ_{k} and {˜ Ψ}_{n} = K \sum k = 1 K \sum ℓ = 1 ℓ \neq k τ_{k}^{*} {˜ V}_{k ℓ \cdot n} τ_{ℓ},

where ${˜ V}_{k \cdot n} = τ_{k} {˜ V}_{n} τ_{k}^{*}$ and ${˜ V}_{k ℓ \cdot n} = τ_{k} {˜ V}_{n} τ_{ℓ}^{*}$ , and we estimate $T$ by

{˜ T}_{n} = {˜ Φ}_{n}^{- 1 / 2} {˜ Ψ}_{n} {˜ Φ}_{n}^{- 1 / 2} .

(13)

Considering the eigenvalues ${˜ ρ}_{1 \cdot n} \geq {˜ ρ}_{2 \cdot n} \dots \geq ˜ ρ_{q \cdot n}$ of ${˜ T}_{n}$ , and ${{˜ β}_{n}^{(1)}, \dots, {˜ β}_{n}^{(q)}}$ an orthonormal basis of $X$ such that ${˜ β}_{n}^{(j)}$ is an eigenvector of ${˜ T}_{n}$ associated with ${˜ ρ}_{j \cdot n}$ , we estimate $ρ_{j}$ by ${˜ ρ}_{j \cdot n}$ , $β^{(j)}$ by ${˜ β}_{n}^{(j)}$ and $α^{(j)}$ by ${˜ α}_{n}^{(j)} = {˜ Φ}_{n}^{- 1 / 2} {˜ β}_{n}^{(j)}$ . This gives a robust MSLCA that we denote by RMSLCA.

3.2 Influence functions

In order to derive the influence functions related to the above estimator of MSLCA, we have to specify the functional that corresponds to it. For doing that, we will first recall the functional associated to the above MCD estimator of covariance operator. Let

E_{γ} = {x \in X, < x, V^{- 1} x >_{X} \leq r^{2} (γ)}

where $r (γ)$ is determined by the equation

\frac{2 π^{q / 2}}{Γ (q / 2)} \int_{0}^{r (γ)} t^{q - 1} h (t^{2}) d t = γ,

$Γ$ being the usual gamma function. The functional $V_{1, γ}$ related to the aforementioned MCD estimator of $V$ is defined in [2] (see also [1], [7]) by

V_{1, γ} (P) = \frac{1}{γ} \int_{E_{γ}} (x - M_{P} (E_{γ})) \otimes (x - M_{P} (E_{γ})) d P (x),

where

M_{P} (B) = \frac{1}{γ} \int_{B} x d P (x) .

It is known that $V_{1, γ} (P_{X}) = σ_{γ}^{2} V$ where

σ_{γ}^{2} = \frac{2 π^{q / 2}}{γ q Γ (q / 2)} \int_{0}^{r (γ)} t^{q + 1} h (t^{2}) d t .

Therefore, the functional $T_{1, γ}$ related to $T$ is defined as

T_{1, γ} (P) = f (V_{1, γ} (P))^{- 1 / 2} g (V_{1, γ} (P)) f (V_{1, γ} (P))^{- 1 / 2}

where $f$ and $g$ are defined in (3). Now, we can give the influence functions related to RMSLCA of $X$ . First, putting

κ_{0} = \frac{π^{q / 2}}{(q + 2) Γ (q / 2 + 1)} \int_{0}^{r (γ)} t^{q + 3} h^{'} (t^{2}) d t,

and $T_{γ} = T_{1, γ} (P_{X})$ , we have:

Theorem 3

We suppose that the assumptions $(A_{1})$ to $(A_{3})$ hold. Then

IF (x; T_{γ}, P_{X})

=

−σ−2γ2κ0\large{1}Eγ(x)IF(x;T,PX),

where $IF (x; T, P_{X})$ is given in (8).

From this theorem we to obtain the following proposition which proves that RMSLCA is robust since the preceding influence function is bounded. We denote by $∥ \cdot ∥_{\infty}$ the usual operators norm defined by $∥ A ∥_{\infty} = {sup}_{x \in X - {0}} (∥ A x ∥_{X} / ∥ x ∥_{X})$ .

Proposition 2

We suppose that the assumptions $(A_{1})$ to $(A_{3})$ hold. Then,

sup x \in X ∥ IF (x; T_{γ}, P_{X}) ∥_{\infty} \leq \frac{σ_{γ}^{- 2}}{2 | κ_{0} |} K (K - 1) (∥ V ∥_{\infty} + 1) ∥ V^{1 / 2} ∥_{\infty}^{2} r^{2} (γ) .

Now, we give in the following theorem, the influence functions related to the canonical coefficients and the canonical directions obtained from RMSLCA. For $j \in {1, \dots, q}$ , denoting by $R_{γ \cdot j}$ (resp. $B_{γ \cdot j}$ ; resp. $A_{γ \cdot j}$ ) the functional such that $R_{γ \cdot j} (P)$ is the $j$ -th largest eigenvalue of $T_{1, γ} (P)$ (resp. the associated eigenvector; resp. $A_{γ \cdot j} (P) = f {(V_{1, γ} (P))}_{1, γ}^{- 1 / 2} B_{γ \cdot j} (P)$ ), we put $ρ_{γ \cdot j} = R_{γ \cdot j} (P_{X})$ , $β_{γ}^{(j)} = B_{j} (P_{X})$ and $α_{γ}^{(j)} = A_{γ \cdot j} (P_{X})$ . Considering

ν_{0} = \frac{2 π^{q / 2}}{Γ (q / 2)} h (r (γ)^{2}) r (γ)^{q - 1} σ_{γ}, ν_{1} = \frac{r (γ)}{σ_{γ}}, ν_{2} = \frac{2 ν_{0} ν_{1}^{3}}{σ_{γ} q (q + 2)} - \frac{2 γ}{σ_{γ}},

κ_{1} = - \frac{r (γ)^{2}}{q γ}, κ_{2} = \frac{σ_{γ} ν_{2} + 2 γ}{q γ σ_{γ} ν_{2}}, κ_{3} = - \frac{2}{σ_{γ} ν_{2}}, κ_{4} = \frac{r (γ)^{2} - q σ_{γ}^{2}}{q},

(14)

we have:

Theorem 4

We suppose that the assumptions $(A_{1})$ to $(A_{3})$ hold. Then, for any $x \in X$ and any $j \in {1, \dots, q}$ , we have:

(i) IF (x; ρ_{γ \cdot j}, P_{X})

=

−σ−2γ2κ0\large{1}Eγ(x)K∑k=1K∑ℓ=1ℓ≠k< β(j)k,xk>k<xℓ−Vℓkxk , β(j)ℓ >ℓ.

(ii) We suppose, in addition, that $ρ_{1} > ρ_{2} > \dots > ρ_{q}$ . Then :

	$IF (x, α_{γ}^{(j)}, P_{X})$	$=$	$−σ−3γ2κ0\large{1}Eγ(x)IF(x;α(j),PX)$
			$+σ−3γ{(12κ0−κ1−κ2∥V−1/2x∥2X)\large{1}Eγ(x)−κ4}β(j),$

where $IF (x; α^{(j)}, P_{X})$ is given in (2).

Remark 2

From this theorem, we recover the results of [5] which gives the influence function of MCD estimator of LCA of two random vectors. Indeed, using the notation of Remark 1 and (12), we deduce from the previous theorem that, when $K = 2$ , we have

IF(X;ρ2j,PX)=−σ−2γ2κ0\large{1}Eγ(x)(2ρjujvj−ρ2ju2j−ρ2jv2j),

what is the result obtained in [5].

4 Asymptotics for RMSLCA

In this section we deal with asymptotic expansion for RMSLCA. We first establish asymptotic normality for ${˜ T}_{n}$ and then we derive the asymptotic distribution of the canonical coefficients.

Theorem 5

Under the assumptions $(A_{1})$ to $(A_{3})$ , $\sqrt{n} ({˜ T}_{n} - T)$ converges in distribution, as $n \to + \infty$ , to a random variable $U_{γ}$

having a normal distribution in

L (X)

, with mean 0 and covariance operator equal to that of the random operator

	$Z_{γ}$	$=$	$σ−2γκ3\large{1}Eγ(X)K∑k=1K∑ℓ=1ℓ≠k{−12(τ∗k(Xk⊗Xk)Vkℓτℓ+τ∗ℓVℓk(Xk⊗Xk)τk)$

where $w : [0, + \infty [\to R$ is the function defined by

w(t)=\large{1}[0,r(γ)](t)(κ1+κ2t2)+κ4

(15)

and $κ_{1}$ , $κ_{2}$ , $κ_{3}$ and $κ_{4}$ are given in (14).

This theorem permits to obtain asymptotic distributions for the canonical coefficients. Let ${(ρ_{j}^{'})}_{1 \leq j \leq s}$ (with $s \in N^{*}$ ) be the decreasing sequence of distinct eigienvalues of $T$ , and $m_{j}$ the multiplicity of $ρ_{j}^{'}$ . Putting $η_{j} = \sum_{k = 0}^{j - 1} m_{k}$ with $m_{0} := 0$ , we clearly have $ρ_{i} = ρ_{j}^{'}$ for any $i \in {η_{j - 1} + 1, \dots, η_{j}}$ . We denote by $Π_{j}$ the orthogonal projector from $X$

onto the eigenspace associated with

ρ_{j}^{'}

, and by

Δ

the continuous map which associates to each self-adjoint operator

A

the vector

Δ (A)

of its eigenvalues in nonincreasing order. For

j \in {1, \dots s}

, we consider the

m_{j}

-dimensional vectors

{^υ}_{j} = ⎛ ⎜ ⎜ ⎜ ⎝ \begin{matrix} \end{matrix}

Robust multiple-set linear canonical analysis based on minimum covariance determinant estimator

Authors

Robustifying multiple-set linear canonical analysis with S-estimator

Influence Function and Robust Variant of Kernel Canonical Correlation Analysis

Robust Kernel (Cross-) Covariance Operators in Reproducing Kernel Hilbert Space toward Kernel Methods

Gene-Gene association for Imaging Genetics Data using Robust Kernel Canonical Correlation Analysis

Nearly Optimal Robust Mean Estimation via Empirical Characteristic Function

Convolution and Correlation Theorems for Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

Significance testing for canonical correlation analysis in high dimensions

1 Introduction

2 Influence in multiple-set canonical analysis

2.1 Multiple-set linear canonical analysis

Definition 2.1

2.2 Estimation based on empirical covariance operator

2.3 Influence functions

Theorem 1

Proposition 1

Theorem 2

Remark 1

3 Robust multiple-set linear canonical analysis (RMSLCA)

3.1 Estimation of MSLCA based on MCD estimator

3.2 Influence functions

Theorem 3

Proposition 2

Theorem 4

Remark 2

4 Asymptotics for RMSLCA

Theorem 5

Robust multiple-set linear canonical analysis based on minimum covariance determinant estimator

Authors

Related Research

Robustifying multiple-set linear canonical analysis with S-estimator

Influence Function and Robust Variant of Kernel Canonical Correlation Analysis

Robust Kernel (Cross-) Covariance Operators in Reproducing Kernel Hilbert Space toward Kernel Methods

Gene-Gene association for Imaging Genetics Data using Robust Kernel Canonical Correlation Analysis

Nearly Optimal Robust Mean Estimation via Empirical Characteristic Function

Convolution and Correlation Theorems for Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

Significance testing for canonical correlation analysis in high dimensions

This week in AI

1 Introduction

2 Influence in multiple-set canonical analysis

2.1 Multiple-set linear canonical analysis

Definition 2.1

2.2 Estimation based on empirical covariance operator

2.3 Influence functions

Theorem 1

Proposition 1

Theorem 2

Remark 1

3 Robust multiple-set linear canonical analysis (RMSLCA)

3.1 Estimation of MSLCA based on MCD estimator

3.2 Influence functions

Theorem 3

Proposition 2

Theorem 4

Remark 2

4 Asymptotics for RMSLCA

Theorem 5