# An Information Geometric Viewpoint on the Detection of Range Distributed Targets

Authors: Mengjiao Tang; Yao Rong; Jie Zhou
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The paper adopts the information geometry, to put forward a new viewpoint on the detection of range distributed targets embedded in Gaussian noise with unknown covariance. The original hypothesis test problem is formulated as the discrimination between distributions of the measurements and the noise. The Siegel distance, which is exactly the well-known geodesic distance between images of the original distributions via embedding into a higher-dimensional manifold, is given as an intrinsic measure on the difference between multivariate normal distributions. Without the assumption of uncorrelated measurements, we propose a set of geometric distance detectors, which is designed based on the Siegel distance and different from the generalized likelihood ratio algorithm or other common criterions in statistics. As special cases, the classical optimal matched filter, Rao test, and Wald test, which have been proven to have the CFAR property, belong to the set. Moreover, it is also accessible to an intuitively geometric analysis about how strongly the data contradict the null hypothesis.


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## Body

## 1. Introduction

Worthy of repetitive thought is the problem of detecting range distributed targets in the presence of Gaussian noise with unknown covariance. In the actual projects, measurements are collected via radar or sonar, which will be modeled as reflections from targets. With the collected data, a hypothesis test aims at distinguishing between target returns plus noise and noise only. In 1986, Kelly firstly proposed his famous generalized likelihood ratio test (GLRT) for detecting point targets in [1]. Later, a lot of advances on the detection of distributed targets are made based on the generalized likelihood ratio algorithm. There is no exception for the recent work that mainly focuses on the research of adaptive detection. In this field, one class is dealing with completely unknown signals [2, 3]. Specifically, paper [2] by Conte et al. provides a valuable statistical tool for ensuring the CFAR property. Others consider detecting a rank-one steering vector [4–6], or a spread-spectrum signal [7, 8], and so forth. All these detectors are reasonably testified to have perfect detection performances in corresponding situations.However, a common problem is that the detectors designed by the generalized likelihood ratio algorithm are limited to the case of determinate signals, due to the given form of distribution under the alternative hypothesis. In this paper, the detection problem with completely unknown target returns is discussed. In addition, there is no assumption whether the measurements are correlated or not. The original hypothesis test problem is formulated as a measure on the difference between distributions of the measurements and the noise, which is intuitively described via the platform of information geometry.The differential geometric approach was first introduced by Rao [9], in order to supply a way for the construction of distances between parametric density functions. Later related papers greatly improved Rao’s concept. It is worth mentioning that Amari presented statistical models with a differential structure in [10, 11], which plays an important role in information geometry. The geodesic distance has been put forward as the shortest length between two distributions on the statistical manifold endowed with the Fisher metric and Levi-Civita connection. A very active field of information geometry is about the manifold of multivariate normal distributions. On the multivariate normal manifold, the explicit expression of the geodesics has been derived by Eriksen [12] and later by Calvo and Oller [13], when the initial distribution and direction are given. However, the geodesic distance between multivariate normal distributions has not yet been obtained, except for on the submanifold of multivariate normal distributions with the same mean [14]. The lack of the explicit expression greatly limits the application of the geodesic distance. In 1990, Calvo and Oller proposed the Siegel distance on the multivariate normal manifold via embedding into the Siegel group [15]. The Siegel distance aims at calculating the geodesic distance between the images of the initial distributions and also provides a lower bound for the geodesic distance. Moreover, it is proven by Calvo et al. in [16] that the Siegel distance shows the similar behavior with the geodesic distance in special cases. This suggests that the Siegel distance is reasonable to be applied to measure the difference between the general multivariate normal distributions.In this paper, the key point is how to calculate the Siegel distance between the distributions of the measurements and the noise. Since the target returns are completely unknown, the possible distributions of measurements form a submanifold of multivariate normal distributions. By calculating the Siegel distanceD from some point on the multivariate normal manifold to the submanifold, a set of detectors are designed with the critical region (1)Wϵ=zt∣D>γϵ,where ϵ denotes the significance level and the constant γϵ is chosen to satisfy (2)Probzt∈Wϵ∣H0≤ϵ.From the viewpoint of information geometry, it also provides an intuitive analysis about how strongly the measurements contradict the null hypothesis. Moreover, the optimal matched filter, Rao test (or modified 2-step GLRT), and Wald test (or 2-step GLRT), which have been proven to have the CFAR property in [2], can also be derived from the set of detectors.The outline of the paper is as follows: Section2 shows the problem formulation. Section 3 reviews some important principles related to the geodesic distance between multivariate normal distributions. Section 4 is devoted to the Siegel distance and the design of geometric distance detectors and provides a discussion on the choice of the nominal noise covariance matrix in Section 5. Some intuitive interpretations are presented in Section 6. Section 7 gives a conclusion.


## 2. Problem Formulation

The problem under investigation is the detection of completely unknown target returns in the presence of Gaussian noise with unknown covariance. We assume the targets are spatially distributed acrossH range cells, from which the primary data {zi}, i=1,…,H, are collected. The primary data consist of the possible target returns plus Gaussian noise. To acquire information about the noises, the secondary data coming from K cells arranged around the targets are supposed to be available. Or rather, the secondary data consist of a training set zi, i=H+1,…,H+K, of which each sample shares the same distribution as the Gaussian noise adherent to the primary data. Therefore, the detection problem is formulated as(3)H0:zi=ni,i=1,…,H+K,H1:zi=pi+ni,i=1,…,H,zi=ni,i=H+1,…,H+K,where, pi, i=1,…,H, denote the signals, and the noises ni, i=1,…,H+K, are independent Gaussian vectors with mean zero and the same unknown covariance matrix M. All pis, nis, and zis are p-dimensional vectors.LetZPT=z1T,…,zHT and ZST=zH+1T,…,zH+KT. In fact, the hypotheses (3) carry out a comparison between the distributions of zi and the noise. As usual, considering the distribution of the whole measurement set ZP is verified to be more effective [4].Suppose the joint probability distribution of the measurementsZP is Gaussian with an unknown covariance matrix Σ. Let ZP0 be the realization of random vector ZP. In the real-time processing such as radar target detection, owing to the unknown mean of ZP, the multivariate normal sampling model for ZP is regarded as (4)NZP0,Σ,Σ∈S+Hp,where S+n denotes the set of n×n symmetric positive-definite matrices. As a comparison, an array of noise with the same size is considered. Let NPT=n1T,…,nHT; then we can specify the sampling distribution of NP as (5)N0Hp,Λ,where Λ∈S+Hp denotes the block diagonal matrix having the same block S∈S+p. The matrix S, which is obtained from the collected data, contains information about the unknown noise covariance. We will have a discussion in Section 5.The remaining problem is how to compare the two distributions above. It is essentially a property of distance and can be intuitively described via the platform of information geometry. In this paper, our major concern is to measure the difference between distributions (4) and (5) with methods of information geometry and also to give out a geometric criterion of how strongly the data contradict the hypothesis H0.Remark 1.
In [2, 4, 5], since the distribution of ZP under H1 is specified as (6)Np1⋮pH,M⋱M,it is limited to discuss the case of determinate signals. Thus, many other cases cannot be described by the process of generalized likelihood ratio algorithm; however, the derived detectors still have perfect performances in the simulations of [2]. For our development, the methods of information geometry enlarge the scope of our discussion, since the covariance Σ in (4) has no constraints on the matrix structure.Remark 2.
It is common in literature works to use distance as a test statistic. Please refer to [14, 15]. In fact, the Mahalanobis distance can also be used to derive a perfect test, but it is not an intrinsic distance in geometry.


## 3. The Riemannian Geometry of the Multivariate Normal Model

We hereafter review some important notions in information geometry, in order to induce a measure of distance on the multivariate normal model.


### 3.1. Statistical Manifold

The statistical model describes a family of parameterized probability distributions:(7)G=px;ξ∣ξ=ξ1,…,ξn∈Ξ,where Ξ is a subset of Rn. Under the condition of C∞ diffeomorphism, the family G is considered as an n-dimensional statistical manifold [11]. Each probability distribution p(x;ξ) is represented by a point on the manifold G with the corresponding coordinate ξ.


### 3.2. The Fisher Metric and the Geodesic Distance

For a statistical manifoldG, the Fisher metric aims at defining an inner product on the tangent space Tξ(G). It is given by the Fisher information matrix G(ξ) widely used in information science as (8)gijξ=Eξ∂∂ξilog⁡⁡px;ξ∂∂ξjlog⁡⁡px;ξ,where Eξ denotes the expectation with respect to the distribution p(x;ξ).With the Fisher metric, the length of a curveξ(t), t∈[a,b], on the manifold is defined in [11] as (9)ξt=∫ab∑i,jgijξ˙iξ˙jdt,where the dot denotes the differentiation with respect to the variable t and the indices indicate the corresponding components in the given coordinate parameterisations. Suppose a curve joining p(x;ξ1) and p(x;ξ2); the length between the endpoints depends on the choice of the curve. The geodesic distance between p(x;ξ1) and p(x;ξ2) is defined as the minimum length over all possible curves connecting the two points on the manifold G [17]; that is, (10)Dpx;ξ1,px;ξ2=Δmin⁡ξt∣ξa=ξ1,ξb=ξ2.We must emphasize that the geodesic distance is not only the intrinsic metric on the manifold of parametric probability distributions but also a genuine metric in the sense that it satisfies the nonnegative, symmetry properties and the triangle inequality. As a measure between probability distributions, the Kullback-Leibler divergence is also popular in information science, which is well-known as the relative entropy [18]: (11)Dklpx;ξ1,px;ξ2=Eξ1log⁡⁡px;ξ1-log⁡⁡px;ξ2.However, it is not a genuine metric, since it fails to be symmetric and satisfies the triangle inequality [19].


### 3.3. Riemannian Connection and Geodesics

Besides the Fisher metric, another fundamental tensor in Riemannian geometry is the Riemannian connection. The Riemannian connection is a symmetric affine connection that defines a linear one-to-one mapping between tangent spaces. For the Fisher metric, the Riemannian connection exists uniquely. The Riemannian connection coefficients are given in the form of the Christoffel symbolsΓijk by the Fisher metric [11] as(12)Γijk=∑l12gkl∂gil∂ξj+∂gjl∂ξi-∂gij∂ξk.Geodesics are defined as the autoparallel curves on the manifold. Usually with the Christoffel symbols, a geodesicξ(t) is calculated by the equations [11]:(13)ξ¨kt+∑i,jξ˙itξ˙jtΓijkξt=0,k=1,…,n.It must be pointed out that, on Riemannian manifold, the geodesic distance between two points equals the length of the shortest geodesic segment between them [14].


### 3.4. Riemannian Geometry of the Multivariate Normal Manifold

The multivariate normal manifold consists ofn-dimensional normal distributions parameterized by the mean and covariance matrix and is denoted by (14)Gn=Nμ,Σ∣μ∈Rn,Σ∈S+n.As detailed in [14], the coordinate system of Gn is given by μi,i=1,…,n;σij,i≤j. Moreover, the tangent space Tξ(Gn) coincides with the product space of n-dimensional vectors and n×n symmetric matrices, which is denoted by Rn×Sn. According to (8), the Fisher metric is given in the form of inner product:(15)η1,η2ξ=η1TΣ-1η2,η1,Aξ=0,A,Bξ=12tr⁡Σ-1AΣ-1B,where ξ denotes the point (μ,Σ) and η1,η2∈Rn,A,B∈Sn. According to (12) and (13), the geodesics on the multivariate normal manifold Gn are calculated by solving the following equations:(16)d2μdt2-dΣdtΣ-1dμdt=0,d2Σdt2+dμdtdμdtT-dΣdtΣ-1dΣdt=0.The geodesic distance is calculated by integral (9) along the shortest geodesic. Specifically, when n=1, the geodesic is explicitly given in [14, 20, 21]. The geodesic distance on the univariate normal manifold is given by(17)DNμ1,σ12,Nμ2,σ22=2cosh⁡-1⁡μ1-μ22+2σ12+σ224σ1σ2.Besides the aforementioned basic properties, the geodesic distance defined on the multivariate normal manifold owns the invariance property under the group transformationRn×GLn(R) which acts as (18)gP,ημ,Σ=Pμ+η,PΣPT,∀P,η∈GLnR×Rn,where GLnR denotes the group of n-dimensional regular matrices.Unfortunately, in the general casen>1, the analytical solutions to the geodesic equations (16) are complicated. The recent development is the achievement of deriving the explicit formula of the geodesic curves in [12, 13, 22], when the initial point and initial direction of the geodesic are given. Nevertheless, it remains unsolved how to obtain the geodesic curves given the endpoints and further to calculate the geodesic distance between them. However, efforts are made on the set of multivariate normal distributions with constant mean vectors, and another well-known geometry is defined.


### 3.5. The Submanifold with Fixed Mean Vector

Consider the submanifoldGμ0n in Gn defined by(19)Gμ0n=Nμ0,Σ∣Σ∈S+n,where μ0∈Rn is a constant vector. The structure of Gμ0n has been studied in [14, 15, 23]. The geodesics have been explicitly given and the geodesic distance has been computed. In particular, it is shown in [14] that Gμ0n is a totally geodesic submanifold of Gn. That means, on the manifold Gn, the geodesic joining two points of Gμ0n will line entirely in Gμ0n.Thus, if givenΣ1,Σ2∈S+n, the geodesic from Nμ0,Σ1 to Nμ0,Σ2 is given in [17], by (20)Σt=Σ11/2⁡exp⁡⁡tlog⁡⁡Σ1-1/2Σ2Σ1-1/2Σ11/2,t∈0,1,and the initial vector is in the direction of (21)Σ˙0=Σ11/2⁡log⁡⁡Σ1-1/2Σ2Σ1-1/2Σ11/2.For any A∈Sn, define AΣ2=A,AΣ=tr⁡[Σ-1A2]. Then the geodesic distance is (22)D=Σ˙0Σ1=tr⁡log⁡2⁡Σ1-1/2Σ2Σ1-1/2=12∑i=1nln⁡2⁡λi,where λi, i=1,…,n, are the eigenvalues of Σ1-1Σ2.


## 3.1. Statistical Manifold

The statistical model describes a family of parameterized probability distributions:(7)G=px;ξ∣ξ=ξ1,…,ξn∈Ξ,where Ξ is a subset of Rn. Under the condition of C∞ diffeomorphism, the family G is considered as an n-dimensional statistical manifold [11]. Each probability distribution p(x;ξ) is represented by a point on the manifold G with the corresponding coordinate ξ.


## 3.2. The Fisher Metric and the Geodesic Distance

For a statistical manifoldG, the Fisher metric aims at defining an inner product on the tangent space Tξ(G). It is given by the Fisher information matrix G(ξ) widely used in information science as (8)gijξ=Eξ∂∂ξilog⁡⁡px;ξ∂∂ξjlog⁡⁡px;ξ,where Eξ denotes the expectation with respect to the distribution p(x;ξ).With the Fisher metric, the length of a curveξ(t), t∈[a,b], on the manifold is defined in [11] as (9)ξt=∫ab∑i,jgijξ˙iξ˙jdt,where the dot denotes the differentiation with respect to the variable t and the indices indicate the corresponding components in the given coordinate parameterisations. Suppose a curve joining p(x;ξ1) and p(x;ξ2); the length between the endpoints depends on the choice of the curve. The geodesic distance between p(x;ξ1) and p(x;ξ2) is defined as the minimum length over all possible curves connecting the two points on the manifold G [17]; that is, (10)Dpx;ξ1,px;ξ2=Δmin⁡ξt∣ξa=ξ1,ξb=ξ2.We must emphasize that the geodesic distance is not only the intrinsic metric on the manifold of parametric probability distributions but also a genuine metric in the sense that it satisfies the nonnegative, symmetry properties and the triangle inequality. As a measure between probability distributions, the Kullback-Leibler divergence is also popular in information science, which is well-known as the relative entropy [18]: (11)Dklpx;ξ1,px;ξ2=Eξ1log⁡⁡px;ξ1-log⁡⁡px;ξ2.However, it is not a genuine metric, since it fails to be symmetric and satisfies the triangle inequality [19].


## 3.3. Riemannian Connection and Geodesics

Besides the Fisher metric, another fundamental tensor in Riemannian geometry is the Riemannian connection. The Riemannian connection is a symmetric affine connection that defines a linear one-to-one mapping between tangent spaces. For the Fisher metric, the Riemannian connection exists uniquely. The Riemannian connection coefficients are given in the form of the Christoffel symbolsΓijk by the Fisher metric [11] as(12)Γijk=∑l12gkl∂gil∂ξj+∂gjl∂ξi-∂gij∂ξk.Geodesics are defined as the autoparallel curves on the manifold. Usually with the Christoffel symbols, a geodesicξ(t) is calculated by the equations [11]:(13)ξ¨kt+∑i,jξ˙itξ˙jtΓijkξt=0,k=1,…,n.It must be pointed out that, on Riemannian manifold, the geodesic distance between two points equals the length of the shortest geodesic segment between them [14].


## 3.4. Riemannian Geometry of the Multivariate Normal Manifold

The multivariate normal manifold consists ofn-dimensional normal distributions parameterized by the mean and covariance matrix and is denoted by (14)Gn=Nμ,Σ∣μ∈Rn,Σ∈S+n.As detailed in [14], the coordinate system of Gn is given by μi,i=1,…,n;σij,i≤j. Moreover, the tangent space Tξ(Gn) coincides with the product space of n-dimensional vectors and n×n symmetric matrices, which is denoted by Rn×Sn. According to (8), the Fisher metric is given in the form of inner product:(15)η1,η2ξ=η1TΣ-1η2,η1,Aξ=0,A,Bξ=12tr⁡Σ-1AΣ-1B,where ξ denotes the point (μ,Σ) and η1,η2∈Rn,A,B∈Sn. According to (12) and (13), the geodesics on the multivariate normal manifold Gn are calculated by solving the following equations:(16)d2μdt2-dΣdtΣ-1dμdt=0,d2Σdt2+dμdtdμdtT-dΣdtΣ-1dΣdt=0.The geodesic distance is calculated by integral (9) along the shortest geodesic. Specifically, when n=1, the geodesic is explicitly given in [14, 20, 21]. The geodesic distance on the univariate normal manifold is given by(17)DNμ1,σ12,Nμ2,σ22=2cosh⁡-1⁡μ1-μ22+2σ12+σ224σ1σ2.Besides the aforementioned basic properties, the geodesic distance defined on the multivariate normal manifold owns the invariance property under the group transformationRn×GLn(R) which acts as (18)gP,ημ,Σ=Pμ+η,PΣPT,∀P,η∈GLnR×Rn,where GLnR denotes the group of n-dimensional regular matrices.Unfortunately, in the general casen>1, the analytical solutions to the geodesic equations (16) are complicated. The recent development is the achievement of deriving the explicit formula of the geodesic curves in [12, 13, 22], when the initial point and initial direction of the geodesic are given. Nevertheless, it remains unsolved how to obtain the geodesic curves given the endpoints and further to calculate the geodesic distance between them. However, efforts are made on the set of multivariate normal distributions with constant mean vectors, and another well-known geometry is defined.


## 3.5. The Submanifold with Fixed Mean Vector

Consider the submanifoldGμ0n in Gn defined by(19)Gμ0n=Nμ0,Σ∣Σ∈S+n,where μ0∈Rn is a constant vector. The structure of Gμ0n has been studied in [14, 15, 23]. The geodesics have been explicitly given and the geodesic distance has been computed. In particular, it is shown in [14] that Gμ0n is a totally geodesic submanifold of Gn. That means, on the manifold Gn, the geodesic joining two points of Gμ0n will line entirely in Gμ0n.Thus, if givenΣ1,Σ2∈S+n, the geodesic from Nμ0,Σ1 to Nμ0,Σ2 is given in [17], by (20)Σt=Σ11/2⁡exp⁡⁡tlog⁡⁡Σ1-1/2Σ2Σ1-1/2Σ11/2,t∈0,1,and the initial vector is in the direction of (21)Σ˙0=Σ11/2⁡log⁡⁡Σ1-1/2Σ2Σ1-1/2Σ11/2.For any A∈Sn, define AΣ2=A,AΣ=tr⁡[Σ-1A2]. Then the geodesic distance is (22)D=Σ˙0Σ1=tr⁡log⁡2⁡Σ1-1/2Σ2Σ1-1/2=12∑i=1nln⁡2⁡λi,where λi, i=1,…,n, are the eigenvalues of Σ1-1Σ2.


## 4. The Design of Geometric Distance Detector

As described in Section2, it is possible to apply the results of geodesic distance to compare the distributions of ZP and NP, further to construct test statistics for problem (3).Letn=H×p. Note Σ∈S+n in (4); thus the possible distributions of ZP constitute a submanifold GZP0n in Gn. We denote the distribution in (5) as the point ξ1 in the manifold Gn. Since in general case n>1, it has been mentioned before that the direct acquisition of the geodesic distance between multivariate normal distributions with different mean vectors is hard. Here we introduce the Siegel distance given by Calvo and Oller in [15], which attempts to calculate the geodesic distance between the images of the original distributions based on embedding into G0n+1:(23)f:Gn⟶G0n+1μ,Σ⟼Σ+μμTμμT1.Then the Siegel distance is defined in [15] as (24)Dsξ1,ξ2=ΔDfξ1,fξ2.Similarly, the Siegel distance betweenξ1 and the submanifold GZP0 is defined as(25)Dsξ1,GZP0n=Δmin⁡⁡Dsξ1,ξ2∣ξ2∈GZP0n=min⁡⁡Dsξ1′,ξ2′∣ξ2′∈G0n=min⁡⁡Dfξ1′,fξ2′∣ξ2′∈G0n,where ξ1′ denotes the normal distribution N-ZP0,Λ and G0n=N0,Σ∣Σ∈S+n. The second equality is due to the invariance properties of the Siegel distance, which is the same as the geodesic distance. The last equality follows from the definition of the Siegel distance in (24).It has been proven thatGn is isometric to f(Gn). However, f(Gn) is not a geodesic submanifold in G0n+1. That is to say, the geodesic curve joining the images f(ξ1) and f(ξ2) in G0n+1 may contain points outside f(Gn). Thus, the Siegel distance provides a lower bound for the geodesic distance between multivariate normal distributions, which has also been verified as a distance measure on the multivariate normal manifold.Givenξ2′∈G0n, it must be mentioned that f(ξ1′) and f(ξ2′) denote two points(26)fξ1′:N0n+1,Λ+ZP0ZP0T-ZP0-ZP0T1,fξ2′:N0n+1,Σ00T1in G0n+1. By (20), the geodesic from f(ξ2′) to f(ξ1′) is given as (27)Σt=Σ1/200T1exp⁡⁡tQΣ1/200T1,t∈0,1,where(28)Q=log⁡⁡Σ-1/2Λ+ZP0ZP0TΣ-1/2-Σ-1/2ZP0-ZP0TΣ-1/21.In expression (25), the initial direction at f(ξ2′) is (29)Σ˙0=Σ1/200T1QΣ1/200T1.The solution to (25) is easy to be found according to the following lemma.Lemma 3 (Pythagorean theorem).
Letξ be a point in G and let G0 be a submanifold of G. A necessary and sufficient condition for a point ξ′∈G0 to be a stationary point of the function D(ξ,·):θ↦D(ξ,θ) restricted on G0 is for the geodesic connecting ξ and ξ′ to be orthogonal to G0 at ξ′.The proof of the Pythagorean theorem is common in differential geometry. Then the lemma follows.Theorem 4.
The pointξ2′ in G0n that achieves the minimum of (25) must satisfy (30)Q=0bbTa,for some a∈R and b∈Rn.Proof.
By the Pythagorean theorem, we have(31)0=Σ˙0,dfξ2′fξ2′=tr⁡Σ-11Σ˙0Σ-11dΣ0=tr⁡Σ-1dΣ0Q.This theorem follows due to the arbitrariness of dΣ.The matrixQ given in (30) has at most two nonzero eigenvalues λ1,λ2, which must satisfy (32)λ1+λ2=tr⁡Q=a,λ12+λ22=tr⁡Q2=2bTb+a2.Note that the eigenvalues λ1 and λ2 are opposite sign while b≠0.Theorem 5.
Ifλ1 and λ2 are a solution of the following equations,(33)λ1-λ2λ1λ2λ2eλ1-λ1eλ2eλ1-eλ22=1ZP0TΛ-1ZP0+1,λ1eλ1-λ2eλ2λ1-λ2=1,then the minimum in (25) is achieved by(34)Σ=Λ+1-λ1-λ2λ1λ2λ2eλ1-λ1eλ2+λ1-λ2eλ1-eλ22·ZP0ZP0T,and the Siegel distance between ξ1 and GZP0n is (35)Dsξ1,GZP0n=λ12+λ22.Proof.
For the eigenvaluesλ1,λ2, the corresponding eigenvectors of Q in (30) can be given as ui=[bT,λi]T, i=1,2. Taking (n+1)×(n+1) orthogonal matrix (36)U=u1u1,u2u2,U0,we have(37)U0U0T=In+1-∑i=121ui2uiuiT.Then from ui2=λi2+bTb=λi2-λ1λ2 and (38)Q=0bbTa=Uλ1λ20n-1UT,we have (39)exp⁡⁡Q=Ueλ1eλ2In-1UT=∑i=12eλiui2uiuiT+U0U0T=In+1+∑i=12eλi-1ui2uiuiT=In+1+g1λ1,λ2bbTg2λ1,λ2bg2λ1,λ2bTg3λ1,λ2,where(40)g1λ1,λ2=∑i=12eλi-1ui2=λ2eλ1-λ1eλ2λ1λ2λ1-λ2+1λ1λ2,g2λ1,λ2=∑i=12eλi-1ui2λi=eλ1-eλ2λ1-λ2,g3λ1,λ2=∑i=12eλi-1ui2λi2=λ1eλ1-λ2eλ2λ1-λ2-1.
From (28) and (39), we can get (41)Σ-1/2Λ+ZP0ZP0TΣ-1/2=In+g1λ1,λ2bbT,(42)-Σ-1/2ZP0=g2λ1,λ2b,(43)0=g3λ1,λ2.Therefore, (44)Σ=Λ+1-g1λ1,λ2g22λ1,λ2ZP0ZP0T,(45)b=-1g2λ1,λ2Σ-1/2ZP0.
From (41)-(42), we have (46)1ZP0TΛ-1ZP0=-1+-1/λ1λ2+g1λ1,λ2g22λ1,λ2.Equations (43) and (46) imply (33). Since G0n+1 is a complete manifold, there exists a solution to (33) of the two variables λ1 and λ2.
Equation (34) follows from (44). By (22), the Siegel distance between ξ1 and GZP0n becomes (47)Tr⁡log⁡2⁡Σ-1/21Λ+ZPZP-ZP-ZPT1Σ-1/21=tr⁡log⁡2⁡Σ-1/2Λ+ZP0ZP0TΣ-1/2-Σ-1/2ZP0-ZP0TΣ-1/21=tr⁡Q2=λ12+λ22.That is (35). This theorem thus holds.The equations in (33) are nonlinear which can be solved numerically, and then we can calculate the Siegel distance Ds by (35). Let T2=(ZP0)TΛ-1ZP0. From Theorem 5, it is easy to find the Siegel distance Ds closely related to T2. As stated in Section 1, with the Siegel distance Ds, the detection problem (3) can be carried out with the critical region defined as (1). Note that the matrix S in Λ is undeterminate, which does not affect the derivation of Ds. Therefore, a set of geometric distance detectors are given via different choices of the matrix S.


## 5. The Choice of Matrix S

Letzi0 be the realization of random vector zi for each i=1,…,H+K. Note that (48)T2=ZP0TΛ-1ZP0=z10⋮zH0TS-1⋱S-1z10⋮zH0=∑i=1Hzi0TS-1zi0.Figure 1 illustrates how the Siegel distance varies with the logarithm of T2 increasing. The curve is obvious to demonstrate a monotonic increase. Thus the critical region in (1) is rewritten as (49)Wϵ=zt∣D=hT2>γϵ=zt∣T2>h-1γϵ,where h is an increasing function of T2.Figure 1
The Siegel distanceD versus the logarithm of T2.If the underlying covarianceM is known, it is natural to take S as M directly. In this case, the optimal matched filter is derived. Otherwise, efficient information about M is given by the maximum likelihood estimate based on the received data. When only with the secondary data, S is replaced by (50)S1=1K∑i=H+1H+Kzi0zi0T,thus the Wald test belongs to the set of geometric distance detectors. In addition, substituting the maximum likelihood estimate based on the whole set of data (51)S2=1H+K∑i=1H+Kzi0zi0Tin place of S, we can find it is the same as the Rao test.In summary, it is verified that the referred three classical tests are members of the set of geometric distance detectors designed based on the Siegel distance.


## 6. Interpretations

In this section, some figures are presented, in order to give intuitive interpretations about how the geometric distance detectors work. As special cases, the optimal matched filter, the Rao test, and Wald test are described in the form of geometric distance detectors.Figure2 illustrates the definition of the geodesic distance from some point on the univariate normal manifold to the submanifold with the same mean. A group of geodesics from N(0,1) are shown. Gradient colors along the geodesics represent different geodesic distances given by (17) from N(0,1) to the current point. The geodesic distance from N(0,1) to G0.551 is given by the integral along the geodesic marked red. As a geometric distance detector, a threshold is required for the hypothesis test problem. H0 is not rejected if the Siegel distance between the distribution of noise and the submanifold determined by the measurements is less than the threshold. The critical region in (1) is obtained by a projection of submanifolds that are outside of the threshold on the observation space. The same principle applies to the Siegel distance on the multivariate normal manifold. As without visualization features, the multidimensional cases are not presented here.Figure 2
Illustration of the geodesic distance fromN(0,1) on the univariate normal manifold to the submanifold G0.551. The blue straight line shows the submanifold with the fixed mean 0.55. The geodesic curves from the initial point N(0,1) with different initial directions are displayed with gradient colored curves, and the one orthogonal to the submanifold G0.551 is marked red.Figure3 shows a simulation with H=1, p=2, since the examples with a higher dimension have similar behaviors but no visualization features. The distributions of noise are displayed at the origin, with the corresponding noise covariance matrix: the underlying covariance M and the sample covariance S1 or S2. All curves were drawn as the projections on the measurement space. It is obvious that the part outside a contour curve represents the critical region of the geometry detector with the corresponding alarm probability listed in the legend. As the intrinsic merit of geometric measure, the magnitude of the Siegel distance reflects how strongly the collected data contradict the null hypothesis H0. In Figure 3(d), contour curves with the same alarm probability α=10-4 are presented. We can observe that the contour curves of the Siegel distance based on S1 and S2 appear to overlap. This is due to the equivalency of the Rao test and Wald test when H=1 [24]. In fact, if we denote the eigenvalues of the p×p matrix Ω=∑i=1HziS1-1ziT as ω1,…,ωp, then, according to [2, 3], the Rao test and Wald test are, respectively, rewritten as (52)TRao2=∑i=1pH+KωiK+ωi,TWald2=∑i=1pωi.When H=1, Ω=z1S1-1z1T has only one nonzero eigenvalue. Thus, both the Rao test and the Wald test coincide to test for the nonzero eigenvalue. However, they yield different Siegel distances because of the distinct covariance matrices which are neglected in projection mapping. The acceptance regions of both the Rao test and the Wald test can be improved by increasing the number of secondary data, so as to approach that of the geometric distance detector based on the underlying noise covariance.Figure 3
Contour curves of the Siegel distance based on (a) the underlying covariance matrixM; (b) the sample covariance matrix S1 or (c) S2. The values marked on the contour curves denote different Siegel distances from the origin to the submanifold specified by the current point. αis listed in the legends denote the corresponding alarm probabilities of the geometric distance detectors; (d) contour curves of the Siegel distances based on different choice of the noise covariance matrix, with the same alarm probability α=10-4. All curves were drawn as the projections on the measurement space.
(a)
(b)
(c)
(d)As a general rule, the Rao test and Wald test have different acceptance regions. Figure4 displays another simulation with H=2, p=2. For flat visualizations, the contour curves of the Siegel distance are projected on the space of Ω.Figure 4
The contour curves of the Siegel distance based on the sample covariance matricesS1 and S2, with the same probability α=10-4. All curves are drawn as the projections on the eigenvalues space of Ω.


## 7. Conclusion

In this paper, an information geometric viewpoint on how to deal with the detection problem of range distributed targets embedded in Gaussian noise with unknown covariance is put forward. More precisely, we have derived a set of geometric distance detectors, of which the optimal matched filter, the Rao test, and Wald test are members. This establishes a link between the information geometry and the hypothesis testing. As a future research, other choices ofS can be tested, and it might also be of interest to find one with a better performance among the set of geometric distance detectors.



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*Source: 930793-2015-05-12.xml*