On The Positive Definiteness of Polarity Coincidence Correlation Coefficient Matrix

Several researchers have investigated the statistical error of PCC as an estimate of bivariate correlation [4], [5]. In multivariate case, using PCC to estimate elements of the covariance matrix does not guarantee a PSD matrix estimator [1], [6]. Devlin et al. [1,Sec. 4.4], referring to a personal communication, claim that element-wise PCC (or “quadrant correlation”), may yield an invalid covariance estimate for p > 3 and real signals.

In this letter, we prove that for real signals with p ≤ 3, and complex signals with p ≤ 2, PCC estimate is PSD. For higher dimensions, counterexamples are presented which yield invalid covariance estimates.

Let x and y be two zero-mean real random variables with correlation coefficient r distributed with elliptical symmetry. It is well known that [6]:

This work was supported by Advanced Communication Research Institute (ACRI), Sharif University of Technology, Tehran, Iran. Authors are with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mails: farzanhaddadi@yahoo.com, Nayebi@sharif.edu, and Aref@sharif.edu).

where sgn(x) = +1 :

Using ( 1), an estimate of r from N iid observations x i , y i , i = 1, . . . , N is given by

where s xi = sgn(x i ). In the complex case, we can define the complex sign function as sgn c (x) sgn(ℜ[x]) + j sgn(ℑ[x]), where ℜ[x] and ℑ[x] are real and imaginary parts of x, respectively. In the Appendix, it is shown that

where (•) * denotes complex conjugate. Similar to (3), an estimate for the complex case is obtained by replacing expectation with the average as

where (•) R and (•) I denote real and imaginary parts, respectively.

Let R p×p be the covariance matrix of p random signals with unit diagonal elements and off-diagonal elements r ij : i, j = 1, • • • , p . For p = 2 case, a valid correlation estimate should satisfy |r| ≤ 1. For the real case of (3), |r| = | sin(•)| ≤ 1. For the complex case, regarding (5) define α and β such that rR = sin(α) and rI = sin(β). Then

(6) and it can be easily checked that the argument of summation in ( 6) belongs to {±2}. This yields α+β ≤ π 2 . In the same manner, we can show that

For p = 3 and real signals, we calculate the valid range of the elements of a 3 × 3 covariance matrix. Then we show that PCC estimate lies in this range.

Let R ∈ R 3×3 be a covariance matrix with unit diagonal elements. Valid range of r 23 should be calculated when r 12 , r 13 ∈ [-1, +1] are fixed. It can be readily

Assume random sign sequences s x , s y , s z with length N . Consider the positions of polarity coincidence with s x as black positions or “+” and elsewhere as white or “-”. Obviously all of the positions in s x is “+” and (s yi , s zi ) have four states of {++, +-, -+, –}. Since the permutation of the samples does not affect the estimate in (3), put the samples of s x , s y , s z from left in the order of {+ + -, + + +, + -+, + –} as in Fig. 1. Then any random sign sequences of s x , s y and s z can be replaced by the model in Fig. 1 with appropriate strip lengths N a i (with a 1 = 1) and relative positions of strips.

Let R s be the covariance matrix of s x , s y , s z with elements r sik , i, k = 1, 2, 3. The maximum of r s12 = +1 occurs in a 2 = 1 and the minimum of

r s12 and r s13 are determined by the values of a 2 and a 3 , r sii = 1, and the possible range of r s23 should be calculated. r s23 depends on the number of polarity coincidences of y and z which is maximum when the strip of z is in the left corner, and minimum when it is in the right corner. After some calculations, the range of r s23 is found as

It should be noted that the effect of finite N is the quantization of the accessible values. Now, it can be readily verified that

and

Therefore, r23 = sin π 2 r s23 satisfies (7). This, besides |r 12 | < 1 and |r 13 | < 1 can be used to show that | R| ≥ 0 (as in (7)) and the assertion is proved that for p = 3 and real data, PCC estimate is a valid covariance matrix.

In this section, some counterexamples are presented to show that PCC covariance estimate is not guaranteed to be PSD in dimensions p > 3 for real signals and p > 2 for complex signals. In real data case with p = 4 and number of observations N = 4, the real sign sequences in Table I results in an invalid covariance estimate. After simple computations, we will have r s12 = r s34 = 0 and r s13 = r s14 = r s23 = r s24 = 0.5. The covariance estimate will be where 0 is the 4 × 1 vector of zeros. As a consequence of the structure of Raug , eigenvalues of R1 are also eigenvalues of Raug . Therefore, Raug is an invalid covariance matrix. This procedure can continue to produce counterexamples for higher

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