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Mean square cross error: performance analysis and applications in nonGaussian signal processing
EURASIP Journal on Advances in Signal Processing volume 2021, Article number: 24 (2021)
Abstract
Most of the cost functions of adaptive filtering algorithms include the square error, which depends on the current error signal. When the additive noise is impulsive, we can expect that the square error will be very large. By contrast, the cross error, which is the correlation of the error signal and its delay, may be very small. Based on this fact, we propose a new cost function called the mean square cross error for adaptive filters, and provide the mean value and mean square performance analysis in detail. Furthermore, we present a twostage method to estimate the closedform solutions for the proposed method, and generalize the twostage method to estimate the closedform solution of the information theoretic learning methods, including least mean fourth, maximum correntropy criterion, generalized maximum correntropy criterion, and minimum kernel risksensitive loss. The simulations of the adaptive solutions and closedform solution show the effectivity of the new method.
Introduction
The mean square error (MSE) is probably the most widely used cost function for adaptive linear filters [1,2,3,4,5]. The MSE relies heavily on Gaussianity assumptions and performs well for Gaussian noise. Recently, information theoretic learning (ITL) has been proposed to process nonGaussian noise. ITL uses the higherorder moments of the probability density function and may work well for nonGaussian noise. Inspired by ITL, some cost functions, such as the maximum correntropy criterion (MCC) [6,7,8,9,10,11], improved least sum of exponentials (ILSE) [12], least mean kurtosis (LMK) [13], least mean fourth (LMF) [14,15,16,17,18,19], generalized MCC (GMCC) [20], and minimum kernel risksensitive loss (MKRSL) criterion [21, 22] have been presented.
The LMK and LMF are robust to subGaussian noise. One typical subGaussian distribution is the uniform distribution. The MCC and ILSE are robust to larger outliers or impulsive noise, which often take relatively more often values that are very close to zero or very large. This means that impulsive noise has a superGaussian distribution [23, 24].
Altogether, the distribution of the additive noise in linear filtering can be divided into three types: Gaussian, superGaussian, and subGaussian. SuperGaussian noise and subGaussian noise are both nonGaussian.
From the viewpoint of performance, for example, the steady error, the MSE, MCC, and LMF work well for Gaussian, superGaussian, and subGaussian noise, respectively. The MSE demonstrates similar performance for the three types of noise under the same signaltonoise ratio (SNR). For Gaussian noise, all the algorithms have similar steady errors. For superGaussian noise, the steady error comparison under the same SNR is MCC < MSE < LMF. For subGaussian noise, the comparison is LMF < MSE < MCC.
Note that the cost functions of the above algorithms all include the square error, which is the correlation of the error signal. When impulsive noise is involved, we can expect that the square error will be very large. By contrast, the cross error (CE), which is the correlation of the error signal itself and its delay, may be very small for impulsive noise.
In our early work [25,26,27], we proposed the mean square cross prediction error to extract the desired signal in blind source separation (BSS), where the square cross prediction error was much smaller than the square prediction error. In this paper, we propose a new cost function called the mean square CE (MSCE) for adaptive filtering to process nonGaussian noise. We expect that the proposed MSCE algorithm will perform well for nonGaussian noise.
Note that the ITL methods can capture higherorder statistics of data. Thus, it is hard to directly obtain the corresponding closedform solutions. We present a twostage method to estimate the closedform solutions for the LMF, MCC, GMCC, MKRSL, and MSCE.
The contributions of this paper are summarized as follows:

i)
We present a new cost function, that is, the MSCE, for adaptive filters, and provide the mean value and mean square performance analysis in detail.

ii)
We propose a twostage method to estimate the closedform solution of the proposed MSCE algorithm.

iii)
We generalize the twostage method to estimate the closedform solution of the LMF MCC, GMCC, and MKRSL algorithms.
The paper is organized as follows: In Section 2, the problem statement is explained in detail. In Section 3, the MSCE algorithm is presented with the adaptive algorithm and closedform solution. In Section 4, the closedform solution of the LMF, MCC, GMCC, and MKRSL are estimated. In Section 5, the mean behavior and mean square behavior of MSCE are analyzed. Simulations are provided in Section 6. Lastly, a conclusion is provided in Section 7.
Problem formulation
The absolute value of the normalized kurtosis may be considered as one measure of nonGaussianity of the error signal. Several definitions for a random variable of zero means are presented as follows:

Definition 1 (normalized kurtosis)
The normalized kurtosis of random variable x is defined as
where x has zero mean.

Definition 2 (subGaussian or platykurtic)
A distribution with negative normalized kurtosis is called subGaussian, platykurtic, or shorttailed (e.g., uniform).

Definition 3 (superGaussian or leptokurtic)
A distribution with positive normalized kurtosis is called superGaussian, leptokurtic, or heavytailed (e.g., Laplacian).

Definition 4 (mesokurtic)
A zerokurtosis distribution is called mesokurtic (e.g., Gaussian).
When the linear filtering problem is considered, there is an input vector u ∈ ℝ^{M}, with unknown parameter w_{o} ∈ ℝ^{M} and desired response d ∈ ℝ^{1}. Data d(i) are observed at each time point i by the linear regression model:
where v is zeromean background noise with variance \( {\sigma}_v^2 \) and L is the length of the sequence. The error signal for the linear filter is defined as
where w is the estimate of w_{o}. The distribution of the additive noise in linear filtering can be divided into three types: Gaussian, superGaussian, and subGaussian. SuperGaussian noise and subGaussian noise are both nonGaussian.
In this research, we made the following assumptions:

A1) The additive noise is white, that is,

A2) Inputs u(t) at different time moments (i, j) are uncorrelated:

A3) The inputs and additive noise at different time moments (i, j) are uncorrelated:
The linear filtering algorithms of the MSE, MCC, and LMF are as follows: the cost function based on the MSE is given by
where E denotes the expectation operator. The gradient is defined as
At the stationary point, E{eu} = 0. The closedform solution denoted by w_{MSE} is given by the Wiener–Hopf equation:
The corresponding stochastic gradient descent, or LMS, algorithm is
where μdenotes the step size and μ > 0.
The cost function based on the LMF is given by
The corresponding stochastic gradient descent algorithm is
The cost function based on the correntropy of the error, also called the MCC, is given by
where σdenotes the kernel bandwidth. The corresponding stochastic gradient descent algorithm is
The cost function based on the GMCC is given by
The corresponding stochastic gradient descent algorithm is
The cost function based on the MKRSL is given by
The corresponding stochastic gradient descent algorithm is
Methods
Adaptive algorithm of the MSCE
The CE can be expressed as e(i)e(iq), where q denotes the error delay. Because the CE may be negative, we provide a new cost function, that is, the MSCE, as
where
The gradient of the MSCE can be derived as
Then, the corresponding stochastic gradient descent algorithm is
Equation (19) may not be robust against outliers. We provide the generalized MSCE (GMSCE) as
where G(x) is an x^{2}like function. The stochastic gradient descent algorithm for the GMSCE is
where g(.) is the derivative of G(.).
A suitable criterion for cost function
Combining the references of ICA [23, 24, 28,29,30] with those of the MCC [6,7,8,9,10,11], ILSE [12], and LMF [14,15,16,17,18,19], we determined that there are three cost functions in the fast ICA [30] algorithm:
G_{2}(u) is used to separate the superGaussian source in ICA, and works as the cost function of the MCC. G_{3}(u) is used to separate the subGaussian source in ICA when there are no outliers, and works as the cost function of the LMF. G_{1}(u) has not been used in adaptive filtering, but cosh(α_{1}u) works as the cost function of the ILSE.
This motivated us to explore ICA or BSS algorithms to determine a suitable criterion for adaptive filtering. Here, we use G_{1}(u) in the proposed GMSCE algorithm:
where 1 ≤ α ≤ 2, and α = 1in the simulations. The derivative of G(x) is
Substituting (28)–(29) into (24) with α = 1, we obtain
Closedform solution of the MSCE and GMSCE
We can estimate the closedform solution of the MSCE from the stationary point ∂J_{MSCE}/∂w = 0:
Substituting (20) into (31), we obtain
It is difficult to solve w from (32) because e^{2}(i) contains the secondorder term of w. We present a twostage method to estimate w.
In the first stage, we estimate e(i) and e(i − q) from (9):
where \( \hat{e}(i) \)and \( \hat{e}\left(iq\right) \) are the estimates of e(i) and e(i − q), respectively.
In the second stage, we estimate w from (32).
If we define
then we can rewrite (33) as
where F_{1}(w) is the estimate of F_{1}(w).
Note that the expectation can be estimated by averaging over the samples using
Equation (34) can be estimated by
Furthermore, we have
If we define R_{du}(q) and R_{uu}(q) as
then we can estimate them as
where R_{du}(q) and R_{uu}(q) are the estimates of R_{du}(q) and R_{uu}(q), respectively.
We can estimate the closedform solution of the MSCE as
Note than tanh(x) ≈ x when x is small. We can estimate the closedform solution of the GMSCE in the same way.
If we define R_{Gdu}(q) and R_{Guu}(q) as
then we can estimate them as
We can estimate the closedform solution of the GMSCE as
Closedform solution of the LMF, MCC, GMCC, and MKRSL
Based on the twostage methods, we can also estimate the closed solution of the LMF, MCC, GMCC, and MKRSL algorithms as follows.
Closedform solution of the LMF
We can estimate the closedform solution of the LMF from the stationary point ∂J_{LMF}/∂w = 0:
In the first stage, we estimate e(i) from (9):
In the second stage, we estimate w from (44).
If we define
then we can rewrite (44) as
where F_{2}(w) is the estimate of F_{2}(w).
With the help of (35), Eq. (47) can be estimated by
If we define R_{du2} and R_{uu2} as
then we can estimate them as
where R_{du2} and R_{uu2} are the estimates of R_{du2} and R_{uu2}, respectively.
We can estimate the closedform solution of the LMF as
Closedform solution of the MCC
We can estimate the closedform solution of the MCC from the stationary point ∂J_{MCC}/∂w = 0:
In the first stage, we estimate e(i) from (45). In the second stage, we estimate w from (52).
If we define
then we can rewrite (52) as
where F_{3}(w) is the estimate of F_{3}(w).
With the help of (35), Eq. (54) can be estimated by
Denote by
We can estimate the closedform solution of the MCC as
Closedform solution of the GMCC
The closedform solution of the GMCC is given by [20]:
In the first stage, we estimate e(i) from (45), and we have
In the second stage, we estimate w from (59). Denote by
We can estimate the closedform solution of the GMCC as
Closedform solution of the MKRSL
The closedform solution of the MKRSL is given by [22]:
In the first stage, we estimate e(i)from (45), and we have
In the second stage, we estimate w from (63). Denote by
We can estimate the closedform solution of the MKRSL as
Performance analysis of MSCE
Mean value behavior
To compare the performance of the MSE and MSCE, we define the total weight error as
where
Substituting (22) into (67), we obtain
where I is the identity matrix.
Substituting (39) into (67), we obtain
Note that R_{uu}(q) is positive definite; thus, (69) is stable for a sufficiently small step size μ.
The eigenvalue decomposition of R_{uu}(q) is R_{uu}(q) = UΛU^{T}. Then, we have the firstorder moment of ε(i)
Let d_{max} denote the maximum eigenvalue of R_{uu}(q). The step size should be selected as
so that the iterations will converge.
Mean square behavior
If we define
where U_{m} is the mth column of U, then we can rewrite (56) as
which is composed of M decoupled difference equations:
The secondorder moment of δ_{m}(i + 1) can be derived from (74) as
From (2), (3), and (67), we obtain
Thus, we have
Substituting (77) into (75) yields
Let
then we have
Substituting (79) into (80) yields
Note that \( {\mathbf{U}}_m^T{\mathbf{R}}_{uu}(q){\mathbf{U}}_m={d}_m \), and we have
Then the secondorder moment of δ_{m}(i + 1) can be rewritten as
Thus, the steady state error of the MSCE algorithm is given by
When μ < < 1/d_{max}, we have
For comparison, we can write the steady state error of the MSE algorithm as
For impulsive noise, its variance E{v^{2}(i)} may be very large, but its MSCE E{v^{2}(i)v^{2}(i − q)} may be small. Thus, the proposed MSCE algorithm may have a smaller steady error than the MSE for impulsive noise.
Selection of delay q
After obtaining the estimate of e(i) using (33), we can estimate the MSCE for q = 1, 2, ⋯, Q:
whereJ_{MSCE} is the estimate of J_{MSCE}. Because the meansquare performance of the MSCE algorithm is proportional to E{v^{2}(i)v^{2}(i − q)} according to (85), we should select q with the smallest J_{MSCE}in (87).
Simulation results and discussion
In this section, the performance of the MSE, MSCE, GMSCE, LMF, MCC, GMCC, and MKRSL will be evaluated by simulations. All the simulation points were averaged over 100 independent runs. The performance of the adaptive solution was estimated by the steadystate meansquare deviation (MSD)
The performance of the closedform solution is
We concluded that the smaller the MSD, the better the performance.
Closedform solutions comparison
The closedform solutions of the MSE, MSCE, GMSCE, LMF, MCC, GMCC, and MKRSL are expressed by (9), (40), (43), (51), (57), (61), and (65), respectively. The GMCC with α = 2, 4, and 6 are denoted by GMCC1, GMCC2, and GMCC3, respectively. The MKRSL with λ = 0.1 and 32 are denoted by MKRSL1 and MKRSL2, respectively.
In the experiments, we compared the MSDs of the closedform solutions of the ten algorithms with different nonGaussian noises. The input filter order was M = 5, and the sample size had length L = 3000. When the SNRs are ranged from − 20 to 20 dB, we obtain similar performance comparisons. Here the SNR was set to 6 dB.
Figures 1 and 2 partly show the four types of sub and superGaussian noise, respectively.
Figure 1a–c shows the periodic noises, and Fig. 1d shows the noise with uniform distribution. The kurtoses of the noises shown in Fig. 1a–d are − 1.5, − 1.0, − 1.4, and − 1.2, respectively.
Figure 2a, b shows the periodic superGaussian noises, and Fig. 2c, d shows impulsive noise. The impulsive noise v(i) is generated as v(i)=b(i)*G(i), where b(i) is Bernoulli process with a probability of success P{b(i)=1}=p. G(i) in Fig. 2c is zeromean Rayleigh noise, and G(i) in Fig. 2d is zeromean Gaussian noise. The kurtoses of the noises shown in Fig. 2a–d are 3.0, 4.1, 14.4, and 7.3, respectively.
The MSDs of the closedfrom solutions for sub and superGaussian noise were shown in Tables 1 and 2, respectively. From the above two tables, we can observe the following three conclusions: firstly, the existing algorithms (MSE, LMF, MCC, GMCC and MKRSL) do not perform better than the MSE method for sub and superGaussian noise simultaneously. The MCC, GMCC1, MKRSL1 and MKRSL2 performs better (worse) than the MSE method for superGaussian (subGaussian) noise, whereas the LMF and GMCC2 perform better (worse) than the MSE for subGaussian (superGaussian) noise. Simulations demonstrated that the proposed MSCE and GMSCE algorithm may perform better than the MSE algorithm both for sub and superGaussian noise. Secondly, the MCC performs as well as the MKRSL, whose parameters,λ and σ, did not influence the MSDs of the closedform solution. Thirdly, the parameters,λ and α, have great influence on the GMCC. When α = 2 and λ = 0.031, GMCC1 performs better than the MSE for superGaussian noise. When α = 4 and λ = 0.005, GMCC2 performs better than the MSE for subGaussian noise.
Adaptive solution for subGaussian noise
In the simulation, the input filter order was M = 5, the sample size had length L = 10,000 and SNR was set to 6 dB. The proposed algorithms (22) and (30) are denoted by MSCE and GMSCE, respectively.
For subGaussian noise shown in Fig. 1a, d, we compared the performance of the LMS, LMF, MCC, GMCC13, MKRSL12, MSCE, and GMSCE. The stepsizes were chosen such that all the algorithms had almost the same initial convergence speed, and other parameters (if any) for each algorithm were experimentally selected to achieve desirable performance.
The comparisons were shown in Figs. 3 and 4. From the two figures we can observe:
First, the GMCC13, LMF, and MSCE performed better than LMS for subGaussian noises. GMCC1 and GMCC2 perform best among the algorithms.
Second, the MKRSL12 and MCC performed worse than the LMS. The performance curves of MKRSL1 and MCC were almost overlapped.
Third, the performance of the adaptive solution was not always consistent with that of the closedform solution. Table 1 showed that the closedform solution of GMCC3 was worse than MSE, but the adaptive solution of GMCC3 was better than MSE. It may be hard for each algorithm to achieve a good tradeoff between the same initial convergence speed and the desirable performance (steadystate error).
Adaptive solution for superGaussian noise
In the simulations, the input filter order was M = 5, the sample size had length L = 10,000 and SNR was set to 6 dB. The stepsizes were chosen such that all the algorithms had almost the same initial convergence speed.
For the superGaussian noise shown in Fig. 2a, d, we compared the performance of the LMS, LMF, MCC, GMCC13, MKRSL12, MSCE, and GMSCE. The comparisons were shown in Figs. 5 and 6. From the two figures we can observe:
First, the proposed MSCE and GMSCE performed much better than other algorithms for the periodic superGaussian noise shown in Fig. 2a. The MSCE performed a litter better than the LMS for impulsive noise shown in Fig. 2d.
Second, the MKRSL1 and MCC had almost the same performance, the two algorithms performed a little better than the LMS.
Third, the LMF, GMCC13, and MKRSL2 performed worse than the LMS, though the closedform solutions of the GMCC2 and MKRSL2 performed better than the LMS.
Combining the above simulations in this section, we can find that each algorithm may have its good points, and no algorithms can perform best for all kinds of noise. Dividing the additive noise into three types will be helpful to select the suitable algorithm for real applications.
Conclusions
This paper proposes a new cost function called the MSCE for adaptive filters, and provided the mean value and mean square performance analysis in detail. We have also presented a twostage method to estimate the closedform solutions for the MSCE method, and generalize the twostage method to estimate the closedform solution of the information theoretic learning methods, such as LMF, MCC, GMCC, and MKRSL.
The additive noise in adaptive filtering is divided into three types: Gaussian, subGaussian, and superGaussian. The existing algorithms do not perform better than the mean square error method for sub and superGaussian noise simultaneously. The MCC, GMCC1, MKRSL1 and MKRSL2 performs better (worse) than the MSE method for superGaussian (subGaussian) noise, whereas the LMF and GMCC2 perform better (worse) than the MSE for subGaussian (superGaussian) noise. Simulations demonstrated that the proposed MSCE and GMSCE algorithm may perform better than the MSE algorithm both for sub and superGaussian noise.
In the future work, the MSCE algorithm may be extended to Kalman filtering, complexvalued filtering, distributed estimation, and nonlinear filtering.
Availability of data and materials
The datasets used during the current study are available from the corresponding author on reasonable request.
Abbreviations
 MSE:

Mean square error
 ITL:

Information theoretic learning
 MCC:

Maximum correntropy criterion
 ILSE:

Improved least sum of exponentials
 LMK:

Least mean kurtosis
 LMF:

Least mean fourth
 GMCC:

Generalized maximum correntropy criterion
 MKRSL:

Minimum kernel risksensitive loss
 SNR:

Signaltonoise ratio
 CE:

Cross error
 BSS:

Blind source separation
 MSCE:

Mean square CE Cross error
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Acknowledgements
Thanks to the anonymous reviewers and editors for their hard work.
Funding
This work was supported by the National Key Research and Development Program of China (Project No. 2017YFB0503400), National Natural Science Foundation of China under Grants 61371182 and 41301459.
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Rui Xue and Gang Wang proposed the original idea of the full text. Rui Xue designed the experiment. Yunxiang Zhang, Yuyang Zhao and Gang Wang performed the experiment and analyzed the results. Yunxiang Zhang and Yuyang Zhao drafted the manuscript. Rui Xue and Gang Wang wrote the manuscript. All authors read and approved this submission.
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Zhang, Y., Zhao, Y., Wang, G. et al. Mean square cross error: performance analysis and applications in nonGaussian signal processing. EURASIP J. Adv. Signal Process. 2021, 24 (2021). https://doi.org/10.1186/s13634021007337
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Keywords
 Adaptive filter
 Mean square error (MSE)
 Maximum correntropy criterion (MCC)
 Least mean fourth (LMF)
 Mean square cross error (MSCE)