Outlier-Robust Truncated Maximum Likelihood Parameter Estimation of Compound-Gaussian Clutter with Inverse Gaussian Texture

Abstract

Compound-Gaussian distributions with inverse Gaussian textures, referred to as the IGCG distributions, are often used to model moderate/high-resolution sea clutter in amplitude. In moderate/high-resolution maritime radars, parameter estimation of the IGCG distributions from radar returns data plays an important role in adaptive target detection. Due to the inevitable existence of outliers of high amplitude in radar returns data from targets and reefs, parameter estimation must be outlier robust. In this paper, an outlier-robust truncated maximum likelihood (TML) estimation method is proposed to mitigate the effect of outliers of high amplitude in data. The data are first transferred into the truncated data by removing a given percentage of the largest samples in amplitude. From the truncated data, the truncated likelihood function is constructed, and its maximum corresponds to the TML estimates of the scale and inverse shape parameters. Further, an iterative algorithm is presented to obtain the TML estimates from data with outliers, which is an extension of the ML estimation method in the case that data contain outliers. In comparison with outlier-sensitive estimation methods and outlier-robust bipercentile estimation methods, the performance of the TML estimation method is close to that of the best ML estimation method in the case that data are without outlier, and it is better in the case that data are with outliers.

1. Introduction

Modeling and parameter estimation of sea clutter play an important role in maritime radar signal processing. Low-resolution sea clutter obeys the complex Gaussian distribution and has an amplitude of Rayleigh distribution [1,2]. Moderate and high-resolution sea clutter exhibits non-Gaussianity and heavy tails in amplitude and intensity [3]. As a combination of positive slow-varying texture and fast-varying complex Gaussian speckle, the compound Gaussian model has been applied to sea clutter modeling [4]. Nowadays, there exist four types of biparametric texture models, including the gamma distributions, inverse gamma distributions, inverse Gaussian distributions, and lognormal distributions, which correspond to the K-distribution model of sea clutter [5,6], the generalized Pareto intensity model [7], the IGCG model [8,9], and the generalized K-distribution lognormal (GK-LN) model [10], respectively.

The IGCG distributions can model well moderate/high-resolution sea clutter and thus their accurate and robust parameter estimation is a necessary prerequisite for adaptive target detection in moderate/high-resolution maritime surveillance radars [11]. For instance, the two recent near-optimal coherent detectors in the IGCG clutter model [12,13] require a known inverse shape parameter of the model. It is well-known that the model selection and parameter estimation of sea clutter are the foundation of high-performance constant false alarm rate (CFAR) detection [12,14]. For the IGCG clutter model, there are many parameter estimation methods, including the methods of moments (MoM estimation method) [8], low-order moment-based estimation method (LOME) [15], the iterative maximum likelihood estimation method (IML estimation method) [16], and the bipercentile (BiP) estimation method [16]. The first three methods are sensitive to outliers in data and can only work in clean clutter data. The BiP estimation method is outlier-robust and can work in real clutter environments with a small percentage of outliers in the data. Besides radar returns of objects and reefs on the sea surface, the range gate pull-off (RGPO) jamming [17] is also a possible source of outliers in data.

It is worth noting that the BiP estimation method uses only two sample percentiles of the data, which brings some loss in performance. The tri-percentile estimation methods in the K-distributions and the GK-LN distributions [18,19] using three sample percentiles and the truncated maximum likelihood (TML) estimation method using truncated data in the generalized Pareto intensity distributions [20] have been recently developed to improve performance. In this paper, the TML estimation method in the IGCG distributions is proposed. For the ICCG distributions, the truncated likelihood function (TLF) is derived, and an iterative algorithm is given to acquire the solution of the truncated maximum likelihood equations. As a result, the outlier-robust TML estimators with different truncation ratios are constructed. Owing to the full utilization of normal samples in the data, the TML estimators obtain better parameter performance than the BiP estimators in the case that data are with outliers, which is verified by simulated data and measured radar data.

This paper is organized as follows: Section 2 reviews existing outlier-sensitive and outlier-robust estimation methods for the IGCG distributions and proposes an outlier-robust TML estimation method for the IGCG distributions. Section 3 reports experimental results using simulated and measured data and provides a comprehensive comparison with other methods. Section 4 concludes our paper.

2. Methods

2.1. Review of Parameter Estimation of IGCG Distributions

The compound-Gaussian model (CGM) with inverse Gaussian distributed texture [7] has been widely used to model moderate/high-resolution sea clutter. The corresponding amplitude distributions are also referred to as the IGCG distributions [11,16]. The expression of an IGCG distribution is a biparametric function as follows:

$$f_r(r;\mu ,\upsilon )=\frac{2\upsilon r{\mathrm{e}}^{1/\upsilon }}{\mu }{\left(1+\frac{2\upsilon r^2}{\mu }\right)}^{-3/2}\left(1+\frac{1}{\upsilon }\sqrt{1+\frac{2\upsilon r^2}{\mu }}\right)\exp \left(-\frac{1}{\upsilon }\sqrt{1+\frac{2\upsilon r^2}{\mu }}\right),r>0$$

(1)

where r represents the amplitude of the clutter, the scale parameter μ > 0 represents the clutter power, and the inverse shape parameter υ ≥ 0 reflects the non-Gaussianity of the clutter. When υ = 0, the IGCG distribution degenerates to the Rayleigh distribution and the compound-Gaussian clutter degenerates to the complex Gaussian clutter. Correspondingly, the cumulative distribution function (CDF) is given as follows:

$$F_r(r;\mu ,\upsilon )=1-e^{1/\upsilon }{\left(1+\frac{2\upsilon r^2}{\mu }\right)}^{-1/2}\exp \left(-\frac{1}{\upsilon }\sqrt{1+\frac{2\upsilon r^2}{\mu }}\right)$$

(2)

When a set of samples {$r_1,r_2,\dots ,r_N$} follow the PDF/CDF with unknown parameters in (1) and (2), the parameter estimation problem is to determine the scale parameter μ and the inverse shape parameter υ resulting in a good fit of the sample data to the model. For the IGCG distributions, there exist two types of estimation methods. The outlier-sensitive estimation methods perform well in pure clutter data, but once data contain outliers, they abruptly deteriorate in performance. The robust-outlier estimation method can maintain a good estimation performance regardless of whether data contain a few outliers or not.

The kth moments of the IGCG distributions are given by as follows [8]:

$$m_k(r)={\mu }^{k/2}{e}^{1/\upsilon }\sqrt{2/(\pi \upsilon )}{K}_{k/2-1/2}(1/\upsilon )\mathsf{\Gamma }(1+k/2),k\in \mathbb{N}$$

(3)

where $K_{\alpha }(x)$ is the second-type modified Bessel function of order α, Γ(x) is the gamma function, and $\mathbb{N}$ stands for the positive integer set. In theory, the two parameters can be represented by the two different moments, which derive the moment-based estimators.

When the second-order and fourth-order sample moments are used, the commonly-used MoM estimator is explicitly given as follows [8]:

$$\begin{array}{l}{\widehat{\mu }}_{MoM}={\widehat{m}}_2(r),{\widehat{\upsilon }}_{MoM}=\frac{{\widehat{m}}_4(r)}{2{\widehat{m}}_2^2(r)}-1,\\ {\widehat{m}}_k(r)=\frac{1}{N}{\displaystyle \sum _{n=1}^N{r}_n^k},k\in \mathbb{N}.\end{array}$$

(4)

The use of the high-order sample moment makes more samples required to obtain high-precision estimates. Therefore, the low-order moment-based (LOM) estimator using the first and second-order sample moments is given in as follows [15]:

$${\widehat{\mu }}_{LOM}={\widehat{m}}_2(r),\sqrt{\frac{{\widehat{m}}_2(r)}{2{\widehat{\upsilon }}_{LOM}}}{e}^{\frac{1}{{\widehat{\upsilon }}_{LOM}}}{K}_0\left({\frac{1}{\widehat{\upsilon }}}_{LOM}\right)={\widehat{m}}_1(r)$$

(5)

When the sample size is given, the LOM estimator attains higher precision than the MoM estimator. The inverse shape parameter is attained by the look-up table method because the nonlinear equation in (5) does not have explicit solution. Besides, the generalized moment-based estimation methods, such as the [zlogz]-based estimators in the K-distributions [21,22], the generalized Pareto distributions [23], and the GK-LN distributions [10], often give better estimates than the MoM estimators. For the IGCG distributions, the [zlogz]-based estimator is not found in the open literature.

The maximum likelihood (ML) estimates of the parameters of the IGCG distributions are the solutions of the ML equations as follows [16]:

$$\begin{array}{l}\frac{1}{N}{\displaystyle \sum _{n=1}^N\frac{a_n^2+3\upsilon (\upsilon +a_n)}{a_n^2(\upsilon +a_n)}{r}_n^2}=\mu ,\\ \frac{1}{N}{\displaystyle \sum _{n=1}^N\frac{a_n^2}{\upsilon +a_n}}=1,\\ a_n=\sqrt{\mu +2\upsilon r_n^2}/\sqrt{\mu },n=1,2,\dots ,N.\end{array}$$

(6)

The system (6) of nonlinear equations does not have explicit solution and an iterative algorithm is given in [16] to obtain the iterative maximum likelihood (IML) estimates, as follows:

$$\begin{array}{l}{\mu }_0,{\upsilon }_0 \mathrm{are} \mathrm{calculated} \mathrm{by} \mathrm{the} \mathrm{explicit} \mathrm{estimators} \mathrm{in} \left(4\right)–\left(6\right),\\ a_n(i)=\sqrt{{\widehat{\mu }}_{i-1}+2{\widehat{\upsilon }}_{i-1}{r}_n^2}/\sqrt{{\widehat{\mu }}_{i-1}},\\ {\widehat{\mu }}_i=\frac{1}{N}{\displaystyle \sum _{n=1}^N\frac{a_n^2(i)+3{\widehat{\upsilon }}_{i-1}({\widehat{\upsilon }}_{i-1}+a_n(i))}{a_n^2(i)({\widehat{\upsilon }}_{i-1}+a_n(i))}{r}_n^2},\\ {\widehat{\upsilon }}_i=\frac{1}{N}{\displaystyle \sum _{n=1}^N(\frac{{\widehat{\upsilon }}_{i-1}^2}{{\widehat{\upsilon }}_{i-1}+a_n(i)}+(a_n(i)-1))},\\ {\widehat{\upsilon }}_{IML}=\underset{i\to +\infty }{\lim } {\widehat{\upsilon }}_i, {\widehat{\mu }}_{IML}=\underset{i\to +\infty }{\lim } {\widehat{\mu }}_i\end{array}$$

(7)

For clean data without outliers, these estimation methods obtain good performance and the IML estimator is the best one, followed by the LOM estimator and the MoM estimator. Because of sample moments or generalized sample moments available, these estimation methods have a sudden performance drop as data contain outliers with certain high amplitude. So, they are called outlier-sensitive estimators, that is, they can only obtain good estimation performance in pure clutter data. Outliers from radar returns of objects and reefs and jamming [17] are unavoidable in maritime surveillance radars and it is necessary to clean data if outlier-sensitive estimators are used. However, it is difficult and time-consuming to distinguish outliers and heavy tails of sea clutter. Therefore, an effective approach is to develop outlier-robust estimation methods.

Based on the important fact that order statistics and sample percentiles are robust to a small percentage of outliers in data, the outlier-robust BiP estimation method is proposed in [16]. A BiP estimator determines the two parameters from two special points in the empirical CDF of the data. Let $\{r_n,n=1,2,\dots ,N\}$ be the amplitude sample set that probably contains a small percentage of outliers of high amplitude. The data are sorted into $\{r_{(n)},n=1,2,\dots ,N\}$ in ascending order. For a pair of given percentile positions 0 < α < β < 1, the two sample percentiles and the inverse shape parameter and scale parameter are estimated by the following [16]:

$$\begin{array}{l}{\widehat{r}}_{\beta }=r_{([N\beta ])},{\widehat{r}}_{\alpha }=r_{([N\alpha ])},\\ {\widehat{\upsilon }}_{BiP}={\mathsf{\Lambda }}_{\alpha ,\beta }\left(\frac{{\widehat{r}}_{\beta }}{{\widehat{r}}_{\alpha }}\right),{\widehat{\mu }}_{BiP}=\frac{2{\widehat{\upsilon }}_{BiP}{\widehat{r}}_{\alpha }^2}{{\widehat{\upsilon }}_{BiP}{}^2{W}^2\left(\frac{e^{1/\widehat{\upsilon }}}{{\widehat{\upsilon }}_{BiP}(1-\alpha )}\right)-1}\end{array}$$

(8)

where [x] is the integer closest to x, ${\widehat{r}}_{\alpha }$ and ${\widehat{r}}_{\beta }$ are the lower and upper sample percentiles, respectively. The function Λ_α,β (x), which is a monotonically increasing function of the inverse shape parameter, is calculated by the look-up table method and the Lambert W-function W(x) is the inverse function of the function xe^x [24]. In the absence of outliers, the BiP estimators exhibit moderate performance in comparison with the outlier-sensitive estimators. In the presence of outliers, the BiP estimators have stronger outlier rejection than outlier-sensitive estimators. However, the BiP estimators also have some defects. Firstly, their performance is poor in small sample sizes due to the intrinsic defect of the empirical CDF of data. Secondly, the estimation performance depends on the position selection of the two sample percentiles. Their position optimization has been considered in the percentile-based estimation methods in other types of amplitude distributions [18,19]. Thirdly, a BiP estimator uses only two points on the empirical CDF and thus information in data is not fully exploited.

2.2. Truncated Maximum Likelihood Estimation Method of IGCG Distributions

Inspired by the TML estimation method in the generalized Pareto intensity distributions [20], we deal with the TML estimation method in the IGCG distributions. The TML estimation method overcomes at least two out of the three defects of the BiP estimation method. In this subsection, the TLF of the IGCG distributions and the TML equations of the IGCG distributions are first given. Moreover, a fast iterative algorithm is proposed to solve the TML equations and outlier-robust TML estimators are constructed.

2.2.1. TLF and TML Equations of IGCG Distributions

Assume that amplitudes $\left\{r_n,n=1,2,\dots ,N\right\}$, which are assumed to be independently and identically distributed (IID), follow the IGCG distribution with unknown scale and inverse shape parameters. Different from some applications [25,26,27] where doubly data truncating or censoring is used to deal with unobtainable samples or outliers. In our application, the sample set probably contains a small percentage of outliers of high amplitude. In addition, outliers are from only the right-hand side and thus the truncated data are used to mitigate the influence of outliers. The sample set is sorted in ascending order to obtain the following:

$$r_{(1)}\le r_{(2)}\le \dots \le r_{(N)}$$

(9)

and the truncated data are given by the following:

$$D_s=\left[r_{(1)},r_{(2)},\dots ,r_{(s)}\right]$$

(10)

where the integer s is relevant to the truncation ratio β in terms of s = [Nβ]. In other words, the (N−s) largest samples are removed from the data.

In terms of the properties of the order statistics [28] and the PDF and CDF in (1) and (2), the TLF of the truncated data D_s is given by the following:

$$\begin{array}{ll}L(\mu ,\upsilon ;D_s)& =\frac{N!}{\left(N-s\right)!}{\displaystyle \prod _{n=1}^s{f}_r\left(r_{(n)};\mu ,\upsilon \right)}{\left[1-F_r\left(r_{(s)};\mu ,\upsilon \right)\right]}^{N-s}\\ & =\frac{{\left(2\upsilon \right)}^s{e}^{N/\upsilon }N!}{{\mu }^s\left(N-s\right)!}{\displaystyle \prod _{n=1}^s\frac{r_{(n)}}{b_{(n)}^3}\exp \left(-\frac{b_{(n)}}{\upsilon }\right)\left(1+\frac{b_{(n)}}{\upsilon }\right)}{\left[\frac{1}{b_{(s)}}\exp \left(-\frac{b_{(s)}}{\upsilon }\right)\right]}^{N-s}\\ & b_{(n)}=\sqrt{1+2\upsilon r_{(n)}^2/\mu },n=1,2,\dots ,s.\end{array}$$

(11)

Correspondingly, the logarithmic TLF of the IGCG distribution is given by the following:

$$\begin{array}{ll}\ln L\left(\mu ,\upsilon ;D_s\right)& =\ln \left(\frac{2^{s}N!}{\left(N-s\right)!}\right)+s\ln \left(\frac{\upsilon }{\mu }\right)+\frac{N}{\upsilon }-\left(N-s\right)\left(\frac{b_{(s)}}{\upsilon }+\ln (b_{(s)})\right)\\ & +{\displaystyle \sum _{n=1}^s\left[\ln \left(r_{(n)}\right)-3\ln \left(b_{(n)}\right)+\ln \left(1+\frac{b_{(n)}}{\upsilon }\right)-\frac{b_{(n)}}{\upsilon }\right]}\\ & =\ln \left(\frac{2^{s}N!}{\left(N-s\right)!}\right)-s\ln \sqrt{\mu }+\Im \left(\upsilon ,{\mu }^{-1/2}{D}_s\right),\\ \Im \left(\upsilon ,{\mu }^{-1/2}{D}_s\right)& =s\ln \upsilon +\frac{N}{\upsilon }-\left(N-s\right)\left(\frac{b_{(s)}}{\upsilon }+\ln (b_{(s)})\right)\\ & +{\displaystyle \sum _{n=1}^s\left[\ln \left(\frac{r_{(n)}}{\sqrt{\mu }}\right)-3\ln \left(b_{(n)}\right)+\ln \left(1+\frac{b_{(n)}}{\upsilon }\right)-\frac{b_{(n)}}{\upsilon }\right]}\end{array}$$

(12)

The operator ${\mu }^{-1/2}{D}_s$ is the power normalization of the data. The TML estimates of the scale and inverse shape parameter using the truncated data D_s is the maximum point of the logarithmic TLF in (12), i.e., the following:

$$\left({\widehat{\mu }}_{TML},{\widehat{\upsilon }}_{TML}\right)=\underset{\mu >0,\upsilon >0}{\arg \max }\left\{-s\ln \sqrt{\mu }+\Im \left(\upsilon ,{\mu }^{-1/2}{D}_s\right)\right\}$$

(13)

The TML estimates of the two parameters is the solution of the TML equations as follows:

$$\begin{array}{l}\frac{\partial \ln L(\mu ,\upsilon ;D_s)}{\partial \upsilon }={{\Im }^{\prime }}_{\upsilon }\left(\upsilon ,{\mu }^{-1/2}{D}_s\right)=\frac{1}{{\upsilon }^2}\left(s\upsilon -\mathsf{\Theta }\left(\upsilon ,{\mu }^{-1/2}{D}_s\right)\right)=0,\\ \frac{\partial \ln L(\mu ,\upsilon ;D_s)}{\partial \mu }=-\frac{s}{2\mu }+{{\Im }^{\prime }}_{\mu }\left(\upsilon ,{\mu }^{-1/2}{D}_s\right)=0,\\ \mathsf{\Theta }\left(\upsilon ,{\mu }^{-1/2}{D}_s\right)\equiv (N-s)\left[(1+\frac{\upsilon }{b_{(s)}})\frac{\upsilon }{b_{(s)}}{\left(\frac{r_{(s)}}{\sqrt{\mu }}\right)}^2-b_{(s)}\right]+N\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{n=1}^s\left[\left(\frac{3\upsilon }{b_{(n)}}+\frac{b_{(n)}}{\upsilon +b_{(n)}}\right)\frac{\upsilon }{b_{(n)}}{\left(\frac{r_{(n)}}{\sqrt{\mu }}\right)}^2-\frac{b_{(n)}^2}{\upsilon +b_{(n)}}\right]},\\ {{\Im }^{\prime }}_{\mu }\left(\upsilon ,{\mu }^{-1/2}{D}_s\right)=(N-s)\left(1+\frac{\upsilon }{b_{(s)}}\right)\frac{{\left(r_{(s)}/\sqrt{\mu }\right)}^2}{\mu b_{(s)}}-\frac{s}{2\mu }+{\displaystyle \sum _{n=1}^s\left[\left(\frac{3\upsilon }{b_{(n)}}+\frac{b_{(n)}}{\upsilon +b_{(n)}}\right)\frac{{\left(r_{(n)}/\sqrt{\mu }\right)}^2}{\mu b_{(n)}}\right]}\end{array}$$

(14)

In (12) and (14), when s = N, the TLF and TML equations have the same form as the likelihood function and ML equations in the IML estimator in [16]. Therefore, the TML estimates of the IGCG distributions are a generalization of the ML estimates in the case that data contain outliers. Differently, the TML estimates are robust to outliers while the ML estimates are not.

2.2.2. Iterative TML Estimators of IGCG Distributions

Similar to the IML estimation method of the IGCG distributions [16], an iterative algorithm is given to obtain the solution of the TML equations from the truncated data, which derives the iterative TML estimators of the IGCG distributions.

The steps of the algorithm are as follows:

(i) Generation of the truncated data. Given the truncation ratio 0 < β < 1 and an amplitude sample set $X=\left\{r_n,n=1,2,\dots N\right\}$, the samples are sorted in ascending order to generate the truncated data, as follows:

$$D_s=\left\{r_{(1)}\le r_{(2)}\le ,\dots ,\le r_{(s)}\right\},s=\left[\beta N\right]$$

(15)

(ii) Initialization of parameters. From the sample set X, the outlier-robust BiP estimator [16] is used to obtain the initial values of the inverse shape parameter and scale parameter by the following:

$$\widehat{\upsilon }(0)={\widehat{\upsilon }}_{BiP}\left(X;\alpha ,\beta \right),\widehat{\mu }(0)={\widehat{\mu }}_{BiP}\left(X;\alpha ,\beta \right).$$

(16)

Note that the upper percentile position in the BiP estimator equals the truncation ratio so that the BiP estimator and the iterative TML estimator have the same outlier-robustness.

(iii) Iteration process. From the values of the two parameters at the (i−1)th iteration, the values at the ith iteration are calculated by the following:

$$\begin{array}{l}{c}_{(n)}(i)=\frac{r_{(n)}}{\sqrt{\widehat{\mu }(i-1)}},{\widehat{b}}_{(n)}(i)=\sqrt{1+2\widehat{\upsilon }(i-1)c_{(n)}{}^2(i)},n=1,2,\dots ,s,\\ \widehat{\upsilon }(i)=\frac{N-s}{s}{\widehat{b}}_{(s)}(i)-\frac{N}{s}+\frac{1}{s}{\displaystyle \sum _{n=1}^s\left[{\widehat{b}}_{(n)}(i)+\frac{{\widehat{\upsilon }}^2(i-1)}{\widehat{\upsilon }(i-1)+{\widehat{b}}_{(n)}(i)}\right]}\\ \widehat{\mu }(i)=\frac{N-s}{s}\left(1+\frac{\widehat{\upsilon }(i)}{{\widehat{b}}_{(s)}(i)}\right)\frac{c_{(s)}^2(i)}{{\widehat{b}}_{(s)}(i)}\widehat{\mu }(i-1)+\frac{1}{s}{\displaystyle \sum _{n=1}^s\left[\left(\frac{3\widehat{\upsilon }(i)}{{\widehat{b}}_{(n)}(i)}+\frac{{\widehat{b}}_{(n)}(i)}{\widehat{\upsilon }(i)+{\widehat{b}}_{(n)}(i)}\right)\frac{c_{(n)}^2(i)}{{\widehat{b}}_{(n)}(i)}\widehat{\mu }(i-1)\right]}\end{array}$$

(17)

(iv) Repeat the iteration process in (iii) until the changes of the parameters at the two next iterations are enough small, i.e., the following:

$$\left|\widehat{\upsilon }(i+1)-\widehat{\upsilon }(i)\right|\le \epsilon ,\left|\widehat{\mu }(i+1)-\widehat{\mu }(i)\right|\le \epsilon .$$

(18)

where $\epsilon$ is a small positive number given in advance.

At the end of the iteration process, the final values of the inverse shape parameter and scale parameter are output as the estimates, denoted by the following:

$${\widehat{\upsilon }}_{TML}=\underset{i\to \infty }{\lim } \widehat{\upsilon }(i), {\widehat{\mu }}_{TML}=\underset{i\to \infty }{\lim } \widehat{\mu }(i)$$

(19)

It is a difficult task to prove that the iterative process converges to the maximum point in (13). In fact, the IML estimator of the IGCG distributions [16], the TML estimator of the generalized Pareto intensity distributions [20], and the iterative ML estimator of Weibull distributions [29] encounter the same difficulty. The imperfection in theory does not impede their applications. It is easily proved that the limitation of the iteration process in (18) is necessarily one solution of the TML Equation (14) or a fixed point of the logarithmic TLF in (12) if the iteration process is convergent. Though it is difficult to prove the convergence of the iteration process, the convergence is verified by all the simulated data and measured data. Besides, when the inverse shape parameter υ is zero, the compound-Gaussian clutter degenerates to the complex Gaussian clutter and the bi-parametric IGCG distribution is reduced to the single parametric Rayleigh distribution. In this case, the TML estimate of the scale parameter is explicitly given by the following:

$${\widehat{\mu }}_{TML}=\frac{1}{s}{\displaystyle \sum _{n=1}^s{r}_{(n)}^2}+\frac{N-s}{s}{r}_{(s)}^2, \upsilon =0.$$

(20)

3. Results and Discussion

In this section, through simulated and measured sea clutter data, the performance of the TML estimation method proposed in this paper is verified and compared with other existing estimation methods. For simulated data, the true values of the parameters are known, and the relative root mean squared errors (RRMSEs) are used to evaluate the estimation performance. For measured sea clutter data, the true values of the parameters are unknown, and the true values are estimated from a large number of samples without outliers. There are several estimated performance evaluation indicators, respectively, the RRMSE, the average Kolmogorov–Smirnov distance (KSD), and the average Kulback–Leibler divergence (KLD) [29].

The RRMSEs are calculated by the following:

$$\mathrm{RRMSE}(\upsilon )=\frac{1}{\upsilon }\sqrt{E\left\{{\left(\widehat{\upsilon }-\upsilon \right)}^2\right\}},\mathrm{RRMSE}(\mu )=\frac{1}{\mu }\sqrt{E\left\{{\left(\widehat{\mu }-\mu \right)}^2\right\}}.$$

(21)

The KSD is calculated by the following:

$$\mathrm{KSD}\left(\upsilon ,\mu ;\widehat{\upsilon },\widehat{\mu }\right)=\underset{r\in \left(0,+\infty \right)}{\max }\left\{\left|F_r\left(r;\upsilon ,\mu \right)-F_r\left(r;\widehat{\upsilon },\widehat{\mu }\right)\right|\right\}$$

(22)

It is worth noting that the KSD mainly reflects the difference between two PDFs on bodies and does not reflect their difference on tails. For instance, the two PDFs have a small KSD ε > 0. In terms of the property of the CDFs, there exists a large positive number R, thus the following:

$$\begin{array}{l}\forall r>R, 1-\epsilon /2\le F_r\left(r;\upsilon ,\mu \right),F_r\left(r;\widehat{\upsilon },\widehat{\mu }\right)<1\\ \hspace{1em}\hspace{1em}\hspace{1em} \left|F_r\left(r;\upsilon ,\mu \right)-F_r\left(r;\widehat{\upsilon },\widehat{\mu }\right)\right|\le \epsilon \end{array}$$

(23)

The inequality (23) means that the KSD keeps invariant when the tails of the two distributions, $\{f(r;\mu ,\upsilon ),f(r;\widehat{\mu },\widehat{\upsilon }),r>R\}$, are altered. Therefore, the KSD does not reflect the difference between the two PDFs on tails. However, the tails of the amplitude distributions of sea clutter are important in the control of the false alarm rate in target detection of maritime surveillance radars.

The KLD of the two PDFs is defined by the following [29]:

$$\mathrm{KLD}\left(\upsilon ,\mu ;\widehat{\upsilon },\widehat{\mu }\right)={\displaystyle {\int }_0^{+\infty }{f}_r\left(r;\upsilon ,\mu \right)}\left|\log f_r\left(r;\upsilon ,\mu \right)-\log f_r\left(r;\widehat{\upsilon },\widehat{\mu }\right)\right|dr$$

(24)

Owing to the logarithmic difference in the integration, the KLD better reflects the difference of the PDFs on tails. Herein, the RMMSE, KSD, and KLD are used in experiments for a full evaluation.

3.1. Performance Evaluation by Using Simulated Data

It is difficult to analyze the RMMSEs of the joint estimates of the two parameters in the TML estimation method (13). Instead, we analyze the RMSME of one of the parameters when the other is known. When the scale parameter is known, the TML estimate of the inverse shape parameter is given by the following:

$${\widehat{\upsilon }}_{TML}=\underset{\upsilon >0}{\arg \max }\left\{\Im \left(\upsilon ,{\mu }^{-1/2}{D}_s\right)\right\}$$

(25)

The Equation (25) shows that the TML estimate at the known scale parameter is equivalent to its estimate as the scale parameter is one. Thus, the RMMSE of the inverse shape parameter is independent of the scale parameter as it is known. Similarly, when the inverse shape parameter is known, the TML estimate of the scale parameter is given by the following:

$$\begin{array}{l}{\widehat{\mu }}_{TML}=\underset{\mu >0}{\arg \max }\left\{\ln L\left(\mu ,\upsilon ,D_s\right)\right\}\\ \hspace{1em}\hspace{1em} =\frac{N-s}{s}\left(1+\frac{\upsilon }{b_{(s)}}\right)\frac{r_{(s)}^2}{b_{(s)}}+\frac{1}{s}{\displaystyle \sum _{n=1}^s\left[\left(\frac{b_{(n)}}{\upsilon +b_{(n)}}+\frac{3\upsilon }{b_{(n)}}\right)\frac{r_{(n)}^2}{b_{(n)}}\right]}\\ when b_{(n)}=\sqrt{1+\frac{2\upsilon r_{(n)}^2}{\mu }}\end{array}$$

(26)

In terms of the definition in (21), the RRMSE of the scale parameter is independent of the scale parameter itself if only the ratio ${\widehat{\mu }}_{TML}/\mu$ is independent of the scale parameter. The ratio is given by the following:

$$\begin{array}{ll}\frac{{\widehat{\mu }}_{TML}}{\mu }& =\frac{{\mathrm{argmax}}_{\mu }\left\{\ln L\left(\mu ,\upsilon ,D_s\right)\right\}}{\mu }\\ & =\frac{N-s}{s}\left(1+\frac{\upsilon }{b_{(s)}}\right)\frac{1}{b_{(s)}}{\left(\frac{r_{(s)}}{\sqrt{\mu }}\right)}^2+\frac{1}{s}{\displaystyle \sum _{n=1}^s\left[\left(\frac{b_{(n)}}{\upsilon +b_{(n)}}+\frac{3\upsilon }{b_{(n)}}\right)\frac{1}{b_{(s)}}{\left(\frac{r_{(s)}}{\sqrt{\mu }}\right)}^2\right]}\\ & =\underset{\mu >0}{\arg \max }\left\{\ln L\left(1,\upsilon ,{\mu }^{-1/2}{D}_s\right)\right\}\end{array}$$

(27)

Note that the normalized data ${\mu }^{-1/2}{D}_s$ are independent of the scale parameter. Thus, the RRMSE of the scale parameter is independent of the scale parameter and depends on the inverse shape parameter. In the TML estimation method, the scale and inverse shape parameters are iteratively estimated, and the two parameters are unknown. When the two parameters are unknown, it is difficult to prove that the RRMSEs of the two parameters are independent of the scale parameter. For other estimation methods [15,16], it is verified to be true by Monte-Carlo tests.

In the first experiment, we examine this point for the TML estimation method and set up the sample number N = 5 × 10³. The inverse shape parameter is taken as 0.1, 1, 4, and 10, and the scale parameter varies from 0.1 to 15, and 10⁵ independent trials to calculate the RRMSE at each parameter setup. As shown in Figure 1, the RRMSEs of the two parameters are almost horizontal lines and thus are independent of the scale parameter. In subsequent simulation experiments, the scale parameter is always taken as one. Moreover, the RRMSEs of the two parameters depend on the inverse shape parameter. As υ gets smaller, its RRMSEs become smaller first and then become larger. The difference is that the RRMSE of the scale parameter decreases with the decrease in υ, which is because the clutter power distribution has a wider range due to the heavier tail of the amplitude distribution, thus resulting in more loss of estimation accuracy.

In the second experiment, we examine the influence of the selection of truncation rate on the TML estimation method in the context of sample data consisting only of pure clutter. In the BiP estimation method, the selection of the upper percentile position is a compromise between estimation accuracy and outlier robustness [18]. The closer to one the upper percentile position is, the more samples are used, and the higher the estimation accuracy is in the case of pure clutter data, but the weaker the robustness to outliers is at the same time. For the TML estimation method, the truncation rate has the same effect as in β of the BiP estimation method. When β = 1, the TML estimator gets the IML estimator, which achieves the highest accuracy in the case of pure clutter data but is sensitive to outliers.

Figure 2 illustrates the RRMSEs of the TML estimation method when β = 0.85, 0.9, 0.95 and 1, and the sample size N varies from 500 to 10⁴, and υ = 2. As the truncation ratio β is close to one, the TML estimation method is similar to the IML estimation method in form and RRMSE curve. The TML estimation method is an extension of the IML estimation method to adapt to a cluttered environment with outliers. Moreover, the straight lines of the RRMSE versus sample size on the logarithmic plot show that the RRMSEs of the TML estimation method are inversely proportional to N^1/2.

The third experiment compares the TML estimator with β = 0.95 with the existing estimators in the case without an outlier. The existing estimators include the MoM estimator [8], the LOME estimator [15], the IML estimator and the BiP estimator with α = 0.5 and β = 0.95. The setup of simulations is the same as in the second experiment. Figure 3 demonstrates the RRMSEs of the two parameters by the five estimators. For the inverse shape parameter, the IML estimator attains the best performance, followed by the TML estimator, the LOME estimator, the BiP estimator, and the MoM estimator. For the scale parameter, the IML estimator keeps the best one, followed by the LOME and MoM estimators, the TML estimator, and the BiP estimator. Note that the scale parameter in the MoM and LOME estimators is estimated by the second-order sample moment of data, which is not affected by the estimates of the inverse shape parameter. For the IML, TML, and BiP estimators, the estimates of the scale parameter are always affected by that of the inverse shape parameter due to the iteration process or the order of the two parameters to be estimated. On the whole, for data without outliers, all five estimators are comparable in performance, though there exist some differences. In other words, when data are without outliers, anyone among the five estimators is acceptable in applications.

In what follows, we examine the influence of the inverse shape parameter on the RRMSEs. In the experiment, we set up the sample size as N = 5000 and the inverse shape parameter ranges from 0.1 to 15, and 10⁴ independent trials to calculate the RRMSEs at each parameter setup. Figure 4 demonstrates the RRMSEs of the five estimators. The RRMSE curves of the five estimators have the same trends. As the inverse shape parameter alters from 0 to 15, the RRMSE of the inverse shape parameter first becomes smaller and then slowly increases, while that of the scale parameter monotonously increases. In the case without an outlier, the IML estimator achieves the smallest RRMSEs, followed by the TML estimator, the LOME estimator, the BiP estimator, and the MoM estimator at the inverse shape parameter. For the scale parameter, the situation is slightly different and the BiP estimator and the TML estimator have poorer performance, and the IML estimator and the MoM and LOME estimators have better performance, which is due to the fact that the errors of the inverse shape parameter propagate into the estimates of the scale parameter in the TML and BiP estimators. Moreover, it is found that the error of the MoM estimator at the inverse shape parameter is much larger than the other four estimators. Therefore, for data without outliers, the MoM estimator is not suggested to be used in applications.

The use of pure clutter data alone is insufficient to illustrate the performance of the TML estimation method. So, in the sequent experiment, we use data with outliers as validation. In this case, the MoM estimator, the LOME estimator, and the IML estimator suffer from quite a severe degradation in performance. Thus, only the outlier-robust BiP estimation method is compared with the TML estimation method. In each trial, δ₁ percentages of the pure clutter data, which two parameters are given are replaced with outliers whose amplitude is δ₂ times the average amplitude of the clutter data, where δ₁ follows the uniform distribution of the interval [0, 0.02N] and δ₂ follows the uniform distribution of the interval [20^0.5, 20], so as to increase the randomicity of outliers in trials. In comparison, the truncation ratio in the TML estimator and the position of the upper percentile in the BiP estimator are taken as β = 0.95 so that they have the same outlier robustness. Figure 5 illustrates the RRMSEs of the two estimators, where υ ranges between 0.1 and 15 and N = 5000. The TML estimator obtains smaller RRMSEs than the BiP estimator. Unlike the BiP estimator, which uses only two special sorted samples in estimation, the TML estimates are obtained by fitting the truncated empirical CDF of the data using the biparametric CDF. Obviously, the curve fitting is better than the simple two-point interpolation.

Besides, the TML estimation method is outlier-robust instead of outlier-free [20]. In other words, outliers also bring the degradation of the estimation performance of the TML estimators even though the percentage of outliers is smaller than one minus the truncation ratio β. The reason is straightforward. The existence of outliers influences the right end of the truncated CDF of the data. This situation is illustrated by the sixth experiment. We compare the TML estimator with the truncation ratio β = 0.80, 0.9, 0.95, and 1 as N = 5000 when there exist outliers of not more than 2% of all the samples, as in the fifth experiment. Figure 6 plots the RRMSE curves of the two parameters as υ alters from 0.1 to 15. Several facts can be observed in Figure 6. Firstly, the IML estimator with β = 1 has the poorest performance for its sensitivity to outliers. Secondly, the TML estimator with β = 0.95 has better performance in the case without outliers than in the case with outliers, which shows that the TML estimation method is outlier-robust rather than outlier-free. In other words, the TML estimation method can avoid sharp degradation from outliers but fails to avoid their effect on performance. Besides, the TML estimator with β = 0.95 is not the best one among the four TML estimators, because outliers severely affect the right end of the truncated CDF of the data. Thirdly, the TML estimator with β = 0.80 attains the best performance. In fact, the selection of the truncation ratio is a tradeoff between the influence of outliers on the truncated CDF of the data and the utilization rate of samples. The TML estimator with β = 0.90 attains almost same performance as one with β = 0.80. In sequent experiments using measured data, the TML estimator with β = 0.90 is used to assure wholly good performance in cases with or without outliers.

3.2. Performance Evaluation Using Measured Data

Two measured radar datasets from the open and recognized IPIX database [30] and CSIR database [31] are used to evaluate the TML estimation method. The dataset from the IPIX database is ‘19980223_184853_ANTSTEP’ with a range resolution of three meters and a pulse repetition frequency of 1000 Hz. It consists of the complex radar returns at the 27 adjacent range cells during one minute. The amplitude map is plotted in Figure 7a, where a deep red stripe on the 20–24th range cells is the radar returns of the target under test, a floating small boat, with an average signal-to-clutter ratio (SCR) of 22.4 dB at the 21st cell. Due to the small grazing angle range of the 27 range cells, the sea clutter data in Figure 7a can be regarded to follow the same amplitude distribution. At first, the sea clutter data at the 7–13th range cells, labeled as area A in Figure 7a, which contains 210,000 samples, are used to calculate the empirical APDF and determine the optimal type from the K-distributions, the IGCG distributions, the generalized Pareto distributions, and the GK-LN distributions. Under each type, the parameters are estimated by the numerical/iterative ML estimators or the tri-percentile estimator. As shown in Figure 7b, the K-distribution fails to model the APDF of the data, and the other three distributions have comparable goodness-of-the-fit. In terms of their KSD and KLD, the IGCG distribution (KSD = 0.0240, KLD = 0.0018) is better than the K distribution (KSD = 0.0951, KLD = 0.0031), the GK-LN distribution (KSD = 0.0319, KLD = 0.0019) and the generalized Pareto distribution (KSD = 0.0241, KLD = 0.0018).

Area B contains half the samples of area A and does not contain outliers. When the IGCG distributions are used to model the data, the empirical APDF and the fitted APDFs by using the five estimators are plotted in Figure 7c. The five fitted APDFs are comparable in goodness-of-the-fit. In other words, for clean data without outliers, all five estimation methods can provide satisfactory parameter estimation. Area C contains 3% outliers that come from target returns. Figure 7d plots the empirical APDF of the data and the fitted IGCG distributions by using the five estimators. The outlier-sensitive MoM, IML, and LOME estimators fail to fit the APDFs of the data due to their poor parameter estimation abilities for data with outliers. The fitted APDFs by using the outlier-robust BiP and TML estimators accord with the empirical APDF of the data.

For a quantitative assessment,

Table 1. Estimation results of the five estimators on a dataset from the IPIX radar database.

Estimators	Area	Inverse Shape Parameter	Scale Parameter (×102)	KSD	KLD
IML [16]	Area A	0.5010	3.9361	0.0240	0.0018
MoM [8]	Area B	0.6951	3.8067	0.0416	0.0027
LOME [15]		0.5269	3.8067	0.0236	0.0026
IML [16]		0.4924	3.7616	0.0197	0.0024
BiP [16]		0.4563	3.6785	0.0230	0.0025
TML		0.4703	3.7377	0.0207	0.0025
MoM [8]	Area C	22.8693	12.8000	0.2811	0.0037
LOME [15]		10.0000	12.8000	0.1399	0.0034
IML [16]		3.2479	8.1615	0.0922	0.0030
BiP [16]		0.5899	4.0802	0.0348	0.0022
TML		0.4998	3.9411	0.0319	0.0018

where the bold fonts denote the minimum KSD and KLD on the area.

lists the estimates of the inverse shape parameter and scale parameters by using the five estimators on the three areas and the KSDs and KLDs of the fitted APDF and the empirical truncated APDF of the data (truncation ratio 0.90). Area A contains a large number of samples and does not contain any outliers. When the IML estimator is used in area A, the error indexes of parameter estimation, including KSD and KLD, are very small. Therefore, the estimates for area A are often used as the true values of the parameters to assess other estimators. In area B without outliers, the errors of the estimates, KSD, and KLD give a consistent evaluation of the five estimators. The IML estimator is the best one, followed by the TML, BiP, LOME, and MoM estimators. In area C with outliers, the estimates obtained from the outlier-sensitive MoM, LOME, and IML estimators severely deviate from the true values of the two parameters. Hence, the three estimators fail to be used in real clutter environments. Differently, the outlier-robust BiP and TML estimators still give high-precision estimates of the two parameters. Besides, the estimates by the TML estimator are much better than those by the BiP estimator, which is owing to the fact that the TML estimator uses most of the points on the empirical APDF while the BiP estimator uses only the two special points on the empirical APDF.

The other dataset comes from the CSIR database [31], an X-band high-resolution sea clutter database, and its filename is ‘TFA10_001.01’. The range resolution of the dataset is fifteen meters, the PRF is 2500 Hz, and it was collected at a high sea state and a small grazing angle. The data consist of the radar returns at 64 range cells during 24 s. The amplitude map on the range-slow time plane is plotted in Figure 8a. The target under test, a moving small boat, is at the 16–17th range cell and the target returns have an average SCR of 19.3 dB. Besides the red stripe from the target returns, there exist some regular orange slant stripes, which originate from the power modulation of long waves on sea clutter at high sea state. It is referred to as the “structural textures in sea clutter” [32]. Sea clutter data on a large-sized area A from the 20th to 36th range of cells is first used to determine the optimal distribution type and parameters. As shown in Figure 8b, the dotted line is the empirical APDF of the data. When the four types of biparametric amplitude distributions are used to fit the empirical APDF, the fitted APDFs are illustrated in Figure 8b. The fitted K-distribution severely deviates from the empirical APDF and is unsuitable to model the data. The other three fitted APDFs can fit well with the empirical APDF of the data. In terms of their KSDs and KLDs, the IGCG distribution provides the best goodness-of-the-fit. The KSD of the KLD of the IGCG distribution is obviously larger than that in the first measured data. In fact, due to the existence of structural textures in the second measured data, one mixture of two compound-Gaussian distributions [32] is a better choice. In applications, the biparametric IGCG model is more suitable, owing to simple parameter estimation and effective adaptive detection under the model.

For area B without an outlier, the empirical APDF of the data and the fitted ICCG distributions using the five estimators are demonstrated in Figure 8c. The fitted APDFs by using the five estimators highly accord with the empirical APDF of the data, and the five estimators are usable for clean sea clutter data. Area C contains about 2% of outliers from target returns. In the case of outliers, the empirical APDF of the data and the fitted IGCG distributions using the five estimators are illustrated in Figure 8d. Obviously, the fitted distributions by using the MoM, LOME, and IML estimators severely deviate from the empirical APDF of the data. The outlier-sensitive estimators fail to use them in the case of outliers. The fitted distributions by using the BiP and TML estimators can better fit the empirical APDF of the data. It is necessary to use outlier-robust estimation methods when data contain outliers.

A quantitative assessment for the five estimators is listed in

Table 2. Experimental results of the five estimation methods on one dataset from the CSIR database.

Estimators	Area	Inverse Shape Parameter	Scale Parameter	KSD	KLD
IML [16]	Area A	0.4039	0.0320	0.0163	0.0804
MoM [8]	Area B	0.4789	0.0301	0.0241	0.1566
LOME [15]		0.4016	0.0301	0.0206	0.1507
IML [16]		0.4038	0.0300	0.0152	0.1483
BiP [16]		0.3721	0.0295	0.0184	0.1496
TML		0.3832	0.0297	0.0175	0.1488
MoM [8]	Area C	14.6806	0.0572	0.3384	0.3610
LOME [15]		3.8466	0.0572	0.1234	0.1690
IML [16]		1.1800	0.0443	0.0416	0.0877
BiP [16]		0.4607	0.0346	0.0167	0.0722
TML		0.3899	0.0341	0.0129	0.0674

where the bold fonts denote the minimum KSD and KLD on the area.

, where the estimates of the two parameters, the KSDs and KLDs of the empirical APDF of the data and the fitted APDFs for each area are included. The IML estimates on area A are used as the true values of the two parameters to assess other estimators, owing to their high precision in the case without outliers and large sample size. In area B without outliers, in terms of the KSDs and KLDs, the IML estimator is the best one, followed by the TML, BiP estimators, the LOME estimator, and the MoM estimator. Particularly, the MoM estimator suffers from large errors in the inverse shape parameter and thus is not recommended for use. In area C with outliers, the estimates by the three outlier-sensitive estimators fully depart from the true values of the two parameters. Particularly, the estimates of the inverse shape parameter have too large errors. The two outlier-robust estimators still give acceptable estimates of the two parameters, though there is some performance loss in comparison with the results in area B. Moreover, the TML estimator behaves much better than the BiP estimator at the inverse shape parameter. In terms of the KSDs and KLDs, the TML estimator is the best one, followed by the BiP estimator, the IML estimator, the LOME estimator, and the MoM estimator.

4. Conclusions

High-resolution maritime surveillance radars need to detect targets from sea clutter of spatial-temporal characteristics. It is necessary to estimate the parameters of sea clutter models from radar returns that often consist of a large number of sea clutter samples and a small number of outliers of high amplitude. This paper proposed the outlier-robust TML estimation method of the IGCG distributions. The experimental results using simulation data and measured radar data show that the TML estimation method gives more robust and precise estimates of the parameters of the IGCG distributions than the existing estimation methods, whether the data contains outliers or not.