Generalized Cumulative Residual Entropy for Distributions with Unrestricted Supports

Abstract

We consider the cumulative residual entropy (CRE) a recently introduced measure of entropy. While in previous works distributions with positive support are considered, we generalize the definition of CRE to the case of distributions with general support. We show that several interesting properties of the earlier CRE remain valid and supply further properties and insight to problems such as maximum CRE power moment problems. In addition, we show that this generalized CRE can be used as an alternative to differential entropy to derive information-based optimization criteria for system identification purpose.

1. Introduction

The concept of entropy is important for studies in many areas of engineering such as thermodynamics, mechanics, or digital communications. An early definition of a measure of the entropy is the Shannon entropy [1, 2]. In Shannon's approach, discrete values and absolutely continuous distributions are treated in a somewhat different way through entropy and differential entropy, respectively. Considering the complementary cumulative distribution function (CCDF) instead of the probability density function in the definition of differential entropy leads to a new entropy measure named cumulative residual entropy (CRE) [3, 4]. In [3, 4], CRE is defined as

ℰ(𝑋)=−_ℝ𝑛+𝑃(|𝑋|>𝑢)log𝑃(|𝑋|>𝑢)𝑑𝑢,(1)where 𝑛 is the dimension of the random vector 𝑋. Clearly, this formula is valid both for a discrete or an absolutely continuous random variable (RV), or with both a discrete and an absolutely continuous part, because it resorts to the CCDF of |𝑋|. In addition, unlike Shannon differential entropy it is always positive, while preserving many interesting properties of Shannon entropy. The concept of CRE has found nice interpretations and applications in the fields of reliability (see [5] where the concept of dynamic CRE is introduced) and images alignment [3].

Shannon entropy can be seen as a particular case of exponential entropy, when entropy order tends to 1. Thus, following the work in [4], a modified version of the exponential entropy, where PDF is replaced by CCDF, has been introduced in [6], leading to new entropy-type measures, called survival entropies.

However, both Rao et al.'s CRE and its exponential entropy generalization by Zografos and Nadarajah lead to entropy-type definitions that assume either positive valued RVs or apply to |𝑋| otherwise. Although the positive case is of great interest for many applications, CRE and exponential entropies entail difficulties when working with RVs with supports that are not restricted to positive values.

In this paper, we show that for an RV 𝑋, (1) remains a valid expression when 𝑃(|𝑋|>𝑢) is replaced by 𝑃(𝑋>𝑢) and integration is performed over ℝ^𝑛, without further hypothesis than in [4]. In addition, some desirable properties are enabled by this CRE definition extension. We also complete the power moment constrained maximum CRE distributions problem that was adressed in [7], for classes of distributions that have lower-unbounded supports. Finally, we illustrate the potential superiority of the proposed generalized CRE (GCRE) against differential entropy in mutual information-based estimation problems.

The paper is organized as follows. Section 2 introduces the GCRE definition. Some properties of GCRE are discussed in Section 3. In Section 4, we introduce cumulative entropy rate and mutual information rate. Section 5 deals with maximum GCRE distributions. With a view to illustrate the potentiality of GCRE, in Section 6, we show on a simple example a possible benefit of GCRE for systems identification.

2. Generalized Cumulative Residual Entropy (GCRE)

We will denote by 𝐹^𝑐𝑋(𝑥) the complementary cumulative distribution function (survival function) of a multivariate RV 𝑋=[𝑋₁,…,𝑋_𝑛]^𝑇 of dimension 𝑛: 𝐹^𝑐𝑋(𝑋)=𝑃(𝑋>𝑥)=𝑃(𝑋_𝑖>𝑥_𝑖,𝑖=1,…,𝑛). We denote by 𝐻_𝐶(𝑋) the GCRE of 𝑋 that we define by

𝐻_𝐶(𝑋)=−_ℝ𝑛𝐹^𝑐𝑋(𝑢)log𝐹^𝑐𝑋(𝑢)𝑑𝑢.(2)Clearly, like the CRE, the GCRE is a positive and concave function of 𝐹^𝑐𝑋. In addition, existence of GCRE can be established without further assumption upon distribution than those assumed for the CRE in [4].

Theorem 1. 𝐻_𝐶(𝑋)<∞ if for some 𝑝>𝑛, 𝐸[|𝑋|^𝑝]<∞.

Proof. First let us remark that from the proof of the existence of CRE in [4], it is sufficient to prove the result when 𝑋 is a scalar RV, that is 𝑛=1, and for 𝑝>1. Then, letting 𝑝⁻¹<𝛼<1, we use the following inequality:
−𝐹^𝑐𝑋(𝑥)log𝐹^𝑐𝑋(𝑥)≤𝑒⁻¹1−𝛼𝐹^𝑐𝑋+1−𝐹_𝑐𝑋(𝑥)^𝛼1I_]0,∞[(𝑥)(𝑥)1I_]−∞,0](𝑥),(3)where 1I_𝐴(𝑥)=1 if 𝑥∈𝐴 and 1I_𝐴(𝑥)=0 otherwise. The existence of ∫_ℝ(𝑒⁻¹/(1−𝛼))[𝐹^𝑐𝑋(𝑥)]^𝛼1I_[0,∞[(𝑥)𝑑𝑥 can be proven just as in [4]. Now, letting 𝑢=−𝑥, we have
1−=𝐹𝐹^𝑐𝑋_𝑋(𝑥)1I_]−∞,0](𝑥)≤1I(𝑥)_[−1,0[1I_]−∞,0](𝑥)≤1I_]0,1](𝑡)+𝐹_𝑋(𝑥)1I_{]−∞,−1[}(𝑥)≤1I_]0,1](𝑢)+𝐹_𝑋(−𝑢)1I_]1,∞](𝑢)≤1I_]0,1](𝑢)+𝐹_−𝑋(𝑢)1I_]1,∞](𝑢)(𝑢)+𝐹^𝑐||_−𝑋||(𝑢)1I^]1,∞]≤1I_]0,1](𝑢)(𝑢)+𝑢^−𝑝𝐸||𝑋||^𝑝1I^]1,∞](𝑢).(4)Thus,
∫_ℝ≤1−∫𝐹^𝑐𝑋(𝑥)1I_]−∞,0](𝑥)𝑑𝑥_ℝ≤1+∫1_∞1I^]0,1](𝑢)+𝑢^−𝑝𝐸||𝑋||^𝑝1I^]1,∞](𝑢)𝑑𝑢<∞.𝑢^−𝑝𝐸||𝑋||^𝑝𝑑𝑢(5) Finally, putting all pieces together one finally proves convergence of right-hand side of (2).

3. A Few Properties of GCRE

Let us now exhibit a few more interesting properties of the GCRE. First, it is easy to check that like Shannon entropy the GCRE remains constant with respect to variable translation:∀𝑎∈ℝ^𝑛,𝐻_𝐶(𝑋+𝑎)=𝐻_𝐶(𝑋).(6) In the same way, it is clear that∀𝑎∈ℝ₊,𝐻_𝐶(𝑎𝑋)=𝑎𝐻_𝐶(𝑋).(7) When 𝑎<0, we do not have such a nice property. However, let us consider the important particular case where the distribution of 𝑋 has a symmetry of the form

∃𝜇,∀𝑥,𝐹^𝑐𝑋(𝜇−𝑥)=1−𝐹^𝑐𝑋(𝜇+𝑥).(8)In this case, we get the following result.

Theorem 2. For an RV 𝑋 that satisfies symmetry property (8), one has ∀𝑎∈ℝ,𝐻_𝐶(𝑎𝑋)=||𝑎||𝐻_𝐶(𝑋).(9)

Proof. Since it is clear that for all 𝑎∈ℝ₊, 𝐻_𝐶(𝑎𝑋)=𝑎𝐻_𝐶(𝑋), we just have to check that 𝐻_𝐶(−𝑋)=𝐻_𝐶(𝑋), which can be established as follows:−𝐻_𝐶(−𝑋)==_ℝ𝐹^𝑐−𝑋(𝑥)log𝐹^𝑐−𝑋(𝑥)𝑑𝑥_ℝ=𝐹^𝑐−𝑋(𝑥−𝜇)log𝐹^𝑐−𝑋(𝑥−𝜇)𝑑𝑥_ℝ=𝐹_𝑋(−𝑥+𝜇)log𝐹_𝑋(−𝑥+𝜇)𝑑𝑥_ℝ=1−𝐹^𝑋(𝑥+𝜇)log1−𝐹_𝑋(𝑥+𝜇)𝑑𝑥_ℝ=𝐹^𝑐𝑋(𝑥+𝜇)log𝐹^𝑐𝑋(𝑥+𝜇)𝑑𝑥_ℝ=−𝐻𝐹^𝑐𝑋_𝐶(𝑥)log𝐹^𝑐𝑋(𝑥)𝑑𝑥(𝑋).(10)

When the entries of vector 𝑋 are independent, it has been shown in [4] that if the 𝑋_𝑖 are nonnegative, then

𝐻_𝐶(𝑋)=_𝑖Π^𝑗≠𝑖𝔼𝑋_𝑗𝐻^𝐶𝑋^𝑖.(11)However, this formula does not extend to RVs with distributions carried by ℝ^𝑛 because 𝐹^𝐶(𝑋) can be integrated over ℝ^𝑛+ in general but never over ℝ^𝑛. However, if the 𝑋_𝑖s are independent and have lower bounded supports with respective lower bounds 𝑚₁,…,𝑚_𝑛,

𝐻_𝐶(𝑋)==_{Π𝑖]𝑚𝑖,∞]}−𝐹^𝑐𝑋(𝑥)log𝐹^𝑐𝑋(𝑥)𝑑𝑥_𝑖Π^𝑗≠𝑖𝔼𝑋^𝑗−𝑚^𝑗𝐻^𝐶𝑋^𝑖,(12)because^∞𝑚_𝑖𝐹^𝑐𝑋_𝑖(𝑢)𝑑𝑢==−𝑢𝑚𝐹^𝑐𝑋_𝑖𝑖(𝑢)^∞𝑚_𝑖+^∞𝑚_𝑖𝑢𝑃_𝑋𝑖(𝑑𝑢)+𝔼𝑋_𝑖.(13)

Conditional GCRE definition is a direct extension of the definition of conditional CRE: the conditional GCRE of 𝑋 knowing that 𝑌 is equal to 𝑦 is defined by

𝐻_𝐶(=𝑋∫||𝑌=𝑦)_ℝ𝑛−𝑃(𝑋>𝑥||𝑌=𝑦)log𝑃(𝑋>𝑥||𝑌=𝑦)𝑑𝑥.(14)We recall an important result from [4] that states that conditioning reduces the entropy.

Theorem 3. For any 𝑋 and 𝑌,𝐻_𝐶(𝑋||𝑌)≤𝐻_𝐶(𝑋)(15)equality holds if and only if 𝑋 is independent of 𝑌.

As a consequence, if 𝑋→𝑌→𝑍 is a Markov chain, we have the data processing inequality for GCRE:𝐻_𝐶(𝑍||𝑋,𝑌)≤𝐻_𝐶(𝑍||𝑋).(16)

4. Entropy and Mutual Information Rates

4.1. Entropy Rate

The GCRE of a stochastic process {𝑋_𝑖} is defined by𝐻_𝐶(𝑋)=lim_𝑛→∞𝐻_𝐶𝑋^𝑛|||𝑋_𝑛−1,𝑋_𝑛−2,…,𝑋₁(17)when the limit exists.

Theorem 4. For stationary processes, the limit exists.

Proof. Consider𝐻_𝐶𝑋^𝑛+1|||𝑋_𝑛,…,𝑋₁≤𝐻^𝐶𝑋^𝑛+1|||𝑋_𝑛≤𝐻_𝐶,…,𝑋₂𝑋^𝑛|||𝑋_𝑛−1,…,𝑋₁.(18)The first line follows from the fact that conditioning reduces entropy and the second follows from the stationarity (see [2] for the equivalent proof in the case of Shannon entropy).

4.2. Mutual Information

Let 𝑋 and 𝑌 be two RVs. We define the cumulative mutual information between 𝑋 and 𝑌 as follows:𝐼_𝐶(𝑋;𝑌)=𝐻_𝐶(𝑋)−𝐻_𝐶(𝑋||𝑌).(19)

Theorem 5. 𝐼_𝐶 is nonnegative and it vanishes if and only if 𝑋 and 𝑌 are independent.

Proof. It is clear that 𝐼_𝐶 is nonnegative because of Theorem 3.

For a random vector 𝐗=(𝑋₁,𝑋₂,…,𝑋_𝑛) of size 𝑛, mutual information is defined by𝐼_𝐶(𝐗)=^𝑛_𝑖=1𝐻_𝐶𝑋^𝑖−𝐻^𝐶𝑋^𝑛|||𝑋_𝑛−1,…,𝑋₁.(20)In the case of stochastic processes {𝑋_𝑖}, we have 𝐻_𝐶(𝑋)=lim_𝑛→∞𝐻_𝐶(𝑋_𝑛|𝑋_𝑛−1,…,𝑋₁) and the limit exists for stationary processes. Then the mutual information rate for {𝑋_𝑖} is defined as𝐼_𝐶(𝑋)=lim_𝑇→∞1𝑇^𝑇_𝑡=1𝐻_𝐶𝑋^𝑡−𝐻^𝐶(𝑋),(21)where 𝐻_𝐶(𝑋_𝑡) is the marginal GCRE of the process 𝑋.

5. Maximum GCRE Distributions

In this section, we only consider the case of one-dimensional RVs (𝑛 = 1). Maximum entropy principle is useful in many scientific areas and most important distributions can be derived from it [8]. The maximum CRE distribution has been studied in [7]. For an RV 𝑋 with a symmetric CCDF in the sense of (8), we are looking for the maximum GCRE distribution, that is, the CCDF that solves the following moment problem:

∫_ℝmax_𝐹𝑐𝐻_𝐶𝐹^𝑐𝑟_𝑖(𝑥)𝑝(𝑥)𝑑𝑥=𝑐_𝑖,𝑖=1,…,𝑚,(22)where 𝑝(𝑥)=−(𝑑/𝑑𝑥)𝐹^𝑐(𝑥), (𝑟_𝑖)_𝑖=1,𝑚, and (𝑐_𝑖)_𝑖=1,𝑚 are fixed 𝐶¹ real valued functions and real coefficients, respectively. The solution of this problem is supplied by the following result.

Theorem 6. When the symmetry property (8) holds, the solution of problem (22), when it can be reached, is of the form 𝐹^𝑐(𝑥)=11+exp∑^𝑚1𝜆_𝑖𝑟^𝑖(𝑥−𝜇)(𝑥≥𝜇).(23)

Proof. −𝐻_𝐶(𝑋)=+^𝜇−∞𝐹^𝑐(𝑥)log𝐹^𝑐(𝑥)𝑑𝑥_∞𝜇=𝐹^𝑐(𝑥)log𝐹^𝑐(𝑥)𝑑𝑥_ℝ−+𝐹^𝑐(𝑥+𝜇)log𝐹^𝑐(𝑥+𝜇)𝑑𝑥_ℝ+=𝐹^𝑐(𝑥+𝜇)log𝐹^𝑐(𝑥+𝜇)𝑑𝑥_ℝ−+1−𝐹^𝑐(𝜇−𝑥)log1−𝐹^𝑐(𝜇−𝑥)𝑑𝑥_ℝ+=𝐹^𝑐(𝑥+𝜇)log𝐹^𝑐(𝑥+𝜇)𝑑𝑥_ℝ++𝐹1_𝑐−𝐹^𝑐(𝜇+𝑥)log1−𝐹^𝑐(𝜇+𝑥)(𝜇+𝑥)log𝐹^𝑐(𝜇+𝑥)𝑑𝑥.(24) Let us define 𝑓 by𝑓𝐹=^𝑐(−𝑥),1−−𝑝𝐹_𝑐−𝐹_𝑐(𝜇+𝑥)log1−𝐹^𝑐(𝜇+𝑥)(𝜇+𝑥)log𝐹^𝑐(𝜇+𝑥)+∑^𝑚₁𝜆_𝑖−𝑝(𝑥)𝑟^𝑖(𝑥).(25) Then, since (𝐹^𝑐(𝑥))^=−𝑝(𝑥), the Euler-Lagrange equation [9] states that the solution 𝐹^𝑐 of problem (22) is a solution of equation𝑑𝑑𝑥𝑓^−𝑝(𝑥)=𝑓_𝐹^𝑐(𝑥),(26)where 𝑓_𝑢 is the partial derivative of 𝑓 with respect to component 𝑢. From (25), we get𝑑𝑑𝑥𝑓^−𝑝𝑓(𝑥)=∑^𝑚₁𝜆_𝑖𝑟^𝑖(𝑥),_𝐹𝑐(𝑥)=log1−𝐹^𝑐(𝜇+𝑥)−log𝐹^𝑐(𝜇+𝑥).(27)Then,log1−𝐹^𝑐(𝜇+𝑥)𝐹^𝑐(𝜇+𝑥)=^𝑚𝐹∑₁𝜆_𝑖𝑟^𝑖(𝑥),_𝑐(𝑥)=11+exp∑^𝑚1𝜆_𝑖𝑟^𝑖(𝑥−𝜇),(28)for 𝑥∈[𝜇,∞[.

5.1. Example

We set the constraints 𝔼[𝑋]=𝜇 and 𝔼[𝑋²]=𝜎². Then the maximum GCRE symmetric solution for the CCDF of X is given by𝐹^𝑐𝑋(𝑥)=11+exp𝜆₁+2𝜆₂𝑥,(29)for 𝑥>0, which is the CCDF of a logistic distribution. The moment constraints lead to 𝜆₁+2𝜆₂𝑥=(𝜎√3/𝜋)(𝑥−𝜇). The corresponding PDF is defined on ℝ by𝑝_𝑋(𝑥)=𝜎√3𝜋exp−(𝜎√3/𝜋)(𝑥−𝜇)1+exp−(𝜎√3/𝜋)(𝑥−𝜇)².(30)

5.2. Positive Random Variables

It has been shown in [7] that the maximum CRE (i.e., the maximum GCRE under additional nonnegative constraint) distribution has CCDF in the form𝐹^𝐶(𝑥)=exp(−^𝑚_𝑖=1𝜆_𝑖𝑟^𝑖(𝑥)),(31)for 𝑥∈[0,∞[. In [7], this result is derived from the log-sum inequality, but of course it can also be derived from the Euler-Lagrange equation along the same lines as in the proof of Theorem 6.

With a positive support constraint and under first and second moment constraints, it comes that the optimum CCDF is of the form 𝐹^𝑐(𝑥)=exp(−𝜆₁−2𝜆₂𝑥), for 𝑥>0. Thus the solution, if it exists, is an exponential distribution. In fact, the first and second power moment constraints must be such that 𝔼[𝑋²]=2(𝔼[𝑋])², otherwise the problem has no exact solution.

6. Simulation Results

With a view to emphasize the potential practical interest of GCRE, we consider a simple system identification problem. Here, we consider an 𝑀𝐴(1) process, denoted by 𝑌=(𝑌_𝑛)_𝑛∈ℤ, generated by a white noise 𝑋=(𝑋_𝑛)_𝑛∈ℤ and corrupted by a white noise 𝑁:𝑌_𝑛=𝑋_𝑛−𝑎𝑋_𝑛−1+𝑁_𝑛.(32) The model input 𝑋 and output 𝑌 are observed and the system model (𝑀𝐴(1)) is assumed to be known. We want to estimate the coefficient 𝑎 without prior knowledge upon the distributions of 𝑋 and 𝑌. Thus, we resort to mutual information (MI) to estimate 𝑎 as the coefficient 𝛼 such that RVs 𝑌^𝛼𝑛=𝑋_𝑛−𝛼𝑋_𝑛−1 and 𝑌_𝑛 show the highest dependence. Shannon MI between 𝑌^𝛼𝑛 and 𝑌_𝑛 is given by 𝑓_𝑆(𝛼)=𝐼_𝑆(𝑌^𝛼𝑛,𝑌_𝑛)=𝐻_𝑆(𝑌^𝛼𝑛)−𝐻_𝑆(𝑌_𝑛|𝑌_𝑛), where 𝐻_𝑆 is Shannon differential entropy. Similarly, for GCRE, MI will be defined as 𝑓_𝐶(𝛼)=𝐼_𝐶(𝑌^𝛼𝑛,𝑌)=𝐻_𝐶(𝑌^𝛼𝑛)−𝐻_𝐶(𝑌^𝛼𝑛|𝑌_𝑛). We compare estimation performance for 𝑎 by maximizing both 𝑓_𝑆(𝛼) and 𝑓_𝐶(𝛼). Since true values of 𝑓_𝑆(𝛼) and 𝑓_𝐶(𝛼) are not available, they are estimated from empirical distributions of (𝑌_𝑛,𝑌^𝛼𝑛).

For simulations, we have chosen 𝑋 Gaussian and 𝑁 with a Laplace distribution: 𝑝_𝑁(𝑥)=(𝜆/2)exp(−𝜆|𝑥|). We consider an experiment with 𝑎=0.5 and noise variance equal to 0.2. Estimation is carried out from observation of (𝑋_𝑛,𝑌_𝑛)_𝑛=1,400. Here, optimization of MIs is realized on a fixed regular grid of 200 points over interval [0,1]. Estimation performance is calculated from 200 successive experiments. Estimation of 𝑎 from Shannon MI leads to bias and standard deviation that are equal to 0.032 and 0.18, respectively, while they are equal to 0.004 and 0.06, respectively, for GCRE MI.

More important, we see on Figure 1(a) that Shannon MI estimates are much more irregular than GCRE MI (Figure 1(b)) estimates because of smoothing brought by density integration in the calculation of CCDF. This difference is important since the use of an iterative local optimization technique would have failed in general to find Shannon's estimated MI global optimum, because of its many local maxima.

(a)

(b)

Figure 1 

Ten estimates of (a) 𝑓𝑆(𝛼) and (b) 𝑓𝐶(𝛼). Dotted line: mean estimate averaged from 200 realizations.

Of course, this drawback can be partly solved by kernel smoothing of the empirical distribution of (𝑌_𝑛,𝑌^𝛼𝑛), for instance by using the method proposed in [10]. However, we have checked that, for the above example, very strong smoothing is necessary and then bias and variance performance remain worse than with GCRE MI estimator.

7. Conclusion

We have shown that the concept of cumulative residual entropy (CRE) introduced in [3, 4] can be extended to distributions with general supports. Generalized CRE (GCRE) shares many nice features of CRE. We also pointed out specific properties of GCRE such as its maximum, moment constrained, and distribution and we have illustrated practical interest of GCRE by showing how it can be used in system identification procedures.