*Article* **Influence of Binomial Crossover on Approximation Error of Evolutionary Algorithms**

**Cong Wang 1, Jun He 2, Yu Chen 1,\* and Xiufen Zou 3,4**

**Abstract:** Although differential evolution (DE) algorithms perform well on a large variety of complicated optimization problems, only a few theoretical studies are focused on the working principle of DE algorithms. To make the first attempt to reveal the function of binomial crossover, this paper aims to answer whether it can reduce the approximation error of evolutionary algorithms. By investigating the expected approximation error and the probability of not finding the optimum, we conduct a case study comparing two evolutionary algorithms with and without binomial crossover on two classical benchmark problems: OneMax and Deceptive. It is proven that using binomial crossover leads to the dominance of transition matrices. As a result, the algorithm with binomial crossover asymptotically outperforms that without crossover on both OneMax and Deceptive, and outperforms on OneMax, however, not on Deceptive. Furthermore, an adaptive parameter strategy is proposed which can strengthen the superiority of binomial crossover on Deceptive.

**Keywords:** binomial crossover; differential evolution; fixed-budget analysis; evolutionary computation; approximation error

**MSC:** 90C15

#### **1. Introduction**

Evolutionary algorithms (EAs) are a family of randomized search heuristics inspired from biological evolution, and many empirical studies demonstrate that crossovers that combine genes of two parents to generate new offspring could be helpful to the convergence of EAs [1–3]. Meanwhile, theoretical results on runtime analysis validate the promising function of crossover in EAs [4–15], whereas there are also some cases that crossover cannot be helpful [16,17].

By exchanging components of target vectors with donor vectors, differential evolution (DE) algorithms implement crossover operations in a different way. Numerical results show that continuous DE algorithms can achieve competitive performance on a large variety of complicated problems [18–21], and its competitiveness is to great extent attributed to the employed crossover operations [22]. However, the binary differential evolution (BDE) algorithm [23], which simulates the working mechanism of continuous DE, is not as competitive as its continuous counterpart. Analysis of the working principle indicates that the mutation and update strategies result in poor convergence of BDE [24], but there were no theoretical results reported on how crossover influences the performance of discretecoded DE algorithms.

This paper is dedicated to investigating the influence of binomial crossover by introducing it to the (1 + 1)EA, excluding the impacts of population and mutation strategies of DE. Although the expected hitting time/runtime is popularly investigated in the theoretical study of randomized search heuristics (RSHs), there is a gap between runtime analysis

**Citation:** Wang, C.; He, J.; Chen, Y.; Zou, X. Influence of Binomial Crossover on Approximation Error of Evolutionary Algorithms. *Mathematics* **2022**, *10*, 2850. https:// doi.org/10.3390/math10162850

Academic Editors: Alexandru Agapie, Denis Enachescu, Vlad Stefan Barbu and Bogdan Iftimie

Received: 24 July 2022 Accepted: 9 August 2022 Published: 10 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and practice because their optimization time to reach an optimum is uncertain and could be even infinite in continuous optimization [25]. Due to this reason, optimization time is seldom used in computer simulation for evaluating the performance of EAs, and their performance is evaluated after running finite generations by solution quality such as the mean and median of the fitness value or approximation error [26]. In theory, solution quality can be measured for a given iteration budget by the expected fitness value [27] or approximation error [28,29], which contributes to the analysis framework named fixed-budget analysis (FBA). An FBA on immune-inspired hypermutations led to theoretical results that are very different from those of runtime analysis but consistent with the empirical results, which demonstrates that the perspective of fixed-budget computations provides valuable information and additional insights for the performance of randomized search heuristics [30].

Accordingly, we evaluate the solution quality of an EA after running finite generations by the expected approximation error and the error tail probability. The former measures the fitness gap between a solution and optimum, and the latter is the probability distribution of the error over error levels, which measures the probability of finding the optimum. An EA is said to outperform another if, for the former EA, its error and tail probability are smaller. Furthermore, an EA is said to asymptotically outperform another if, for the former EA, its error and tail probability are smaller after a sufficiently large number of generations.

The research question of this paper is whether the binomial crossover operator can help reduce the approximation error of EA. As a pioneering work on this topic, we investigate a (1 + 1)*EAC* that performs the binomial crossover on an individual and an offspring generated by mutation, and compare a (1 + 1)*EA* without crossover and its variant (1 + 1)*EAC* on two classical problems, OneMax and Deceptive. By splitting the objective space into error levels, the analysis is performed based on the Markov chain models [31,32]. Given the two EAs, the comparison of their performance is drawn from the comparison of their transition probabilities, which are estimated by investigating the bits preferred by evolutionary operations. Under some conditions, (1 + 1)*EAC* with binomial crossover outperforms (1 + 1)*EA* on OneMax, but not on Deceptive; however, by adding an adaptive parameter mechanism arising from theoretical results, (1 + 1)*EAC* with binomial crossover outperforms (1 + 1)*EA* on Deceptive too.

This work presents the first study on how binomial crossover influences the expected runtime and tail probability of randomized search heuristics. Meanwhile, we also propose a feasible routine to get adaptive parameter settings of EAs from theoretical results. The rest of this paper is organized as follows. Section 2 reviews related theoretical work. Preliminary contents for our theoretical analysis are presented in Section 3. Then, the influence of the binomial crossover on transition probabilities is investigated in Section 4. Section 5 conducts an analysis of the asymptotic performance of EAs. To reveal how binomial crossover works on the performance of EAs for consecutive iterations, the OneMax problem and the Deceptive problem are investigated in Sections 6 and 7, respectively. Finally, Section 8 presents the conclusions and discussions.

#### **2. Related Work**

#### *2.1. Theoretical Analysis of Crossover in Evolutionary Algorithms*

To understand how crossover influences the performance of EAs, Jansen et al. [4] proved that an EA using crossover can reduce the expected optimization time from superpolynomial to a polynomial of small degree on the function Jump. Kötzing et al. [5] investigated crossover-based EAs on the functions OneMax and Jump and showed the potential speedup by crossover when combined with a fitness-invariant bit shuffling operator in terms of optimization time. For a simple GA without shuffling, they found that the crossover probability has a drastic impact on the performance on Jump. Corus and Oliveto [6] obtained an upper bound on the runtime of standard steady-state GAs to hillclimb the OneMax function and proved that the steady-state EAs are 25% faster than their mutation-only counterparts. Their analysis also suggests that larger populations may

be faster than populations of size 2. Dang et al. [7] revealed that the interplay between crossover and mutation may result in a sudden burst of diversity on the Jump test function and reduce the expected optimization time compared to mutation-only algorithms such as (1 + 1) EA. For royal road functions and OneMax, Sudholt [8] analyzed uniform crossover and k-point crossover and proved that crossover makes every (*μ* + *λ*) EA at least twice as fast as the fastest EA using only standard bit mutation. Pinto and Doerr [9] provided a simple proof of a crossover-based genetic algorithm (GA) outperforming any mutationbased black-box heuristic on the classic benchmark OneMax. Oliveto et al. [10] obtained a tight lower bound on the expected runtime of the (2 + 1) GA on OneMax. Lengler and Meier [11] studied the positive effect of using larger population sizes and crossover on Dynamic BinVal.

For non-artificial problems, Lehre and Yao [12] proved that the use of crossover in the (*μ* + 1) steady-state genetic algorithm may reduce the runtime from exponential to polynomial for some instance classes of the problem of computing unique input–output (UIO) sequences. Doerr et al. [13,14] analyzed EAs on the all-pairs shortest path problem. Their results confirmed that the EA with a crossover operator is significantly faster in terms of the expected optimization time. Sutton [15] investigated the closest string problem and proved that a multi-start (*μ* + 1) GA required less randomized fixed-parameter tractable (FPT) time than that with disabled crossover.

However, there is some evidence that crossover is not always helpful. Richter et al. [16] constructed Ignoble Trail functions and proved that mutation-based EAs optimize them more efficiently than GAs with crossover. The later need exponential optimization time. Antipov and Naumov [17] compared crossover-based algorithms on RealJump functions with a slightly shifted optimum, which increases the runtime of all considered algorithms on RealJump. The hybrid GA fails to find the shifted optimum with high probability.

#### *2.2. Theoretical Analysis of Differential Evolution Algorithms*

Most existing theoretical studies on DE are focused on continuous variants [33]. By estimating the probability density function of generated individuals, Zhou et al. [34] demonstrated that the selection mechanism of DE, which chooses mutually different parents for the generation of donor vectors, sometimes does not work positively on the performance of DE. Zaharie and Micota [35–37] investigated the influence of the crossover rate on both the distribution of the number of mutated components and the probability for a component to be taken from the mutant vector, as well as the influence of mutation and crossover on the diversity of the intermediate population. Wang and Huang [38] attributed the DE to a one-dimensional stochastic model, and investigated how the probability distribution of population is connected to the mutation, selection, and crossover operations of DE. Opara and Arabas [39] compared several variants of the differential mutation using characteristics of their expected mutants' distribution, which demonstrated that the classic mutation operators yield similar search directions and differ primarily by the mutation range. Furthermore, they formalized the contour fitting notion and derived an analytical model that links the differential mutation operator with the adaptation of the range and direction of search [40].

By investigating the expected runtime of BDE, Doerr and Zhang [24] performed a first fundamental analysis on the working principles of discrete-coded DE. It was shown that BDE optimizes the important decision variables, but is hard to find the optima for decision variables with a small influence on the objective function. Since BDE generates trial vectors by implementing a binary variant of binomial crossover accompanied by the mutation operation, it has characteristics significantly different from classic EAs or estimation-of-distribution algorithms.

#### *2.3. Fixed-Budget Analysis and Approximation Error*

To bridge the wide gap between theory and application, Jasen and Zarges [27] proposed an FBA framework of RSHs, by which the fitness of random local search and (1 + 1)

EA were investigated for given iteration budgets. Under the framework of FBA, Jasen and Zarges [41] analyzed the any-time performance of EAs and artificial immune systems on a proposed dynamic benchmark problem. Nallaperuma et al. [42] considered the well-known traveling salesperson problem (TSP) and derived the lower bounds of the expected fitness gain for a specified number of generations. Based on the Markov chain model of RSHs, Wang et al. [29] constructed a general framework of FBA, by which they found the analytic expression of approximation error instead of asymptotic results of expected fitness values. Doerr et al. [43] built a bridge between runtime analysis and FBA, by which a huge body of work and a large collection of tools for the analysis of the expected optimization time could meet the new challenges introduced by the new fixed-budget perspective.

Noting that hypermutations tend to be inferior to typical example functions in terms of runtime, Jansen and Zarges [30] conducted an FBA to explain why artificial immune systems are popular in spite of these proven drawbacks. It was shown that the inversely fitness-proportional mutation (IFPM) and the somatic contiguous hypermutation (CHM) could perform better than the single point mutation on OneMax while FBA is performed by considering different starting points and varied iteration budgets. It indicates that the traditional perspective of expected optimization time may be unable to explain the observed good performance, which is due to the limited length of runs. Therefore, the perspective of fixed-budget computations provides valuable information and additional insights.

#### **3. Preliminaries**

*3.1. Problems*

Considering a maximization problem

$$\max f(\mathbf{x}), \quad \mathbf{x} = (\mathbf{x}\_1, \dots, \mathbf{x}\_n) \in \{0, 1\}^n,$$

denote its optimal solution by **x**∗ and optimal objective value by *f* ∗. The quality of a solution **x** is evaluated by its approximation error *e*(**x**) :=| *f*(**x**) − *f* ∗ |. The error *e*(**x**) takes finite values, called error levels:

$$e(\mathbf{x}) \in \{e\_0, e\_1, \dots, e\_L\}, \quad 0 = e\_0 \le e\_1 \le \dots \le e\_{L'}$$

where *L* is a non-negative integer. **x** is called *at the level i* if *e*(**x**) = *ei*, *i* ∈ {0, 1, ... , *L*}. The collection of solutions at level *i* is denoted by X*i*.

We investigate the optimization problem in the form

$$\max f(|\ge|),$$

where | **x** |:= ∑*n i*=1 *xi*. Error levels of (1) take only *n* + 1 values. Two instances, the unimodal OneMax problem and the multi-modal Deceptive problem, are considered in this paper.

**Problem 1** (OneMax)**.**

$$\max f(\mathbf{x}) = \sum\_{i=1}^{n} \mathbf{x}\_{i\prime} \quad \mathbf{x} = (\mathbf{x}\_1, \dots, \mathbf{x}\_n) \in \{0, 1\}^n.$$

**Problem 2** (Deceptive)**.**

$$\max f(\mathbf{x}) = \begin{cases} \sum\_{i=1}^{n} \mathbf{x}\_{i\prime} & \text{if } \sum\_{i=1}^{n} \mathbf{x}\_{i} > n - 1, \\\ n - 1 - \sum\_{i=1}^{n} \mathbf{x}\_{i\prime} & \text{otherwise,} \end{cases} \quad \mathbf{x} = (\mathbf{x}\_{1}, \dots, \mathbf{x}\_{n}) \in \{0, 1\}^{n}.$$

For the OneMax problem, both exploration and exploitation are helpful to the convergence of EAs to the optimum, because exploration accelerates the convergence process and exploitation refines the precision of approximation solutions. However, for the Deceptive problem, local exploitation leads to convergence to the local optimum, but it in turn increases the difficulty to jump to the global optimum. That is, exploitation hinders convergence to the global optimum of the Deceptive problem, thus, the performance of EAs is dominantly influenced by their exploration ability.

#### *3.2. Evolutionary Algorithms*

For the sake of analysis on binomial crossover excluding the influence of population and mutation, the (1 + 1)*EA* presented in Algorithm 1 is taken as the baseline algorithm in our study. Its candidate solutions are generated by the bitwise mutation with probability *pm*. The binomial crossover is appended to (1 + 1)*EA*, getting (1 + 1)*EAC* which is illustrated in Algorithm 2. The (1 + 1)*EAC* first performs bitwise mutation with probability *qm*, and then applies binomial crossover with rate *CR* to generate a candidate solution for selection.

The EAs investigated in this paper can be modeled as homogeneous Markov chains [31,32]. Given the error vector

$$\tilde{\mathbf{e}} = (e\_0, e\_1, \dots, e\_L)',\tag{2}$$

and the initial distribution

$$\mathfrak{q}^{[0]} = (q\_0^{[0]}, q\_1^{[0]}, \dots, q\_L^{[0]})' \tag{3}$$

the transition matrix of (1 + 1)*EA* and (1 + 1)*EAC* for the optimization problem (1) can be written in the form

$$\tilde{\mathbf{R}} = (r\_{\hat{i},\hat{j}})\_{(L+1)\times(L+1)\prime} \tag{4}$$

where

$$r\_{i,j} = \Pr\{\mathbf{x}\_{t+1} \in \mathcal{X}\_i \mid \mathbf{x}\_t \in \mathcal{X}\_j\}, \quad i, j = 0, \dots, L.$$

#### **Algorithm 1** (1 + 1)*EA*

2: randomly generate a solution **x**0 = (*x*1,..., *xn*);

3: **while** the stopping criterion is not satisfied **do**

4: generate the mutant **y***t* = (*y*1,..., *yn*) by bitwise mutation:

$$\text{for } i = 1, \ldots, n\_{\prime} \quad y\_{i} = \begin{cases} 1 - \mathbb{x}\_{i\prime} & \text{if } rnd\_{i} < p\_{m\prime} \\ \mathbb{x}\_{i\prime} & \text{otherwise,} \end{cases} \quad rnd\_{i} \sim \mathcal{U}[0, 1]; \tag{5}$$

5: **if** *f*(**y**) ≥ *f*(**x***t*) **then** 6: **x***t*+1 = **y***t*; 7: **else** 8: **x***t*+1 = **x***t*; 9: **end if** 10: *t* = *t* + 1; 11: **end while**

1: counter *t* = 0;

**Algorithm 2** (1 + 1)*EAC*

1: counter *t* = 0;

$$\text{for } i = 1, \ldots, n\_{\prime} \quad v\_{i} = \begin{cases} 1 - \chi\_{i\prime} & \text{if } rnd1\_i < q\_{m\prime} \\ \chi\_{i\prime} & \text{otherwise,} \end{cases} \quad rnd1\_i \sim \mathcal{U}[0, 1]; \tag{6}$$

$$\text{for } i = 1, \dots, n\_{\prime} \quad y\_{i} = \begin{cases} v\_{i\prime} & \text{if } i = rndi \text{ or } rnd2\_i < \mathbb{C}\_{\mathbb{R}\_{\prime}} \\ \mathbb{x}\_{i\prime} & \text{otherwise,} \end{cases} \\ \text{rnd2}\_i \sim \mathcal{U}[0, 1]; \qquad (7)$$

7: **if** *f*(**y**) ≥ *f*(**x***t*) **then** 8: **x***t*+1 = **y***t*; 9: **else** 10: **x***t*+1 = **x***t*; 11: **end if** 12: *t* = *t* + 1; 13: **end while**

Recalling that the solutions are updated by the elitist selection, we know **R˜** is an upper triangular matrix that can be partitioned as

$$
\tilde{\mathbf{R}} = \begin{pmatrix} 1 & \mathbf{r}\_0 \\ \mathbf{0} & \mathbf{R} \end{pmatrix}.
$$

,

where **r**0 represents the probabilities to transfer from non-optimal statuses to the optimal status, and **R** is the transition submatrix depicting the transitions between nonoptimal statuses.

#### *3.3. Transition Probabilities*

Transition probabilities can be confirmed by considering generation of a candidate **y** with *f*(**y**) ≥ *f*(**x**), which is achieved if "*l* preferred bits" of **x** are changed. If there are multiple solutions that are better than **x**, there could be multiple choices for both the number *l* and the location of "*l* preferred bits".

**Example 1.** *For the OneMax problem, e*(**x**) *equals to the amount of '0'-bits in* **x***. Denoting e*(**x**) = *j and e*(**y**) = *i, we know* **y** *replaces* **x** *if and only if j* ≥ *i. Then, to generate a candidate* **y** *replacing* **x***, "l preferred bits" can be confirmed as follows.*

If an EA flips each bit with an identical probability, the probability of flipping *l* bits are related to *l* and independent of their locations. Denoting the probability of flipping *l* bits by *P*(*l*), we can confirm the connection between the transition probability *ri*,*j* and *P*(*l*).

As presented in Example 1, transition from level *j* to level *i* (*i* < *j*) results from flips of *j* − *i* + *k* '0'-bits and *k* '1'-bits. Then, transition probabilities for OneMax are confirmed as

$$r\_{i,j} = \sum\_{k=0}^{M} \mathbb{C}\_{n-j}^{k} \mathbb{C}\_{j}^{k + (j - i)} P(2k + j - i), \tag{8}$$

where *M* = min{*n* − *j*, *i*}, 0 ≤ *i* < *j* ≤ *n*.

According to definition of the Deceptive problem, we get the following map from | **x** | to *e*(**x**).

$$\begin{array}{ccccccccc}\hline \mathbf{x} & \text{:} & \mathbf{0} & \mathbf{1} & \cdots & n-1 & n\\ \mathbf{z}(\mathbf{x}) & \text{:} & \mathbf{1} & \mathbf{2} & \cdots & n & \mathbf{0} \\ \hline \end{array} \tag{9}$$

Transition from level *j* to level *i* (0 ≤ *i* < *j* ≤ *n*) is attributed to one of the following cases.

Accordingly, transition probabilities for Deceptive are confirmed as

$$r\_{i,j} = \begin{cases} \sum\_{k=0}^{M} \mathbb{C}\_{n-j+1}^{k} \mathbb{C}\_{j-1}^{k+(j-i)} P(2k+j-i), & i \ge 1, \\ P(n-j+1), & i=0, \end{cases} \tag{10}$$

where *M* = min{*n* − *j* + 1, *i* − 1}.

#### *3.4. Performance Metrics*

To evaluate the performance of EAs, we propose two metrics for a given iteration budget, the expected approximation error (EAE) and the tail probability (TP) of EAs for *t* consecutive iterations.

**Definition 1.** *Let* {**x***t*, *t* = 1, 2 . . . } *be the individual sequence of an individual-based EA.*

*(1) The expected approximation error (EAE) after t consecutive iterations is*

$$\varepsilon^{[t]} = \mathbb{E}[\varepsilon(\mathbf{x}\_t)] = \sum\_{i=0}^{L} \varepsilon\_i \Pr\{\varepsilon(\mathbf{x}\_t) = \mathbf{e}\_i\}. \tag{11}$$

*(2) Given i* > 0*, the tail probability (TP) of the approximation error that e*(**x***t*) *is greater than or equal to ei is defined as*

$$p^{[t]}(e\_i) = \Pr\{e(\mathbf{x}\_l) \ge e\_i\}.\tag{12}$$

EAE is the fitness gap between a solution and the optimum. It measures solution quality after running *t* generations. TP is the probability distribution of a found solution over non-optimal levels where *i* > 0. The sum of TP is the probability of not finding the optimum.

Given two EAs A and B, if both EAE and TP of Algorithm A are smaller than those of Algorithm B for any iteration budget, we say Algorithm A outperforms Algorithm B on problem (1).

**Definition 2.** *Let* A *and* B *be two EAs applied to problem (1).*

	- *e* [*t*] A − *e* [*t*] B ≤ 0*,* ∀ *t* > 0*;*
	- *p* [*t*] A (*ei*) − *p* [*t*] B (*ei*) ≤ 0*,* ∀ *t* > 0*,* 0 < *i* < *L.*
	- lim*t*→∞ *e* [*t*] A − *e* [*t*] B ≤ 0*;*
	- lim*t*→+∞ *p* [*t*] A (*ei*) − *p* [*t*] B (*ei*) ≤ 0*.*

The asymptotic outperformance is weaker than the outperformance.

#### **4. Comparison of Transition Probabilities of Two EAs**

In this section, we compare transition probabilities of (1 + 1)*EA* and (1 + 1)*EAC*. According to the connection between *ri*,*j* and *P*(*l*), a comparison of transition probabilities can be conducted by considering the probabilities of flipping "*l* preferred bits".

#### *4.1. Probabilities of Flipping Preferred Bits*

Denote probabilities of (1 + 1)*EA* and (1 + 1)*EAC* to flip "*l* preferred bits" by *P*1(*l*, *pm*) and *P*2(*l*, *CR*, *qm*), respectively. By (5), we know

$$P\_1(l, p\_m) = (p\_m)^l (1 - p\_m)^{n-l}.\tag{13}$$

Since the mutation and the binomial crossover in Algorithm 2 are mutually independent, we can get the probability by considering the crossover first. When flipping "*l* preferred bits" by (1 + 1)*EAC*, there are *l* + *k* (0 ≤ *k* ≤ *n* − *l*) bits of **y** set as *vi* by (7), the probability of which is

$$P\_{\mathbb{C}}(l+k,\mathbb{C}\_{\mathbb{R}}) = \frac{l+k}{n} (\mathbb{C}\_{\mathbb{R}})^{l+k-1} (1-\mathbb{C}\_{\mathbb{R}})^{n-l-k}.$$

If only "*l* preferred bits" are flipped, we know,

$$\begin{split} P\_2(l, \mathbb{C}\_R, q\_m) &= \sum\_{k=0}^{n-l} \mathbb{C}\_{n-l}^k P\_\mathbb{C}(l+k, \mathbb{C}\_R) (q\_m)^l (1 - q\_m)^k \\ &= \frac{1}{n} [l + (n-l)\mathbb{C}\_R - nq\_m \mathbb{C}\_R] (\mathbb{C}\_R)^{l-1} (q\_m)^l (1 - q\_m \mathbb{C}\_R)^{n-l-1} . \end{split} \tag{14}$$

Note that (1 + 1)*EAC* degrades to (1 + 1)*EA* when *CR* = 1, and (1 + 1)*EA* becomes the random search while *pm* = 1. Thus, we assume that *pm*, *CR*, and *qm* are located in (0, 1). A fair comparison of transition probabilities is investigated by considering the identical parameter setting

$$p\_m = \mathbb{C}\_{\mathbb{R}} q\_m = p\_\prime \quad 0 < p < 1. \tag{15}$$

Then, we know *qm* = *p*/*CR*, and Equation (14) implies

$$P\_2(l, \mathbb{C}\_R, p/\mathbb{C}\_R) = \frac{1}{n} \left[ (n - l) + \frac{l - np}{\mathbb{C}\_R} \right] p^l (1 - p)^{n - l - 1}. \tag{16}$$

Subtracting (13) from (16), we have

$$P\_2(l, \mathbb{C}\_{\mathcal{R}}, p/\mathcal{C}\_{\mathcal{R}}) - P\_1(l, p) = \left\{ \frac{1}{n} \left[ (n - l) + \frac{l - np}{\mathbb{C}\_{\mathcal{R}}} \right] - (1 - p) \right\} p^l (1 - p)^{n - l - 1}$$

$$= \left( \frac{1}{\mathbb{C}\_{\mathcal{R}}} - 1 \right) \left( \frac{l}{n} - p \right) p^l (1 - p)^{n - l - 1}. \tag{17}$$

From the fact that 0 < *CR* < 1, we conclude that *P*2(*l*, *CR*, *p*/*CR*) is greater than *P*1(*l*, *p*) if and only if *l* > *np*. That is, the introduction of the binomial crossover in (1 + 1)*EA* leads to the enhancement of the exploration ability of (1 + 1)*EAC*. We get the following theorem for the case that *p* ≤ 1 *n* .

**Theorem 1.** *While* 0 < *p* ≤ 1 *n , it holds for all* 1 ≤ *l* ≤ *n that P*1(*l*, *p*) ≤ *P*2(*l*, *CR*, *p*/*CR*). **Proof.** The result can be obtained directly from Equation (17) by setting *p* ≤ 1 *n* .

For the popular setting where the mutation probability of (1+1)EA is set as 1/*n*, the introduction of binomial crossover does increase the ability to generate new candidate solutions. Then, we investigate how this improvement contributes to change of transition probabilities.

#### *4.2. Comparison of Transition Probabilities*

To validate that algorithm A is more efficient than algorithm B, it is assumed that the probability of A to transfer to promising statuses could be not smaller than that of B.

**Definition 3.** *Let* A *and* B *be two EAs with an identical initialization mechanism.* **A˜** = (*ai*,*j*) *and* **B˜** = (*bi*,*j*) *are the transition matrices of* A *and* B*, respectively. It is said that* **A˜** *dominates* **B˜** *, denoted by* **A˜ B˜** *, if it holds that*

*1. ai*,*j* ≥ *bi*,*j*, ∀ 0 ≤ *i* < *j* ≤ *L; 2. ai*,*j* > *bi*,*j*, ∃ 0 ≤ *i* < *j* ≤ *L.*

Denote the transition probabilities of (1 + 1)*EA* and (1 + 1)*EAC* by *pi*,*j* and *si*,*j*, respectively. For the OneMax problem and Deceptive problem, we get the relation of transition dominance on the premise that *pm* = *CRqm* = *p* ≤ 1 *n* .

**Theorem 2.** *For* (1 + 1)*EA and* (1 + 1)*EAC, denote their transition matrices by* **P˜** *and* **S˜***, respectively. On the condition that pm* = *CRqm* = *p* ≤ 1 *n , it holds for problem (1) that* **S˜ P˜***.*

**Proof.** Denote the collection of all solutions at level *k* by S(*k*), *k* = 0, 1, ... , *n*. We prove the result by considering the transition probability

$$r\_{i,j} = \Pr\{\mathbf{y} \in \mathcal{S}(i) \mid \mathbf{x} \in \mathcal{S}(j)\}, \quad (i < j).$$

Since the function values of solutions are merely related to the number of '1'-bits, the probability to generate a solution **y** ∈ S(*i*) by performing mutation on **x** ∈ S(*j*) depends on the Hamming distance *l* = *H*(**x**, **y**). Given **x** ∈ S*j*, S(*i*) is partitioned as S(*i*) = *L l*=1 S*l*(*i*), where S*l*(*i*) = {**y** ∈ S(*i*) | *H*(**x**, **y**) = *l*}, and *L* is a positive integer that is smaller than or equal to *n*.

Accordingly, the probability to transfer from level *j* to *i* is confirmed as

$$r\_{i,j} = \sum\_{l=1}^{L} \Pr\{\mathbf{y} \in \mathcal{S}\_l(i) \mid \mathbf{x} \in \mathcal{S}(j)\} = \sum\_{l=1}^{L} |\ \mathcal{S}\_l(i) \mid P(l), l$$

where | S*l*(*i*) | is the size of S*l*(*i*), *P*(*l*) the probability to flip "*l* preferred bits".

**Example 2** (**Comparison of transition probabilities for the OneMax problem**)**.** *Let pm* = *CRqm* = *p* ≤ 1 *n . By (8), we have*

$$p\_{i,j} = \sum\_{k=0}^{M} \mathbb{C}\_{n-j}^{k} \mathbb{C}\_{j}^{k + (j - i)} P\_1(2k + j - i, p), \tag{21}$$

$$s\_{i,j} = \sum\_{k=0}^{M} \mathbb{C}\_{n-j}^{k} \mathbb{C}\_{j}^{k + (j-i)} P\_{2}(2k + j - i, \mathbb{C}\_{R}, p/\mathbb{C}\_{R}). \tag{22}$$

*where M* = min{*n* − *j*, *i*}*. Since p* ≤ 1/*n, Theorem 1 implies that*

$$P\_1(2k+j-i,p) \le P\_2(2k+j-i, \mathcal{C}\_{\mathbb{R}^\nu} p/\mathcal{C}\_{\mathbb{R}})\_{\nu}$$

*and by (21) and (22) we have pi*,*j* ≤ *si*,*j*, ∀ 0 ≤ *i* < *j* ≤ *n*.

**Example 3** (**Comparison of transition probabilities for the Deceptive problem**)**.** *Let pm* = *CRqm* = *p* ≤ 1 *n . Equation (10) implies that*

$$p\_{i,j} = \begin{cases} \sum\_{k=0}^{M} \mathbb{C}\_{n-j+1}^{k} \mathbb{C}\_{j-1}^{k+(j-i)} P\_1(2k+j-i, p), & i > 0, \\ P\_1(n-j+1, p), & i = 0, \end{cases} \tag{23}$$

$$s\_{i,j} = \begin{cases} \sum\_{k=0}^{M} \mathbb{C}\_{n-j+1}^{k} \mathbb{C}\_{j-1}^{k+(j-i)} P\_2(2k+j-i, \mathbb{C}\_{R'} \frac{p}{\mathbb{C}\_R}), & i > 0, \\ P\_2(n-j+1, \mathbb{C}\_{R'} p/\mathbb{C}\_R), & i = 0, \end{cases} \tag{24}$$

*where M* = min{*n* − *j* + 1, *i* − 1}*. Similar to the analysis of Example 2, we get the conclusion that pi*,*j* ≤ *si*,*j*, ∀ 0 ≤ *i* < *j* ≤ *n.*

The results demonstrate that when *p* ≤ 1/*n*, the introduction of binomial crossover leads to transition dominance of (1 + 1)*EAC* over (1 + 1)*EA*. In the following section, we would like to answer if transition dominance leads to outperformance of (1 + 1)*EAC* over (1 + 1)*EA*.

#### **5. Analysis of Asymptotic Performance**

In this section, we will prove that (1 + 1)*EAC* asymptotically outperforms (1 + 1)*EA* using the average convergence rate [25,32].

**Definition 4.** *The average convergence rate (ACR) of an EA for t generation is*

$$R\_{EA}(t) = 1 - \left(e^{\left[t\right]} / e^{\left[0\right]}\right)^{1/t}.\tag{25}$$

**Lemma 1** ([32], Theorem 1)**.** *Let* **R** *be the transition submatrix associated with a convergent EA. Under random initialization (i.e., the EA may start at any initial state with a positive probability), it holds*

$$\lim\_{t \to +\infty} R\_{EA}(t) = 1 - \rho(\mathbf{R})\_\prime \tag{26}$$

*where ρ*(**R**) *is the spectral radius of* **R***.*

Lemma 1 presents the asymptotic characteristics of the ACR, by which we get the result on the asymptotic performance of EAs.

**Proposition 1.** *If* **A˜ B˜** *, there exists T* > 0 *such that 1. e*[*t*] A ≤ *e* [*t*] B *,* ∀ *t* > *T;*

*2. p*[*t*] A (*ei*) ≤ *p* [*t*] B (*ei*)*,* ∀ *t* > *T*, 1 ≤ *i* ≤ *L.*

**Proof.** By Lemma 1, we know ∀  > 0, there exists *T* > 0 such that

$$e^{[0]} (\rho(\mathbf{R}) - \epsilon)^t < e^{[t]} < e^{[0]} (\rho(\mathbf{R}) + \epsilon)^t, \quad t > T. \tag{27}$$

From the fact that the transition submatrix **R** of an RSH is upper triangular, we conclude

$$\rho(\mathbf{R}) = \max\{r\_{1,1}, \dots, r\_{L,L}\}. \tag{28}$$

Denote

$$\mathbf{A} = (a\_{i,j}) = \begin{pmatrix} 1 & \mathbf{a}\_0 \\ \mathbf{0} & \mathbf{A} \end{pmatrix}, \quad \mathbf{B} = (b\_{i,j}) = \begin{pmatrix} 1 & \mathbf{b}\_0 \\ \mathbf{0} & \mathbf{B} \end{pmatrix}.$$

While **A˜ B˜** , it holds

$$a\_{j,j} = 1 - \sum\_{i=0}^{j-1} a\_{i,j} < 1 - \sum\_{i=0}^{j-1} b\_{i,j} = b\_{j,j\prime} \ 1 \le j \le L.$$

Then, Equation (28) implies that

$$
\rho(\mathbf{A}) < \rho(\mathbf{B})\,.
$$

Applying it to (27) for  < 1 2 (*ρ*(**B**) − *ρ*(**A**)), we have

$$e^{[t]}\_{\mathcal{A}} < e^{[0]}(\rho(\mathbf{A}) + \epsilon)^{t} < e^{[0]}(\rho(\mathbf{B}) - \epsilon)^{t} < e^{[t]}\_{\mathcal{B}},\tag{29}$$

which proves the first conclusion.

Noting that the tail probability *p*[*t*] (*ei*) can be taken as the expected approximation error of an optimization problem with an error vector

$$\mathbf{e} = (\underbrace{0, \dots, 0}\_{i} 1, \dots, 1)^{\prime}\_{\prime\prime}$$

by (29) we have

$$p\_{\mathcal{A}}^{[t]}(e\_i) \le p\_{\mathcal{B}}^{[t]}(e\_i), \quad \forall \ t > T, \ 1 \le i \le L.$$

The second conclusion is proven.

By Definition 2 and Proposition 1, we get the following theorem for comparing the asymptotic performance of (1 + 1)*EA* and (1 + 1)*EAC*.

**Theorem 3.** *If CR* = *CRqm* = *p* ≤ 1 *n , the* (1 + 1)*EAC asymptotically outperforms* (1 + 1)*EA on problem (1).*

**Proof.** The proof can be completed by applying Theorem 2 and Proposition 1.

On condition that *CR* = *CRqm* = *p* ≤ 1 *n* , Theorem 3 indicates that after sufficiently many number of iterations, (1 + 1)*EAC* can performs better on problem (1) than (1 + 1)*EA*. A further question is whether (1 + 1)*EAC* outperforms (1 + 1)*EA* for *t* < +∞. We answer the question in next sections.

#### **6. Comparison of the Two EAs on OneMax**

In this section, we show that the outperformance introduced by binomial crossover can be obtained for the uni-modal OneMax problem based on the following lemma [29].

**Lemma 2** ([29], Theorem 3)**.** *Let*

$$
\tilde{\mathbf{e}} = (e\_0, e\_1, \dots, e\_L)'\_{\prime} \quad \tilde{\mathbf{v}} = (v\_0, v\_{1\prime}, \dots, v\_L)'\_{\prime\prime}
$$

*where* 0 ≤ *ei*−1 ≤ *ei*, *i* = 1, . . . , *L, vi* > 0, *i* = 0, 1, . . . , *L. If transition matrices* **R˜** *and* **S˜** *satisfy*

$$s\_{j,j} \ge r\_{j,j}, \tag{30}$$

$$\forall \ 1 \le j \le L,\tag{30}$$

$$\sum\_{l=0}^{i-1} (r\_{l,j} - s\_{l,j}) \ge 0, \quad \qquad \qquad \forall \ 0 \le i < j \le L,\tag{31}$$

$$\sum\_{l=0}^{i} (s\_{l,j-1} - s\_{l,j}) \ge 0, \quad \qquad \qquad \forall \ 0 \le i < j - 1 < L,\tag{32}$$

*it holds*

$$\mathfrak{e}' \mathfrak{R}^t \mathfrak{v} \le \mathfrak{e}' \tilde{\mathbf{S}}^t \tilde{\mathbf{v}}\_{\cdot \cdot}$$

For the EAs investigated in this study, conditions (30)–(32) are satisfied thanks to the monotonicity of transition probabilities.

**Lemma 3.** *When p* ≤ 1/*n (n* ≥ 3*), P*1(*l*, *p*) *and P*2(*l*, *CR*, *p*/*CR*) *are monotonously decreasing in l.*

**Proof.** When *p* ≤ 1/*n*, Equations (13) and (14) imply that

$$\frac{P\_1(l+1,p)}{P\_1(l,p)} = \frac{p}{1-p} \le \frac{1}{n-1},\tag{33}$$

$$\frac{P\_2(l+1,\mathbb{C}\_R,p/\mathbb{C}\_R)}{P\_2(l,\mathbb{C}\_R,p/\mathbb{C}\_R)} = \frac{(l+1)(1-\mathbb{C}\_R) + n\mathbb{C}\_R(1-p/\mathbb{C}\_R)}{l(1-\mathbb{C}\_R) + n\mathbb{C}\_R(1-p/\mathbb{C}\_R)} \overset{p}{=} \frac{l+1}{l} \frac{p}{1-p} \le \frac{l+1}{l} \frac{1}{n-1} \tag{34}$$

all of which are not greater than 1 when *n* ≥ 3. Thus, *P*1(*l*, *p*) and *P*2(*l*, *CR*, *p*/*CR*) are monotonously decreasing in *l*.

#### **Lemma 4.** *For the OneMax problem, pi*,*j and si*,*j are decreasing in j.*

**Proof.** We validate the monotonicity of *pi*,*j* for (1 + 1)*EA*, and that of *si*,*j* can be confirmed in a similar way.

 By (21) we know

$$p\_{i,j+1} = \sum\_{k=0}^{M} \mathbb{C}\_{n-j-1}^{k} \mathbb{C}\_{j+1}^{i-k} P\_1(2k+j+1-i, p),\tag{35}$$

$$p\_{i,j} = \sum\_{k=0}^{M} \mathbb{C}\_{n-j}^{k} \mathbb{C}\_{j}^{i-k} P\_1(2k+j-i, p), \tag{36}$$

where *M* = min{*n* − *j* − 1, *i*}. Moreover, (33) implies that

$$\frac{C\_{j+1}^{i-k}P\_1(2k+j+1-i,p)}{C\_j^{i-k}P\_1(2k+j-i,p)} = \frac{j+1}{(j+1)-(i-k)}\frac{p}{1-p} \le \frac{j+1}{2}\frac{1}{n-1} < 1,$$

and we know

$$\mathbb{C}\_{j+1}^{i-k} P\_1(2k+j+1-i,p) < \mathbb{C}\_j^{i-k} P\_1(2k+j-i,p). \tag{37}$$

Note that

$$\min\{n-j-1,i\} \ge \min\{n-j,i\}, \quad \mathbb{C}^k\_{n-j-1} < \mathbb{C}^k\_{n-j}.\tag{38}$$

From (35)–(38) we conclude that

$$p\_{i,j+1} < p\_{i,j}, \quad 0 \le i < j < n.$$

Similarly, we can validate that

$$s\_{i,j+1} < s\_{i,j}, \quad 0 \le i < j < n.$$

In conclusion, *pi*,*j* and *si*,*j* are monotonously decreasing in *j*.

**Theorem 4.** *On condition that pm* = *CRqm* = *p* ≤ 1 *n , it holds for the OneMax problem that*

$$(1+1)EA\_{\mathbb{C}} \overset{\succ}{\sim} (1+1)EA\_{\mathbb{C}}$$

**Proof.** Given the initial distribution **q˜**[0] and transition matrix **R˜** , the level distribution at iteration *t* is confirmed by

$$
\tilde{\mathbf{q}}^{[t]} = \tilde{\mathbf{R}}^t \tilde{\mathbf{q}}^{[0]}.\tag{39}
$$

Denote

$$\mathfrak{S} = (\mathfrak{e}\_{0\prime}\mathfrak{e}\_{1\prime}\dots\mathfrak{e}\_{L})^{\prime}\_{\prime} \quad \mathfrak{S}\_{i} = (\underbrace{0\cdots \dots 0}\_{i}1, \dots, 1)^{\prime}\_{\cdot \cdot}.$$

By premultiplying (39) with **e˜** and **o˜***i*, respectively, we get

$$e^{|t|} = \tilde{\mathbf{e}}^{t} \mathbf{R}^{t} \tilde{\mathbf{q}}^{|0|},\tag{40}$$

$$p^{|t|}(e\_i) = \Pr\{e(\mathbf{x}\_l)\} \ge e\_i\} = \mathfrak{d}\_i^\prime \tilde{\mathbf{R}}^\dagger \tilde{\mathbf{q}}^{|\mathbf{0}|}.\tag{41}$$

Meanwhile, by Theorem 2 we have

$$q\_{j,j} \stackrel{<}{\leq} s\_{j,j} \stackrel{<}{\leq} p\_{j,j},\tag{42}$$

$$\sum\_{l=0}^{i-1} (q\_{l,j} - s\_{l,j}) \ge 0, \quad \sum\_{l=0}^{i-1} (s\_{l,j} - p\_{l,j}) \ge 0, \quad \forall \ i < j. \tag{43}$$

and Lemma 4 implies

$$\sum\_{l=0}^{i} (s\_{l,j-1} - s\_{l,j}) \ge 0, \quad \sum\_{l=0}^{i} (p\_{l,j-1} - p\_{l,j}) \ge 0 \tag{44} \qquad \forall \ i < j - 1. \tag{44}$$

Then, (42)–(44) validate satisfaction of conditions (30)–(32), and by Lemma 2 we know

$$\begin{aligned} \mathfrak{s}' \mathfrak{S}^t \mathfrak{S}^t \mathfrak{q}^{[0]} &\leq \mathfrak{e}' \mathfrak{P}^t \mathfrak{q}^{[0]}, & \forall t > 0;\\ \mathfrak{o}'\_i \tilde{\mathbf{S}}^t \tilde{\mathbf{q}}^{[0]} &\leq \mathfrak{o}'\_i \mathbf{P}^t \tilde{\mathbf{q}}^{[0]}, & \forall t > 0, \ 1 \leq i < n. \end{aligned}$$

Then, we get the conclusion by Definition 2.

The above theorem demonstrates that the dominance of transition matrices introduced by the binomial crossover operator leads to the outperformance of (1 + 1)*EAC* on the uni-modal problem OneMax.

#### **7. Comparison of the Two EAs on Deceptive**

In this section, we show that the outperformance of (1 + 1)*EAC* over (1 + 1)*EA* may not always hold on Deceptive. Then, we propose an adaptive strategy of parameter setting arising from the theoretical analysis, with which (1 + 1)*EAC* performs better in terms of tail probability.

#### *7.1. Numerical Demonstration for Inconsistency between the Transition Dominance and the Algorithm Outperformance*

For the Deceptive problem, we first present a counterexample to show even if the transition matrix of an EA dominates another EA, we cannot conclude that the former EA outperforms the latter.

**Example 4.** *We construct two artificial Markov chains as the models of two EAs. Let EA*R *and EA*S *be two EAs starting with an identical initial distribution*

$$\mathbf{p}^{[0]} = \left(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n}\right)^t.$$

*and the respective transition matrices are*

$$
\tilde{\mathbf{R}} = \begin{pmatrix}
1 & \frac{1}{n^3} & \frac{2}{n^3} & \cdots & \frac{n}{n^3} \\
& 1 - \frac{1}{n^3} & \frac{1}{n^2} \\
& & 1 - \frac{1}{n^2} - \frac{2}{n^3} & \ddots \\
& & & \ddots & \frac{n-1}{n^2} \\
& & & & 1 - \frac{1}{n}
\end{pmatrix}
$$

*and*

$$\mathbf{S} = \begin{pmatrix} 1 & \frac{2}{n^3} & \frac{4}{n^3} & \cdots & \frac{2n}{n^3} \\ & 1 - \frac{2}{n^3} & \frac{1}{n^2} + \frac{1}{2n} \\ & & 1 - \frac{n^2 + 2n + 8}{2n^3} & \ddots \\ & & & \ddots & \frac{n-1}{n^2} + \frac{n-1}{2n} \\ & & & & 1 - \frac{n^2 + n + 2}{2n^2} \end{pmatrix}.$$

*Obviously, it holds* **S˜ R˜** *. Through computer simulation, we get the curve of EAE difference of the two EAs in Figure 1a and the curve of TPs difference between the two EAs in Figure 1b. From Figure 1b, it is clear that EA*R *does not always outperform EA*S *because the difference of TPs is negative at the early stage of the iteration process but later positive.*

**Figure 1.** Simulation results on the difference of EAEs and TPs for the counterexample. (**a**) Difference of expected approximation errors (EAEs). (**b**) Difference of tail probabilities (TPs).

Now we turn to discuss (1 + 1)*EA* and (1 + 1)*EAC* on Deceptive. We demonstrate (1 + 1)*EAC* may not outperform (1 + 1)*EA* over all generations although the transition matrix of (1 + 1)*EAC* dominates that of (1 + 1)*EA*.

**Example 5.** *In* (1 + 1)*EA and* (1 + 1)*EAC, set pm* = *CRqm* = 1/*n. For* (1 + 1)*EAC, let qm* = 1 2 *, CR* = 2 *n . The numerical simulation results of EAEs and TPs for 5000 independent runs are depicted in Figure 2. It is shown that when n* ≥ 9*, both EAEs and TPs of* (1 + 1)*EA could be smaller than those of* (1 + 1)*EAC. This indicates that the dominance of the transition matrix does not always guarantee the outperformance of the corresponding algorithm.*

**Figure 2.** Numerical comparison for (1 + 1)*EA* and (1 + 1)*EAC* applied to the Deceptive problem, where *n* refers to the problem dimension. (**a**) Numerical comparison of expected approximation errors (EAEs). (**b**) Numerical comparison of tail probabilities (TPs).

With *pm* = *CRqm* = *p* ≤ 1/*n*, although the binomial crossover leads to transition dominance of (1 + 1)*EAC* over (1 + 1)*EA*, the enhancement of exploitation plays a governing role in the iteration process. Thus, the imbalance of exploration and exploitation leads to poor performance of (1 + 1)*EAC* at some stage of the iteration process. As shown in the previous two examples, the outperformance of (1 + 1)*EAC* cannot be drawn from the dominance of transition matrices.

The fitness landscape of Deceptive confirms that global convergence of EAs on Deceptive is principally attributed to the direct transition from level *j* to level 0, quantified

by the transition probability *r*0,*j*. By investigating the impact of binomial crossover on the transition probability *r*0,*j*, we arrive at an adaptive strategy for the regulation of the mutation rate and the crossover rate, by which performance of both (1 + 1)*EA* and (1 + 1)*EAC* are enhanced.

#### *7.2. Comparisons on the Probabilities to Transfer from Non-Optimal Statuses to the Optimal Status*

A comparison between *p*0,*j* and *s*0,*j* is performed by investigating their monotonicity. Substituting (13) and (14) into (23) and (24), respectively, we have

$$p\_{0,j} = P\_1(n-j+1, p\_m) = (p\_m)^{n-j+1}(1-p\_m)^{j-1},\tag{45}$$

$$\begin{split} s\_{0,j} &= P\_3(n-j+1, \mathbb{C}\_{\mathbb{R}}, q\_m) \\ &= \frac{1}{n} [(j-1)(1-\mathbb{C}\_{\mathbb{R}}) + n\mathbb{C}\_{\mathbb{R}}(1-q\_m)] \mathbb{C}\_{\mathbb{R}}^{n-j}(q\_m)^{n-j+1} (1-q\_m\mathbb{C}\_{\mathbb{R}})^{j-2} . \end{split} \tag{46}$$

We first investigate the maximum values of *p*0,*j* to get the ideal performance of (1 + 1)*EA* on the Deceptive problem.

**Theorem 5.** *While*

$$p\_m^\* = \frac{n-j+1}{n},\tag{47}$$

*p*0,*j gets its maximum values pmax* 0,*j* = *n*−*j*+1 *n n*−*j*+1 *j*−1 *n j*−1 *.*

**Proof.** By (45), we know

$$\frac{\partial}{\partial p\_m} p\_{0,j} = (n - j + 1 - np\_m) p\_m^{n-j} (1 - p\_m)^{j-2}.$$

While *pm* = *n*−*j*+1 *n* , *p*0,*j* gets its maximum value

$$p\_{0,j}^{\max} = P\_1(n-j+1, \frac{n-j+1}{n}) = \left(\frac{n-j+1}{n}\right)^{n-j+1} \left(\frac{j-1}{n}\right)^{j-1}.$$

$$\mathbf{0}$$

Influence of the binomial crossover on *s*0,*j* is investigated on condition that *pm* = *qm*. By regulating *CR*, we compare *p*0,*j* with the maximum value *smax* 0,*j* of *s*0,*j*.

**Theorem 6.** *On condition that pm* = *qm, the following results hold.*

**Proof.** Note that (1 + 1)*EAC* degrades to (1 + 1)*EA* when *CR* = 1. Then, if the maximum value *smax* 0,*j* of *s*0,*j* is obtained by setting *CR* = 1, we have *smax* 0,*j* = *p*0,*j*; otherwise, it holds *smax* 0,*j* > *p*0,*j*.

1. For the case that *j* **= 1**, Equation (46) implies

$$s\_{0,1} = q\_m^n (\mathbb{C}\_R)^{n-1}.$$

Obviously, *s*0,1 is monotonously increasing in *CR*. It gets the maximum value while *C R* = 1. Then, by (45) we get *smax* 0,1 = *p*0,1.

2. While *j* **= 2**, by (46) we have

$$\frac{\partial s\_{0,2}}{\partial \mathbb{C}\_R} = \frac{n-1}{n} q\_m^{n-1} (\mathbb{C}\_R)^{n-3} (n - 2 + (1 - nq\_m)\mathbb{C}\_R).$$

$$C\_R^\star = \frac{n-2}{nq\_m - 1}.\tag{48}$$

Then, we have *smax* 0,2 > *p*0,2.

3. For the case that **3** *≤ j ≤ n −* **1**, we denote

$$s\_{0,j} = \frac{n-j+1}{n} q\_m^{n-j+1} I\_1 + \frac{(j-1)(1-q\_m)}{n} q\_m^{n-j+1} I\_{2,j}$$

where

$$\begin{aligned} I\_1 &= (\mathcal{C}\_R)^{n-j} (1 - q\_m \mathcal{C}\_R)^{j-1}, \\ I\_2 &= (\mathcal{C}\_R)^{n-j+1} (1 - q\_m \mathcal{C}\_R)^{j-2}. \end{aligned}$$

Then,

$$\begin{aligned} \frac{\partial I\_1}{\partial \mathbb{C}\_R} &= (\mathbb{C}\_R)^{n-j-1} (1 - q\_m \mathbb{C}\_R)^{j-2} (n - j - (n - 1) q\_m \mathbb{C}\_R), \\\frac{\partial I\_2}{\partial \mathbb{C}\_R} &= (\mathbb{C}\_R)^{n-j} \left( 1 - \frac{\mathbb{C}\_R}{n} \right)^{j-3} (n - j + 1 - (n - 1) q\_m \mathbb{C}\_R). \end{aligned}$$

$$\mathbb{C}\_{\mathbb{R}}^{\star} \in \left( \frac{n-j}{(n-1)q\_{m}}, \frac{n-j+1}{(n-1)q\_{m}} \right). \tag{49}$$

Accordingly, we know *smax* 0,*j* > *p*0,*j*.

• If *n*−*j n*−1 < *qm* < *n*−*j*+1 *n*−1 , *I*1 gets its maximum value when *CR* = *n*−*j* (*n*−1)*qm* , and *I*2 is monotonously increasing in *CR*. Then, *s*0,*j* get its maximum value *smax* 0,*j* at some

$$C\_R^\star \in \left(\frac{n-j}{(n-1)q\_m}, 1\right],\tag{50}$$

and we know *smax* 0,*j* > *p*0,*j*.

4. While *j* **=** *n*, Equation (46) implies that

$$\frac{\partial \mathfrak{s}\_{0,n}}{\partial \mathbb{C}\_R} = (n-1)(1 - q\_m \mathbb{C}\_R)^{n-3}(1 - 2q\_m - (n - 1 - nq\_m)q\_m \mathbb{C}\_R).$$

Denoting

$$g(q\_{m\prime}C\_{\mathbb{R}}) = 1 - 2q\_m - (n - 1 - nq\_m)q\_mC\_{\mathbb{R}\prime}$$

we can confirm the sign of *∂s*0,*n*/*∂CR* by considering

$$\frac{\partial}{\partial \mathcal{C}\_R} \lg(q\_{m\prime} \mathcal{C}\_R) = -(n - 1 - nq\_m)q\_m.$$

• While 0 < *qm* ≤ *n*−1 *n* , *g*(*qm*, *CR*) is monotonously decreasing in *CR*, and its minimum value is

$$\lg(q\_{m'}1) = (nq\_m - 1)(q\_m - 1).$$

The maximum value of *g*(*qm*, *CR*) is

$$\lg(q\_{m\prime}0) = 1 - 2q\_m\ldots$$

(a) If 0 < *qm* ≤ 1 *n* , we have

$$\lg(q\_{m\prime}C\_R) \ge \lg(q\_{m\prime}1) > 0.$$

Thus, *∂s*0,*n ∂CR* ≥ 0, and *s*0,*n* is increasing in *CR*. For this case, *s*0,*n* get its maximum value when *C R* = 1, and we have *smax* 0,*n* = *p*0,*n*.

(b) If 1 *n* < *qm* ≤ 1 2 , *s*0,*n* gets the maximum value *smax* 0,*n* when

$$\mathcal{C}\_{\mathbb{R}}^{\star} = \frac{1 - 2q\_m}{q\_m(n - 1 - nq\_m)}.$$

Thus, *smax* 0,*n* > *p*0,*n*.

$$\lg(q\_{m'}1) = (nq\_m - 1)(q\_m - 1) < 0.$$

Then, *s*0,*n* is monotonously decreasing in *CR*, and its maximum value is obtained by setting *C R* = 0. Accordingly, we know *smax* 0,*n* > *p*0,*n*.

In summary, *smax* 0,*n* > *p*0,*n* while *qm* > 1 *n* ; otherwise, *smax* 0,*n* = *p*0,*n*.

Theorems 5 and 6 present the "best" settings to maximize the transition probabilities from non-optimal statuses to the optimal level, by which we get a parameter adaptive strategy that greatly enhances the exploration of compared EAs.

#### *7.3. Parameter Adaptive Strategy to Enhance Exploration of EAs*

Since the level index *j* is equal to the Hamming distance between **x** and **x**∗, improvement of level index *j* is bounded by reduction of the Hamming distance obtained by replacing **x** with **y**. Then, while the local exploitation leads to a transition from level *j* to a non-optimal level *i*, the practically adaptive strategy of parameters can be obtained according to the Hamming distance between **x** and **y**.

When (1 + 1)*EA* is located at the solution **x** at status *j*, Equation (47) implies that the "best" setting of mutation rate is *p m*(*j*) = *n*−*j*+1 *n* . Once it transfers to solution **y** at status *i*(*i* < *j*), the "best" setting changes to *p m*(*i*) = *n*−*i*+1 *n* . Then, the difference of "best" settings is *j*−*i n* , bounded from above by *H*(**x**,**y**) *n* . Accordingly, the mutation rate of (1 + 1)*EA* can be updated to

$$p'\_m = p\_m + \frac{H(\mathbf{x}, \mathbf{y})}{n}.\tag{51}$$

For (1 + 1)*EAC*, the parameter *qm* is adapted using the strategy consistent to that of *pm* to focus on influence of *CR*. That is,

$$q\_m' = q\_m + \frac{H(\mathbf{x}, \mathbf{y})}{n}.\tag{52}$$

Since *s*0,*j* demonstrates different monotonicity for varied levels, one cannot get an identical strategy for the adaptive setting of *CR*. As a compromise, we would like to consider the case that 3 ≤ *j* ≤ *n* − 1, which is obtained by random initialization with overwhelming probability.

According to the proof of Theorem 6, we know *CR* should be set as great as possible for the case *qm* ∈ (0, *n*−*j n*−1 ]; while *qm* ∈ ( *n*−*j n*−1 , 1], *C R* is located in intervals whose boundary values are *n*−*j* (*n*−1)*qm* and *n*−*j*+1 (*n*−1)*qm* , given by (49) and (50), respectively. Then, while *qm* is updated by (52), the update strategy of *CR* can be confirmed to satisfy that

$$\mathbf{C}'\_{R}q'\_{m} = \mathbf{C}\_{R}q\_{m} + \frac{H(\mathbf{x}, \mathbf{y})}{n-1}.$$

Accordingly, the adaptive setting of *CR* could be

$$\mathbf{C}'\_{R} = \left(\mathbf{C}\_{R}q\_{m} + \frac{H(\mathbf{x}, \mathbf{y})}{n-1}\right) / q'\_{m'} \tag{53}$$

where *q m* is updated by (52).

By incorporating the adaptive strategy (51) to (1 + 1)*EA*, we compare the performance of its adaptive variant with the adaptive (1 + 1)*EAC* that regulates its mutation rate and crossover rate by (52) and (53), respectively. For 13–20 dimensional Deceptive problems, numerical simulation of the tail probability is implemented by 10,000 independent runs. The initial value of *pm* is set as 1 *n* . To investigate the sensitivity of the adaptive strategy on initial values of *qm*, the mutation rate *qm* in (1 + 1)*EAC* is initialized with values √1 *n* , 3 2 √*n* and √2 *n* , and the corresponding variants are denoted by (1 + 1)*EA*1 *C*, (1 + 1)*EA*2 *C* and (1 + 1)*EA*3 *C*, respectively.

The converging curves of averaged TPs are illustrated in Figure 3. Compared to the EAs with fixed parameters during the evolution process, the performance of the adaptive EAs on Deceptive has been significantly improved. Furthermore, we also note that the converging curves of adaptive (1 + 1)*EAC* are not sensitive to the initial mutation rate. Although transition dominance does not necessarily lead to outperformance of (1 + 1)*EAC* over (1 + 1)*EA*, the proposed adaptive strategy can greatly enhance global exploration of (1 + 1)*EAC* to a large extent, and consequently, we get the improved adaptive (1 + 1)*EAC* that is not sensitive to initial mutation rates.

**Figure 3.** *Cont.*

**Figure 3.** Numerical comparison on tail probabilities (TPs) of adaptive (1 + 1)*EA* and (1 + 1)*EAC* applied to the Deceptive problem, where *n* is the problem dimension. (1 + 1)*EA*1 *C*, (1 + 1)*EA*2 *C*, and (1 + 1)*EA*3 *C* are three variants of (1 + 1)*EAC* with *qm* initialized as √1 *n* , 3 2 √*n* , and √2 *n* , respectively.

#### **8. Conclusions and Discussions**

Under the framework of fixed-budget analysis, we conduct a pioneering analysis of the influence of binomial crossover on the approximation error of EAs. The performance of EAs after running finite generations is measured by two metrics: the expected value of the approximation error and the error tail probability, by which we make a case study by comparing the performance of (1 + 1)*EA* and (1 + 1)*EAC* with binomial crossover.

Starting from the comparison of the probability of flipping "*l preferred bits*", it is proven that under proper conditions, incorporation of binomial crossover leads to the dominance of transition probabilities, that is, the probability of transferring to any promising status is improved. Accordingly, the asymptotic performance of (1 + 1)*EAC* is superior to that of (1 + 1)*EA*.

It is found that the dominance of transition probability guarantees that (1 + 1)*EAC* outperforms (1 + 1)*EA* on OneMax in terms of both expected approximation error and tail probability. However, this dominance does lead to the outperformance on Deceptive. This means that using binomial crossover may improve the performance on some problems but not on other problems.

For Deceptive, an adaptive strategy of parameter setting is proposed based on the monotonicity analysis of transition probabilities. Numerical simulations demonstrate that it can significantly improve the exploration ability of both (1 + 1)*EAC* and (1 + 1)*EA*, and superiority of binomial crossover is further strengthened by the adaptive strategy. Thus, a problem-specific adaptive strategy is helpful for improving the performance of EAs.

Our future work will focus on a further study for the adaptive setting of crossover rate in population-based EAs on more complex problems, as well as the development of adaptive EAs improved by the introduction of binomial crossover.

**Author Contributions:** Conceptualization, J.H. and X.Z.; formal analysis, C.W.; writing—original draft preparation, C.W.; writing—review and editing, Y.C. and J.H.; funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Fundamental Research Funds for the Central Universities grant number WUT:2020IB006.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**4. Mitochondrial dehydrogenases under diabetic conditions**

A vast body of literature covers the functional modifications of mitochondria under diabetic conditions. Because cardiovascular insults remain the leading cause of death for diabetic patients [38], most of the attention has been focused on the modifications occurring in cardiac mitochondria to provide a better rationale for the morphological and functional modifications observed in diabetic hearts. Further, the incidence and severity of cardiovascular complications are markedly increased in both male and female diabetic patients, with the latter exhibiting a greater incidence than the diabetic male counterparts [39].

In humans, our understanding of the disease is complicated by the presence of two pathological conditions: diabetes type 1, in which insulin production is lacking, and diabetes type 2, in which insulin is still produced, sometimes to a greater extent than under normal conditions, and yet it is not functionally efficient in controlling the glycemic state of the patient. An additional confounding parameter is that to a large extent type 2 diabetic patients are overweight or obese, with an altered lipid status both in serum and within tissues (lipotoxicity).

This variety of conditions is also reflected in the heterogeneity of animal experimental models utilized to investigate the disease. Also here, models of type 1 diabetes are conceptually simpler in that animals are injected with agents that damage pancreatic beta-cells and abolish insulin secretion, fully mimicking the human condition. Models for type 2 diabetes are more heterogeneous, spanning from genetically conditioned animals (KO animals for specific proteins, receptors, or signaling molecules) to genetically inbred animals (e.g. Koletsky rats), to animals that develop diabetes spontaneously, to animals fed assorted high fat diets with or without sucrose, and injected with subliminal doses of beta-cell damaging agents that limit but not completely abolish the endogenous production of insulin, or treated with various oral antidiabetic agents [40]. Because of this plethora of models, it is not simple to determine to which extent modifications of mitochondrial dehydrogenase are a primary or secondary insult in the pathogenesis of the diabetic condition and its complications.

The development of diabetic cardiomyopathy has been etiologically attributed to several factors including metabolic [41], biochemical [42] and ultra-structural [43] modifications within the cardiac myocytes. Scrutiny of mitochondrial function in cardiac mitochondria from streptozotocin-treated rats has shown a decline in respiration and oxygen consumption, more pronounced for state 3 than state 4, which translates into a decrease in ATP production [44]. This energy deficiency has been considered a key factor in the development of diabetes-related cardiac dysfunctions, although changes in cardiac microvasculature [45], metabolic and hormonal disturbances [46] and concurring hyperglycemia [47] can certainly act as co-factors. Combining defective mitochondrial respiration with increased ROS formation, studies from our laboratory have indicated that succinate dehydrogenase [48] and pyruvate dehydrogenase [49] are highly sensitive to ROS-induced damage, forming stable, non-functional adducts within the mitochondrion of streptozotocin-treated rats. As mentioned in the previous section, the consequences of this dysfunction are three-fold: 1) the inability of the electron transport chain to operate properly is compromised, resulting in decreased ATP production; 2) the production of ROS is enhanced, further damaging macromolecules within the mitochondrion and its surroundings, and 3) forcing the cardiac myocytes to depend on glycolysis to produce ATP. Interestingly, these changes were observed predominantly in cardiac and liver tissue but not in the kidney of the diabetic animals [49], rising the intriguing question of which protective mechanism(s) may operate in the latter organ under diabetic conditions.


**6.1 Antisense oligonucleotides (AMO)**

Antisense inhibition of miRNA function has been an important tool for uncovering miRNA biology and potential therapeutics [31]. Synthetic oligonucleotides can be used therapeutically when miRNA dysregulation contributes to pathophysiology. These oligonucleotides are known as anti-miRNA oligonucleotides (miRNA inhibitors). To improve functional potency and to provide protection against nuclease degradation, they are often chemically modified [31, 32]. An ideal modification should increase binding affinity to the extent that specificity is compromised and should be non-toxic. There are four most common oligonucleotide modifications: (1) 2'-O-methyl groups, (2) phosphorothioate, (3) locked nucleic acid (LNA) anti-miRNA constructs, and (4) N,N-diethyl-4-(4-nitronaphthalen-1-ylazo) phenylamine (ZEN).

2'-O-methyl groups are the first generation of AMOs. 2'-O-methyl modifications can help AMOs to increase nuclease resistance and facilitate binding affinities to miRNA by the addition of an O-methyl group to the 2'-C atom. Phosphorothioate, compared to the 2'-O-methyl analogs, bonds at both the 3′ and 5′ ends to prevent nuclease degradation, and a 3′ cholesterol tail to help with cell uptake [33]. These modifications help the AMO to penetrate into tissues and

**99**

*Therapeutic Implication of miRNA in Human Disease DOI: http://dx.doi.org/10.5772/intechopen.82738*

LNA-AMOs [39].

**6.2 miRNA sponges**

phenomenon in human and animals.

**6.3 miRNA masking**

organs and significantly increase their half-lives in the target tissues. However, the first generation of AMOs have relative low potencies to be effective in animal model [34]. Second generation AMOs contain other modifications at the 2′ sugar position. Locked nucleic acid (LNA) modifications which are bicyclic nucleic acids that tether the 2'O to the 4'C via methylene bridge locking sugar into a 3′ endo conformation have been shown to have the best binding affinity and nuclease resistance. This group of AMO has been widely used in experimental animals [35–37]. Currently, the most advanced miRNA targeting therapy is SPC3649 (miravirsen), which is a locked nucleic acid-modified oligonucleotide antagonizing miR-122. This is the first miRNA-targeted drug to enter human clinical trials [38]. Despite LNA modifications have higher binding affinity, these modifications can lead to off-target effects which may cause toxicity *in vivo* [35]. Recently, a new compound called N,N-diethyl-4-(4-nitronaphthalen-1-ylazo)-phenylamine (ZEN), when is included at each end of the AMO, led to increased binding affinity to the miRNA and inhibited exonuclease degradation. Recent studies have shown that this group of AMOs (ZEN-AMOs) have higher potency and less toxicity than

miRNA sponges are transcripts that contain multiple (typically 4–10 separated by a few nucleotides) tandem-binding sites to a miRNA of interest and are transcribed from mammalian expression vectors. The use of miRNA sponges in mammalian cells was introduced by Ebert and colleagues [40]. miRNA sponges have been found to occur naturally as long non-coding RNA in plants and animals. Synthetic miRNA sponges are usually plasmid or viral vectors which contain tandemly arrayed miRNA binding sites, separated with a small nucleotide spacer and inserted into a 3'UTR of the reporter gene driven by an RNA polymerase II promoter [40, 41]. miRNA sponges have the ability to inhibit an entire family of miRNA by using the common seed sequence, and can therefore inhibit multiple miRNAs at once. Some of the endogenous circular RNAs have been founded to function as nature miRNA sponges. For example, circRNA7 has been shown to be functions as a miRNA sponge for miRNA-7 in the mouse tissues [42]. The authors have further demonstrated that the testis-specific circRNA, sex-determining region Y (Sry), serves as a miR-138 sponge [42]. These finding suggest that circRNA functioning as a miRNA sponge to regulate miRNA expression may be a common

MicroRNA-masking antisense oligonucleotide technology (miR-mask) is another strategy for miRNA-based therapeutics. In contrast to miRNA sponges, miR-masks consist of single-stranded 2'-O-methyl modified antisense oligonucleotides that are fully complementary to the expected miRNA binding site in the 3'-UTR of target mRNA [43]. A miR-mask does not directly interact with its target miRNA but binds to the binding site of that miRNA in the 3' UTR of the target mRNA by fully complementary mechanism. Therefore, the miR-mask blocks the access of its target miRNA to the binding site so as to rescue its target mRNA via blocking the action of its target miRNA. miR mask is designed to be fully complementary to the target mRNA sequence of a miRNA, which suggests that the antimiRNA action of a miR-mask is gene-specific. The strategy of miRNA masking has been used to disrupt miRNA function and involves masking the target site on target mRNA using a modified single-stranded RNA complementary to the target

*Therapeutic Implication of miRNA in Human Disease DOI: http://dx.doi.org/10.5772/intechopen.82738*

organs and significantly increase their half-lives in the target tissues. However, the first generation of AMOs have relative low potencies to be effective in animal model [34]. Second generation AMOs contain other modifications at the 2′ sugar position. Locked nucleic acid (LNA) modifications which are bicyclic nucleic acids that tether the 2'O to the 4'C via methylene bridge locking sugar into a 3′ endo conformation have been shown to have the best binding affinity and nuclease resistance. This group of AMO has been widely used in experimental animals [35–37]. Currently, the most advanced miRNA targeting therapy is SPC3649 (miravirsen), which is a locked nucleic acid-modified oligonucleotide antagonizing miR-122. This is the first miRNA-targeted drug to enter human clinical trials [38]. Despite LNA modifications have higher binding affinity, these modifications can lead to off-target effects which may cause toxicity *in vivo* [35]. Recently, a new compound called N,N-diethyl-4-(4-nitronaphthalen-1-ylazo)-phenylamine (ZEN), when is included at each end of the AMO, led to increased binding affinity to the miRNA and inhibited exonuclease degradation. Recent studies have shown that this group of AMOs (ZEN-AMOs) have higher potency and less toxicity than LNA-AMOs [39].

## **6.2 miRNA sponges**

*Antisense Therapy*

let-7c, let-7f, and miR-200c were significantly decreased in pancreas cancer patients whereas miR-486-5p and miR-451 were significantly increased in those patients compared with the normal people, which suggests that these miRNAs can be served as signature biomarkers for pancreas cancer [30]. Additionally, circulating miRNAs can be employed in diagnostic strategies to detect for alterations associated with either disorder affliction or treatment response. In the miRNA-based therapeutics, the miRNA expression is altered in different diseases and it is now feasible to correct miRNA expression by injecting miRNAs similar to the use of antisense mRNAs and RNAi. For example, because the activation of onco-miRNA genes could cause development of cancer, artificial antisense miRNAs could be synthesized and used

There are at least two main strategies to target miRNA expression for prevention and potential treatment of disease. The first strategies is the use of oligonucleotides or virus-based constructs to either directly block the expression of a disease-

associated signature miRNA or to directly substitute for the loss of expression of the miRNA. The second strategies is indirectly employing drugs to alter miRNA expression by targeting their transcription and processing. Blocking miRNA expression can be achieved by the use of antisense oligonucleotides, miRNA sponges, miRNAmask and small RNA inhibitors. Restoring downregulated miRNA expression can be achieved by using synthetic miRNA (miRNA mimic) or by inserting genes coding for miRNA into viral constructs. At the transcriptional level, small-molecule miRNA inhibitors can be employed to prevent the transitions from DNA transcript to pri-miRNA and pre-miRNA. Antisense oligonucleotides can be employed at the mature miRNA level to induce degradation or revert the mature miRNA into a duplex form with the antisense oligonucleotide. At the functional level, miRNA masks can bind complementarily to the 3' UTR region of target mRNA, competing for bindings with endogenous miRNAs for the specific target. miRNA sponges can be employed to bind target miRNA via complementary mRNA binding sites,

Antisense inhibition of miRNA function has been an important tool for uncovering miRNA biology and potential therapeutics [31]. Synthetic oligonucleotides can be used therapeutically when miRNA dysregulation contributes to pathophysiology. These oligonucleotides are known as anti-miRNA oligonucleotides (miRNA inhibitors). To improve functional potency and to provide protection against nuclease degradation, they are often chemically modified [31, 32]. An ideal modification should increase binding affinity to the extent that specificity is compromised and should be non-toxic. There are four most common oligonucleotide modifications: (1) 2'-O-methyl groups, (2) phosphorothioate, (3) locked nucleic acid (LNA) anti-miRNA constructs, and (4) N,N-diethyl-4-(4-nitronaphthalen-1-ylazo)-

2'-O-methyl groups are the first generation of AMOs. 2'-O-methyl modifications can help AMOs to increase nuclease resistance and facilitate binding affinities to miRNA by the addition of an O-methyl group to the 2'-C atom. Phosphorothioate, compared to the 2'-O-methyl analogs, bonds at both the 3′ and 5′ ends to prevent nuclease degradation, and a 3′ cholesterol tail to help with cell uptake [33]. These modifications help the AMO to penetrate into tissues and

to inhibit their targeted miRNAs to treat or prevent the cancer.

decreasing expression levels of target miRNAs (see **Figure 1**).

**6.1 Antisense oligonucleotides (AMO)**

**6. miRNA therapeutics: strategies**

**98**

phenylamine (ZEN).

miRNA sponges are transcripts that contain multiple (typically 4–10 separated by a few nucleotides) tandem-binding sites to a miRNA of interest and are transcribed from mammalian expression vectors. The use of miRNA sponges in mammalian cells was introduced by Ebert and colleagues [40]. miRNA sponges have been found to occur naturally as long non-coding RNA in plants and animals. Synthetic miRNA sponges are usually plasmid or viral vectors which contain tandemly arrayed miRNA binding sites, separated with a small nucleotide spacer and inserted into a 3'UTR of the reporter gene driven by an RNA polymerase II promoter [40, 41]. miRNA sponges have the ability to inhibit an entire family of miRNA by using the common seed sequence, and can therefore inhibit multiple miRNAs at once. Some of the endogenous circular RNAs have been founded to function as nature miRNA sponges. For example, circRNA7 has been shown to be functions as a miRNA sponge for miRNA-7 in the mouse tissues [42]. The authors have further demonstrated that the testis-specific circRNA, sex-determining region Y (Sry), serves as a miR-138 sponge [42]. These finding suggest that circRNA functioning as a miRNA sponge to regulate miRNA expression may be a common phenomenon in human and animals.

#### **6.3 miRNA masking**

MicroRNA-masking antisense oligonucleotide technology (miR-mask) is another strategy for miRNA-based therapeutics. In contrast to miRNA sponges, miR-masks consist of single-stranded 2'-O-methyl modified antisense oligonucleotides that are fully complementary to the expected miRNA binding site in the 3'-UTR of target mRNA [43]. A miR-mask does not directly interact with its target miRNA but binds to the binding site of that miRNA in the 3' UTR of the target mRNA by fully complementary mechanism. Therefore, the miR-mask blocks the access of its target miRNA to the binding site so as to rescue its target mRNA via blocking the action of its target miRNA. miR mask is designed to be fully complementary to the target mRNA sequence of a miRNA, which suggests that the antimiRNA action of a miR-mask is gene-specific. The strategy of miRNA masking has been used to disrupt miRNA function and involves masking the target site on target mRNA using a modified single-stranded RNA complementary to the target

sequence [44, 45]. The miRNA-masking method, in which only specific mRNA is masked, may lead to more specific and safer therapeutic strategies.


*4.2. Temporal Trend*

Trend analyses were conducted for the annual index images using the aforementioned OLS regression and F-statistics. On the basis of the calculated slopes and significant levels, the trends of changes in the drought indices were classified into seven categories: (1) highly significant decrease (*SLOPE* < 0, *p* ≤ 0.01), (2) significant decrease (*SLOPE* < 0, 0.01 < *p* ≤ 0.05), (3) moderately significant decrease (*SLOPE* < 0, 0.05 < *p* ≤ 0.1), (4) no significant change (*p* > 0.1), (5) moderately significant increase (*SLOPE* < 0, 0.05 < *p* ≤ 0.1), (6) significant increase (*SLOPE* > 0, 0.01 < *p* ≤ 0.05), and (7) highly significant increase (*SLOPE* > 0, *p* ≤ 0.01) [35], as shown in Figure 4. Figure 4 shows a significant PCI decrease for 8.8% of the study area and a significant increase for 13.6% of the area. The remaining area exhibited no significant change during the last 18 years. The area with a significant increase in the PCI is mainly located near Mudanjiang, Jilin, and Yanji City, while the area with significantly decreasing PCI is mainly distributed near Liaoyang, Anshan, and Dandong City. For TCI, 3.3% of the area experienced a significant increase and 3.3% of the area witnessed a significant decrease. The area with a significant decrease was distributed near the north of Shuangyashan and Qitaihe City and south of Anshan City, suggesting that these areas experienced very severe droughts in the recent 18 years. The positive trend was mainly observed in Mudanjiang and Tonghua City. With regard to the VCI, a significant increase was seen for 5.3% of the study area, whereas a significant decrease was seen for 21.3%, indicating that a larger area tended to experience very severe droughts recently. The areas with a significant decrease in the VCI are mostly located in the southern coastal region, Yanji in Jilin, and the northern part of Shuangyashan. The rest of Shuangyashan and Baishan exhibited an increase in the VCI. VHI trend analysis showed that 4.9% of the GCM area experienced a significant drought alleviation, whereas 14.1% experienced a significant drought aggravation. The spatial pattern of the VHI trend is similar to that of the VCI trend. SDCI trend analysis revealed that about 11% of the central area of the GCM experienced a significant increase (i.e., drought condition is relieving); meanwhile, almost an identical extent in the southwest experienced drought aggravation. TVDI trend analysis showed that a sparse area of 9.0% of the GCM experienced a significant increase, whereas 3.5% of the area experienced a significant decrease. The region with the decreasing TVDI was mostly distributed near Mudanjiang, Liaoyuan, and Tonghua.

Thus, the trend analyses indicate di fferent drought trends for di fferent indices. For the GCM, both PCI and SDCI show similar patterns that the central area is getting wetter and the southwest area is getting drier. VCI and VHI exhibit similar patterns showing sparse areas with decrease (drying) and overwhelming increase (wetting).

**Figure 4.** Slopes of drought indices during 2001–2018. (**a**) PCI, (**b**) TCI, (**c**) VCI, (**d**) VHI, (**e**) SDCI, and (**f**) TVDI.

#### *4.3. Correlations between Drought Indices and Meterological Factors*

The annual average precipitation and temperature were collected from 14 prefecture-level cities in the GCM. Six annual drought indices were plotted for these cities, as shown in Figures 5 and 6. As shown in Figure 5, PCI and SDCI both show similar annual patterns with precipitation. This is because precipitation is an important input for both these indices. The other four indices (VCI, TCI, TVDI, and VHI) do not show any clear relation with annual precipitation. TCI mainly reflects the surface temperature variation, while for the other three indices, NDVI is an important input.

**Figure 5.** Relationship between annual drought indices and precipitation in the GCM. (**a**–**n**) are the 14 prefecture-level cities. (**a**) Anshan; (**b**) Baishan; (**c**) Benxi; (**d**) Dandong; (**e**) Fushun; (**f**) Jilin; (**g**) Jixi; (**h**) Liaoyang; (**i**) Liaoyuan; (**j**) Mudanjiang; (**k**) Qitaihe; (**l**) Shuangyashan; (**m**) Tonghua; (**n**) Yanji.

**Figure 6.** Relationship between annual drought indices and temperature in the GCM. (**a**–**n**) are the 14 prefecture-level cities. (**a**) Anshan; (**b**) Baishan; (**c**) Benxi; (**d**) Dandong; (**e**) Fushun; (**f**) Jilin; (**g**) Jixi; (**h**) Liaoyang; (**i**) Liaoyuan; (**j**) Mudanjiang; (**k**) Qitaihe; (**l**) Shuangyashan; (**m**) Tonghua; (**n**) Yanji.

Dandong has a relatively larger precipitation volume with high variations from 2001 to 2018 because of its special geolocation. Its forest coverage is as high as 65%, endowing the region with high water-holding capacity. Precipitation-based drought indices in this area indicate clearly humid characteristics. For example, the precipitation was very high in 2012 and 2013, and the average annual temperature was lower (Figure 6d). Therefore, most of the indices identified the lack of a drought condition in these 2 years. Benxi, which is a city neighboring Dandong, had slightly lower annual average temperature and precipitation. Its drought condition was similar to Dandong according to di fferent indices. Anshan and Liaoyang are located inland and receive less precipitation than Dandong and Benxi. In particular, in 2014, the precipitation of Liaoyang was only 300 mm, and the average temperature was higher than 10 ◦C, which was significantly higher than that in the other years. PCI and SDCI clearly indicate the drought situation in Liaoyang. Anshan received precipitation as low as approximately 400 mm in 2014, and the average annual temperature was higher than 11 ◦C. PCI and SDCI also reflected the drought situation of Anshan correctly. Fushun received an annual precipitation exceeding 1,000 mm in 2010 and 2013, and its average annual temperature was between 5.5 ◦C and 6 ◦C; meanwhile, in 2011 and 2014, it received a very low precipitation of 500 mm, and the PCI and SDCI values clearly reflect these changes. Baishan and Tonghua are located close to the Changbai Mountain Nature Reserve. The annual precipitation and temperature were relatively stable from 2001 to 2018. The drought indices also tended to vary smoothly. Jilin and Liaoyuan experienced similar annual precipitation variations in the recent years. In 2011, their annual precipitation was the lowest, only about 500 mm, which was captured by PCI and SDCI. Yanji, Jixi, Mudanjiang, Qitaihe, and Shuangyashan are located inland north of the Changbai Mountain Nature Reserve and have high latitudes, low annual average temperatures, and annual precipitations less than 800 mm. Their PCI, TCI, VCI, VHI, and SDCI values are lower than those in other regions. In the GCM, the TVDI, VCI, TCI, and VHI values also exhibit distinct annual patterns that do not appear to be related to the annual precipitation and temperature. This may because these indices do not consider precipitation, and indicate droughts based on the LST anomalies and vegetation health. The annual average temperature is usually not sensitive enough to reflect LST anomalies; hence, LST-based indices fail to show consistency with annual average temperature variations. Further, vegetation health may be sometimes a ffected by factors besides drought, and there is usually a time lag before drought can cause deterioration of vegetation health. This may be the reason why vegetation-based indices also fail to show consistency with the precipitation and temperature patterns.

#### *4.4. Correlations between Drought Indices and Land Cover Types*

An examination of the annual land cover data from 2001 to 2018 revealed very little land cover change over these years. Therefore, in this study, we assumed that there was no land cover change, and we used the land cover map of 2018 as the current condition of land cover to investigate the correlations between drought indices and land cover types. According to the 2018 land cover map, in the GCM, deciduous broadleaf forests, grasslands, and savannas accounted for 46.8%, 30.4%, and 18.1%, respectively. The other seven land cover types account for less than 5%.

The distribution of vegetation types has a strong relationship with regional climatic factors, and surface temperature is an important climatic factor that a ffects the zonal distribution of vegetation types. Temperature is an important limiting factor for plant growth in the GCM; forest vegetation types, in particular, have distinct vertical zonal distribution characteristics. NDVI represents vegetation characteristics, and there exists a clear relationship between NDVI and LST. Le Page et al. [36] found that the negative correlation between NDVI and LST in agricultural area is due to drastic evaporation that decreases LST. However, there exists a positive correlation between NDVI and LST in the northeastern part of the study area, and this correlation is explained by the simultaneous forest leaf loss and fall in surface temperature (the coldest months). Figure 7a,b present the mean values of the six drought indices for di fferent land cover types in 2018 and 2001. For the NDVI-based index (i.e., VCI) and the LST-based index (TCI), when the land cover changes from deciduous needleleaf forests to unvegetated

lands, VCI decreases, while TCI increases. These two indices exhibit distinct patterns in di fferent land cover types. The variability in the slope of the inverse LST–NDVI relationship in association with local topographic and environmental conditions has been assessed in previous studies [37]. The validity of VHI as a drought detection tool relies on the assumption that the NDVI and LST at a given pixel vary inversely over time with variations in VCI and TCI driven by local moisture conditions. However, over vast areas and long periods, the LST–NDVI relationship is nonunique and often nonnegative [37–39]. According to [37–39], NDVI and LST are positively related usually in energy-limited ecosystems, which implies that high temperature promotes the growth of vegetation. In this case, VHI and TVDI may not be appropriate for indicating drought conditions. However, for water-limited ecosystems, where high temperature may inhibit vegetation growth, NDVI has a negative relation with LST, conforming with the assumption of TVDI. In this case, TVDI and VHI are applicable for indicating drought conditions. Since the water-holding capacity of deciduous broadleaf forests is stronger than that of grasslands, deciduous broadleaf forests generally are more humid than grasslands. This is reflected by almost all the drought indices. Savannas comprise a mixed forest–grassland type of vegetation and have drought indices intermediate between those of the deciduous forests and grasslands.

**Figure 7.** Drought indices and their changing slopes for different land cover types: (**a**) Drought indices in 2018; (**b**) drought indices in 2001; and (**c**) slopes (×100) of drought indices from 2001 to 2018.

We examined the zonal statistic to the slope value based on land cover types. Figure 7c shows the mean *SLOPE* values (scaled by 100) of drought indices from 2001 to 2018 for di fferent land cover types. The TVDI *SLOPE* values of evergreen needleleaf forests is the highest among all land cover types; the *SLOPE* values of the other five indices are low and negative. This suggests that evergreen needleleaf forests in the GCM clearly have experienced severe drought conditions. Savannas and deciduous needleleaf forests also have similar but weaker changing patterns with positive slope for TVDI but negative slope for the other indices, which is similar to Zribi et al. [40], who analyzed drought affection on vegetation coverage based on time series Vegetation Anomaly Index (VAI). Our analysis of slope and land cover types showed that VCI and VHI have negative values with similar variation patterns across different land cover types. This means that the VCI and VHI decreased from 2001 to 2018, indicating an overall drying trend. The sharpest trend occurs in the case of unvegetated Lands. Since PCI is a precipitation-based index, when the precipitation increases, the slope of PCI is positive, indicating a wetting situation. Broadleaf croplands and deciduous broadleaf forests, which together account for a large proportion of the GCM, have high water-holding capacity. Figure 7c shows that the slope values for these types is positive, indicating a wetting situation.


**13. Conclusion**

Diabetes mellitus is a metabolic disorder in the endocrine system. It is known to be a dreadful disease that is found in all parts of the world with a serious threat to the health of mankind. Diabetes mellitus affect most of the people of both developed and developing countries. There are lots of synthetic drugs that have been used to control and to treat diabetic patients with partial recovery from this dreaded disease. Alternative to these synthetic agents, plants provide a potential source of hypoglycemic drugs and are widely used in several traditional systems of medicine to prevent diabetes. Several medicinal plants have been investigated for their beneficial use in different types of diabetes. Several phytonutrients have been identified from medicinal plants and this presents an exciting opportunity for the development of new types of therapeutics for diabetes mellitus. Most abundant phytonutrients present in medicinal plants are the alkaloids, terpenes, and phenolics. Phytomedicine has been used since ancient time in many parts of the world where access to modern medicine is limited. Despite consid‐ erable progress in the treatment of diabetes by oral hypoglycemic agents, search for newer drugs continues because the existing synthetic drugs have several limitations as shown in table 1. The treatment of diabetes with synthetic drugs in the developing countries is expensive due to poverty and lack of access to Medicare. Hence, phytotherapy has significant role to play in the developing countries compared to synthetic drugs because it is safe, less expensive and available as a gift of nature.


**1. Introduction**

402 Sociological Landscape – Theories, Realities and Trends

Zagema B., Lobbyist S. (2011). Land and Power. The growing scandal surrounding the new

The relationship between sociology and history is complex and far from harmonious; it has aspects that make it particularly difficult, leading to theoretical and methodological uncertainty while creating a number of "friction" topics. While some authors consider historical sociology an established field, and other consider it difficult to distinguish between history and sociology, communication between historical and sociological disciplines is notably limited. One indication of this is the paucity of university departments devoted to the subject. Despite certain expectations of the conjunction of these two disciplines, communication between their respective practitioners is frequently loaded with mutual ignorance, prejudices, bias, and even opinion-extremes. From the outset we should say that, intending to deal with the complicated relationship between sociology and history, we do so in the knowledge that our approach must necessarily be - given the thematic breadth of the problem - selective, and, moreover, limited by coming from the sociological perspective. It is very likely that if a paper on the same topic were written by a historian, other issues would be emphasised. Opinions on specific issues would no doubt diverge, and accents be placed on different themes.


Preface

Worldwide, more than 400 million tons of plastic are produced annually. These plastics break down into small particles and those with a length of < 5 mm are called microplastics (MPs). The World Health Organization (WHO) has identified MPs as an emergency pollutant for the environment and human health. MPs hold toxic chemicals (such as plasticizers and colorings) that are used as ingredients during the polymerization process. Moreover, MPs act as transport vectors of other hazardous substances (pesticides, heavy metals, and biofouling), which can accumulate in the environment and induce toxicity to biodiversity. In addition to the inhalation of MPs, their existence in drinks and foods is the main pathway for entering the human body. The accumulation of MPs in the human body places people at risk for respiratory problems, cytotoxicity, and inflammatory as well as autoimmune illnesses. Therefore, it is crucial to identify the safest and most effective strategies for removing MPs from the environment. This book provides a comprehensive summary of MPs and discusses advances and challenges in removing them from our environment.

We would like to express our gratitude to all the contributing authors. We also wish to thank the publishing process manager at IntechOpen for being generously helpful.

> **El-Sayed Salama** Lanzhou University, Lanzhou, China


**1. Introduction**

Corrosive ingestion is a medical emergency that is especially prevalent in developing countries such as Thailand [1–9]. Since 2020, the COVID-19 has had an enormous impact on many sectors worldwide, and it had affected the trend of rising incidence and severity of diseases [10]. However, the actual incidence should not be precise as the tip of the iceberg phenomenon is probably under-report [4, 8]. Currently, various studies on this topic are still being developed for medical knowledge to the achievement goal of the best practice. Perforation and stricture are complications of corrosive ingestion which are currently being researched and which are discussed in this chapter.