Abstract
Logistic-based sample assumption is proposed in this paper, with a research on different random distributions through this system. It provides an assumption system of logistic-based sample, including its sample space structure. Moreover, the influence of different random distributions for inputs has been studied through this logistic-based sample assumption system. In this paper, three different random distributions (normal distribution, uniform distribution, and beta distribution) are used for test. The experimental simulations illustrate the relationship between inputs and outputs under different random distributions. Thereafter, numerical analysis infers that the distribution of outputs depends on that of inputs to some extent, and this assumption system is not independent increment process, but it is quasistationary.
1. Introduction
Since 1990s, chaos, as a new subject with its characteristics of internal randomness, initial condition sensitivity, and a series of similar cryptography, has become one of the greatest hot points in information security and secure communication. During these two decades, researches on chaos almost divide into four parts, namely, chaotic synchronization [1, 2], chaotic control [3, 4], chaotic secure communication [5, 6], and chaotic characteristics with cryptography [7, 8]. For the researches on chaotic signal, there are also a lot of papers, devoted to study the characteristics of chaotic sequences, which focused on its complicacy, weak key analysis, statistical or randomness tests, and so forth [9–12], and they also paid much attention to self- or cross-correlation so that a new improved chaotic signal can be made [13]. All of their studies have significant contributions to studying and using chaos indeed. However, to be different, some papers paid attention to chaos and randomness, although they are different phenomenon generated by different systems. And they provided the innovative point on studying chaos. Bucolo et al. had discussed whether chaos works better than noise [14]. Wang referred to the fact that so far chaos has been described by deterministic equations in mathematics, without any mathematical description by randomness [15]. In the study of random variables and random process [16], an interesting idea hits our mind and we are excited to understand chaotic signal generated by logistic mapping through this new viewpoint [17]. Due to iteration in logistic mapping, the relationship between one value and its next value should have relevance [18]. Among the data, they are not i.i.d., for they are not independent with each other. It is easy to find that there is a correlation between one value and its next value, which satisfy the logistic mapping, but if we select the data from another way, do they also have some regular rules or some interesting results? That is the key question for which this paper wants to show the answer.
In the practical engineering application, randomness is the common research object and it is of great importance to do researches on chaos and randomness. During the studies, we wondered about the characteristics of outputs when there are different random distributions data inputting into this system; therefore, our previous paper had a simple searching on it [19]. And now, we provide a deep discussion of output characteristics and the influence on different random distribution inputs in this paper.
Section 2 provides the preparation of mathematic theory and then logistic-based sample model assumption and its selection methods are respectively shown in Section 3. In Section 4, data under different random distributions are used as inputs for sample space, with the outputs simulation through logistic-based sample model assumption system in Section 5. Numerical analysis and summary are proposed in Section 6. Finally, we come to a conclusion of this paper. At present, our treatment is heuristic. To some extent, it is really an interesting exploration of such research.
2. Preparation of Mathematical Theory
2.1. Random Vector
is -dimension random vector, where each component of random vector is one-dimensional random variable like . The distribution of is named marginal distribution of ( is the distribution of ).
2.2. Classifications of Stochastic Process
In this paper, we provide four kinds of stochastic process for preparation of mathematical theory, and the definitions of them are given below [20].
2.2.1. Independent Increment Process
Let ; if are independent of each other, then it is defined that is the independent increment process.
Moreover, if for any of all , only depends on (), then it is named that has stationary increments. The independent process of stationary increments is defined as the independent stationary increment process.
2.2.2. Markov Process
If, for arbitrary , , , there is , , then it is defined that is Markov process. It is inferred that status of process only depends on current moment and is unrelated to the past moments, which is the most important characteristic of Markov process with the name of stability ineffectiveness theory.
Meanwhile, is defined as the probability function of the transition, which means the probability when translates to .
2.2.3. Gauss Process
If, for arbitrary positive integer and is -dimensional normal random variable, then it is defined that random process is the normal process or Gauss process.
2.2.4. Stationary Process
If for arbitrary constant and positive integer , and , there is the same simultaneous distribution between and , then it is defined that random process is the stationary process.
Moreover, if and is a constant and, for arbitrary, , its covariance like exists and is unrelated to , then it is defined that random process is wide station ary process.
3. Logistic-Based Sample Model Assumption
The well-known logistic mapping is shown as below:
Actually, it is iterated from one initial value through such chaotic mapping, and it can obtain a series of sequences like . According to chaotic dynamics, when is in the interval of [3.5699456, 4], logistic map can generate chaos and the outputs are nonperiodic and nonconvergent. When , the map is full shot in the unit interval [0, l]. Therefore, the following deduction is under the condition of , and logistic map is surjection in (0, 1).
In this paper, a logistic-based sample model assumption is proposed, considered as a system, through which a series of sets of data can be obtained. Since the initial value is of the great importance in chaotic systems, and for logistic map there is no exception. Like the definition in discrete-time random process, it can be considered as the mapping from the test sample space to the discrete-time signal set (universe). In the sample space , there is a discrete-time signal corresponding to every test input , shown in Figure 1.
In the description and analysis of random process, it is convenient to consider the random process as a sequence of random variables with indexes. That is to say, for a certain value , taking , for example, the signal value is a random variable defined in sample space . Meanwhile, according to each , it has a value of . Therefore, random process can be expressed as the following random variable sequence with indexes , , , , .
For logistic-based sample model assumption, where we assumed that there is a sample space set consisting of the initial values, and the iterated data through logistic mapping can also constitute a lot of collection sets. The assumption model is shown in Figure 2.
Firstly, using different initial values to construct a sample 0, then each of them in sample 0 can iterate a group of sequences like , ; through logistic mapping. For this assumption system, sample 0 can seem as the random vector , each component of which can seem as one-dimensional random variable.
Secondly, it is considered the iterated outputs with number index as the other samples from sample 1 to sample , and all of them, constitute the logistic-based sample space.
Therefore, we can obtain different output sets sample from random vector sample 0, which is also the initial value set in logistic mapping, that is, , for each sample output. Finally, unfolding the whole, we can obtain a matrix; that is,
We found that if we chose the sample 0 randomly, it can be considered that one-dimensional random variable of random vector is mutually independent. Therefore, for every independent selected initial value , it can be under different random distributions. If we chose them as the inputs, we wonder what the characteristics of other samples ranging from 1 to should be. It means that we used the column of matrix like to test, that is . In every sample, each value is iterated from different independent initial values in sample 0 and they are also mutually independent apparently. It is important and necessary for us to analyze its characteristics, such as its distributions after inputting data under different random distributions.
There is the probability-distribution function of data shown in (3):
The probability-density function is shown in (4) for each r.v.:
According to ensemble average, the mean value of a sequence is defined as in (5) and the variance value is defined as in (6):
For discrete-time random process is the sequence of random variables with indexes, its mean value can be calculated to form a (certain) mean sequence, which is defined as the mean value of the discrete-time random process. Similarly, the variances of each random variable can form a (certain) variance sequence, which is defined as the variance value of the discrete-time random process. Therefore, both of them are the first-order ensemble average value of random process, commonly depending on . Mean value defines the average of the random signal, and variance value defines the average of deviation from the mean value in the square of signal.
In our research, under the logistic-based sample model assumption, we deduced the statistical values through the formula given before. The results in theoretical analysis are shown below:
For outputs samples , we can get a set . It can form an ensemble average . In calculation, we need to get the mean value of every row of the matrix and put them in order by column number . Therefore, such is the ensemble average mean value. Description in math langue is given below: where . The matrix with row and column can be calculated to obtain the ensemble average mean value sequence .
Similarly, variance can be calculated and form a variance sequence . Description in math langue is given below: where . The matrix with row and column can be calculated to obtain the ensemble average variance .
With the condition of surjection of logistic map, that is, when , we can easily get the mean value and variance; thus it is easy to get the ensemble average mean value . Similarly, with the surjection of logistic map, it can easily calculate the ensemble average variance .
4. Different Random Distributions
In this paper, we focused on what kind of distributions outputs can appear if inputs are under different random distributions. Therefore, we firstly provided three typical different random distributions in this section.
4.1. Normal Distribution
In Figure 2, there is an index set , which is also one of the samples in the logistic-based sample model assumption space, and it is also the set of random vector .
Firstly, the following shows the testing data under normal distribution; we simply chose arrays as the input; it obeys (0, 1) normal distribution; that is, mean value is 0 and variance is 1 and its probability density function is shown as follows:
Use “random (“normal,” 0.5, 1, 1, 1000)” to obtain such data.
Special instruction: for logistic mapping, it is requested that initial value should be in the interval (0, 1). That is to say, we should select the inputting data in the range of 0 to 1, so the inputs under random distribution cannot be inputted to the system directly. Therefore, after getting the data under normal distribution, we selected the outputs results of them to satisfy the initial condition of logistic map, and the figure is shown below in Figure 3.
4.2. Uniform Distribution
Secondly, it shows the testing results of data under uniform distribution. It simply chose arrays as the inputs, in order to compare them with others; it obeys (0, 1) uniform distribution; that is, mean value is 0 and variance is 1, and its probability density function is shown in (11):
Use “random (“uniform,” 0, 1, 1, 1000)” to obtain such data. Therefore, its time-domain figure and distribution figure are shown in Figure 4.
4.3. Beta Distribution
In this part, it shows the testing results of data under beta distribution. It simply chose arrays as the inputs, in order to compare them with others, it obeys (0, 1) beta distribution. There are two parameters in beta distribution and they are and . Its pdf is in (12) as below:
For beta distribution, the formulas of and are given as follows:
Use “random (“beta,” 2, 3, 1, 1000)” to obtain such data. Therefore, its time-domain figure and distribution figure are shown in Figure 5.
5. Simulation of the Outputs Distribution
According to inputs data under different random distributions given before, we made the simulations of outputs data and plot the figures of their distributions. Due to Figure 2, the outputs data are from other sets, respectively.
5.1. Outputs Distribution Simulation under Normal Distribution Inputs
For the testing data as inputs obeying normal distribution like what we described in Section 3, in this part we got the outputs distribution simulations as below. After iteration for many times, we picked up results of other samples to test the distribution of outputs. They are shown in Figure 6, in which we find that the distribution of outputs looks like that of inputs under normal distribution, especially when it is nearly the middle of iterated numbers.
5.2. Outputs Distribution Simulation under Uniform Distribution Inputs
As it was tested before, the testing data under uniform distribution are used to be the values of sample 0, which are also the initial values for logistic map. After iteration for many times, we picked up some results of other samples in sample space to test the distribution of outputs. They are shown in Figure 7, in which we find that the distribution of outputs looks like that of inputs.
5.3. Outputs Distribution Simulation under Beta Distribution Inputs
Thirdly, for the testing data under beta distribution like description in Section 3, we got the distribution simulations of outputs as below. After iteration for many times, we picked up results of other samples to test the distribution of outputs. They are shown in Figure 8.
6. Numerical Analysis and Summary
In this section, we did some numerical analysis and summarized this paper.
6.1. Comparison with Figures
Firstly, we compared with the simulation results through their distribution figures of output data.
Through qualitative analysis, it can be simply concluded that the output data under normal distribution and uniform distribution seem similar to those of input data, but they are different when input data are under beta distribution. The comparison table is shown in Table 1.
From Table 1, we can safely obtain the elementary conclusion of different random distribution research on logistic-based sample assumption. Basically, the distribution of outputs data tends to be similar to that of the inputs through the so far experiments. Exceptionally, if the random distribution inputs change to beta distribution, the results are not the same like before. However, it is interesting that the outputs distributions seem to be similar among each other no matter what random distribution data inputs, so we make a bold assumption that chaotic system and its characteristics may be the cause to this phenomenon, but it also needs to be verified and maybe it is our future work.
6.2. Comparison with Similar Iterated Numbers
Secondly, we compared the simulation results through similar iterated numbers in Table 2, that is, through similar samples in sample space . Moreover, the selection range of samples is also provided through the results, but it also needs exact theory formulated.
Through many simulations we made in Matlab, there is an interesting discovery; that is, we found that the distribution of outputs looks like that of inputs under different distributions, especially when it is nearly the middle of iterated numbers, with the inputs under normal distribution and uniform distribution. But it is not universality, because when inputs are under beta distribution, the outputs distribution in the middle of iterated numbers appears to have similar characteristics like other ones. If we want to know more about such phenomenon, we should have further research on it through both theoretical derivation and experimental simulations.
6.3. Analysis on Classification of Stochastic Process
6.3.1. Independent Increment Analysis
For sample space , let , , and we can get its increment description . Let , , , and make an analysis of its independent.
For this system is logistic-based sample assumption system to some extent it satisfied the equation . Therefore, , , and ; thus we can know that the increment of this system can be described as below:
Through the concept in Section 2.2.1, if depends on , they are not mutually independent. Therefore, we can get the conclusion that this logistic-based sample assumption is not independent increment process.
6.3.2. Stationary Analysis
In our simulation, the contact is the same; that is, . Let positive integer , and in sample space , there are and . We can get the similar simultaneous distribution between () and (). Therefore, through the stationary analysis of experimental point, we can obtain the conclusion that this logistic-based sample assumption system is quasistationary process.
7. Conclusion
This paper studied a logistic-based random distribution discussion, which provided the logistic-based sample model assumption and some results of it, and the main part of this paper is, the research on solving a problem, that is, “what the characteristic of the outputs distributions with input data under different random distributions is.” We performed some experiments to study whether inputs data under different random distributions can generate outputs data under the same random distribution and the influence or relationship between them. The experimental results inferred that, for logistic-based sample model assumption, the distribution of outputs data through this system would appear to have almost similar distribution like the inputs data correspondingly. In particular, for given random distribution inputs, such system will generate the outputs which can obey the same distribution as its inputs through the numerical analysis and summary.
Carrying on such research may have many advantages, but at present our treatment is heuristic and this work also needs an exact theory to be formulated in future.
APPENDIX
Acronyms
cdf: Cumulative distribution function i.i.d.: S-independent and identically distributed pdf: Probability density function r.v.: Random variable r.p.: Random process qr.v.: Quasi-random variable qr.p.: Quasi-random variable s-: Statistical(ly) Dt.r.p: Discrete-time random process.
Notation
: Number index of is : s-mean value of : s-variance value of : pdf of : cdf. of : Ensemble average mean value of matrix : Ensemble average variance value of matrix : Gamma function of : Beta function with interval () : Uniform function in () : Normal function with .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (no. 61072072), Innovated Team Project of “Modern Sensing Technology” in colleges and universities of Heilongjiang Province (no. 2012TD007), and Graduate Student Academic Exchange Project of Heilongjiang Province.