One of the basic problems of quickest change-point detection is the design of an optimal sequential test. If an alarm is too quickly, there is a high risk that the detection is false. And at the same time, if an alarm is too slowly, the detection delay would be too large. So an optimal sequential test is important, which is expected to have the smallest detection delay among all tests without exceeding a certain false alarm rate.
Moreover, an optimal sequential test is useful in earthquake monitoring, traffic monitoring, etc. And these data are often in the form of network data. There are many network data in our daily life, such as social networks, the representations are Facebook, wechat, etc. Hence, it is useful to analyze network data, such as the change detection of network data, which is in Amini A A , Nguyen X L.(2013), Keshavarz H, Michailidis G, Atchade Y. (2018) and Wei, C. , Wiesel, A. , and Blum, R. S. . (2012). These papers discuss the change detection problem of the Gaussian graphic model. For Gaussian graphic model, we can get accurate distribution conveniently, which makes the change point problem easy to deal with. The Gaussian graphic model is a very special case.
However there are many challenges and difficulties in dealing with general large network data. When the network is large, it is difficult to obtain the distribution of network data, because that the normalization coefficient is not easy to be calculated, since it needs exponential computation complexity. For example, the distribution of exponential family of random graph models(ERGMs), also known as models, which is introduced by Frank and Strauss(1986), Frank(1991) and Wasserman and Pattison(1996), is , represents for a network, are the features of the network, the normalization coefficient is difficult to obtain especially for large network. And this phenomenon also exsits for the Markov networks in probabilistic graphical model which is introduced by Koller D, Friedman N(2009). It is common in network data, which urges us to find a method to overcome this difficulty.
At the same time, in the literature, there are mainly four kinds of optimal sequential tests in infinite independent observation sequence: Shiryaev test , SLR (sum of the log likelihood ratio) test , the conventional CUSUM test introduced by Page (1954) and Shiryaev-Roberts test for an infinite independent observation sequence. The tests , , , , and were proven by Shiryaev (1963, 1978), Chow, Robbins and Siegmund (1971, P.108), Frisén (2003), Moustakides (1986) and Polunchenko and Tartakovsky (2010), to be optimal, and the control line are all constants. All of the optimal tests are based on likelihood ratio statistics, and the analyzed observation sequences are one-dimensional. Moreover, to obtain infinite sequnce of networks is not realistic. So it is necessary to deal with the change-point problem for finite sequences of networks about how to give the corresponding optimal sequential tests.
In this paper, in order to overcome the above difficulties, we give a finite measure space to replace the traditional probability space by discarding the normailization coefficient. And a new measure of detection delay, which is based on the corresponding measure space, is proposed. The likelihood ratio statistics are replaced by measure ratio statistics. In a word, the method we proposed are based on the finite measure not the probability space. Moreover, we know that the difference between measure space and probability space is the normalization coefficient, so traditional likelihood ratio statistics are special cases of our method. Note that, we consider unweighted network sequences here.
The rest of this paper is organized as follows. Section 2 states the method we use in this paper. Section 2.1 presents a performance measure for finite network data, , to evaluate how well a sequential test detects changes in a finite network observations. Section 2.2 gives the optimal sequential test under the performance measure we defined. Our results are illustrated by comparison of the numerical simulations of different sequential tests in Section 3, and the optimal sequential test we give obviously have better peformance than other 3 sequential tests. Section 3.1 is the case for independent and dependent finite sequences for networks in finite measure space. Section 3.2 is the case for independent and dependent finite sequences for networks in traditional probability space. Section 4 includes some concluding remarks. Proofs of the theorems are given in the Appendix.
2 Optimal sequential tests for finite sequences of network under finite measure
In this section, we first briefly describe the differences between probability space and measure space. Then we construct the optimal sequential tests for network observations.
Consider the cases such as the exponential family of random graph models(ERGMs) which is proposed in Frank and Strauss(1986), Frank(1991) and Wasserman and Pattison(1996).
represents a network, are the crucial features of the network.
When the normalization coefficient is difficult to deal with, in the probability space, the statistics, such as CUSUM-type, cannot be used. To avoid this, we consider finite measure space , we just discard the normalization coefficient, since there is no need to guarantee that in finite measure space.
And as in the probabiliy space, we also have measure density functions and conditional measure density functions for network objects as follows.
Suppose that there is a network set , let be a finite measure, assume that is differentiable in the usual sense in calculus. Let be the derivative of .
is the network object, is -field. In the discrete case, we use to replace . Then is the corresponding measure density function.
The conditional measure density functions
Moreover, in this paper, we consider unweighted network sequences. The joint meansure density function is similar. Suppose that there are finite dependent network observations, denoted by . Then let denotes the number of nodes in each network observation. In this paper, we consider the change-point problem for network sequences as follows
All the networks in sequences have a common number of nodes , which is a constant. Moreover, without loss of generality, we assume .
Pre-change and post-change measure density functions are and , and conditional measure density functions of the -th observation for are and , which are known. And the measures above are all finite.
Let () be the change-point time, then the joint measure density function can be written as
where and are respectively the pre-change and the post-change conditional measure densities of the -th observation for . When , i.e., a change never occurs in observations . And the corresponding joint measure function is denoted by .
2.1 Performance measures of sequential tests
There are some different performance measures for sequential test in Shiryaev (1963, 1978), Chow, Robbins and Siegmund (1971, P.108), Frisén (2003), Moustakides (1986) and Polunchenko and Tartakovsky (2010). These performance measures relate to the probability distributionand the probability expectation in probability space. So in finite measure space, we also have and when the change-point time ().
is the finite measure expectation, which is a first order moment based on joint measure density function. That is
where is the joint measure when .
Futhermore, denotes the measure expectation with the pre-change joint measure density .
It is easy to known that the measure expectations have the similar properties as in probability space.
Let be a sequential test, the definition is
where , is a set of all of the sequential tests satisfying and for . Following the idea of the randomization probability of the change time and the definition of describing the average detection delay proposed by Moustakides (2008), we define a performance measure , which is based on measure expectations, to evaluate the detection performance of each sequential test in the following
where are weights, which are functions of .
For the CUSUM test statistics, which is widely used in quality detection. Moreover, it is simple, which is as follows
for , where . And it is a special case with .
The performance measure in (2), denotes the detection delay. Note that when , or, , that is, we will newly detect the change of network starting from . So it is reasonable to let the weight of the detection delay as . At the same time, if we set the weight , then the numerator and denominator of the meansure in (2) can be regarded as the generalized out-of-control average run length and in-control average run length . For the numrical simulation in Section 3, we choose the CUSUM test statistics with .
2.2 Optimal sequential tests
In order to give the optimal network sequential tests, we fisrt give a definition about the optimal criterion of any sequential tests for finite observations under finite measure space. After defining a new performance measures under finite measure, we give the optimal sequential test, which is related to the control limit , so in this section, we show the optimal control limit under the performance measure we defined.
A sequential test with is optimal under the performance measure if
where satisfies .
Before giving the optimal sequential test, we need to show the corresponding conditional measure expectations w.r.t. , , which is similar as in probability space. More details are as follow:
be an integrable random variable on. Let be a sub--field of . The conditional measure expectation of given under measure , denoted by , is the a.s.-unique random variable satisfying the following two conditions:
is measurable in ;
It is easy to obtain that the conditional measure expectations have the similar properties as in probability space. In the proofs of theorems, we need to use these properties.
Motivated by Chow, Robbins and Siegmund (1971) we present a nonnegative random dynamic control limit that is defined by the following recursive equations
for , where is a constant,
Now, by using the test statistics and the control limits we define a sequential test as follows
It is clear that , and for . The positive number can be regarded as an adjustment coefficient of the random dynamic control limit, as is increasing on with and for . And c control the value of .
The following theorem shows that the sequential test is optimal under the measure .
Let be a positive number satisfying . There exists a positive number such that is optimal in the sense of (2) with ; that is,
In particular, if satisfies , that is, and , then
Here, the random dynamic control limit of the optimal control chart can be called an optimal dynamic control limit.
From Theorem 1 we know that for every measure corresponding to the weights , we can construct an optimal control chart under the measure .
While Markov process is common, so we give a simple version of Theorem 1 when the at most order of Markov process is fixed. Since and are measurable with respect to , there are functions , and such that
for . Therefore, the optimal control limit in (5) can be written as
Note that the optimal control limit depends on the sequence of networks, for . We let the sequence of networks be at most a p-order Markov process; that is, both the pre-change network observations and the post-change network observations are -order and -order Markov processes with transition measure density functions and , respectively, where and
for and . We known that , means that the sequence of networks are mutually independent.
Let the sequence of networks be at most a p-order Markov process for and the weights and for . Then
Let . The optimal control limit can be written as
Let and , then we have
Theorem 2 is a special case of Theorem 1 when the observation sequence is a Markov process and at most order of this Markov process is fixed.
3 Comparison and analysis of simulation results
Suppose that there is a finite sequence of network and . In section 3.1, under finite measure space, we first give an example for independent samples condition by comparing four different sequential tests, with constant control limits, with linear decrease control limits, with linear increase control limits, and with optimal control limits which is defined in this paper. Then we give an example for dependent samples condition, which is 1-order Markov process and we also compare the four different sequential tests. All of the above is based on CUSUM-type statistics and the number of nodes .
Note that traditional probability space is a special case, so when the normalization coefficient is computable for a network distribution, then we can use the probability space, we also give examples in Section 3.2 about independent and dependent Erdos-Rényi network sequences.
Moreover, in simulation, the performance measure with and for . And note that ME is proportitional to E, the ratio is fixed and unknown. For any test ,
is easily estimated by sample mean through simulation. So in the third column of tables, we replacewith . For all of the results, , and the four tests all have adjustment coefficient c to guarantee , which is denoted by in tables.
The explanation of the simulation results is at the end of this section.
3.1 Comparison of simulation of indenpedent network observations in finite measure space
Let be an independent network observations sequence with a pre-change measure density function and a post-change measure density function for . The function is a term used in Exponential Family Random Graph Models, which means the number of edges in a network.
There are many terms in ERGMs and many linear combination of these terms. The example we give is the simplest. The ratio of the pre-change and post-change measure density functions and is
for . We compare the performance of four tests, the first one is with constant control limits
The second test, denotes a test with decrease control lmits as follows
And denotes a test with increase control lmits as follows
The last one is the optimal test with .
The simulation result for independent finite ERGM network observations is shown in Table 1. All of the results are based on repititions.
Table 1 Simulation of four tests for independent finite network observations.
3.2 Comparison of simulation of depedent network observations in finite measure space
Let be a dependent network observation sequence. We note that if there is no cross term of and in , then the network observations are independent obviously. So, here we let pre-change measure density function and a post-change measure density function for . The ratio of the pre-change and post-change measure density functions is
for . We compare the performance of four tests, which the same as indenpendent case above.
The simulation result of dependent finite ERGM network observations is shown in Table 2. All of the results are based on repititions.
Table 2 Simulation of four tests for dependent finite network observations.
3.3 Comparison of simulation of indepedent network observations in probability space
Let be an independent network observation sequence. Moreover, before the change point, the networks are Erdos-Rényi networks with link probability 0.5, and the link probability equals to 0.6 after the change point. We compare the performance of four tests, the first one is with constant control limits
The second test, denotes a test with decrease control lmits as follows
And denotes a test with increase control lmits as follows
The last one is the optimal test with
The simulation result of independent finite Erdos-Rényi network observations is shown in Table 3. All of the results are based on repititions.
Table 3 Simulation of four tests for independent finite network observations.
3.4 Comparison of simulation of depedent network observations in probability space
Let be a dependent network observation sequence. Given , is a Erdos-Rényi network. Before the change point, the link probability is , and the link probability equals to after the change point.
The simulation result of dependent finite Erdos-Rényi network observations is shown in Table 4. All of the results are based on repititions.
Table 4 Simulation of four tests for dependent finite network observations.
Next, we give a explanation about the numerical simulation results in the above four tables. The fist two tables are for general independent and dependent ERGMs. The last two tables are for specific Erdos-Rényi random networks. The first column shows that we compare four different sequential tests, the adjustment coefficient is shown in the second column to guarantee that , in tables we use to replace it. The fourth column is the focus of our attention, the smallest of , the better of the sequential test. We find that the optimal sequential tests in four tables all have the smallest , which is consistent with Theorem 1. Note that the method in this paper can be applied in many networks, as long as we know the distribution of the networks either in probability space or in general finite measure space. Futher, the sequence of networks in Section 3.2 and Section 3.4 can be -order, while in simulation we choose 1-order networks. And in simulation, we try to make sure , but in Table 1 and Table 3, for the constant control limits, , that’s because if we adjust the constant c a little bit, the corresponding is far from 40. There are many different control limits, in this paper we choose relatively representative control limits.
The usual CUSUM control chart we use in quality detection is constant control limits. From the simulation, we can conclude that under the performance measure we defined, constant control limits maybe not good enough, somtimes the increase control limits and decrease control limits have better performance, which indicates that finding an optimal dynamic control limits is necessary. The optimal sequential test we defined is the guidance for how to give the suitable or optimal dynamic control limits. Indeed we can conclude that the optimal control limits are a.s. decrease.
4 Concluding remarks
For the ERGMs models, and probabilistic graphical models, the normalization coefficient is difficult to compute, there are many method to approximate them, which needs complexty computations. We choose a finite measure to define corresponding measure ratio statistics for sequential test problem is a method to solve the difficulty. Measure ratio statistics can also show the difference between pre-change and post-change distribution. And we can also give optimal sequential test under finite measure. All of the definition about finite measure space comes from Shao, J. (2003).
When the pre-change and post-change measure density functions are known, it is simple. However, if the post-change measure density function is unknown, the problem may be more difficult, and would be considered later. If the number of parameters in ERGMs is large, it maybe results in dimension curse, which is also worthy to be considered.
The proofs of Thoerem in Appendix refers to Dong Han, Fugee Tsung, and Jinguo Xian(2019). The expectations and conditional expectations for the change-point k() are extend to measure expectations and conditional measure expectations .
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