Sample size estimation device, sample size estimation method, and sample size estimation program

Abstract

This sample size estimation device includes an input unit, a cost function approximation unit, a network construction unit, a minimum convex cost flow problem solving unit, an estimation control unit, and an output unit. The input unit receives input of observed aggregate data and potential information of a graphical model on a path graph. From the potential information and the aggregate data, the cost function approximation unit linearly approximates a concave function portion of an objective function of a MAP estimation problem including a sample size. The network construction unit creates an instance of a minimum convex cost flow problem from the linearly approximated objective function. The minimum convex cost flow problem solving unit derives the optimal solution for the minimum convex cost flow problem instance. The estimation control unit controls / executes MAP estimation overall by issuing commands while exchanging data with the cost function approximation unit, the network construction unit, and the minimum convex cost flow problem solving unit. The estimation control unit also derives a MAP estimated solution from the optimal solution for the instance. The output unit outputs the derived MAP estimated solution.

Description

[Technical Field]

[0001] The present invention relates to a sample size estimation device, a sample size estimation method, and a sample size estimation program. [Background technology]

[0002] Advances in sensors and information and communication technologies have enabled the collection of a wide variety of data on a large scale. However, due to privacy concerns and the difficulty of observation, only aggregated data, rather than individual data, is often available. For example, human location information obtained from GPS and other sources is sometimes provided as time-based area population data that prevents tracking of individuals due to privacy concerns. Here, time-based area population data refers to information on the number of people in each area at each time step. The Collective Graphical Model (CGM) (Non-Patent Document 3) is a widely used model that enables the extraction of deeper information from data by performing interpolation, estimation, and learning based on probabilistic modeling, even in situations where only aggregated data is available.

[0003] One of the operations in Collective Graphical Models (CGM) is the operation of MAP (maximum a posteriori) estimation of the true contingency table of the graphical model when the observed aggregate data and the underlying graphical model potential are given. This MAP estimation is essential for interpolation and estimation, and is also used as a subroutine for learning, so it is a very important operation and has been the subject of much research.

[0004] These will be explained using mathematical formulas. First, a brief explanation of CGM will be given.

[0005] Let H=(N,A) be an undirected graph, and consider a graphical model expressed by the following probability mass function:

[0006]

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[0007] However, φ ij (x i ,x j ) is the variable (X i ,X j ) and Z(θ) is the normalization constant (partition function). Also, the random variable X i is a finite set X i Assume that t takes values of t.

[0008] Let x be a sample from this graphical model. (1) ,···,x (M) Let the contingency table for the vertices be n i =(n i (x i )|x i ∈X i ) and the contingency table n for the edges ij =(n ij (x i ,x j )|x i ∈X i ,x j ∈X j ) is defined as follows:

[0009]

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[0010] In this case, the distribution of n (called the CGM distribution) can be written as follows: i is the degree of vertex i.

[0011]

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[0012] Also, the observed value y is assumed to be generated from some probability distribution p(y|n) that represents the observation noise. Specifically, at each vertex, the value n i is observed according to the following observation noise model:

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[0014] However, -logp i ,x i (y i (x i )|n i (x i )) is n i (x i ) is a convex function. The posterior distribution of n is given by p(n|y) ∝ p(n)·p(y|n). Then, the MAP estimation problem is max n p(n|y). For simplicity, for any i=1,2,...,|N|, let X i ={1,2,...,R}. The following discussion can be easily extended to other cases.

[0015] In this invention, we consider the case where graph H is a path graph. A path graph is an undirected graph whose vertex set is V = [T] and whose edge set is E = {(t, t+1)|t∈[T-1]}, where [n]:={1, 2, , n} for a natural number n. We transform the MAP estimation problem on a path graph. Considering it as a minimization problem of -logp(n|y), we solve the following optimization problem.

[0016]

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[0017] However, f tij (z):=logz!-z·logφ tij , g(z):=-logz!, h ti (z):=-logp ti (y ti |z) and ntx i :=nt(x i ), n tij :=nt,t+1(x i ,x j ), φtij :=φt,t+1(x i ,x j ) was set.

[0018] The MAP estimation problem (9) is generally known to be NP-hard, making it extremely difficult to solve efficiently. Therefore, methods have been proposed, such as a method of continuously relaxing the problem using Stirling approximation and solving it by message passing (Non-Patent Document 4), and a method of solving it by applying the discrete DC (Difference of Convex) algorithm (Non-Patent Document 1). [Prior art documents] [Non-patent literature]

[0019] [Non-Patent Document 1] Yasunori Akagi, Naoki Marumo, Hideaki Kim, Takeshi Kurashima, and Hiroyuki Toda. Non-approximate inference for collective graphical models on path graphs via discrete difference of convex algorithm. In NeurIPS, 2021. [Non-patent document 2] Takanori Maehara and Kazuo Murota. A framework of discrete DC programming by discrete convex analysis. Mathematical Programming, 152(1-2):435-466, 2015. [Non-patent document 3] Daniel R. Sheldon and Thomas G. Dietterich. Collective graphical models. In NIPS, pages 1161-1169, 2011. [Non-patent document 4] Tao Sun, Daniel Sheldon, and Akshat Kumar. Message passing for collective graphical models. In ICML, pages 853-861, 2015. Summary of the Invention [Problem to be solved by the invention]

[0020] In conventional techniques, the sample size M is assumed to be given in advance. However, in actual applications, M is rarely given in advance, and it is necessary to estimate the sample size M by some means. However, there is currently no means available for estimating the sample size M.

[0021] An object of the present invention is to provide a sample size estimation device, a sample size estimation method, and a sample size estimation program that are capable of appropriately estimating a sample size M from observations. [Means for solving the problem]

[0022] One aspect of the present invention is a sample size estimation device. The sample size estimation device includes an input unit, a cost function approximation unit, a network construction unit, a minimum convex cost flow problem solving unit, an estimation control unit, and an output unit. The input unit receives potential information of a graphical model on a path graph and observed aggregate data. The cost function approximation unit linearly approximates the concave function portion of the objective function of the MAP estimation problem, including the sample size, from the potential information of the graphical model and the aggregate data input to the input unit. The network construction unit creates an instance of the minimum convex cost flow problem from the objective function linearly approximated by the cost function approximation unit. The minimum convex cost flow problem solving unit finds an optimal solution to the instance of the minimum convex cost flow problem created by the network construction unit. The estimation control unit controls and executes the entire MAP estimation by exchanging data with the cost function approximation unit, the network construction unit, and the minimum convex cost flow problem solving unit and issuing commands to them, and ultimately finds a MAP estimation solution from the optimal solution of the instance of the minimum convex cost flow problem found by the minimum convex cost flow problem solving unit. The output unit outputs the found MAP estimation solution.

[0023] One aspect of the present invention is The computer runs This is a sample size estimation method that inputs potential information of a graphical model on a path graph and observed aggregated data, linearly approximates the concave function part of the objective function of the MAP estimation problem, including the sample size, from the input potential information of the graphical model and aggregated data, creates an instance of a minimum convex cost flow problem from the linearly approximated objective function, finds an optimal solution to the created instance of the minimum convex cost flow problem, and finds a MAP estimation solution from the found optimal solution to the instance of the minimum convex cost flow problem.

[0024] One aspect of the present invention is a sample size estimation program that causes a computer to execute the functions of each component of the sample size estimation device described above. [Effects of the Invention]

[0025] According to the present invention, there are provided a sample size estimation device, a sample size estimation method, and a sample size estimation program that are capable of appropriately estimating the sample size M from observations. [Brief explanation of the drawings]

[0026] [Figure 1] FIG. 1 is a block diagram illustrating an example of the functional configuration of a sample size estimation device according to an embodiment. [Figure 2] FIG. 2 is a block diagram illustrating an example of a hardware configuration of the sample size estimation device according to the embodiment. [Figure 3] FIG. 3 is a flowchart illustrating sample size estimation performed by the sample size estimation device according to the embodiment. [Figure 4] FIG. 4 is a diagram illustrating an example of formulation of a minimum cost flow problem in the embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0027] Hereinafter, an embodiment of the present invention will be described with reference to the drawings.

[0028] (Functional configuration) First, the functional configuration of a sample size estimation device according to an embodiment will be described. Fig. 1 is a block diagram showing an example of the functional configuration of a sample size estimation device 10 according to an embodiment. The sample size estimation device 10 estimates the sample size M at the same time as outputting a MAP estimated solution of a Collective Graphical Model (CGM) on a path graph. In other words, the sample size estimation device 10 outputs a MAP estimated solution including the sample size M.

[0029] The sample size estimation device 10 includes an operation unit 11, an input unit 12, an estimation control unit 13, a cost function approximation unit 14, a network construction unit 15, a minimum convex cost flow problem solving unit 16, an output unit 17, a potential storage unit 18, a summary data storage unit 19, and a MAP estimated solution storage unit 20.

[0030] The operation unit 11 is an interface for performing operations from the outside. The operation unit 11 enables operations such as inputting data by operating the input unit 12, storing and modifying the input data, starting MAP estimation by issuing a command to the estimation control unit 13, and outputting the MAP estimation results by issuing a command to the output unit 17. The input data is potential information of the graphical model and observed aggregated data.

[0031] Data is input from the outside to the input unit 12. The input unit 12 stores data in the potential storage unit 18 and the aggregate data storage unit 19 and corrects the data.

[0032] The estimation control unit 13 controls and executes the entire MAP estimation by exchanging data with the cost function approximation unit 14, network construction unit 15, and minimum convex cost flow problem solving unit 16 and issuing commands to them, and ultimately finds the MAP estimation solution.

[0033] The cost function approximation unit 14 linearly approximates the concave function part of the objective function.

[0034] The network construction unit 15 creates an instance of the minimum convex cost flow problem.

[0035] The minimum convex cost flow problem solving unit 16 finds an optimal solution to the instance of the minimum convex cost flow problem created by the network building unit 15 .

[0036] The output unit 17 reads and outputs the MAP estimated solution stored in the MAP estimated solution storage unit 20.

[0037] The potential storage unit 18 stores the potential information of the graphical model input to the input unit 12.

[0038] The aggregated data storage unit 19 stores the aggregated data input to the input unit 12 .

[0039] The MAP estimated solution storage unit 20 stores the MAP estimated solutions obtained by the estimation control unit 13, the cost function approximation unit 14, the network construction unit 15, and the minimum convex cost flow problem solving unit 16.

[0040] (Hardware configuration) Next, the hardware configuration of the sample size estimation device 10 will be described with reference to Fig. 2. The sample size estimation device 10 is configured as a computer. For example, the sample size estimation device 10 is configured as a personal computer, a server computer, or the like. Fig. 2 is a block diagram showing an example of the hardware configuration of the sample size estimation device 10 according to the embodiment.

[0041] 2, the sample size estimation device 10 includes an input device 31, a CPU 32, a main memory device 35, an auxiliary memory device 38, and an output device 39. In addition to these, the sample size estimation device 10 may also include other peripheral devices.

[0042] The input device 31, CPU 32, main memory device 35, auxiliary memory device 38, and output device 39 are electrically connected to one another via a bus 40, and exchange data and commands via the bus 40.

[0043] The input device 31 is a device that receives signals from the outside, converts them into data, and passes them to the CPU 32, the main memory device 35, and the auxiliary memory device .

[0044] The output device 39 is a device that receives data from the CPU 32, the main memory device 35, and the auxiliary memory device 38, converts the data into signals, and outputs the signals.

[0045] The main memory device 35 stores programs and data necessary for the processes executed by the CPU 32. The CPU 32 reads out the necessary programs and data from the main memory device 35 and executes them to perform various processes.

[0046] The main storage device 35 has a non-rewritable read only memory (ROM) 36 and a rewritable random access memory (RAM) 37. The RAM 37 and the auxiliary storage device 38 exchange programs and data with each other.

[0047] The ROM 36 is a non-volatile memory that stores a program (BIOS) that controls the CPU 32 at startup.

[0048] The RAM 37 is a volatile memory that temporarily stores programs and data required for processing by the CPU 32.

[0049] The auxiliary storage device 38 stores programs and data supplied via an external device or a network, and provides the programs and data temporarily required for processing by the CPU 32 to the RAM 37. For example, the auxiliary storage device 38 is a non-volatile memory such as an HDD (Hard Disk Drive) or an SSD (Solid State Drive).

[0050] The CPU 32 is a processor, which is hardware that processes data and instructions, and includes a control unit 33 that issues commands and an arithmetic unit 34 that processes data.

[0051] The control device 33 controls the input device 31 , the arithmetic unit 34 , the main memory device 35 , the auxiliary memory device 38 , and the output device 39 .

[0052] The arithmetic unit 34 reads the program and data from the RAM 37 , executes the program to process the data, and provides the processed data to the RAM 37 .

[0053] In this hardware configuration, the input device 31 constitutes the operation unit 11. The output device 39 constitutes the output unit 17. The main memory device 35 and the auxiliary memory device 38 constitute the potential storage unit 18, the aggregated data storage unit 19, and the MAP estimated solution storage unit 20. The CPU 32 and the main memory device 35 constitute the input unit 12, the estimation control unit 13, the cost function approximation unit 14, the network construction unit 15, and the minimum convex cost flow problem solving unit 16.

[0054] For example, the CPU 32 loads a program that executes the functions of the estimation control unit 13, the cost function approximation unit 14, the network construction unit 15, and the minimum convex cost flow problem solving unit 16 from the auxiliary storage device 38 into the RAM 37, and executes the loaded program to perform the operations of the estimation control unit 13, the cost function approximation unit 14, the network construction unit 15, and the minimum convex cost flow problem solving unit 16.

[0055] (Sample size estimation process) Next, a description will be given of the sample size estimation process executed by the sample size estimation device 10. Fig. 3 is a flowchart showing an example of the procedure and process contents of the sample size estimation process executed by the sample size estimation device according to the embodiment.

[0056] In step S1, potential information and aggregated data of the graphical model are accumulated. Specifically, upon receiving a command from the operation unit 11, the input unit 12 receives the potential information and aggregated data of the graphical model from outside, modifies the potential information and aggregated data if necessary, and stores the potential information in the potential accumulation unit 18 and the aggregated data in the aggregated data accumulation unit 19.

[0057] In step S2, the concave function portion of the objective function is linearly approximated. More specifically, upon receiving an instruction to start estimation from the operation unit 11, the estimation control unit 13 issues an instruction to linearly approximate the concave function portion of the objective function to the cost function approximation unit 14. Accordingly, the estimation control unit 13 reads potential information from the potential storage unit 18 and also reads aggregated data from the aggregated data storage unit 19, and passes the potential information and aggregated data to the cost function approximation unit 14. The cost function approximation unit 14 linearly approximates the potential information and aggregated data, and returns the linearly approximated potential information and aggregated data to the estimation control unit 13.

[0058] In steps S1 and S2, the input unit 12 may receive a command to receive the potential information and aggregated data of the graphical model and a command to start estimation from the operation unit 11, and pass the command to start estimation, the potential information, and the aggregated data to the estimation control unit 13.

[0059] In step S3, an instance of the minimum convex cost flow problem is created. More specifically, the estimation control unit 13 issues a command to the network construction unit 15 to create an instance of the minimum convex cost flow problem. In response, the estimation control unit 13 passes the potential information linearly approximated by the cost function approximation unit 14 and the aggregated data to the network construction unit 15. The network construction unit 15 creates an instance of the minimum convex cost flow problem from the received potential information and aggregated data, and returns the created instance of the minimum convex cost flow problem to the estimation control unit 13.

[0060] In step S4, an optimal solution to the instance of the minimum convex cost flow problem is found. More specifically, the estimation control unit 13 issues a command to the minimum convex cost flow problem solving unit 16 to find an optimal solution to the instance of the minimum convex cost flow problem. In response, the estimation control unit 13 passes the instance of the minimum convex cost flow problem created by the network construction unit 15 to the minimum convex cost flow problem solving unit 16. The minimum convex cost flow problem solving unit 16 finds an optimal solution to the received instance of the minimum convex cost flow problem and returns the found optimal solution to the estimation control unit 13. The estimation control unit 13 finds a MAP estimated solution including the optimal solution sample size M for the received instance of the minimum convex cost flow problem and stores it in the MAP estimated solution storage unit 20.

[0061] Thereafter, upon receiving a command from the operation unit 11, the output unit 17 reads the MAP estimation solution of the MAP estimation problem including the sample size M from the MAP estimation solution storage unit 20 and outputs it to the outside.

[0062] The following describes in detail the estimation of the sample size M performed by the sample size estimation device 10. The sample size estimation device 10 estimates the sample size M by performing MAP estimation simultaneously with the contingency table n.

[0063] (MAP estimation problem) To perform MAP estimation for sample size M, we also set a prior distribution p(M) for sample size M. However, we assume that -logp(M) is a convex function. p(M) can be a uniform distribution (10), Poisson distribution (11), Gaussian distribution (12), etc.

[0064]

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[0065] In this case, the probability distribution p(n,M|y) is as follows:

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[0067] The MAP estimation problem is min n,M Since it is -logp(n,M|y), by rearranging the equation, the MAP estimation problem including M becomes as follows, similar to (9).

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[0069] Here, k1(M):=-logp(M)+MlogZ, k2(M):=-logM! The value of logZ can be calculated using dynamic programming.

[0070] (Solution to MAP estimation problem) As in Non-Patent Document 1, the above optimization problem is solved by the discrete DC algorithm (Non-Patent Document 2) and the minimum convex cost flow algorithm.

[0071] (DC algorithm applied) Here, we will explain the linear approximation of the concave function part of the objective function executed by the cost function approximation unit 14. The linear approximation of the concave function part of the objective function is performed by applying the DC algorithm.

[0072] The DC algorithm is a function that can be solved by min(n) where Q(n) is a convex function and R(n) is a concave function. n∈D It is a framework for solving optimization problems of the form P(n) = Q(n) + R(n). The DC algorithm finds a sequence n of feasible solutions such that P(n(1)) ≥ P(n(2)) ≥ ≥ P(n(s)). (1) ,···,n (s) The optimization is performed by generating

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[0078] In particular, the cost function c ij If all are discrete convex functions, this problem is called the minimum convex cost flow problem, and it is known that efficient algorithms exist.

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[0080] An instance of the minimum cost flow problem constructed for |T|=3, R=2 is shown in Figure 4.

[0081] Since all of the cost functions on the edges of the minimum cost flow problem constructed above satisfy discrete convexity, this problem is a minimum convex cost flow problem. It is known that there are algorithms for the minimum convex cost flow problem, such as shortest path iteration and capacity scaling, which can efficiently find the optimal solution.

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[0085] (Overall view of the algorithm) To summarise the above, the overall algorithm is as follows:

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[0087] The algorithm guarantees that the objective function value is monotonically decreasing and that the algorithm stops after a finite number of iterations.

[0088] [effect] According to the embodiment, it becomes possible to appropriately estimate the sample size M from the observation.

[0089] The present invention is not limited to the above-described embodiments, and various modifications can be made in the implementation stage without departing from the spirit of the invention. Furthermore, the embodiments may be implemented in appropriate combinations, in which case the combined effects can be obtained. Furthermore, the above-described embodiments include various inventions, and various inventions can be extracted by combining selected elements from the disclosed elements. For example, if the problem can be solved and the desired effect can be obtained even if some elements are deleted from all elements shown in the embodiments, the configuration from which these elements are deleted can be extracted as an invention. [Explanation of symbols]

[0090] 10...Sample size estimator 11...Operation unit 12...Input section 13...Estimation control unit 14...Cost function approximation section 15...Network Construction Department 16...Minimum convex cost flow problem solving part 17...Output section 18...Potential storage section 19...Aggregated data storage section 20…MAP estimated solution storage unit 31...Input device 32...CPU 33...Control device 34...Arithmetic device 35…Main memory 36...ROM 37...RAM 38…Auxiliary storage device 39...Output device 40...Bus

Claims

1. an input unit to which potential information of the graphical model on the path graph and observed aggregate data are input; a cost function approximation unit that linearly approximates a concave function part of an objective function of a MAP estimation problem including a sample size from the potential information and aggregated data of the graphical model input to the input unit; a network construction unit that creates an instance of a minimum convex cost flow problem from the objective function linearly approximated by the cost function approximation unit; a minimum convex cost flow problem solving unit that finds an optimal solution to an instance of the minimum convex cost flow problem created by the network building unit; an estimation control unit that controls and executes the entire MAP estimation by exchanging data with the cost function approximation unit, the network construction unit, and the minimum convex cost flow problem solving unit and issuing commands to the unit, and that finds a MAP estimation solution from an optimal solution of an instance of the minimum convex cost flow problem found by the minimum convex cost flow problem solving unit; an output unit that outputs the obtained MAP estimation solution; Sample size estimator.

2. With respect to the sample size M, we set a prior distribution p(M) that is a uniform distribution, where −logp(M) is a convex function. The MAP estimation problem including the sample size is expressed as follows: [Equation 1] The sample size estimation device of claim 1 .

3. the cost function approximation unit performs linear approximation of a concave function part of the objective function by applying a DC algorithm; The sample size estimation device of claim 1 .

4. The network construction unit creates a minimum convex cost flow problem on a graph g=(ν, ε) by the following procedure: [Equation 2] The sample size estimation device of claim 1 .

5. a potential storage unit that stores potential information of the graphical model input to the input unit; an aggregated data storage unit that stores the aggregated data input to the input unit; a MAP estimated solution storage unit that stores the MAP estimated solutions found by the estimation control unit, the cost function approximation unit, the network construction unit, and the minimum convex cost flow problem solving unit, the input unit stores the input potential information of the graphical model in a potential storage unit, and stores the input aggregated data in an aggregated data storage unit; the estimation control unit reads potential information of the graphical model from the potential storage unit, reads aggregated data from the aggregated data storage unit, and stores a MAP estimated solution in the MAP estimated solution storage unit; the output unit reads the MAP estimated solution from the MAP estimated solution storage unit and outputs it to an external device. The sample size estimation device of claim 1 .

6. the input unit corrects the input potential information of the graphical model and corrects the input aggregate data. The sample size estimation device of claim 1 .

7. Input the potential information of the graphical model on the path graph and the observed aggregate data, From the input potential information and aggregated data of the graphical model, a linear approximation is performed for the concave function part of the objective function of the MAP estimation problem, including the sample size. Create an instance of the minimum convex cost flow problem from the linearly approximated objective function, Find the optimal solution of the created instance of the minimum convex cost flow problem, Find the MAP estimated solution from the optimal solution of an instance of the minimum convex cost flow problem. A computer-implemented method for estimating sample size.

8. A sample size estimation program that causes a computer to execute the functions of each component of the sample size estimation device according to claim 1.

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