Information processing device, information processing method, and program

Abstract

An information processing device according to one aspect of this invention comprises a processor and a storage unit. The storage unit comprises a first storage area and a first storage area. The first storage area stores event occurrence data about a location of occurrence of an event to be analyzed. The second storage area stores covariate data that varies with time. The processor has a kernel function designation unit, a calculation method designation unit, and an intensity function estimation unit. The kernel function designation unit accepts designation of a kernel function in a Gaussian process. The calculation method designation unit accepts designation of a calculation method of an equivalent kernel function. The intensity function estimation unit calculates an equivalent kernel function on the basis of the designated kernel function and calculation method, and uses the calculated equivalent kernel function to estimate a probability of occurrence of an event with respect to the covariate.

Description

Information processing device, information processing method, and program

[0001] One aspect of the present invention relates to an information processing device, an information processing method, and a program.

[0002] Consider measuring the time until a point event occurs probabilistically on a time axis where covariates are defined at any point. For example, (covariate, event) = (presence or absence of medication, patient death), (room temperature and humidity, equipment failure), (amount of debt, company bankruptcy), etc. Survival analysis analyzes the time from the start of observation to the occurrence of an event to estimate the probability of the event occurring for the covariate, and also predicts the time of future event occurrence.

[0003] Zhang et al., "Time-varying covariates and coefficients in Cox regression models", Annals of Translational Medicine, 6(7), 2018.Baddeley et al., "Nonparametric estimation of the dependence of a spatial point process on spatial covariates", Statistics and Its Interface, 5, pp.221-236, 2012.

[0004] The Cox proportional hazards model is a representative technique for survival analysis. In the Cox proportional hazards model, the relationship between the logarithm of the probability of an event occurring and a covariate is given by a linear regression model (see, for example, Non-Patent Document 1). Therefore, it is not possible to consider the non-linear effect of a covariate on the probability of an event occurring.

[0005] For example, in the relationship between room temperature and equipment failure, the probability of equipment failure is thought to increase when the room temperature is either too low or too high. However, the Cox proportional hazards model can only represent intensity functions that monotonically increase or decrease with respect to room temperature, so it cannot correctly estimate the probability of an event occurring.

[0006] Non-Patent Document 2 discloses a technique for estimating a non-linear relationship between an intensity function and a covariate from data. However, Non-Patent Document 2 cannot be used for survival analysis because it cannot consider cases where the observation end time depends on the occurrence time of an event.

[0007] The present invention has been made in light of the above circumstances, and aims to provide a technique that enables estimation of a nonlinear relationship between an intensity function and a covariate in survival analysis.

[0008] An information processing device according to one aspect of the present invention includes a processor and a storage unit. The storage unit includes a first storage area and a second storage area. The first storage area stores event occurrence data related to the occurrence position of an event to be analyzed. The second storage area stores time-varying covariate data. The processor includes a kernel function designation unit, a calculation method designation unit, and an intensity function estimation unit. The kernel function designation unit accepts designation of a kernel function in a Gaussian process. The calculation method designation unit accepts designation of a calculation method for an equivalent kernel function. The intensity function estimation unit calculates an equivalent kernel function based on the designated kernel function and calculation method, and estimates the occurrence probability of an event for a covariate using the calculated equivalent kernel function.

[0009] According to one aspect of the present invention, a technique can be provided that can estimate a nonlinear relationship between an intensity function and a covariate in survival analysis.

[0010] Fig. 1 is a functional block diagram showing an example of an information processing apparatus according to an embodiment. Fig. 2 is a functional block diagram showing an example of the information processing apparatus 1 shown in Fig. 1. Fig. 3 is a flowchart showing an example of a processing procedure of the information processing apparatus 1 shown in Fig. 1.

[0011] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. <Configuration> Fig. 1 is a functional block diagram showing an example of an information processing device according to an embodiment. The information processing device 1 is a computer equipped with a processor and a memory. The information processing device 1 includes a processor 11, an input / output interface 12, and a storage unit 13. The processor 11, the input / output interface 12, and the storage unit 13 are communicably connected to each other via a bus. The processor 11 controls the information processing device 1. The processor 11 is an arithmetic processing device such as a CPU (Central Processing Unit) or an MPU (Micro Processing Unit).

[0012] The input / output interface 12 is an interface that enables transmission and reception of information between the input device 2 and the output device 3. The input / output interface 12 may include a wired or wireless communication interface. That is, the information processing device 1, the input device 2, and the output device 3 may transmit and receive information via a network such as a LAN or the Internet.

[0013] The storage unit 13 is a storage medium. The storage unit 13 is configured by combining a nonvolatile memory that can be written to and read from at any time, such as a hard disk drive (HDD) or a solid state drive (SSD), a nonvolatile memory such as a read-only memory (ROM), and a volatile memory such as a random access memory (RAM). The storage unit 13 has a storage area including a program storage area and a data storage area. The program storage area stores an operating system (OS), middleware, and application programs required to execute various processes.

[0014] The input device 2 includes, for example, a keyboard, a pointing device, etc., which are used by the owner of the information processing device 1 (for example, an assignor, an administrator, or a supervisor) to input instructions to the information processing device 1. The input device 2 may also include a reader for reading data to be stored in the storage unit 13 from a memory medium such as a USB memory, or a disk device for reading such data from a disk medium. The input device 2 may also include an image scanner.

[0015] The output device 3 includes a display that displays output data to be presented to the owner from the information processing device 1, a printer that prints the output data, etc. The output device 3 may also include a writer that writes data to be input to another information processing device 1 such as a PC or a smartphone onto a memory medium such as a USB memory, and a disk device that writes such data onto a disk medium.

[0016] FIG. 2 is a functional block diagram showing an example of the information processing device 1 shown in FIG. 2 . In FIG. 2 , the storage unit 13 stores a program 10 that causes the processor 11 to function as the information processing device 1. The storage unit 13 further includes a first storage area 131, a second storage area 132, and a third storage area 133. The first storage area 131 stores event occurrence data 100. The event occurrence data 100 is data related to the occurrence locations of events to be analyzed, and includes at least the number of observed events, a sequence of event locations, and an observation area. The second storage area 132 stores time-varying covariate data 101. The third storage area 133 stores an intensity function distribution 105 calculated by the processor 11.

[0017] The processor 11 includes, as processing functions according to the embodiment, a kernel function designation unit 102, a calculation method designation unit 103, an intensity function estimation unit 112, and an output control unit 114. The kernel function designation unit 102, the calculation method designation unit 103, the intensity function estimation unit 112, and the output control unit 114 are functional processes realized by the arithmetic processing of the processor 11 based on the program 10.

[0018] The kernel function designation unit 102 accepts designation of a kernel function in a Gaussian process. The kernel function is designated by a user, for example, by operating the input device 2. The calculation method designation unit 103 accepts designation of a calculation method for an equivalent kernel function. The calculation method may also be designated, for example, by a user operating the input device 2. The intensity function estimation unit 112 calculates an equivalent kernel function based on the designated kernel function and calculation method. Furthermore, the intensity function estimation unit 112 estimates the occurrence probability of an event for a covariate using this calculated equivalent kernel function. The intensity function distribution 105 is stored in a third storage area 133.

[0019] The output control unit 114 outputs the intensity function distribution 105 to the output device 3 via the input / output interface 12. The output device 3 visualizes and displays the calculated intensity function distribution 105 on a display, for example.

[0020] Next, the operation of the above configuration will be described. <Operation> (Overview) First, an overview of the operation will be described. In this embodiment, the processor 11 mainly executes the processes (1) to (4), thereby realizing estimation of an intensity function for a covariate based on a Bayesian estimation method that uses a Gaussian process as a prior distribution.

[0021] (1) For variables that follow a Gaussian process defined in covariate space, the square of the variable is defined as the intensity function. In this way, the estimate of the square root of the intensity function that maximizes the posterior probability (maximum a posteriori probability estimate or MAP estimate) is given as the solution to a system of equations with N unknowns, where N is the number of observed data. This can be expressed as the representative theorem being true. This fact makes it easy to numerically solve the estimate of the square root of the intensity function.

[0022] (2) The estimation error of the square root of the intensity function is calculated using Laplace approximation. That is, the Hessian matrix of the MAP estimate of the logarithmic posterior probability distribution to which the square root of the intensity function follows is calculated. The inverse matrix of this Hessian matrix is ​​then multiplied by -1 to obtain the covariance matrix of the estimate of the square root of the intensity function.

[0023] (3) Under the Laplace approximation in (2), the gamma distribution that the estimated value of the intensity function follows is obtained. Obtaining the probability distribution for this estimated value is the final goal of intensity function estimation.

[0024] (4) The hyperparameters required for estimating the intensity function are estimated from observed data using the empirical Bayes method. The empirical Bayes method is a method that estimates the hyperparameters that maximize the marginal likelihood. A typical example of a hyperparameter is the parameter of the kernel function in a Gaussian process.

[0025] Fig. 3 is a flowchart showing an example of a processing procedure of the information processing device 1 shown in Fig. 1. In Fig. 3, the processor 11 accepts a user's specification of a kernel function in a Gaussian process (step SST21). Next, the processor 11 accepts a user's specification of a calculation method for an equivalent kernel function (step SST22).

[0026] Next, the processor 11 calculates an equivalent Kernel function based on the specified Kernel function and calculation method (step ST23).Furthermore, the processor 11 estimates an intensity function for the covariate using the calculated equivalent Kernel function (step S24).

[0027] (Details) Next, the details of the operation will be explained with reference to mathematical expressions.

[0028] [Event occurrence data] Data on the occurrence time of the event to be analyzed is given as input. The event occurrence data consists of the number of units to be observed (denoted as <<1>>), the observation end time for each unit (denoted as <<2>>), and a binary flag (denoted as <<3>>) that determines whether observation ended due to the occurrence of an event for each unit. Note that the binary flag takes a value of 1 if observation ended due to the occurrence of an event, and a value of 0 if no event occurred within the observation period.

[0029] Regarding [Covariate Data]: The covariate data observed for each unit are given as a function (denoted as [5]) that takes any point [4] within the observation period as input and outputs the covariate. In many applications, information about the covariate is only available at a finite number of points within the observation domain [6]. In such cases, it is assumed that the function [7] is constructed using a regression model or an interpolation technique such as kriging.

[0030] [Specifying the kernel function in Gaussian processes] To use Gaussian processes, a function called a kernel function, which determines the smoothness of the function to be modeled, is specified. The values ​​of the parameters (referred to as hyperparameters) included in the function are also specified at the same time. Note that in this device, the function to be modeled is the intensity function for the covariates. The kernel function for any two points (8) in the covariate space is denoted as (9).

[0031] Examples of kernel functions include the Gaussian kernel given by (1) and the kernel (2) expressed by the inner product of finite-dimensional feature mapping vectors (10). However, the selection of kernel functions is not limited to these.

[0032] Regarding [Specifying the calculation method of the equivalent kernel function], the method for calculating the equivalent kernel function defined in [Estimation of the intensity function] described later is given as input. The method for calculating the equivalent kernel function consists of the type of calculation method and the number of Monte Carlo integration points for each unit (denoted as <<11>>).

[0033] The calculation method can be selected from Type 1 and Type 2. Type 2 can be selected only when the kernel function is given by the inner product of finite-dimensional feature mapping vectors. Details of the calculation method are explained in [Estimation of Intensity Function].

[0034] [Estimation of Intensity Function] Based on the information given above, processor 11 calculates an equivalent kernel function (denoted as <<12>>) and uses that equivalent kernel function to estimate an intensity function (denoted as <<13>>) for the covariate.

[0035] First, the equivalent kernel function 12 is defined as the solution of the integral equation (3).

[0036] In preparation for numerically solving the integral equation (3), the integral part of equation (3) is approximated by Monte Carlo integration as shown in (4).

[0037] where 14 is the covariate at the j-th sample point in each unit. For convenience of explanation, we rewrite equation (4) as equation (5). However, (6) holds true.

[0038] When type 1 is specified as the type of calculation method, equation (5) is solved as a matrix equation for the column vector function <<15>> to obtain equation (7), which is the equivalent kernel function (denoted as <<16>>).

[0039] Here, {17} is a {19}-dimensional diagonal matrix with {18} as its diagonal component, {20} is a Gram matrix, and {21} is a vertical vector {22}.

[0040] When type 2 is specified as the type of calculation method, the equivalent kernel function (denoted as <<23>>) is obtained from equation (5) in the form of equation (8), assuming that the kernel function is given by the inner product of finite-dimensional feature mapping vectors as in equation (2). Here, {24} is the number of dimensions of the feature mapping vector, {25} is the {26}-dimensional unit matrix, and {27} is the column vector {28}.

[0041] Next, the MAP estimate of the square root of the intensity function is calculated using the equivalent kernel function <<12>> using equation (9). However, {29} can be obtained by solving the simultaneous equation (10).

[0042] Next, under the Laplace approximation, it is assumed that the square root of the intensity function follows a normal distribution with the MAP estimated value as the mean, and the covariance matrix (denoted as <<30>>) is calculated using equation (11). However, (12) holds true.

[0043] Finally, the probability distribution that the estimated value of the intensity function at each covariate value y follows is calculated as a gamma distribution whose scale parameter (denoted as 31) and shape parameter (denoted as 32) are given by equation (13).

[0044] In the process of [Estimating the Intensity Function], the validity of the hyperparameters specified in [Specifying the Kernel Function in the Gaussian Process] can also be evaluated based on the marginal likelihood function. When optimizing the hyperparameters, the hyperparameters that maximize the marginal likelihood function are searched for, and the value is used to recalculate Equation (13).

[0045] Regarding the [Estimated Intensity Function Distribution], the probability distribution of the intensity function calculated in [Estimation of Intensity Function] is output. What is output is a function that outputs a gamma distribution value with the scale and shape parameters given by Equation (13) for any covariate value y.

[0046] As described above, according to the embodiment, when data on the location of an event and a time-varying (non-stationary) covariate are given, it is possible to estimate the probability of an event occurring with respect to the covariate. In turn, it is possible to provide a technique that can estimate a nonlinear relationship between an intensity function and a covariate in survival analysis.

[0047] It should be noted that the present invention is not limited to the above-described embodiment. For example, the kernel function is not limited to the Gaussian kernel or the kernel of equation (2).

[0048] Furthermore, in the implementation stage, this invention can be embodied by modifying the components within the scope of the embodiments. Furthermore, various inventions can be formed by appropriately combining multiple components disclosed in the above embodiments. For example, some components may be deleted from all the components shown in the embodiments. Furthermore, components across different embodiments may be appropriately combined.

[0049] That is, this invention is not limited to the above-described embodiments, and the components can be modified and embodied in practice without departing from the spirit of the invention. Furthermore, various inventions can be formed by appropriately combining multiple components disclosed in the above-described embodiments. For example, some components may be omitted from all the components shown in the embodiments. Furthermore, components from different embodiments may be appropriately combined.

[0050] 1... Information processing device 2... Input device 3... Output device 10... Program 11... Processor 12... Input / output interface 13... Memory unit 131... First memory area 132... Second memory area 133... Third memory area 100... Event occurrence data 101... Covariate data 102... Kernel function designation unit 103... Calculation method designation unit 105... Intensity function distribution 112... Intensity function estimation unit 114... Output control unit.

Claims

1. An information processing device comprising a processor and a memory unit, wherein the memory unit comprises a first memory area for storing event occurrence data relating to the occurrence position of an event to be analyzed, and a second memory area for storing time-varying covariate data, and the processor comprises a kernel function designation unit that accepts designation of a kernel function in a Gaussian process, a calculation method designation unit that accepts designation of a calculation method for an equivalent kernel function, and an intensity function estimation unit that calculates the equivalent kernel function based on the designated kernel function and calculation method, and uses the calculated equivalent kernel function to estimate the occurrence probability of an event for a covariate.

2. The information processing device according to claim 1, wherein the event occurrence data includes at least the number of units to be observed, the observation end time for each of the units, and a flag indicating whether observation has ended for each of the units.

3. The information processing device according to claim 1, wherein the kernel function designation unit further receives designation of hyperparameter values ​​of the kernel function.

4. The information processing device according to claim 1, wherein said calculation method designation unit accepts designation of at least the type of calculation method for said equivalent Kernel function and the number of points for Monte Carlo integration.

5. An information processing method for an information processing device having a memory unit that stores event occurrence data relating to the occurrence position of an event to be analyzed and time-varying covariate data, the information processing method including the steps of: the processor accepting a specification of a kernel function in a Gaussian process; the processor accepting a specification of a calculation method for an equivalent kernel function; the processor calculating the equivalent kernel function based on the specified kernel function and calculation method; and the processor estimating the occurrence probability of an event for a covariate using the calculated equivalent kernel function.

6. A program that causes a computer to function as each of the units of the information processing device according to any one of claims 1 to 4.