JP2013149204A - Signal extraction device, method, and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To extract a long-term non-steady signal and a local non-steady signal from time series data.SOLUTION: In a state estimation unit 22, according to Normalized UFK and on the basis of time series data on an observed value, a state vector X including a cyclical variation component, a nonlinear trend component, and an event component is estimated sequentially and an estimated value of the observed value is calculated. In a signal extraction unit 23, from the state vector X sequentially estimated by the state estimation unit 22, a signal indicating each of the cyclical variation component, the nonlinear trend component, and the event component is extracted.

Description

  The present invention relates to a signal extraction apparatus, method, and program, and more particularly, to a signal extraction apparatus, method, and program for extracting a signal from time series data.

Conventionally, a model called m ^th -order stochastic trend has been proposed in order to analyze trends and cycles of economic time series data (Non-patent Document 1). In this model, when m = 1, it matches the normal random walk model, and when m ≧ 2, the trend slope component is modeled by random walk. A model called n ^th -order stochastic cycle has been proposed for periodic signals. The Wiener-Kolmogorov filter is derived as the minimum variance estimator for each model.

Andrew C. Harvey, et al., “General Model-Based Filters for Extracting Cycles and Trends in Economic Time Series”, The Review of Economics and Statistics, 2003.

  An object of the present invention is to provide a signal extraction apparatus, method, and program capable of extracting a long-term unsteady signal and a local unsteady signal from time series data.

  In order to achieve the above object, the signal extraction apparatus according to the present invention updates the state vector X including the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component in a non-linear manner using the system noise. And a state space model defined using an observation equation indicating the relationship between the state vector X and the observed value using observation noise, and a covariance matrix of the estimated error variance of the state vector X A normalization estimation error variance covariance matrix normalized by the sum of the diagonal components of the covariance matrix of the system noise and the variance of the observed noise, and a normalization system normalizing the covariance matrix of the system noise by the sum Generation of a sigma point sequence using a noise covariance matrix and a normalized observation noise variance obtained by normalizing the variance of the observed noise with the sum, updating the state of the state vector X, and In accordance with a normalized UFK (Unscented Kalman Filter) that performs measurement update, state estimation means for sequentially estimating the state vector X based on time-series data of the observed values, and from the state vector X sequentially estimated by the state estimation means And a signal extracting means for extracting a signal indicating each of the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component.

  A signal extraction method according to the present invention is a signal extraction method in a signal extraction device including a state estimation unit and a signal extraction unit, and the signal extraction device uses the state estimation unit to perform a cyclic fluctuation component and a long-term unsteady component. And a state update expression for nonlinearly updating the state vector X including the local non-stationary component using the system noise, and an observation equation indicating the relationship between the state vector X and the observed value using the observation noise. Normalized estimation error obtained by normalizing the covariance matrix of the estimated error variance of the state vector X with the sum of the diagonal components of the system noise covariance matrix and the observed noise variance value for the state space model determined using A variance covariance matrix, a normalized system noise covariance matrix obtained by normalizing the covariance matrix of the system noise with the sum, and a variance value of the observation noise Based on normalized UFK (Unsented Kalman Filter) that performs generation of sigma point sequence using normalized observation noise variance normalized in step, state update of state vector X, and observation update, based on the time series data of the observed values The state vector X is sequentially estimated, and the cyclic fluctuation component, the long-term unsteady component, and the local non-stationary component are obtained from the state vector X sequentially estimated by the state estimation unit by the signal extraction unit. And a step of extracting a signal indicating each of the stationary components.

  A program according to the present invention is a state update formula for updating a computer in a non-linear time manner using a system noise for a state vector X including a cyclic fluctuation component, a long-term unsteady component, and a local unsteady component. And the covariance matrix of the estimated error variance of the state vector X for the state space model defined using the observation equation indicating the relationship between the state vector X and the observed value using the observation noise, and the system noise covariance matrix Normalized estimation error variance covariance matrix normalized by the sum of the diagonal components of and the observed noise variance, and the normalized system noise covariance matrix obtained by normalizing the system noise covariance matrix by the sum, and Performs sigma point sequence generation, state vector X state update, and observation update using normalized observation noise variance obtained by normalizing the observed noise variance with the sum According to a normalized UFK (Unscented Kalman Filter), state estimation means for sequentially estimating the state vector X based on time-series data of the observed values, and the cyclic variation from the state vector X sequentially estimated by the state estimation means It is a program for functioning as a signal extraction means for extracting a signal indicating each of a component, the long-term unsteady component, and the local unsteady component.

  According to the present invention, the state update equation for non-linearly updating the state vector X including the cyclic variation component, the long-term unsteady component, and the local unsteady component using the system noise by the state estimation unit. , And a state space model defined using an observation equation indicating the relationship between the state vector X and the observation value using the observation noise, the covariance matrix of the estimated error variance of the state vector X is expressed as the system noise covariance A normalized estimation error variance covariance matrix normalized by the sum of the diagonal components of the matrix and the variance of the observed noise, and a normalized system noise covariance matrix obtained by normalizing the system noise covariance matrix by the sum. N that performs generation of a sigma point sequence, state update of the state vector X, and observation update using the normalized observation noise variance obtained by normalizing the variance value of the observation noise with the sum. According rmalized UFK (Unscented Kalman Filter), sequentially estimating the state vector X on the basis of the time-series data of the observations. Then, a signal extraction unit extracts a signal indicating each of the circulation fluctuation component, the long-term unsteady component, and the local unsteady component from the state vector X sequentially estimated by the state estimation unit. To do.

  Thus, the state vector X including the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component is sequentially estimated according to the normalized UFK, and the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component are estimated. By extracting a signal indicating each unsteady component, a long-term unsteady signal and a local unsteady signal can be extracted from the time-series data.

  As described above, according to the signal extraction apparatus, method, and program of the present invention, the state vector X including the cyclic fluctuation component, the long-term non-stationary component, and the local non-stationary component is sequentially generated according to the Normalized UFK. Estimate and extract signals indicating each of the cyclic fluctuation component, long-term unsteady component, and local unsteady component, thereby extracting long-term unsteady signal and local unsteady signal from the time series data. Can be extracted.

It is the schematic which shows the structure of the signal extraction apparatus which concerns on embodiment of this invention. It is a figure for demonstrating the time series data of an observed value. It is a figure which shows output data. It is a flowchart which shows the content of the signal extraction process routine in the signal extraction apparatus which concerns on embodiment of this invention. It is a graph which shows the time series data of an observed value. It is a graph which shows the signal of the extracted circulation fluctuation component. It is a graph which shows the signal of the extracted trend component. It is a graph which shows the signal of the extracted event component. It is a graph which shows the predicted value of an observed value. It is a figure which shows the evaluation result of the performance of each model.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

<Outline of the invention>
An actual time series represented by marketing time series data greatly fluctuates with time, and is often difficult to analyze without prior stabilization. Such a time series is called non-stationary, and the present invention enables analysis, modeling, and signal extraction for non-stationary time series.

  The state estimation algorithm used in the present invention is a Normalized Unscented Kalman Filter (nUKF) that estimates a time-series state modeled in a nonlinear Gaussian type with high accuracy. In the time series model, nonlinear trend components and dynamic event components are newly introduced to model non-stationary time series with high accuracy. These components show the characteristics of time series data. In particular, event components that take signals of sales and campaign effects provide useful information in the analysis of marketing time series.

<System configuration>
The signal extraction apparatus 100 according to the embodiment of the present invention receives time series data of observation values and extracts a signal. This signal extraction device 100 is constituted by a computer including a CPU, a RAM, and a ROM storing a program for executing a signal extraction processing routine described later, and is functionally configured as follows. Yes. As shown in FIG. 1, the signal extraction device 100 includes an input unit 10, a calculation unit 20, and an output unit 30.

The input unit 10 receives time-series data of input observation values. The observed value is a scalar value, that is, the time series data is a one-dimensional time series. For example, as shown in FIG. 2, it is time series data composed of a set (n = 1,..., N) of an epoch t _n and an observed value (t _n ).

  The calculation unit 20 includes a state space modeling unit 21, a state estimation unit 22, and a signal extraction unit 23.

  The state space modeling unit 21 defines a nonlinear Gaussian model in which time-series data is expressed in the state space, as will be described below, based on the definition of the state space model set in advance.

  First, nonlinear Gaussian modeling shown in the following equations (1) and (2) is performed.

However, _Xt is a state vector at time t, and normally an unobserved quantity is taken. And an object thereof is to estimate using the time-series data y _t the state vector X _t. w is a system noise vector, which is a white noise vector following a normal distribution. μ is observation noise, and is white noise according to a normal distribution. As shown in equation (2), y _t the observation value at time t, generated by the linear transformation H state X _t. The above equation (2) shows the relationship between the state vector Xt and the observation value yt using the observation noise. f () is a non-linear part, and the type of f () varies depending on the model. State vector X _t is expressed by the following equation (3).

Where x _t , ..., x _{t−M + 1} , a ₁ , ..., a _M are variables related to the cyclic fluctuation component expressed in the autoregressive process, T _t , n _t , β _t This is a variable related to the nonlinear trend component. E _t and β ′ _t are variables relating to the event component. Therefore, the parameters to be estimated are 2 × M + 5 in total. F (X _t-1 ) and w of the state update equation using this state vector X _t are expressed by the following equation (4).

  Here, H in the observation equation (2) above is a vector of 1 row (2 × M + 5) columns, the 1st, (2 × M + 1) th, (2 × M + 4) th Element is 1 and others are 0. The state update formulas for the nonlinear trend term and event term are derived from the following formulas (5) to (7), respectively.

  This is called the first-order Gaussian Markov approximation for the time update of unknown variables, and is defined as a continuous system. If the equations (5) and (6) are solved simultaneously, a state update equation for the nonlinear trend component is derived, and solving the equation (7) leads to a state update equation for the event component. The first order Gaussian Markov approximation is similar to the Dynamic Model Compensation (DMC) method. The nonlinear trend component is a component that models a long-term unsteady component, and is expressed by the following equations (8) and (9).

However, (DELTA) shows the data interval of time series data. Here, by ^setting e− ^αΔ≡β , an equation similar to the above equation (4) is obtained. nt represents the slope of the trend component T, which changes with time. Therefore, unlike the local constant trend (random walk model) and the local linear trend (constant slope), which are often adopted as trend models, it is possible to realize a non-linear change in trend with a small amount of noise. The event component is a component that models a local unsteady component, and is represented by the following equation (10).

Here, by ^setting e− ^γΔ≡β ′, an equation similar to the above equation (4) is obtained. β ′ is set to take a different value at the event time and at the normal time. For example, the value of β ′ is set according to the following equation (11).

This β 'value setting means that β' takes the boundary between the unsteady and steady regions in the first-order AR coefficient, and it becomes possible to take a signal with long-term memory (correlation) at the event. Usually, it means that no signal is taken. Switching between event and normal is done automatically. Switching to the event from the normal, using the technique of outlier detection using a normal distribution, the start of the event when a y _t is detected as outliers. Switching to normal time from the event, beta 'restless 0.9 near the minimum value of the range, carried out at the point where E _t approaches zero. For example, when β ′ becomes a value in the vicinity of 0.9, which is the minimum value in the range, for a predetermined number of times, the mode is switched to the normal time.

  As described above, the state space modeling unit 21 defines a type of f () as shown in the above-described equation (4) and defines H in the equation (2).

  In addition, the state space modeling unit 21 sets appropriate values for the diagonal components of the normalized system noise covariance matrix, which will be described later, and the observed noise variance.

The state estimation unit 22 performs sequential estimation of state vectors based on the input time-series data and the state space model modeled by the state space modeling unit 21. Since the model is a nonlinear, it performs sequential estimation of the state vector X _t by Normalized UKF was Unscented Kalman Filter and (UKF) based. A normal UKF algorithm includes generation of a sigma point sequence represented by the following equations (12) to (22), state update, and observation update (filter).

  Here, λ is a predetermined constant, and L is the number of dimensions of the state vector X. ￣y is the predicted value of the observed value, and ￣y = H￣X. v is a prediction error, and P is a covariance matrix of estimated error variance of the state vector X.

  In Normalized UKF used in this embodiment, ￣P, ^ P existing in the above normal algorithm is the sum of the variance values of all noises, that is, the sum of the diagonal components of the covariance matrix Q of the system noise w. Divide by the sum traceQ + r of the variance value r of noise μ. For example, ￣P in the above equation (22) is replaced with ￣P 'defined by the following equation (23).

  By this operation, the system noise covariance matrix Q in the equation (18) and the observed noise variance r in the equation (19) are normalized as shown in the following equations (24) and (25). . That is, the elements of the system noise covariance matrix Q and the observed noise variance r are all 1 or less. In addition, the sum of the elements of the system noise covariance matrix Q and the variance value of the observation noise is 1.

  Usually Q and r are unknown, and the setting of the variance of each noise largely depends on the analyst's experience. In general, the variance value of noise has a large effect on the estimation result of the state. Therefore, this amount is limited to 1 or less and the setting is facilitated to enable stable analysis.

  Also, the shape of the algorithm is appropriately changed by this series of normalization operations. However, by reducing the set value of α included in the above equations (15) and (16), the algorithm can be matched with the normal UKF algorithm. be able to. This is shown as follows.

Here, by setting α so that M · α ² becomes 1, the influence of the normalized number M on the generation of the sigma point sequence disappears. That is, the following equations (30) and (31) are obtained.

  The above equation (31) is normal sigma point sequence generation when it is assumed that the state follows a normal distribution. In this case, L + κ = 3 is often set.

  As described above, the state estimation unit 22 uses the normal time-series data of the input observation values and the covariance matrix ￣P ′, ^ of the estimated error variance of the normalized state vector X in the normal UFK algorithm. Sequential estimation of state vector X according to Normalized UKF algorithm (generation of sigma point sequence, state update, observation update) replaced with P ', normalized system noise covariance matrix Q', and observed noise variance r ' And a predicted value ￣y (= H￣X) of the observed value is calculated.

  The signal extraction unit 23 extracts a signal from the state vector X sequentially estimated by the state estimation unit 22.

Here, the state vector X _t is expressed by the following equation (32).

Among the elements of the state vector, x _t is a cyclic variation component, T _t is a nonlinear trend component, and _Et is an event component. The signal extraction unit 23 extracts these three components as characteristic signals in the target time series. By analyzing the extracted signal, it is possible to grasp in detail the characteristics and fluctuations of the time series. In the marketing time series, event components often coincide with signals from promotions such as sales and campaigns. The marketer needs detailed analysis of the promotional effectiveness and the event component contains this important information.

  The nonlinear trend component is an example of a long-term unsteady component, and the event component is an example of a local unsteady component.

As shown in FIG. 3, the output unit 30 outputs data indicating the extraction result and the predicted value of the signal extracted by the signal extraction unit 23 to the user. For example, time-series data consisting of a set (n = 1,..., N) of an epoch t _n , a predicted value (t _n ), an extracted signal A (t _n ), and a signal B (t _n ) is output. Is done.

<Operation of signal extraction device>
Next, the operation of the signal extraction device 100 according to the present embodiment will be described. First, when time-series data composed of observation values at times t _{1 to} t _N is input to the signal extraction device 100, the input time-series data is stored in a memory (not shown) by the signal extraction device 100. The Then, the signal extraction processing routine shown in FIG.

  First, in step S101, time-series data of input observation values is acquired. In step S102, a nonlinear Gaussian model of the above equations (1) and (2) is set based on the definition of a preset state space model, and the normalized system noise covariance matrix Q is set. 'And the variance value r of the observed noise are set to appropriate values.

  In step S103, using the nonlinear Gaussian model set in step S102, the normalized system noise covariance matrix Q ′, and the observed noise variance r ′, the above step is performed according to the Normalized UFK algorithm. Using the time-series data of the observed values acquired in S101, generation of the sigma point sequence, state update, and observation update are sequentially performed to sequentially estimate the state vector X and calculate the predicted value of the observed value.

  In the next step S104, a signal indicating each of the cyclic fluctuation component, the nonlinear trend component, and the event component is extracted from the state vector X sequentially estimated in step S103, and the signal extraction result and the calculation in step S103 are performed. Data indicating the predicted value is generated and output by the output unit 30, and the signal extraction processing routine is terminated.

<Experimental result>
Next, experimental results using the signal extraction method described in this embodiment will be described. An experiment was conducted to extract signals indicating each of the cyclic fluctuation component, nonlinear trend component, and event component from time series data related to daily sales at an EC site.

  From the sales time series (daily, about 2 years) of a certain EC site as shown in FIG. 5, the circulation fluctuation component around the trend as shown in FIG. 6, the trend component as shown in FIG. 7, as shown in FIG. The event component was extracted. Overall, signal extraction was good.

  Further, as shown in FIG. 8, it can be seen that the event component captures the sale campaign component that is performed in summer and winter every year with a long-term correlation signal. Further, FIG. 9 shows the prediction results of the first term of the model, and FIG. 10 shows the results of evaluating the performance. The model 3 corresponding to the state space model used in the present embodiment has a lower AIC (Akaike information amount) than the baseline model (model 1) that takes into account circulation fluctuations and normal trend terms. It was confirmed that the modeling accuracy was improved by the state space model used in the embodiment.

  As described above, according to the signal extraction device according to the present embodiment, the state vector X including the cyclic fluctuation component, the non-linear trend component, and the event component is sequentially estimated according to the normalized UFK, and the cyclic fluctuation component, the non-linear trend By extracting a signal indicating each of the component and the event component, a long-term unsteady signal and a local unsteady signal can be extracted from the time-series data.

  In addition, the modeling accuracy of non-stationary time series in which the behavior of fluctuations changes with time is improved, and in particular, signals indicating nonlinear trend components and event components can be extracted. Moreover, the trend component and the event component indicate the characteristics of the time series data, and in particular, the event component that takes a promotion effect signal such as a sale or a campaign can provide useful information in the analysis of the marketing time series.

  In addition, by sequentially estimating the state vector X in accordance with the normalized error variance covariance matrix normalized, the system noise covariance matrix, and the normalized UFK using the observed noise variance values, the state of the nonlinear model can be determined. A vector can be estimated.

  In addition, the state estimation algorithm used in this embodiment is based on the Unscented Kalman Filter (UKF) that accurately estimates the time-series state modeled in a nonlinear Gaussian type, and the total noise variance value that exists in the model A method of normalizing the sum of to 1 is used. Normally, since the variance value of noise is unknown, it is almost always set based on the experience and intuition of the analyst, which greatly affects the state estimation result. In the present embodiment, stable analysis and state estimation are enabled by suppressing the total sum of all noise variance values present in the model to 1.

  Note that the present invention is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the gist of the present invention.

  For example, a value of 0 to 1 is assigned to the normalized system noise covariance matrix and the observed noise variance value, and the normalized system noise covariance matrix and the observed noise variance value are assigned. Using each of these combinations, it is possible to sequentially estimate the state vector and specify an optimal variance value. In this case, a signal may be extracted from a state vector estimated under an optimal variance value.

  In the present specification, the embodiment has been described in which the program is installed in advance. However, the program can be provided by being stored in a computer-readable recording medium.

DESCRIPTION OF SYMBOLS 10 Input part 20 Calculation part 21 State space modeling part 22 State estimation part 23 Signal extraction part 30 Output part 100 Signal extraction apparatus

Claims (3)

A state update expression for nonlinearly updating the state vector X including the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component using the system noise, and the state vector X using the observation noise For a state space model defined using an observation equation indicating a relationship with an observed value, a covariance matrix of the estimation error variance of the state vector X, a diagonal component of the covariance matrix of the system noise, and a variance value of the observed noise Normalized estimation error variance covariance matrix normalized by the sum of, normalized system noise covariance matrix normalized by the sum of the system noise covariance matrix, and variance of the observed noise normalized by the sum Normalized UFK (Uncennes) that performs generation of sigma point sequence using normalized normalized observation noise variance, state update of state vector X, and observation update according to ted Kalman Filter), state estimation means for sequentially estimating the state vector X based on the time-series data of the observed values;
Signal extraction means for extracting signals indicating each of the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component from the state vector X sequentially estimated by the state estimating means;
A signal extraction device. A signal extraction method in a signal extraction device including state estimation means and signal extraction means,
The signal extraction device includes:
A state update equation for updating the time vector nonlinearly with the system noise using the system noise and the observation noise using the state fluctuation means, the long-term unsteady component, and the local unsteady component using the state estimation means. For the state space model defined using the observation equation indicating the relationship between the state vector X and the observed value, the covariance matrix of the estimated error variance of the state vector X is the diagonal component of the covariance matrix of the system noise and Normalized estimation error variance covariance matrix normalized by the sum of the observed noise variance, normalized system noise covariance matrix normalized by the sum of the system noise covariance matrix, and the observed noise variance Normalize for generating a sigma point sequence using the normalized observation noise variance normalized by the sum, updating the state of the state vector X, and updating the observation According UFK (Unscented Kalman Filter), a step of sequentially estimating the state vector X on the basis of the time-series data of the observations,
A step of extracting a signal indicating each of the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component from the state vector X sequentially estimated by the state estimating means by a signal extracting means; When,
A signal extraction method comprising: Computer
A state update expression for nonlinearly updating the state vector X including the cyclic fluctuation component, the long-term unsteady component, and the local unsteady component using the system noise, and the state vector X using the observation noise For a state space model defined using an observation equation indicating a relationship with an observed value, a covariance matrix of the estimation error variance of the state vector X, a diagonal component of the covariance matrix of the system noise, and a variance value of the observed noise Normalized estimation error variance covariance matrix normalized by the sum of, normalized system noise covariance matrix normalized by the sum of the system noise covariance matrix, and variance of the observed noise normalized by the sum Normalized UFK (Uncennes) that performs generation of sigma point sequence using normalized normalized observation noise variance, state update of state vector X, and observation update ted Kalman Filter), state estimation means for sequentially estimating the state vector X based on time-series data of the observed values, and the cyclic variation component from the state vector X sequentially estimated by the state estimation means, A program for functioning as signal extraction means for extracting a signal indicating each of a long-term unsteady component and the local unsteady component.