JP2014211827A - Derivation device, derivation method and derivation program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a derivation device, a derivation method and a derivation program for, on the basis of observation data of m points observed in time series for each of n-dimensional items related to would-be which changes with time, deriving time series data at a point of time not included in the observation data.SOLUTION: A derivation device 1 acquires the observation data of m points arranged in time series for each of n-dimensional items (S101), and accepts the designation of an item to be derived (S103), and derives a conversion map for searching, as mapping from a start region matrix consisting of n rows×m columns in which items are arranged so as to be aligned in a row direction and time series are arranged so as to be aligned in a column direction with the acquired observation data as elements, an end region matrix consisting of L rows×m columns in which the observation data of m+1-i (i is a natural number) points and the derivation data of i-1 points are arranged in a time sequential order in the i-th row with the observation data and derivation data of the designated item as elements (S105 to S108), and derives the derivation data by the derived conversion map (S109).

Description

  The present invention relates to a derivation device and a derivation method for deriving data at a time point not included in the observation data based on observation data of m points observed in time series for each of n-dimensional items related to events that change over time. And a derivation program.

  Predicting time-series changes in time-varying events such as financial markets such as stock prices, exchange rates, prices, weather, electricity consumption, biomolecular progress data, and complicated disease progress It is important not only in scientific research. Conventional time series prediction methods are generally developed on the assumption that a sufficient number of samples exist. As a typical time series model, an AR model, an MA model, and an ARMA model can be exemplified. There is also a method of performing time series prediction by reconstructing an attractor by applying Takens' embedding theory (Non-Patent Document 1). In such time series prediction, in order to construct a prediction model, long-term time series data including a sufficient number of samples is required.

  On the other hand, in recent years, data measurement technology has made remarkable progress, and it has become easy to obtain enormous items of data from a small number of samples at high speed. For example, in high-throughput biodata used for genetic research, that is, microarray data, it is possible to simultaneously measure tens of thousands of gene probes from one tens of samples on one chip. Examples of events that can observe data of many items from a small sample include various statistical reports, weather information, financial market information, astronomical observation information, and the like.

Takens, F., “In Dynamical Systems and Turbulence, Springer-Verlag New York, 1981, pp. 366-381.

  However, the conventional time-series prediction method obtains long-term time-series data in an actual system even though long-term time-series data that is sufficiently long is necessary to construct a prediction model. There is a problem that is often difficult. For example, in molecular biology experiments, due to various limitations, generally only a few tens of time series data can be obtained, and it is difficult to obtain long-term time series data. In addition, in the case of stock data in the market, a large number of data can be obtained, but because of the degree of condition dependence or time-varying model, only the most recent data is useful for forecasting because of time-dependent data. Is equivalent to typical data.

  On the other hand, short-term time-series data having many items is not often used for time-series prediction. The reason is that short-term time-series data is considered to lack information to be used for event prediction.

  The present invention has been made in view of such circumstances, and uses high-dimensional time-series data having many items for event prediction. The inventors of the present application have abundant interaction information between variables because high-dimensional time-series data having many items even in the short term reflects various aspects of the event. From this point of view, I think that it reflects the dynamic characteristics accumulated in the event. From such an idea, the present inventors extended the Takens embedding theory to convert high-dimensional time-series data into long-term low-dimensional time-series data, that is, the inverse of the conversion by Takens embedding theory. I found a conversion method. By using this method, it is possible to predict the dynamic characteristics of some items of an event by converting high-dimensional time-series data into long-term low-dimensional time-series data. That is, the present invention embodies such an idea, and provides a derivation device, a derivation method, and a derivation program capable of estimating unknown time-series data based on event observation data. With the goal.

  In order to achieve the above object, the deriving device according to the present invention provides m points (m) obtained by observing each item of n dimensions (n is an integer of 2 or more) in time series regarding events that change over time. Is a derivation device for deriving data at a time point not included in the observation data as derived data based on observation data of 2 or more), and obtaining m-point observation data for each n-dimensional item An acquisition unit that receives the designation of an item to be derived, and observation data acquired by the acquisition unit as elements, n rows arranged such that items are arranged in a row direction and time series are arranged in a column direction × A recording unit for recording an m-column start matrix, and observation data and derived data of a specified item as elements, and derivation of observation data of m + 1-i points and i-1 points in the i-th row (i is a natural number). Data in chronological order An L row × m column (L is an integer of 2 or more) placed end region matrix is recorded in the recording unit by means for deriving a conversion map that is a mapping from the start region matrix, and the derived conversion map. It is characterized by comprising means for deriving the end region matrix as a map from the start region matrix, and an output unit for outputting the derived data included in the derived end region matrix.

  In addition, the deriving device according to the present invention provides m points (m is an integer equal to or greater than 2) obtained by observing each of n-dimensional (n is an integer equal to or greater than 2) items relating to events that change over time. A deriving device for deriving, as derived data, data at a time point not included in the observation data based on the observed data, an acquisition unit for acquiring m-point observation data for each of n-dimensional items, and a deriving device Item accepting means for accepting designation of an item to be performed, n coefficients and observation data, and m + 1-i point observation data (i = 1, 2,..., L) and i-1 point for the designated item Formula generating means for generating m + 1-i formulas composed of variables related to the derived data, derivation means for deriving derived data based on the generated formulas, and an output unit for outputting the derived data And with It is characterized by that.

  In the derivation device according to the present invention, the mathematical expression generation means is a linearity composed of a product of n coefficients and n items of observation data, and variables relating to observation data and derivation data of specified items. I is generated, and the derivation means derives derived data related to the variable by using the values of n coefficients obtained from the mathematical expression generated by the mathematical expression generation means. .

  Further, the derivation device according to the present invention includes a function designating unit that receives designation of a non-linear function and an expansion order of the designated non-linear function, and the mathematical expression generating means uses the non-linear function and the expansion order designated by the function designating unit. On the basis of this, n nonlinear functions are generated as i equations, each of which has observation data and derived data as variables.

  In the derivation device according to the present invention, the derivation data is data for L-1 points after the m-th observation data, or data for L-1 points before the first observation data. It is characterized by.

  In the derivation method according to the present invention, observation of m points (m is an integer of 2 or more) obtained by observing each of n-dimensional (n is an integer of 2 or more) items in time series regarding events that change over time. A derivation method using a derivation device that derives data at a time point not included in the observation data as derivation data based on data, the derivation device comprising m points of observation data for each of n-dimensional items , Determining the item to be derived, n coefficients and observation data, and observation data (L = 1, 2,..., I) of m + 1-L points for the specified item and A step of generating i formulas composed of variables related to the derived data of L-1 points and a step of deriving derived data based on the generated formulas are performed.

  The derivation program according to the present invention allows a computer to obtain m points (m is an integer greater than or equal to 2) obtained by observing each of n-dimensional items (n is an integer greater than or equal to 2) relating to an event that changes over time. ) Is a derivation program for deriving data at a time point not included in the observation data as derivation data, and the computer obtains m-point observation data for each of the n-dimensional items. Determining items to be derived, n coefficients and observation data, and m + 1-L point observation data (L = 1, 2,..., I) and L−1 points for the specified item. Generating i mathematical formulas composed of variables related to the derived data, and deriving derived data based on the generated mathematical formulas. The

  The derivation device, derivation method, and derivation program having the features described above can estimate future or past unknown data for a specified item by deriving unknown derived data from high-dimensional observation data. is there.

  According to the present invention, unknown point-in-time data can be derived based on m-point observation data obtained by observing each n-dimensional item related to an event that changes over time in time series. As a result, it is possible to derive data at an unknown time point from the relationship between the observation data of n-dimensional items, and to obtain an excellent effect such as being able to estimate an event.

It is explanatory drawing which shows notionally the reconstruction of the dynamic system by Takens embedding theory. It is a block diagram which shows the structural example of the derivation | leading-out apparatus based on this invention. It is a flowchart which shows an example of the derivation | leading-out process of the linearity time series data of the derivation | leading-out apparatus based on this invention. It is a flowchart which shows an example of the derivation | leading-out process of the nonlinear time series data of the derivation | leading-out apparatus based on this invention. It is explanatory drawing which shows an example of the observation data which the derivation device concerning the present invention acquires. It is explanatory drawing which shows an example of the derivation data which the derivation | leading-out apparatus based on this invention outputs.

<Estimation of time series data>
First, time series data will be described as a premise of the present invention. When an event that changes over time is regarded as a dynamic system (system) including items of n dimensions (n is an integer of 2 or more), the dynamic system represents the following equation (1) indicating an n-dimensional state vector: Can be expressed as

x (t) = (x ₁ (t), x ₂ (t),..., x _n (t)) ^T ∈ R ⁿ (1)

In Expression (1), x _j (t), (j = 1, 2,..., N) are state variables of the dynamic system, n is the dimension of the system, and t is a time point such as an observation time. ing. R ⁿ represents an n-dimensional Euclidean space. The state variable x _j (t) is a variable that changes with time. In the case of a continuous system, the change in the state variable is expressed as dx (t) / dt = F (x (t)). The change of the state variable is expressed as x (t + 1) = F (x (t)).

When factors such as the structure, properties, and external influences of a dynamic system are clear and can be mathematically expressed, the dynamic characteristics of the system (state variable x _j (t) ( The value of t = 0 → T)) can be derived. On the other hand, when factors such as the configuration, properties, and external influences of the dynamic system are unclear and cannot be mathematically expressed, the dynamic characteristics of the system cannot be derived. Therefore, a part or all of the data of the state variable x _j (t) observed in the past fixed period (t = t−τ, t−2τ,..., T−kτ) (τ is a time interval) Based on the estimation of time series data in a dynamic system that estimates data of a specific state variable x _s (t) for a certain period in the future (t = t + τ, t + 2τ,..., T + (L−1) τ) Is done.

In addition, the time series data of the dynamic system can be classified into categories such as long-term low-dimensional time series data and short-term high-dimensional time series data based on the period and the number of dimensions. Long-term low-dimensional time-series data means that a relatively small number of state variables (eg, x ₁ (t), x ₂ (t), x ₃ (t)) t = t ₁ , t ₂ ,..., t _k ) ([x ₁ (t ₁ ), x ₁ (t ₂ ),... x ₁ (t _k )] ^T , [x ₂ (t ₁ ), x ₂ (t ₂ ), ... x ₂ (t _k )] ^T , [x ₃ (t ₁ ), x ₃ (t ₂ ), ... x ₃ (t _k )] ^T ) is there. Short-term high-dimensional time-series data means that a relatively small number of state variables (eg, x ₁ (t), x ₂ (t),..., X _n (t)) Data ([x ₁ (t ₁ ), x ₁ (t ₂ ),... X ₁ (t _m ) at a time point (for example, t = t ₁ , t ₂ ,..., T _m ) ( _m << _n ) )] ^T , [x ₂ (t ₁ ), x ₂ (t ₂ ), ... x ₂ (t _m )] ^T , ..., [x _n (t ₁ ), x _n (t ₂ ), ... it is a ^{_{x n (t m)] T}} ).

  Conventional prediction of time-series data is a method for estimating future time-series data for a specific state variable using long-term low-dimensional time-series data observed in the past. On the other hand, the present invention is suitable for estimating future time series data of a specific state variable using short-term high-dimensional time series data observed in the past. Note that time-series data can be estimated not only for future time-series data after the period when the state variable is observed, but also for past time-series data before the period when the state variable is observed. is there. That is, it is suitable for converting known short-term high-dimensional time-series data into long-term low-dimensional time-series data including unknown time-series data. Moreover, it is possible to estimate unknown time-series data for all the state variables by changing the state variables to be estimated.

<Relationship between short-term high-dimensional time-series data and long-term low-dimensional time-series data>
Next, the relationship between short-term high-dimensional time-series data and long-term low-dimensional time-series data will be described. FIG. 1 is an explanatory diagram conceptually showing reconstruction of a dynamic system based on Takens' embedding theory. Takens' embedding theory shows that in dynamic systems, it is possible to reconstruct dynamic properties from long-term low-dimensional time-series data.

FIG. 1A conceptually shows an attractor of a three-dimensional dynamic system composed of state variables x ₁ , x ₂ and x ₃ . For the dynamic system shown in FIG. 1A, time series data obtained by observing x ₁ of the three state variables during a period T having a sufficient length is expressed as y = x ₁ ( What is shown as t) is the graph shown in FIG. That is, FIG. 1B shows long-term low-dimensional time series data obtained by observing a three-dimensional dynamic system.

Based on the observed long-term low-dimensional time-series data shown in FIG. 1B, the time-series data of the period “T-2τ” generated by delay embedding for the time delay amount “τ” is shown. It is a graph shown in FIG.1 (c). FIG. 1C shows three-dimensional time-series data represented as z ₁ (t) = y (t), z ₂ (t) = y (t + τ) and z ₃ (t) = y (t + 2τ). Yes.

As a phase space composed of the three-dimensional time series data (z ₁ , z ₂ , z ₃ ) shown in FIG. 1C, Pn = (z ₁ (t _n ), z ₂ (t _n ), z ₃ (t _The conceptual diagram shown in FIG. 1 (d) is obtained by reconstructing the attractor of the dynamic system shown as _n )). FIG. 1 (d) shows Pn = (z ₁ (t _n ), z in a phase space consisting of three-dimensional time series data z ₁ (t _n ), z ₂ (t _n ), z ₃ (t _n ). _{2 (t n), z 3} (t n)) conceptually illustrates the attractor dynamic system shown as. The attractor shown in FIG. 1 (d) is topologically in phase with the attractor shown in FIG. 1 (a).

  As described with reference to FIG. 1, Takens' embedding theory provides a method for deriving short-term high-dimensional time-series data from long-term low-dimensional time-series data. The present application discloses a method using a conversion map for converting high-dimensional time-series data, for example, short-term high-dimensional time-series data into long-term low-dimensional time-series data. The transformation map can extract the interaction information between each variable, and further convert the interaction information into time-series information, so that long-term low-dimensional time-series data is derived from short-term high-dimensional time-series data. It becomes possible. Thus, in the present application, for example, as a reverse conversion of conversion from long-term low-dimensional time-series data to short-term high-dimensional time-series data performed using Takens embedding theory, short-term that has not been realized in the past Disclosed is a method that enables conversion from static high-dimensional time-series data to long-term low-dimensional time-series data.

<Outline of reverse embedding theory (inverse transformation)>
The conversion map Φ used for converting time series data will be described. The flow F in the open subset U of the Euclidean space R ⁿ , the compact subset A of the open subset U whose box dimension is d, and the integer L (L> 2d) are defined, and the time interval τ (τ> 0) is Appropriate values shall be taken. In this case, almost all smooth functions y and resulting delay coordinates z = (y (x), y (F _T (x)),..., Y (F _{(L−1) T} (x ))) For ^T (x∈A), there is a transformation map Φ: R ⁿ → ^RL (subscripts T and (L-1) T denote τ and (L-1) τ) .) At this time, Φ (x) = z is one-to-one, and is an inset on an arbitrary compact subset of a smooth manifold included in the compact subset A. Then, the transformation map Φ shows the state x included in the compact subset A along the flow F and the future state y (F _iT (x)), (i = 1, 2,..., L−1). (Subscript iT indicates i τ).

The conversion from short-term high-dimensional time-series data to long-term low-dimensional time-series data based on reverse embedding theory will be described in detail. In the system x (t) of the dynamic system shown as an n-dimensional state vector using the above equation (1), there is an attractor represented by a compact subset A of the box dimension d. Further, y (t) is defined as a function using the state variable x _i (t). Here, for simplicity, y (t) = x _i (t). Further, z ₁ (t) = y (t), z ₂ (t) = y (t + τ),..., Z _L (t) = y (t + (L−1) τ) (τ: delay constant) By performing delay embedding of y (t) as in the following, a new system represented by the following formula (2) can be constructed.

z (t) = (z ₁ (t), z ₂ (t),..., z _L (t)) ^T ∈ R ^L (2)

Z (t) in Equation (2) shows a state variable of the system ∈R ^L reconstructed, L is shows the dimensions of the system.

According to Takens' embedding theory and the generalized embedding theory for Sauer, Yorke and Casdagli's fractal attractor shown as Non-Patent Document 1, the attractor of the reconstructed system z (t) is the original system when L> 2d. Topologically in phase with x (t). Therefore, a map Φ (Φ (x (t))) = z (t)) exists between the Euclidean space R ⁿ of the system x (t) and the Euclidean space R ^L of the system z (t). The map Φ is a transformation matrix that maps the system x (t) and transforms it into the system z (t). Under the map Φ, from the viewpoint of topologically in-phase, the one-dimensional time series data y (t) has the same attractor information as the data at the point m of x (t). Thus, by the map [Phi, system x (t) ∈R ⁿ are converted to the system z (t) ∈R ^L. That is, the map Φ is a mapping from the start region x (t) εR ⁿ to the end region z (t) εR ^L.

The map Φ will be further described in detail. The n-dimensional time series data observed at m points at time interval τ is ^expressed as x _j = x (t + (j−1) τ), (j = 1, 2,...) Using state variable x (t) ∈R ⁿ . .., M), the map Φ can be obtained as Φ (x ₁ , x ₂ ,..., X _m ) or the following equation (3).

In the above equation (3), x _ij = x _i (t + (j−1) τ), z _ij = z _i (t + (j−1) τ), and z _ij = y (t + (i + j−2)) Since τ), the map Φ can be rewritten as the following equation (4).

  Φ: X → Y (4)

  In Equation (4), the matrices X and Y are as follows.

  As described above, X is converted to Y by the map Φ. As shown in Expression (4), X is a high-dimensional matrix composed of observation data at the time of each m point of the n-dimensional state variable x (t). Y is a matrix of L × m in form, and the data at (m + L−1) time points of any one state variable y (t) of the state variables x (t) The matrix is arranged by shifting the time points forward by one. Y arranged in this way can be regarded as a collection of a plurality of one-dimensional time-series data indicated as the state variable y (t). That is, it can be said that the map Φ has a property as a map for converting the high-dimensional time-series data X into the one-dimensional (low-dimensional) time-series data Y.

  The state variable y (t) is a long-term one-dimensional time series derived for any one dimension (item) selected from observation data of a plurality of dimensions (items) indicated as the state variable x (t). It is data. It is also possible to select a state variable x (t) of any other dimension (item) and reconstruct a new system z (t). Therefore, it is possible to convert X into a plurality of Ys, and theoretically, it is possible to derive the one-dimensional time series data Y for all the state variables x (t) included in X.

  When the number of points obtained by time series observation of the state variable x (t) indicating the observation data for the original system is m, it is included in the converted one-dimensional time series data y (t). The number of elements is (m + L−1). As described above, when L> 2d, the attractor of the reconstructed system is topologically in phase with the attractor of the original system. For this reason, the number of elements included in the converted time-series data y (t) is larger than the number of observation data related to the state variable x (t). Accordingly, the obtained short-term high-dimensional time series data observed at the time of m points for the n state variables by the map Φ is the long-term low dimension at the time of m + L−1 points for one state variable. It can be said that it can be converted into time-series data. In the present application, such conversion is referred to as “reverse embedding”.

<Determination method of map Φ>
Next, a method for determining the map Φ will be described. In order to perform the conversion based on the reverse embedding theory described above, the map Φ must be determined. The map Φ is determined by the values x _j = x (t + (j−1) τ), (j = 1, 2,...) Observed at the point m of each variable of the system x (t) ∈R ^n. , M), Φ ₁ (x _j ) = y _j , (j = 1, 2,..., M) and Φ _i (x _j ) = Φ _i−1 (x _{j + 1} ), Under (m + (L−1) × (m−1)) constraint conditions (i = 2,..., L), (j = 1,..., M−1), Φ = ( Φ ₁ , Φ ₂ , ... , Φ _L ) ^T. In the present application, in order to increase the accuracy of prediction, depending on the properties of the system x (t) εR ⁿ , the case where the map Φ is a linear map having linearity and the case of being a nonlinear map having no linearity A method for determining each map Φ will be described.

<Determination of linear map Φ>
When the change of the system x (t) εR ⁿ has linearity or can be linearly approximated, the linear map Φ having linearity is used as the transformation map Φ. The mapping Φ (X) by the linear map Φ can be expressed by the following equation (5) as a matrix A of L rows × n columns.

  Φ (X) = AX = Y (5)

  In the equation (5), the matrices X and Y are as shown as the above equation (4), and the matrix A is as follows.

X is a matrix of n rows × m columns composed of observation data observed at m points of each state variable x (t) ∈R ⁿ . Y is an L row × m column in which data of m + L−1 points of any one state variable y (t) in x (t) is arranged by shifting the time point forward by one for each column. It is a matrix. Furthermore, AX = Y shown as the equation (5) can be expressed as the following equation (6).

In equation (6), a ⁱ = (a _i1 , a _i2 ,..., A _in ), (i = 1, 2,..., L), x _j = (x _1j , x _1j ,... , X _1j ) ^T , (j = 1, 2,..., M). From equation _{(6), x 1, x} 2, ..., observations were obtained for x _m data y _1, y _2, ..., with respect to y _m, m pieces of linear equations a ⁱ · x _j = Y _j , (j = 1, 2,..., M) and (m−1) × (L−1) linear equations a ⁱ · x _j −a ⁱ⁻¹ · x _{j + 1} = 0, (i = 2, 3,..., N, j = 1, 2,..., M−1) can be obtained. In _{y (t), y 1,} y 2, ..., y m are known observation _{data, y m + 1, y m} + 2 ..., and the target of y _{L + m-1} is presumed This is unknown time-series data. Therefore, by calculating unknown time-series data included in y (t) from the equation shown as equation (6), the state variable y (t) selected from x (t) is later than time m. Time series data can be calculated. That is, it is possible to derive unknown time-series data of the state variable x (t) related to the selected dimension (item), and it is possible to estimate the state change of the observation target based on the derived time-series data. It becomes.

Next, a calculation algorithm based on the linear map Φ will be described. First, the observed time series data x ₁ , x ₂ ,..., X _m are acquired, and the variable y indicating the time series data is added to the observed data related to the state variable x _s (t) related to the selected item. _1, y _2, ..., allocates y _m. A mathematical formula shown in the following formula (7) is generated from these time series data, and a linear equation for a ¹ using the generated mathematical formula is solved to obtain a ¹ (procedure Sa1).

a ¹ · x _j = y _j (7)
Where j = 1, 2,..., M

Then, i = 2,3, ..., based on the equation (8) below shows the L, and the primary equation for a ^{i i} = 2,3, ..., generated in the order of L, The procedure for solving the linear equation for a ⁱ using the generated mathematical expression is repeated to obtain a ⁱ (procedure Sa2).

a ⁱ · x _j = y _{i + j−1} Expression (8)
Where j = 1, 2,..., M−1.

Using the coefficients a ⁱ , (i = 1, 2,..., L) calculated based on the equations (7) and (8), the time series data y _{m + i−1} at the time after m is expressed as follows: Calculation is performed according to equation (9) (procedure Sa3).

y _{m + i-1} = a ⁱ · x _m (9)

The time series data ym _{+ i-1} calculated by the equation (9) is unknown time series data to be estimated.

  In such a calculation algorithm based on the linear map Φ, there are (m−1) × L + 1 linear equations for n × L unknowns, and iterative operations are performed to solve these linear equations. The When the short-term high-dimensional time-series data X is used as the time-series data X, n >> m and n × L >> (m−1) × L + 1. Usually, if the number of unknown variables is much larger than the number of linear equations, a solution cannot be found. In this application, since the dimension of the original system is high and the data at the point m is generally observed from the asymptotic attractor of the system, the dimension d is low and applied to a system where d << n. It is assumed that Since L is an integer slightly larger than d, L << n. Therefore, there are few variables that are linearly independent of n × L unknown variables. Therefore, a solution can be obtained if a certain condition is satisfied.

  Next, conditions for obtaining a solution will be described. The above equation (8) can be expressed as the following equation (10).

a ⁱ · x _j = y _{i + j -1} = a ^{i -1} · x _{j +1} (10)

In equation (10), a ^i-1 includes L variables to be determined. Based on the theory of linear equations, the rank of the coefficient matrix determines whether or not the solution of the linear equations can be obtained. Here, as shown in the following formula (11), X ~ (X ~ is a tilde above X, the same in the following description) is defined.

  In the above calculation algorithm, when rank (X ~)> L, a solution cannot be obtained. On the other hand, when rank (X˜) ≦ L, a solution can be obtained. In particular, when rank (X˜) = L, the solution can be obtained accurately. It is possible to obtain a solution by setting the number m of observation time points to an appropriate value so as to satisfy the condition. In particular, by setting m = L + 1, the linear map Φ can be accurately set. It becomes possible to ask for.

<Determination of nonlinear map Φ>
System x (t) ∈R ⁿ without linearity in a change in, and for the highly nonlinear system to the extent that they can not be linearly approximated, even using a linear map Φ described above, a highly accurate prediction It is difficult to do. A method of determining a nonlinear map Φ that can be applied to such a highly nonlinear system will be described. Each element of the nonlinear map Φ is expressed using the following equation (12).

In Expression (12), g _l (x), l = 1, 2,..., K is a basis function in the nonlinear map Φ, and a _li , (l = 1, 2,..., K, i = 1, 2,..., L) are the respective coefficients. As the basis function g _l (x), various nonlinear functions such as a multidimensional equation, a trigonometric function, a function related to a Fourier series, an approximate function of a neural network, a function related to a fuzzy set, and various fractal functions can be used.

  By using the equation (12), the nonlinear map Φ can be obtained by applying the above-described method for obtaining the linear map Φ. However, since the number of k tends to be much larger than the number of observation points m, it may be difficult to obtain the nonlinear map Φ based on the equation (12). Even in such a case, it is possible to obtain the nonlinear map Φ. In the present application, the following method is disclosed as a method for obtaining the nonlinear map Φ.

A calculation algorithm based on the nonlinear map Φ will be described. First, the observed time series data x ₁ , x ₂ ,..., X _m are acquired, and the variable y indicating the time series data is added to the observed data related to the state variable x _s (t) related to the selected item. _1, y _2, ..., allocates y _m. Then, a basis function g _l (x), l = 1, 2,..., K and its expansion order k are selected (step Sb1).

Next, x _j = [g ₁ (x _j ), g ₂ (x _j ),..., G ₃ (x _j ) so that Φ _i (x _j ) = _x˜j ^T · α _i. ,] ^T (where j = 1, 2,..., M) and α _i are constructed (procedure Sb2).

Next, the mathematical formula shown in the following formula (13) is generated by the compression sensing technique, and α ₁ is obtained by solving the generated mathematical formula (procedure Sb3).

x ~ _j ^T · α ₁ = y _j Expression (13)
Where j = 1, 2,..., M

Next, based on the following formula (14) shown for i = 2, 3,..., L, formulas for α _i are generated in the order of i = 2, 3,. The procedure for solving the mathematical expression for α _i using the above mathematical formula is repeated by the compression sensing technique to obtain α _i (procedure Sb4).

^{_{x ~ j T · α i =}} y i + j-1 ··· formula (14)
Where j = 1, 2,..., M−1.

Using the coefficient α _i calculated based on the equation (14), time series data ym _{+ i−1} at the time point after _{m is} calculated by the following equation (15) (procedure Sb5).

y _{m + i-1} = x ~ _m ^T · α _i (15)

The time series data ym _{+ i-1} calculated by the equation (15) is unknown time series data to be estimated. In this way, the nonlinear map Φ can be obtained.

<Deriving device>
The method for estimating unknown time-series data described in detail above can be embodied as an industrially applicable invention by applying it as a derivation device using a computer. The derivation device according to the present invention observes m points (m is an integer equal to or greater than 2) obtained by observing each of n-dimensional (n is an integer equal to or greater than 2) items related to events that change over time. Based on the data, data at a time point not included in the observation data is derived as derived data. FIG. 2 is a block diagram illustrating a configuration example of the derivation device according to the present invention. The derivation device 1 shown in FIG. 2 is realized using a computer such as a personal computer, a server computer, a client computer connected thereto, and other various computers. The deriving device 1 includes various mechanisms such as a control unit 10, a recording unit 11, a storage unit 12, an input unit 13, an output unit 14, and an acquisition unit 15.

  The control unit 10 is configured using a circuit such as a CPU (Central Processing Unit), and is a mechanism that controls the entire detection apparatus 1.

  The recording unit 11 is a non-volatile auxiliary recording mechanism such as a magnetic recording mechanism such as an HDD (Hard Disk Drive) or a non-volatile semiconductor recording mechanism such as an SSD (Solid State Disk). In the recording unit 11, various programs and data such as the derivation program 11a according to the present invention are recorded.

  The storage unit 12 is a volatile main storage mechanism such as an SDRAM (Synchronous Dynamic Random Access Memory) or an SRAM (Static Random Access Memory).

  The input unit 13 is an input mechanism including hardware such as a keyboard and a mouse, and software such as a driver.

  The output unit 14 is an output mechanism including hardware such as a monitor and a printer, and software such as a driver.

  The acquisition unit 15 is an interface mechanism including a LAN board that acquires data such as observation data from the outside, various memory slots, wireless communication devices, hardware such as an image reading device, and software such as a driver. The input unit 13 can be used as the acquisition unit 15.

  Then, by storing the derivation program 11a recorded in the recording unit 11 in the recording unit 12 and executing it based on the control of the control unit 10, the computer executes various procedures related to the derivation program 11a. Functions as a derivation device 1 for For the sake of convenience, the recording unit 11 and the storage unit 12 are distinguished from each other, but both have the same function of recording various types of information, and which mechanism is used for recording according to the specifications of the apparatus, the operation mode, and the like. Can be determined as appropriate. Similarly, a register provided in the control unit 10 may be used as the recording unit 11.

  Next, the process of the derivation device 1 according to the present invention will be described. FIG. 3 is a flowchart showing an example of the derivation process of the linearity time series data of the derivation device 1 according to the present invention. The control unit 10 of the derivation device 1 uses the acquisition unit 15 to acquire observation data of m points arranged in time series for each of the n-dimensional items (S101). As described above, the observation data is acquired by the acquisition unit 15 or the input unit 13. However, the observation data is recorded in the recording unit 11 in advance, and the observation data is read by reading the observation data from the recording unit 11. It is good as acquisition. As observation data, it is possible to use various data such as image data in addition to data obtained by measurement by various measuring devices and economic data such as stock prices. When image data is used as the observation data, for example, with respect to representative values of each pixel constituting the image or a plurality of pixels constituting a predetermined region, brightness, R (Red) value, G (Green) value, and B (Blue) ) The tone value of the value can be acquired and used as observation data. The image data itself may be acquired by receiving an input from the outside, or may be acquired by reading an image using an acquisition unit 15 such as a scanner having an image reading function.

  The control unit 10 records the acquired observation data in the recording unit 11 (S102). The recording unit 11 includes an n-row × m-column matrix X (starting matrix) in which the observation data acquired by the acquisition unit 15 is an element, items are arranged in the row direction, and time series are arranged in the column direction. To be recorded. The actual recording format in the recording unit 11 does not necessarily have to be recorded in a matrix format, and an address indicating a row and a column, or an identification code substituted for it, is attached so that it can be arranged in the matrix format. It ’s fine.

After recording the read observation data, the operator operates the input unit 13 to specify an item to be estimated as an item to be derived, and further specifies the number L of points to be estimated. The control unit 10 accepts designation of items to be derived from the input unit 13 and the number of time points to be derived (S103). In step S103, designation of an item x _s (t) designated from among a plurality of items x (t) related to observation data is accepted. This process corresponds to the selection of the specific state variable x _s (t) described above. The designation of the number of time points is a process of receiving the number of time points to be predicted, that is, the value of L described above. As the processing in step S103, the designation may be accepted by reading preset items and the number of points in time. Further, there may be a plurality of items to be specified, and all items may be specified.

The control unit 10 assigns observation data and derived data to variables (S104). In step S104, for the observed data, variables y _1, y _2, ..., by assigning y _m recorded in the recording unit 11. Also, variables ym _{+ 1} , ym _{+ 2} ,..., Ym _{+ L} corresponding to the time points m + 1, m + 2,..., M + L-1 corresponding to the number L of time points specified in step S102. Define _-1 . The defined variables y _{m + 1} , y _{m + 2} ,..., Y _{m + L-1} are recorded in the recording unit 11 as variables to which the derived data is assigned. Variable column y _1, y ₂ which are recorded in the recording unit _{11, ..., y m, y} m + 1, y m + 2, ..., y m + L-1 is the subsequent processing, 1 Treated as a dimensional matrix Y (final matrix).

The control unit 10 generates a mathematical expression related to the element a ₁ in the first row of the linear map Φ from the observation data and the derived data assigned to the variable (S105), and calculates the element a ₁ based on the generated mathematical expression (S105). S106). In steps S105 to S106, processing corresponding to the procedure Sa1 is executed among the methods described as the determination of the linear map Φ described above.

Based on a ₁ calculated in step S106, the control unit 10 sequentially generates a mathematical formula related to the element a _i in the i-th row of the linear map Φ (S107), and sequentially generates the element a _i based on the generated mathematical formula. Calculate (S108). In steps S107 to S108, a process corresponding to the procedure Sa2 is executed among the processes described as the determination of the linear map Φ. The elements a _i are sequentially calculated by repeating the generation and calculation of mathematical expressions in the order of a ₂ , a ₃ ,..., That is, by repeating the processes of steps S107 to S108.

The control unit 10 derives time-series data y _{m + i-1} by calculation based on the linear map Φ having the calculated a _i as an element and the observation data x _m (S109). In step S109, the process corresponding to the procedure Sa3 is executed among the methods described as the determination of the linear map Φ.

  The control unit 10 determines whether time-series data has been derived for all items specified in step S103 (S110).

  If it is determined in step S110 that there is an item for which time-series data has not been derived (S110: NO), the control unit 10 returns to step S104, and repeats the subsequent processing to determine the items that have not been derived. Deriving time series data.

If it is determined in step S110 that time series data has been derived for all items (S110: YES), the control unit 10 outputs time series data ym _{+ i-1} for each derived item from the output unit 14. (S111). The output in step S111 is performed by various methods such as display on a monitor, printing from a printer, transmission to an external apparatus, recording on the recording unit 11, and the like.

  As described above, the derivation process of the linearity time series data is executed.

  Next, a process for deriving nonlinear time-series data will be described. FIG. 4 is a flowchart showing an example of the derivation process of the nonlinear time series data of the derivation device 1 according to the present invention. The control unit 10 of the derivation device 1 acquires m-point observation data for each of the n-dimensional items by the acquisition unit 15 (S201), and records the acquired observation data in the recording unit 11 (S202).

  After recording the read observation data, the operator operates the input unit 13 to specify an item to be estimated as an item to be derived, and further specifies the number L of points to be estimated. The control unit 10 accepts designation of items to be derived from the input unit 13 and the number of points to be derived (S203).

Further, the operator operates the input unit 13 and receives designation of the basis function g ₁ (x) and the order k (S204). In step S204, the process corresponding to the procedure Sb1 is executed among the methods described as the determination of the nonlinear map Φ. In addition, as a process of step S204, it is good also as designation | designated reception by reading the basis function and order set beforehand.

Control unit 10 constructs a matrix x ~ _j ^T and coefficients alpha _i of the nonlinear map [Phi _i basis functions constituting the _{(x j) g i (x} ) (S205). In step S205, the process corresponding to the procedure Sb2 is executed among the methods described as the determination of the nonlinear map Φ.

The control unit 10 assigns observation data and derived data variables (S206). Further, the control unit 10 generates a mathematical formula related to the element α ¹ in the first row of the nonlinear map Φ from the observation data and the derived data assigned to the variable (S207), and calculates the element α ¹ based on the generated mathematical formula. (S208). In steps S207 to S208, processing corresponding to the procedure Sb3 is executed among the methods described as the determination of the nonlinear map Φ.

Based on α ¹ calculated in step S208, the control unit 10 sequentially generates mathematical expressions related to the element α ⁱ in the i-th row of the nonlinear map Φ (S209), and sequentially generates the elements α ⁱ based on the generated mathematical expressions. Calculate (S210). In steps S209 to S210, the process corresponding to the procedure Sb4 is executed among the methods described as the determination of the nonlinear map Φ. The element α ⁱ is sequentially calculated by repeating the generation and calculation of mathematical expressions in the order of α ² , α ³ ,..., That is, by repeating the processing of steps S209 to S210.

The control unit 10 derives time-series data y _{m + i-1} by calculation based on the nonlinear map Φ having the calculated α ⁱ as an element and the observation data x _m (S211). In step S211, the process corresponding to the procedure Sb5 is executed among the methods described as the determination of the nonlinear map Φ described above.

  The control unit 10 determines whether time-series data has been derived for all items specified in step S203 (S212).

  In step S212, when it is determined that there is an item for which time-series data has not been derived (S212: NO), the control unit 10 returns to step S206, and repeats the subsequent processing to determine the items that have not been derived. Deriving time series data.

If it is determined in step S212 that time series data has been derived for all items (S212: YES), the control unit 10 outputs time series data ym _{+ i-1} for each derived item from the output unit 14. (S212).

  As described above, the process of deriving nonlinear time series data is executed.

<Example of application to an actual dynamic system>
As described above in detail, the deriving device 1 of the present invention can estimate unknown time-series data as the time-series data deriving process. The derivation of unknown time-series data executed by the derivation device 1 of the present invention is based on time-series data having n-dimensional items to be converted over time, particularly on the basis of short-term high-dimensional time-series data. Useful for estimating time-series data of dimensions. Its application range is wide, and it can be applied to various fields such as financial markets such as stock prices, exchange rates, prices, weather, power consumption, biomolecular level progress data, and complicated disease progress.

  An example in which the present invention is applied to weather prediction will be described. FIG. 5 is an explanatory diagram showing an example of observation data acquired by the deriving device 1 according to the present invention. FIG. 5 is a visible image obtained by imaging the vicinity of the southwestern Pacific with a meteorological satellite, and it is possible to confirm a situation where a typhoon is generated. The deriving device 1 acquires pixel data as observation data from the visible image illustrated in FIG. The pixel data to be acquired can be set as appropriate, such as all pixels constituting the visible image, pixels at specific coordinates in the image, and representative values (for example, average values) of pixels for each preset region. The numerical values used for the pixel data can be set as appropriate, such as gradation values for monochrome images and gradation values for R, G, and B color components for color images.

  This time, as a pseudo test according to the present invention, gradation values classified into 256 levels for each of the R value, G value, and B value of the pixels corresponding to the coordinate positions of 300 points included in the visible image illustrated in FIG. Obtained as 900 points of observation data. Further, the X-coordinate and Y-coordinate indicating the center of the typhoon vortex shown in FIG. That is, a visible image from a meteorological satellite was regarded as a system including a 902-dimensional state variable, and 902 items of observation data included in the system were acquired. In addition, a similar process for acquiring observation data was performed on 21 visible images obtained by imaging at predetermined time intervals. By such a method, multi-dimensional observation data at 21 points in time was acquired from a system that changes over time (dynamic system) at regular time intervals. Then, as the items to be derived, the X coordinate and the Y coordinate indicating the position of the typhoon eye are specified, and the specified X coordinate and Y coordinate for the eight time points after the time point related to the observation data based on the linear map Φ. The value of was derived and output.

  FIG. 6 is an explanatory diagram showing an example of derived data output by the deriving device 1 according to the present invention. FIG. 6 is an explanatory diagram showing coordinates indicating the position of the typhoon eye on the map with respect to 21 points of observation data, 8 points of derived data, and 8 points of actual data. In order to clarify the position on the map, the coastline is shown for the land. The land shown in the lower left of the figure is the Philippine Islands such as Luzon, and the land shown in the upper left is Taiwan Island. In addition, the point indicated by “△” in the figure is the actual position of the typhoon eye indicated by the observation data, the point indicated by “◯” is the estimated position of the typhoon eye indicated by the derived data, and “◇” The indicated points indicate the actual positions of the actual typhoon eyes at the time corresponding to the derived data.

  As shown in FIG. 6, the temporal change in the position of the typhoon center indicated by the eight derived data derived from the 21 observation data, that is, the predicted course of the typhoon approximates the actual course. Can be confirmed. In the current numerical prediction related to weather, the course of typhoon is predicted based on weather data such as temperature, humidity, atmospheric pressure, and wind direction at each altitude at each point. On the other hand, the present invention shows that it is possible to predict the course of a typhoon from only a visible image from a weather satellite. When actually applying the present invention, various meteorological data are used as observation data. Furthermore, these meteorological data and data obtained from image data such as visible images and infrared images are used in combination as observation data. By doing so, it is possible to expect a more accurate course prediction.

  In the above embodiment, only a part of the infinite number of examples of the present invention is disclosed, and it can be applied to various methods for deriving (estimating) time series data using the concept of the map Φ. It is. The calculation procedure for calculating the map Φ is not limited to the method described above, and various methods can be applied. For example, an algorithm for calculating a plurality of elements at a time may be applied to the generation of mathematical formulas and the calculation of elements related to the elements of the map Φ instead of sequentially calculating each line.

DESCRIPTION OF SYMBOLS 1 Derivation apparatus 10 Control part 11 Recording part 11a Derivation program 12 Storage part 13 Input part 14 Output part 15 Acquisition part

Claims (7)

Based on the observation data of m points (m is an integer of 2 or more) obtained by observing each of n-dimensional items (n is an integer of 2 or more) in time series regarding events that change over time, the observation data A derivation device for deriving data at a time not included in
an acquisition unit for acquiring observation data of m points for each of n-dimensional items;
Item accepting means for accepting designation of an item to be derived;
A recording unit that records observation data acquired by the acquisition unit as an element, records an n-row × m-column start matrix arranged such that items are arranged in a row direction and a time series is arranged in a column direction;
L rows × m columns (with m + 1-i point observation data and i−1 point derivation data arranged in chronological order in the i-th row (i is a natural number) with the observation data and derivation data of the specified item as elements. Means for deriving a transformation map in which L is an integer of 2 or more) and a mapping from the start matrix;
Means for deriving the end-region matrix as a mapping from the start-region matrix recorded in the recording unit by the derived transformation map;
An derivation device comprising: an output unit that outputs derivation data included in the derived end-region matrix. Based on observation data of m points (m is an integer of 2 or more) obtained by observing each of n-dimensional items (n is an integer of 2 or more) in time series regarding events that change over time, the observation data A derivation device for deriving data at a time not included in
an acquisition unit for acquiring observation data of m points for each of n-dimensional items;
Item accepting means for accepting designation of an item to be derived;
Consists of n coefficients and observation data, and m + 1-i point observation data (i = 1, 2,..., L) for the specified item and i-1 point derived data. formula generating means for generating m + 1-i formulas;
Derivation means for deriving derived data based on the generated mathematical formula;
An derivation device comprising: an output unit that outputs the derived data. The derivation device according to claim 2, wherein
The mathematical expression generating means generates i mathematical expressions having linearity composed of a product of n coefficients and observation data of n items, and variables relating to observation data and derived data of designated items,
The derivation device, wherein the derivation unit derives the derivation data related to the variable using the values of n coefficients obtained from the mathematical formula generated by the mathematical formula generation unit. The derivation device according to claim 2, wherein
A function specification unit that receives the specification of the nonlinear function and the expansion order of the specified nonlinear function;
The mathematical expression generating means generates n nonlinear functions using the observation data and the derived data as variables, respectively, as i mathematical expressions based on the nonlinear function and the expansion order specified by the function specifying unit. A derivation device. The derivation device according to any one of claims 1 to 4, wherein
The derivation device, wherein the derivation data is data for L-1 points after the m-th observation data or data for L-1 points before the first observation data. Based on observation data of m points (m is an integer of 2 or more) obtained by observing each of n-dimensional items (n is an integer of 2 or more) in time series regarding events that change over time, the observation data A derivation method using a derivation device for deriving data at a time not included in the derivation data,
The derivation device includes:
obtaining observation data at m points for each of the n-dimensional items;
Determining the item to derive;
n coefficients and observation data, and m + 1-L point observation data (L = 1, 2,..., i) for the specified item and L-1 point derived data variables. generating i mathematical expressions;
And a step of deriving derived data based on the generated mathematical formula. Based on observation data of m points (m is an integer of 2 or more) obtained by observing the computer in time series for each item of n dimensions (n is an integer of 2 or more) related to events that change over time, A derivation program for deriving data at a time point not included in the observation data as derived data,
On the computer,
obtaining observation data at m points for each of the n-dimensional items;
Determining the item to derive;
n coefficients and observation data, and m + 1-L point observation data (L = 1, 2,..., i) for the specified item and L-1 point derived data variables. generating i mathematical expressions;
And a step of deriving derived data based on the generated mathematical formula.