JP2005242580A - Parameter estimation method, data prediction method, parameter estimation device, data prediction device, and computer program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for certainly estimating a parameter of a time series comprising data obtained from a system accompanying input, a method for predicting realistic data, a parameter estimation device, a data prediction device, and a computer program. <P>SOLUTION: This data prediction device (parameter estimation device) sets a cross-correlation function of a present sample time series äx<SB>n</SB>} and a future input time series äu<SB>n</SB>} as 0 to calculate data (S2), generates white noise äe<SB>n</SB>} (S3), calculates the dispersion σ<SB>e</SB><SP>2</SP>of äe<SB>n</SB>} (S4), sets a cross-correlation function of äu<SB>n</SB>} and äe<SB>n</SB>} as 0 and sets a cross-correlation function of present äx<SB>n</SB>} and future äe<SB>n</SB>} as 0 to calculate data (S5), sets an autocorrelation function of äe<SB>n</SB>} as σ<SB>e</SB><SP>2</SP>at lag 0, and as 0 at the other lags, and calculates the parameter of an ARMAX model (S6). The data prediction device decides convergence of the parameter (S7), calculates new white noise when not converging (S8), and repeats the calculation until the parameter converges. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention realizes a method for estimating parameters from a time series of data, a method for predicting new data using estimated parameters, a parameter estimation device, a data prediction device, and a computer as a parameter estimation device or a data prediction device. Relates to a computer program.

  Various methods have been proposed for predicting new data values by analyzing a time series of data obtained from a certain system. One method is to apply a time series of data to a discrete-time linear model, analyze the parameters, estimate the parameters that indicate the relationship between the data, and based on the discrete-time linear model configured using the estimated parameters. There are methods for predicting new data values. As the discrete-time linear model, an AR (Autoregressive: Autoregressive) model or an ARMA (Autoregressive Moving Average) model is used.

Assuming that the time series of data {x _n : n = 1, 2,...} Is a sample time series of a stationary ergodic normal process x (t), the (p, q) -order ARMA model is expressed as It is expressed by a formula.

Here, {e _n } is normal white noise with an average value of 0 and a variance σ _e ² . {A _i : i = 1, 2,..., P} and {b _j : j = 1, 2,..., Q} are parameters of the ARMA model. In addition, {a _i } and {b _j } are such that A (Z ⁻¹ ) and B (Z ⁻¹ ) given by the following equations satisfy the steady-state condition, the reversible condition, and the strongly positive real condition, respectively.
A (Z ⁻¹ ) = 1 + a ₁ Z ⁻¹ +... + A _p Z ^−p
B (Z ⁻¹ ) = 1 + b ₁ Z ⁻¹ +... + B _q Z ^−q

As a method for estimating the parameters {a _i } and {b _j } of the equation (1) from the sample time series {x _n }, a pseudo linear regression method is conventionally used. In the conventional quasi-linear regression method, white noise {e _n } is generated using random numbers and the like, and {a _i } and {b _j } are calculated from the generated {e _n } and the time series {x _n }. and, using the calculated {a _i} and the {b _j} (1) formula to calculate the white noise {e _n}, the calculation is repeated until the parameters {a _i} and {b _j} converges. By this method, the parameters of the ARMA model are estimated. Non-Patent Document 1 proposes a technique that facilitates convergence of values of the parameters {a _i } and {b _j } of the ARMA model.

  In the ARMA model as described above, the value of one data included in the time series of data is determined from the past data value and the past noise. In reality, however, the data obtained from the system is often influenced by the input to the system, such as when the sales data of the product is affected by the temperature. Therefore, there is a problem that the ARMA model cannot cope with the actual problem.

As a discrete-time linear model considering the input to the system, an ARMAX (Autoregressive Moving Average with exogenous input) model is conventionally used. If a sample time series {x _n : n = 1, 2,...} Consisting of a plurality of data obtained from the system is a sample time series of a stationary ergodic normal process x (t), (p, m, q ) The next ARMAX model is expressed by the following equation (2).

Here, {e _n } is a normal white noise with mean value 0 and variance σ _e ² , and {u _n } is an input time series for the system. {A _i : i = 1, 2,..., P}, {c _i : i = 1, 2,..., M} and {b _i : i = 1, 2,. It is. Further, it is assumed that the equation (2) satisfies a steady condition. In modern control theory, it is well known that q ≦ p and m ≦ p. In the following, for simplicity, the discussion proceeds with q = m = p.
The vector Z _n and the vector θ are defined as follows.
Z _n = [-x _n-1 , ...,-x _np , u _n-1 , ..., u _np , e _n-1 , ..., e _np ] ^T
θ = [a ₁ , ..., a _p , c ₁ , ..., c _p , b ₁ , ..., b _p ] ^T
When formula (2) is rewritten to formula for x _n , formula (2) can be expressed by formula (3) below.
x _n = θ ^T Z _n + e _n (3)

When a sample time series {x _n : n = 1, 2,..., N} and an input time series {u _n : n = 1, 2,. In order to obtain from the time series by the method of least squares, θ that minimizes the right side of the following equation (4) obtained from equation (3) may be obtained.

Therefore, the right side of the equation (4) is partially differentiated by θ and equal to 0, so that the vector θ estimated from the sample time series {x _n } and the input time series {u _n } composed of N data is obtained. The estimated value θ _N is given by the following equation (5).

If k ≧ 0 and the autocorrelation function of {x _n } with a lag of k is R _k , R _k and R _−k can be expressed by the following equations according to the definition of the autocorrelation function. Here, E [y _n ] is an expected value of the data string {y _n : n = 1, 2,.
R _k = E [x _n x _{n + k} ], R _−k = R _k

Similarly, if the autocorrelation function of {u _n } is W _k , W _k and W _−k can be expressed by the following equations.
W _k = E [u _n u _{+ k} ], W _−k = W _k

Similarly, if the autocorrelation function of {e _n } is S _k , S _k and S _−k can be expressed by the following equations.
S _k = E [en _{n + k} ], S _−k = S _k

For the cross-correlation function between {x _n } and {e _n }, T _k and T _−k are defined by the following equations.
_{T k = E [e n x} n + k], T -k = E [e n x nk]
At this time, the respective cross-correlation functions of {x _n } and {e _n } are expressed by the following equations.
E [x _n e _nk ] = E [e _n x _{n + k} ] = T _k
E [x _n e _{n + k} ] = E [e _n x _nk ] = T _−k

For the cross-correlation function between {x _n } and {u _n }, V _k and V _−k are defined by the following equations.
_{V k = E [u n x} n + k], V -k = E [u n x nk]
At this time, the respective cross-correlation functions of {x _n } and {u _n } are expressed by the following equations.
E [x _n u _nk ] = E [u _n x _{n + k} ] = V _k
E [x _n u _{n + k} ] = E [u _n x _nk ] = V _−k

Also for the cross-correlation function between {u _n } and {e _n }, Q _k and Q _−k are defined by the following equations.
_{Q k = E [e n u} n + k], Q -k = E [u n e n + k]
At this time, the respective cross-correlation functions of {u _n } and {e _n } are expressed by the following equations.
E [u _n e _nk ] = E [e _n u _{n + k} ] = Q _k
E [u _n e _{+ k} ] = E [e _n u _nk ] = Q _−k

Vectors a, b, and c are defined by the following equations.
a = [a ₁ , a ₂ ,..., a _p ] ^T
b = [b ₁ , b ₂ ,..., b _p ] ^T
c = [c ₁ , c ₂ ,..., c _p ] ^T

If N → ∞ in equation (5), the left side of equation (5) is θ _N → θ, and the right side of equation (5) is the autocorrelation function of {x _n }, {u _n }, {e _n } and It can be described by a cross-correlation function. Therefore, the formula (5) can be expressed by the following formula (6).

  Here, R, V ′, T ′, W, S ′, and Q ′ are p-by-p matrix, and (7), (8), (9), (10), and (11) below. , (12). Further, r, v, and t are p rows and columns, and are expressed by the following equation (13).

When the equation (6) is rewritten, the following simultaneous equations are obtained.
Ra−V′c−T′b = −r (14)
−V ′ ^T a + Wc + Q′b = v (15)
−T ′ ^T a + Q ′ ^T c + S′b = t (16)
The vectors a, b, and c can be calculated by solving the simultaneous equations (14) to (16).

In order to obtain the vectors a, b, and c that are estimated values of the parameters of the ARMAX model using the pseudo linear regression method, white noise {e _n } is generated using random numbers and the generated {e _n } and From {x _n } and {u _n }, vectors a, b, and c are calculated according to equations (14) to (16), and a new white color is calculated from the calculated vectors a, b, and c using equation (2). Noise {e _n } is calculated, and the calculation is repeated until the vectors a, b, and c converge.
Kazuhiro Takeyasu, 3 others, “Robust Bootstrap System Identification Algorithm”, Journal of Signal Processing, Signal Processing Society of Japan, March 2003, Vol. 7, No. 2, p. 167-176

  The pseudo linear regression method as described above has a problem that, depending on the nature of the problem, the vectors a, b, and c, which are parameters to be obtained, do not converge and the parameters cannot be estimated. Further, in the above-described method, it is necessary to perform matrix calculation using p-by-p matrix R, V ′, T ′, W, S ′, Q ′ in order to calculate vectors a, b, c. There was a problem that the calculation time was long.

  The present invention has been made in view of such circumstances, and an object of the present invention is to facilitate convergence of parameter values of the ARMAX model taking external input into account while simultaneously reducing calculation time. A parameter estimation method, a data prediction method, a parameter estimation device, a data prediction device, and a computer program are provided.

The parameter estimation method according to the first aspect of the present invention uses a computer including a storage unit and a calculation unit, and a sample time series {x _n : n = 1, 2,...} And an input time series {u _n consisting of a plurality of data. In the method of estimating parameters when n = 1, 2,...} Is applied to a (p, p, p) order (where p is a natural number) ARMAX model, the sample time series and the input time series are Storing in the storage unit, storing the matrix R, W, V, r, v defined by the following formula in the storage unit;

White noise {e _n : n = 1, 2,...} Composed of a plurality of random data having an average value of 0 is generated by the arithmetic unit, and the variance σ _e ² of the white noise generated by the arithmetic unit is The calculation unit calculates the matrices T and t defined by the following formulas in the calculation unit,

Vector a = [a ₁ , a ₂ ,..., A _p ] ^T with _p parameters related to the sample time series in the ARMAX model as matrix elements, and p parameters related to the input time series as matrix elements C = [c ₁ , c ₂ ,..., C _p ] ^T and a matrix b = [b ₁ , b ₂ ,..., B _p ] ^T having _p parameters related to the white noise as matrix elements, Calculated by the calculation unit based on the following formula,

  Substituting the parameters of the ARMAX model included in the vectors a, b and c calculated by the arithmetic unit, the sample time series and the input time series value into the formula of the ARMA model, and calculating a new white noise The new vectors a, b and c are calculated by the calculation unit using the new white noise calculated by the calculation unit, and the vectors a, b and c are converged until the vectors a, b and c converge. The calculation is repeated in the calculation unit.

A data prediction method according to a second aspect of the present invention is a sample time series {x _n : n = 1, 2,...} Composed of a plurality of data obtained from a system with input using a computer having a storage unit and a calculation unit. , And a method for predicting new data to be added to the sample time series from an input time series {u _n : n = 1, 2,... And storing the input time series in the storage unit, and using the parameter estimation method according to the first aspect of the invention, the calculation unit calculates an estimated value of the parameter when the sample time series and the input time series are applied to the ARMX model. Substituting the estimated parameter values calculated by the calculation unit, the sample time series and the input time series into the formula of the ARMA model, and predicting new data to be added to the sample time series. And calculating the value by the arithmetic unit.

The parameter estimation device according to the third aspect of the present invention provides a sample time series {x _n : n = 1, 2,...} And an input time series {u _n : n = 1, 2,. p, p) In an apparatus for estimating parameters when applied to an ARMAX model of the order (where p is a natural number), means for storing the sample time series and the input time series, and a matrix R defined by the following equation , W, V, r, v for storing,

Means for generating white noise {e _n : n = 1, 2,...} Composed of a plurality of random data having an average value of 0; and means for calculating a variance σ _e ² of the white noise generated by the means; Means for calculating matrices T and t defined by:

The vector a = [a ₁ , a ₂ ,..., A _p ] ^T having matrix elements as estimated values of p parameters related to the sample time series in the ARMAX model, and p parameters related to the input time series A matrix c = [c ₁ , c ₂ ,..., C _p ] ^{T having} an estimated value as a matrix element and a matrix b = [b ₁ , b ₂ having an estimated value of p parameters related to the white noise as matrix elements. , ..., b _p ] ^T , based on the following formula:

  Substituting the estimated values of the parameters of the ARMAX model included in the vectors a, b and c calculated by the means, the sample time series and the input time series values into the formula of the ARMA model, new white noise is obtained. Means for calculating, means for calculating new vectors a, b and c using the new white noise calculated by the means, and calculating vectors a, b and c until the vectors a, b and c converge. And means for repeating.

A data prediction apparatus according to a fourth aspect of the present invention comprises a sample time series {x _n : n = 1, 2,...} Composed of a plurality of data obtained from a system with input, and a plurality of data input to the system. Means for predicting new data to be added to the sample time series from the input time series {u _n : n = 1, 2,...}, A means for storing the sample time series and the input time series; A parameter estimation apparatus according to the invention; means for calculating an estimated value of the parameter when the sample time series and the input time series are applied to an ARMAX model using the parameter estimation apparatus; and the parameter calculated by the means And means for substituting the sample time series and the input time series into the formula of the ARMA model to calculate a predicted value of new data to be added to the sample time series. And butterflies.

A computer program according to the fifth aspect of the present invention includes a sample time series {x _n : n = 1, 2,...} And an input time series {u _n : n = 1, 2,. In a computer storing the defined matrices R, W, V, r, v,

A computer program for estimating parameters when the sample time series and the input time series are applied to an (MAX) model of (p, p, p) order (where p is a natural number), and the computer has an average value of 0 A procedure for generating white noise {e _n : n = 1, 2,...} Consisting of a plurality of random data, a procedure for causing a computer to calculate a variance σ _e ² of the generated white noise, A procedure for calculating matrices T and t defined by the following formulas:

The computer has a vector a = [a ₁ , a ₂ ,..., A _p ] ^{T having} _p parameter estimated values related to the sample time series in the ARMAX model, p pieces related to the input time series. Matrix c = [c ₁ , c ₂ ,..., C _p ] ^{T having} the estimated values of the parameters in the matrix element and matrix b = [b ₁ having the estimated values of the p parameters related to the white noise as matrix elements. , B ₂ ,..., B _p ] ^T based on the following equation:

  Substituting the estimated values of the parameters of the ARMAX model included in the calculated vectors a, b, and c, the sample time series and the input time series values into the formula of the ARMA model, and calculating a new white noise. A procedure for calculating, a procedure for causing the computer to calculate new vectors a, b and c using the calculated new white noise, and a vector for the vectors a, b and c until the vectors a, b and c converge. And a procedure for repeating the calculation.

A computer program according to a sixth aspect of the present invention is a sample time series {x _n : n = 1, 2,...} Composed of a plurality of data obtained from a system with input, and an input composed of a plurality of data input to the system A computer program for storing a time series {u _n : n = 1, 2,...} For predicting new data to be added to the sample time series. , A procedure for calculating an estimated value of a parameter when the sample time series and the input time series are applied to an ARMAX model, and a computer that calculates the estimated value of the parameter, the sample time series, and the input Substituting the time series into the formula of the ARMA model and calculating a predicted value of new data to be added to the sample time series. And butterflies.

The present inventor tried to improve the conventional quasi-linear regression method by introducing a priori knowledge derived from the construction of the formula of the ARMAX model. First, consider the autocorrelation function of normal noise {e _n }. Since the elements included in the normal white noise {e _n } are not related to each other, the autocorrelation function S _k of {e _n } can be theoretically expressed by the following equation.

  Therefore, a matrix S obtained by improving the matrix S ′ by introducing a theoretical value into the matrix S ′ defined by the equation (11) is represented by the following equation, where I is a unit matrix.

Next, consider the cross-correlation function between {x _n } and {e _n }. In the ARMAX model expressed by equation (2), the current value of x _n is related to the past value of e _n . However, the current value of x _n is because there is no relevance to the value of future e _n, the cross-correlation function between e _n the future than the data and x _n can be expressed by the following equation.
E [x _n e _{n + k} ] = 0
Similarly, the following equation holds.
_{E [e n x nk] =} 0
Therefore, T _−k = 0. Further, from the equation (2), since the current value of x _n and the current value of e _n are related, T ₀ ≠ 0. Accordingly, the matrix T obtained by improving the matrix T ′ by introducing theoretical values into the matrix T ′ defined by the equation (9) is expressed by the following equation.

Similarly, for the cross-correlation function between {x _n } and {u _n }, the current x _n value is related to the past u _n value, but the current x _n value is the future u _n value. Since it is not related to the value of _n, the following equation holds.
E [x _n u _{n + k} ] = E [u _n x _nk ] = V _−k = 0
Further, since the current value of x _n is not related to the current value of u _n from the equation (2), V ₀ = 0. Accordingly, the matrix V obtained by improving the matrix V ′ by introducing theoretical values into the matrix V ′ defined by the equation (8) is expressed by the following equation.

Furthermore, since the input time series {u _n } and the normal white noise {e _n } should be uncorrelated, the cross-correlation functions between {u _n } and {e _n } are all as shown in the following equation. 0.
E [e _n u _{n + k} ] = E [u _n e _nk ] = Q _k = 0
E [u _{n nk} ] = E [u _n e _{+ k} ] = Q _−k = 0
Accordingly, the matrix Q obtained by improving the matrix Q ′ by introducing a theoretical value into the matrix Q ′ defined by the equation (12) is represented by the following equation, where the zero matrix is O.
Q = O (21)

Each autocorrelation is obtained when a sample time series {x _n : n = 1, 2,..., N} and an input time series {u _n : n = 1, 2,. When the function and each cross-correlation function are calculated, since the value of N is finite, the values of the cross-correlation function and the autocorrelation function are different from the theoretical values. Of the actually calculated values, the part where the theoretical value is obtained is replaced with the theoretical value, so that the expression (6) for obtaining the parameters of the ARMAX model can be expressed by the following expression. Become.

When the equation (22) is rewritten, the following simultaneous equations are obtained.
Ra-Vc-Tb = -r (23)
−V ^T a + Wc = v (24)
-T ^T a + σ _e ² b = t (25)
From the equations (25) and (24), the following equations (26) and (27) are obtained.
b = (t + T ^T a) / σ _e ² (26)
c = W ⁻¹ (v + V ^T a) (27)
By substituting the equations (26) and (27) into the equation (23), the following equation (28) is obtained.

The vector a can be calculated from the equation (28). By substituting the calculated vector a into the equations (26) and (27), the vector b and the vector c can be calculated. In order to obtain the vectors a, b, and c that are estimated values of the parameters of the ARMAX model, white noise {e _n } is generated using random numbers and the like, and the generated {e _n }, {x _n }, and {u since the _n}, the vector a, b, and c (26) were calculated to (28), computed vectors a, b, calculates the new white noise using the c (2) wherein {e _n} The calculation is repeated until the vectors a, b, and c converge.

In the first, third and fifth inventions, a sample time series {x _n } composed of a plurality of data obtained from a system with input and an input time series {u _n } composed of a plurality of data input to the system Are applied to the ARMAX model, the estimated parameter values are input time series autocorrelation function, white noise autocorrelation function, sample time series and input time series cross correlation function, sample time series and white noise mutual Calculation is performed according to equations (26) to (28) using theoretical values for the correlation function and the cross-correlation function between the input time series and white noise.

In the second, fourth and sixth inventions, the estimated value of the new data to be added to the sample time series {x _n } is calculated using the estimated value of the parameter of the ARMAX model calculated by the equations (26) to (28). calculate.

In the first, third, and fifth inventions, the sample time series {x _n } composed of a plurality of data obtained from a system with input and the input time series {u} composed of a plurality of data input to the system _n }, when calculating the estimated parameter values when applied to the ARMAX model, autocorrelation function of white noise, cross-correlation function between sample time series and input time series, cross-correlation function between sample time series and white noise , And by calculating the cross-correlation function between the input time series and white noise using theoretical values, it is difficult to converge the parameter estimates using the conventional quasi-linear regression method. The convergence of the estimated value becomes easy. In addition, since the probability that the parameter converges is improved, the calculation time for parameter estimation can be shortened.

  In the second, fourth, and sixth inventions, using the estimated value of the parameter of the ARMAX model, the value of the data obtained from the system is determined from the past data value, noise, and further the input to the system. Data obtained from systems that are affected by external inputs, such as the value of a stock price that is affected by the trading volume of a stock or the sales volume of a product that is affected by the arrival volume of a product, by calculating a predicted value of new data Can be estimated. Therefore, the present invention has an excellent effect, such as predicting the stock price or the sales volume of the product more realistically.

Hereinafter, the present invention will be specifically described with reference to the drawings showing embodiments thereof.
FIG. 1 is a block diagram showing the configuration of the data prediction apparatus of the present invention. The data prediction device 1 also has a function as a parameter estimation device of the present invention, and is configured using a general-purpose computer. The data prediction apparatus 1 includes a CPU (arithmetic unit) 11 that performs computation, a RAM (storage unit) 12 that stores temporary information generated along with the computation, an external storage device 13 such as a CD-ROM drive, And an internal storage device 14 such as a hard disk. The CPU 11 reads the computer program 20 of the present invention from the recording medium 2 such as a CD-ROM by the external storage device 13 and stores the read computer program 20 in the internal storage device 14. The computer program 20 is loaded from the internal storage device 14 to the RAM 12 as necessary, and the CPU 11 executes processing necessary for the data prediction device 1 based on the loaded computer program 20. The data prediction device 1 includes an input device 15 such as a keyboard or a mouse and an output device 16 such as a liquid crystal display or a CRT display, and is configured to accept operations from an operator such as data input. .

The internal storage device 14 includes a sample time series {x _n : n = 1, 2,...} Composed of a plurality of data such as stock prices or product sales volume, and a sample time series such as stock transaction volume or product arrival volume. Are stored as input time series {u _n : n = 1, 2,...

  The computer program 20 may be loaded from the external server device (not shown) connected to the communication network N to the state monitoring device 1 according to the present invention and stored in the internal storage device 14.

FIG. 2 is a flowchart showing parameter estimation processing and data prediction processing performed by the data prediction apparatus 1 of the present invention. The CPU 11 of the data prediction apparatus 1 performs the following processing according to the computer program 20 loaded into the RAM 12. The CPU 11 of the data prediction apparatus 1 reads the sample time series {x _n } and the input time series {u _n } stored in the internal storage device 14 into the RAM 12 (S1). CPU11 then, of {x _n} and autocorrelation function using the value of the data contained in the _{{u n} {x n}} R k = E [x n x n + k], {u n} the autocorrelation function _{W k = E [u n u} n + k], to calculate the cross-correlation between {x _n} and {u _n} function _{V k = E [u n x} n + k], (7) Matrixes R, W, V, r, and v defined by equations (10), (20), and (13) are calculated (S2). CPU11 then, by a method such as generating a random number the value of time as an initial value using routine random number generation, the average value of 0 in a plurality of white noise consisting of random data {e _{n: n} = 1, 2,...} Are generated (S3), and the variance σ _e ² of the generated white noise {e _n } is calculated (S4).

CPU11 then, {x _n} and {e _n} with the value of the data contained in {x _n} and {e _n} and the cross-correlation function T _k = E of [e _n x _{n +} k] And the matrices T and t defined by the equations (19) and (13) are calculated (S5). Next, the CPU 11 calculates the vector a by the equation (28), and calculates the vector b and the vector c by substituting the vector a into the equations (26) and (27), thereby obtaining {x _n } and {u _n} the consisting estimates of the parameters of ARMAX model when fitted to ARMAX model vector a, b, calculates the c (S6).

Next, the CPU 11 determines whether or not the calculated vectors a, b, and c have converged (S7). At this time, for example, the CPU 11 has vectors a ′ = [a ′ ₁ , a ′ ₂ ,..., A ′ _p ] ^T , b which are the latest vectors a, b, c and the previously calculated vectors a, b, c. _{'= [b' 1, b} '2, ..., b' p] T, c '= [c' 1, c '2, ..., c' p] is defined by the following equation (29) using a ^T D is calculated, and it is determined that the vectors a, b, and c have converged when a state where D <ε is continuously established a predetermined number of times for a predetermined minute amount ε. In addition, you may use the process which determines the convergence of vector a, b, c using another method.

If it is determined in step S7 that the vectors a, b, c have not converged (S7: NO), the vectors a, b, c and {x _n } and {u _n } are _expressed as m = q = Substituting p into equation (2), new white noise {e _n : n = 1, 2,...} is calculated (S8), and the process returns to step S4. The CPU 11 repeats the processes in steps S4 to S8 until it is determined in step S7 that the vectors a, b, and c have converged.

When it is determined in step S7 that the vectors a, b, and c have converged (S7: YES), the CPU 11 calculates the calculated vectors a, b, c, {x _n }, {u _n }, By substituting {e _n } into the equation (2), the predicted value of data to be added to the sample time series {x _n }, that is, the predicted data such as the stock price or the sales amount of the product is calculated (S9). In step S9, new data added to the input time series or a predicted value of the data may be used as necessary. Through the above processing, the parameters of the ARMAX model are estimated, and new data to be added to the sample time series can be predicted.

As described above in detail, in the present invention, the sample time series {x _n } composed of a plurality of data obtained from a system with input and the input time series {u _n } composed of a plurality of data input to the system are used. When calculating parameter estimates when applied to the ARMAX model, white noise autocorrelation function, cross-correlation function between sample time series and input time series, cross-correlation function between sample time series and white noise, and input A theoretical value is used for the cross-correlation function between the time series and white noise. Accordingly, even when it is difficult to converge the parameter estimation value by the conventional pseudo linear regression method, the parameter estimation value can be easily converged. In addition, since the probability that the parameter converges is improved, the calculation time for parameter estimation can be shortened.

  In the present invention, the value of the data obtained from the system is determined from the past data value, noise, and the input to the system, and the ARMAX model is used as a discrete-time linear model. It is possible to estimate a value of data obtained from a system affected by an external input, such as an affected stock price value or a sales amount of a product affected by the arrival amount of the product. Accordingly, it is possible to make a prediction of the stock price or the sales volume of the product more realistically.

In the present embodiment, the data prediction apparatus of the present invention shows a mode in which the sample time series {x _n } and the input time series {u _n } are stored therein, but the present invention is not limited to this. Instead, it may be configured to include means for receiving data included in the sample time series {x _n } and the input time series {u _n } from the outside, and to perform a process of calculating prediction data for the received data.

  Further, in the present embodiment, the data prediction apparatus 1 of the present invention has a form that also has the function of the parameter estimation apparatus of the present invention. However, the present invention is not limited to this, and the predicted value of the parameter of the ARMAX model The parameter estimation device for calculating the data and the data prediction device for predicting the data by the ARMAX model may be configured by separate computers, and the parameter estimation and the data prediction may be performed by inputting / outputting information to / from each other.

It is a block diagram which shows the structure of the data prediction apparatus of this invention. It is a flowchart which shows the parameter estimation process and data prediction process which the data prediction apparatus of this invention performs.

Explanation of symbols

1 Data prediction device (parameter estimation device)
11 CPU (calculation unit)
12 RAM (storage unit)
2 Recording medium 20 Computer program

Claims (6)

A sample time series {x _n : n = 1, 2,...} And an input time series {u _n : n = 1, 2,. In a method of estimating parameters when applied to an (MAX) model of (p, p, p) (where p is a natural number),
Storing the sample time series and the input time series in the storage unit;
A matrix R, W, V, r, v defined by the following equation is stored in the storage unit,

Generating white noise {e _n : n = 1, 2,...} Composed of a plurality of random data having an average value of 0 in the arithmetic unit;
The white noise variance σ _e ² generated by the calculation unit is calculated by the calculation unit,
The calculation unit calculates the matrices T and t defined by the following equations,

Vector a = [a ₁ , a ₂ ,..., A _p ] ^T with _p parameters related to the sample time series in the ARMAX model as matrix elements, and p parameters related to the input time series as matrix elements C = [c ₁ , c ₂ ,..., C _p ] ^T and a matrix b = [b ₁ , b ₂ ,..., B _p ] ^T having _p parameters related to the white noise as matrix elements, Calculated by the calculation unit based on the following formula,

Substituting the parameters of the ARMAX model included in the vectors a, b and c calculated by the arithmetic unit, the sample time series and the input time series value into the formula of the ARMA model, and calculating a new white noise Calculated in part
New vectors a, b and c are calculated using the new white noise calculated by the calculation unit.
A parameter estimation method characterized by repeating the calculation of the vectors a, b, and c in the computing unit until the vectors a, b, and c converge. Using a computer including a storage unit and a calculation unit, a sample time series {x _n : n = 1, 2,...} Composed of a plurality of data obtained from a system with input, and a plurality of data input to the system In a method for predicting new data to be added to the sample time series from an input time series {u _n : n = 1, 2,.
Storing the sample time series and the input time series in the storage unit;
Using the parameter estimation method according to claim 1, an estimated value of a parameter when the sample time series and the input time series are applied to an ARMX model is calculated by the arithmetic unit,
Substituting the estimated value of the parameter calculated by the calculation unit, the sample time series and the input time series into the formula of the ARMA model, and calculating a predicted value of new data to be added to the sample time series A data prediction method characterized by calculating with A sample time series {x _n : n = 1, 2,...} And an input time series {u _n : n = 1, 2,...} Composed of a plurality of data are (p, p, p) order (where p is In an apparatus for estimating parameters when applied to an ARMAX model of (natural number)
Means for storing the sample time series and the input time series;
Means for storing matrices R, W, V, r, v defined by the following equations:

Means for generating white noise {e _n : n = 1, 2,...} Composed of a plurality of random data having an average value of 0;
Means for calculating the variance σ _e ² of the white noise generated by the means;
Means for calculating the matrices T and t defined by:

The vector a = [a ₁ , a ₂ ,..., A _p ] ^T having matrix elements as estimated values of p parameters related to the sample time series in the ARMAX model, and p parameters related to the input time series A matrix c = [c ₁ , c ₂ ,..., C _p ] ^{T having} an estimated value as a matrix element and a matrix b = [b ₁ , b ₂ having an estimated value of p parameters related to the white noise as matrix elements. , ..., b _p ] ^T , based on the following formula:

Substituting the estimated values of the parameters of the ARMAX model included in the vectors a, b and c calculated by the means, the sample time series and the input time series values into the formula of the ARMA model, new white noise is obtained. Means for calculating;
Means for calculating new vectors a, b and c using the new white noise calculated by the means;
Means for repeating the calculation of the vectors a, b and c until the vectors a, b and c converge. A sample time series {x _n : n = 1, 2,...} Composed of a plurality of data obtained from a system with input, and an input time series {u _n : n = 1 composed of a plurality of data input to the system , 2,... From a device for predicting new data to be added to the sample time series,
Means for storing the sample time series and the input time series;
The parameter estimation device according to claim 3,
Means for calculating an estimated value of a parameter when the sample time series and the input time series are applied to an ARMX model using the parameter estimation device;
Means for substituting the estimated value of the parameter calculated by the means, the sample time series and the input time series into an equation of the ARMA model, and calculating a predicted value of new data to be added to the sample time series; A data prediction apparatus comprising: Sample time series {x _n : n = 1, 2,...} And input time series {u _n : n = 1, 2,...} And a matrix R, W, V defined by the following equations , R, v in a computer storing

A computer program for estimating parameters when the sample time series and the input time series are applied to an (MAX) model of (p, p, p) order (where p is a natural number),
A procedure for causing a computer to generate white noise {e _n : n = 1, 2,...} Composed of a plurality of random data having an average value of 0;
A procedure for causing the computer to calculate the variance σ _e ² of the generated white noise,
A procedure for causing a computer to calculate matrices T and t defined by the following formulas:

The computer has a vector a = [a ₁ , a ₂ ,..., A _p ] ^{T having} _p parameter estimated values related to the sample time series in the ARMAX model, p pieces related to the input time series. Matrix c = [c ₁ , c ₂ ,..., C _p ] ^{T having} the estimated values of the parameters in the matrix element and matrix b = [b ₁ having the estimated values of the p parameters related to the white noise as matrix elements. , B ₂ ,..., B _p ] ^T based on the following equation:

Substituting the estimated values of the parameters of the ARMAX model included in the calculated vectors a, b, and c, the sample time series and the input time series values into the formula of the ARMA model, and calculating a new white noise. The procedure to calculate,
A procedure for causing a computer to calculate new vectors a, b and c using the calculated new white noise;
A computer program comprising: causing a computer to repeat the calculation of vectors a, b, and c until the vectors a, b, and c converge. A sample time series {x _n : n = 1, 2,...} Composed of a plurality of data obtained from a system with input, and an input time series {u _n : n = 1 composed of a plurality of data input to the system , 2, ...}, a computer program for predicting new data to be added to the sample time series,
A procedure for causing a computer to calculate an estimated value of a parameter when the sample time series and the input time series are applied to an ARMAX model using the parameter estimation method according to claim 1;
A step of causing a computer to calculate a predicted value of new data to be added to the sample time series by substituting the calculated estimated value of the parameter, the sample time series and the input time series into the formula of the ARMA model; A computer program comprising:

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