JPS5935463B2 - Nonlinear estimator - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION When a reference variable indicating the state of a system is expressed numerically, the present invention provides a method for determining which numerical value the state of the system takes from a combination of several explanatory variables that are considered to define the state of the system. Or,
Regarding devices for statistical estimation, see Table 1.

If the state of the system does not change statistically in the future, this can directly serve as a system state prediction device. Traditionally, methods for statistically estimating the state of a system include numerical estimation methods such as multiple regression analysis and filtering (see Figure 1 and Table 1 below), and pattern estimation methods using quantification theory (see Figure 1 and Table 1 below). (see figure and Table 2 below).

The former assumes that the state of the system is expressed by a number of fixed numbers, and performs operations on those numbers themselves.
In this sense, it is called a deterministic method, and in the latter, the state of the system is expressed by the probability that the system has assumed various numbers and patterns, or in other words, the probability of the system's existence for various numbers and patterns. We will call this a probabilistic method in the sense that it assumes that it will occur and then performs operations on its probability of existence. Figure 1 is a conceptual diagram of the conventional deterministic numerical estimation method, in which a regression coefficient is statistically created from the numerical data obtained up to the point of estimation (on the left side of the figure), and the regression coefficient is used. indicates that the numerical value at the desired point in time (on the right side of the figure) is to be estimated. Figure 2 is a conceptual diagram of the conventional probabilistic pattern estimation method, in which the turn data between each pattern (the transition probability on the left side of the figure is calculated statistically) from the parameters obtained up to the time of estimation. Indicates that probability is used to estimate the pattern when the estimation is desired.

Historical method xl () 1a1m+0...a1m+n namm+10゜0amm+n In this case, do something like this.

If we assume that this is m-1 and X1 is the sulfur oxide concentration of 1q of air pollution, then equation (1) is multiplied by This means that the sulfur oxide concentration at that point in time is weighted (regression coefficient) for estimation. [Remarks] Here, in pattern A, the value of X1 is XlA,...
, the value of Xnl is X. Assume that the state of pattern A that takes the value nlA is expressed by the following equation.

For example, let Xk be defined as each point k(k-1,...,m
), Equation (13) shows a concentration distribution pattern in which the concentration at each point is XkA.

Furthermore, the existence probability PA(t) for pattern A is defined as time t
The transition probability PAB from pattern B to pattern A is the probability that the state of pattern A shown by equation (13) occurs.・This is the probability of moving to the state of pattern A.

Furthermore, the transition probability PAB from pattern B to pattern A depends on the existence probability Pu of other patterns U.

For example, when patterns A and B are considered to be patterns related to the sulfur oxide concentration distribution, pattern U corresponds to a pattern related to the wind speed distribution, etc., which is considered to have an influence on changes in the sulfur oxide concentration distribution.

Deterministic methods cannot capture the hidden state of the system as a pattern.

D. Since several numbers representing the state of the system in the next step are calculated by linear or nonlinear combinations of several quantities representing the state of the system, it is necessary to average the effects of various patterns. It takes shape. Therefore, although it can follow gradual changes, it has the disadvantage that it is not well suited for estimating phenomena that change rapidly. For example, time series estimation has the disadvantage that estimated values are subject to time advances and lags. The probabilistic method has the disadvantage that it cannot be estimated using continuous numerical values because it estimates phenomena by classifying them into several patterns.

The purpose of the present invention is to eliminate the above-mentioned drawbacks, to be suitable for estimating rapidly changing phenomena (in the case of time-series estimation, there is little time lag in estimated values), and to provide continuous numerical values. An object of the present invention is to provide a nonlinear estimating device that can perform estimation.

In order to achieve the above object, in the present invention, as shown in FIG. b. Next, the transitions of these patterns are estimated probabilistically, and finally, the estimated pattern is returned to the equation 1 value based on the correspondence relationship.

That is, the concept of the conventional probabilistic pattern estimation method shown in FIG. 2 is used as is.

This eliminates the drawbacks of conventional deterministic numerical estimation methods. As mentioned above, this alone can only estimate a specific, fixed pattern, and cannot estimate continuous numerical values. By the way, if you think about estimation using continuous numerical values, it can be said that if you can estimate not only a specific fixed pattern but also any intermediate pattern, you can use that as a substitute. So, book 2. In the invention, the similarity between any intermediate pattern (corresponding to continuous numerical values) and a specific pattern is simply determined by relating the similarity by looking at the numerical vector (corresponding to the right side of equation (13)) that defines each pattern. A method is used to estimate an arbitrary pattern using the estimation results of a specific pattern.The relationship is made by linear interpolation between elements of numerical vectors.This allows continuous numerical estimation. This makes it possible to eliminate the shortcomings of conventional probabilistic pattern estimation methods.The algorithm consists of the following four steps.

(1) Patterning: A reference variable (described later as 1) and an explanatory variable (described later as 2) are discretized.

(2) Nonlinear transformation: convert the patterned reference variables and explanatory variables into V
). Rearrange in L-dimensional space.

(3) Linear extrapolation and L-dimensional space arrangement: Linearly interpolate new values of explanatory variables within the L-dimensional space, and arrange the corresponding reference variables within the L-dimensional space of the reference variables.

(4) Determination of patterns and linear interpolation: It is determined how close to which pattern the reference variable corresponding to the new value of the explanatory variable is located in the L-dimensional space, and linear interpolation is performed accordingly.

here 1 Criterion variable: A variable that is an indicator of a phenomenon.

This is the subject of explanation using explanatory variables (variables that are the subject of estimation, also known as estimated variables). 2 explanatory variables:
A variable for explaining a phenomenon (a variable for estimating, also called an estimated variable).

3 References: Chikio Hayashi, 2 others, Clear report processing and statistical mathematics:
Sangyo Tosho (1970).

Hereinafter, the present invention will be explained in detail with reference to Examples.

The device according to the invention has a program diagram shown in FIG.

Moreover, a conceptual diagram is shown in FIG. First, Zn(n-1, . . . , N
): The past value of the reference variable (for example, the value of sulfur oxide concentration at time n), where: and the value measured by the explanatory variable measuring device 2, −−αゝ4−1ふ1Ut′已−1 [−−−−] 1,,,
N: Past n-th de of the α-th explanatory variable (for example, wind speed (α-1) and atmospheric temperature (α = 2) at time n
) here, the relationship with (see Figure 5a),
(For example: Value of the t-th category of the reference variable... (51) Where, Nt: Number of past data belonging to the t-th category of the reference variable.・・・・・・・・・(52)ZH( Nt=
1,...,N'): Value of past Nt-th data belonging to the t-th category of the reference variable.
・・・・・・(53): Value of the kth category of the αth explanatory variable ・・・・・・・・・(54) Here, Nk: The kth category of the αth explanatory variable Number of past data belonging to the category ・・・・・・・・・(55)x'
″a(Nr'=1,...,Nr'): Past Nth item belonging to the kth category of the αth explanatory variable
The value of the k-th data is as shown in (56)), thereby categorizing it (relationship (see Figure 5b)), where T: the category of the reference variable. Number of pieces ・・・・・・・・・(
27) K: Number of categories α for the α-th explanatory variable (28) -t Also, the symbol -t in z indicates the average of z for k in category t. , x indicates the mean of X within α category k.

α and the nonlinearly transformed relationship (see Figure 5 C, d), where L: number of dimensions of the space created by nonlinearly transformed (30).

Next, 1x: αth α measured by the explanatory variable measuring device 2
i-th new α value 16 of the th explanatory variable...
...(31) is input to the linear interpolation device 4, and α(l) is linearly interpolated according to the value of 1x (see Figure 5e). Value 18...
...(32) Here, calculate and convert it into ``y(,) -.Σ. 1a Arashi: The value obtained by converting 1a Arashi 20...・After conversion to (33) (Fig. 5f
), distance calculation device 6, "D'T.Hl:"y
The distance 21 (34) from (,?, to the perpendicular line perpendicular to the interpolation line of y(;) is calculated (see Figure 5g), and the , 1t:"D" The category T.t-1 22 (35) that gives the minimum value of "D" is determined (see FIG. 5h).

Finally, the linear interpolation device 4 performs linear interpolation between 1t and 1t-'' according to the position of the foot of the perpendicular line (see Figure 5i), and 1z: estimated value of the reference variable 13...・・・・・・
(36) is calculated.

In addition, It is.

In the nonlinear estimation device described above, a sulfur oxide concentration meter is used as the reference variable measuring device 1, a sulfur oxide concentration is adopted as the reference variable, and an atmospheric pressure meter, an anemometer, an atmospheric Using a thermometer, we use the magnitude of the pressure gradient vector, the angle of the pressure gradient vector, the wind speed, -wind direction, and the temperature difference between upper floors and the ground as explanatory variables, which are processed using atmospheric pressure, wind speed, and atmospheric temperature. This is an estimation and prediction device for air pollutant concentration (sulfur oxide concentration).

In the present invention, a linear extrapolation device is used, but for example, a nonlinear extrapolation device that performs interpolation using a quadratic curve can also be used in place of the linear extrapolation device b.

At the final stage of estimation using the nonlinear estimation device according to the present invention, the position 3 of the reference variable in the L-dimensional space corresponding to the new value 1x (in the case of α=1) of the explanatory variable is determined as shown in FIG. (L-
Assume that you have reached point 1 in case 2).

In this case, a conventional category estimating device that performs discrimination by considering only the distance from each t has determined that the new value 1 belongs to the nearest category t-4. However, in the device according to the present invention, since point 1 is closest to line segment W7l, point 1 is
The value z of the reference variable corresponding to x yields the following result.

If the distances from point 1 to line segment 6-7 and line segment 3-4 are the same, care must be taken.

In this device, the effects of 6-7 and 3-4 are not averaged.

When averaging is performed for t1 or Z, the line segment 6 becomes TZ5.5, iZzZt (t=5.3), respectively.
This is because the phenomenon is completely different from the phenomena above in -7 and 3-4. In this case, do as follows, and attach the probability that the phenomenon will occur.

As explained above, the nonlinear estimation device according to the present invention has
Estimation and prediction of nonlinear phenomena that occur due to complex relationships between a wide variety of variables, such as air pollution concentration, river flow rate, and water and sewage load, as well as phenomena expressed by multivalued functions (geup phenomenon and uncertain phenomena) ) can be applied to estimation and prediction.

It is especially suitable for handling a large number of variables and large amounts of data that are too large for humans to process.

[Brief explanation of the drawing]

FIG. 1 is a conceptual diagram of a conventional deterministic numerical estimation method, showing that numerical values are estimated from numerical values using regression coefficients.

Claims (1)

[Claims] 1. A method for determining the correspondence between the numerical value of arbitrary new data and a specific pattern that has been classified and extracted in advance by linear interpolation, and probabilistically estimating the transition of the specific pattern. A nonlinear estimating device comprising means for converting a pattern back into a numerical value based on the correspondence relationship.