Method for construction of long-term prediction intervals and its structural learning for gaseous system in steel industry

Abstract

The present invention belongs to the field of information technology, involving the techniques of fuzzy modeling, reinforcement learning, parallel computing, etc. It is a method combining granular computing and reinforcement learning for construction of long-term prediction interval and determination of its structure. Adopting real industrial data, the present invention constructs multi-layer structure for assigning information granularity in unequal length and establishes corresponding optimization model at first. Then considering the importance of the structure on prediction accuracy, Monte-Carlo method is deployed to learn the structural parameters. Based on the optimal multi-layer granular computing structure along with implementing parallel computing strategy, the long-term prediction intervals of gaseous generation and consumption are finally obtained. The proposed method exhibits superiority on accuracy and computing efficiency which satisfies the demand of real-world application. It can be also generalized to apply on other energy systems in steel industry.

Claims

1. A method for construction of long-term prediction intervals and its structural learning approach for generation and consumption of an industrial energy system, the method comprising steps of: step 1: data pre-processing collecting data of generation and consumption units of the industrial energy system from real-time relational database, and implement essential noise elimination, filtering and imputation; step 2: Fuzzy C-Means (FCM) dividing the data into segments with equal length, i.e., Z={z.sub.1, z.sub.2, . . . , Z.sub.N}, where Z.sub.i ∈.sup.n, n denotes the number of data points in each segment, and N is the number of segments; implementing FCM clustering algorithm so as to obtain the prototype matrix V={v.sub.1, v.sub.2, . . . , v.sub.c} and the corresponding fuzzy membership grades U={u.sub.1, u.sub.2, . . . , u.sub.N}, where V.sub.i ∈.sup.n, u.sub.i ∈.sup.c, C denotes the dimension of the prototype matrix; step 3: establishment of the multi-layer granular computing model assigning information granularity α.sub.i,j and β.sub.i from bottom to top on the prototype matrix V={v.sub.1, v.sub.2, . . . , v.sub.c}, where i=1, 2, . . . , m; j=1, 2, . . . , n.sub.i and n.sub.1≠n.sub.2≠ . . . ≠n.sub.m; as such, the numeric prototypes are successfully extended into the intervals; in order to optimize the above parameters, this method defines coverage cov and specificity spec, as follows: $\begin{array}{cc}\mathrm{cov}\stackrel{.}{=}\frac{1}{T}\underset{i=1}{\overset{T}{.Math.}}{\lambda }_i& \left(1\right)\end{array}$ $\begin{array}{cc}\mathrm{spec}\stackrel{.}{=}\mathrm{range}-\frac{1}{T}\underset{i=1}{\overset{T}{.Math.}}.Math.{\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i.Math.& \left(2\right)\end{array}$ where T denotes the number of data points in a sample; λ.sub.i is a marker variable, which will be tagged as 1 if the constructed prediction interval covers the data point, otherwise it will be tagged as 0; range denotes the difference between the maximum and minimum value of the data points; z.sub.i and z.sub.i respectively represent the upper and lower bounds of the constructed prediction intervals; maximizing the cov and spec; wherein, the cov should be greater than or equal to the prescribed confidence level (1−ρ)×100%, where ρ∈[0,1] denotes the level of significance; considering Eq. (1) as the constraints, meaning that the cov should be greater than or equal to the objective confidenece interval; the direction for optimizing the information granularities is opposite with the one for assignment; the optimization models are established as follows: (1) 2.sup.nd layer $\begin{array}{cc}\max {\mathrm{range}}^{\left(2\right)}-\frac{1}{m}\underset{i=1}{\overset{m}{.Math.}}.Math.{\stackrel{\_}{z}}_i^{\left(2\right)}-{\underset{\_}{z}}_i^{\left(2\right)}.Math.s.t.{\sigma }_{\min }^{\left(2\right)}\epsilon \le \frac{1}{m}\underset{i=1}{\overset{m}{.Math.}}{\beta }_i\le {\sigma }_{\max }^{\left(2\right)}\epsilon \frac{1}{m}\underset{i=1}{\overset{m}{.Math.}}{\lambda }_i^{\left(2\right)}\ge \left(1-\rho \right)\times 100\%& \left(3\right)\end{array}$ where range.sup.(2) denotes the difference between the maximum and minimum value of data points in 2.sup.nd layer; z.sub.i.sup.(2) and z.sub.i.sup.(2) respectively represent the upper and lower bounds of the intervals; ε is a hyper-paramter to control the overall information granulartiy; σ.sub.min.sup.(2) and σ.sub.max.sup.(2) are set to control β.sub.i for not excessively far from ε; λ.sub.i.sup.(2) is a marker variable which is similar with λ.sub.i, which is tagged as 1 if the constructed prediction interval covers the data point in 2.sup.nd layer, otherwise tagged as 0; (2) 1.sup.st layer being different with 2.sup.nd layer, the 1.sup.st layer is to solve a number of m optimization problems, each of which can be described as follows: $\begin{array}{cc}\max {\mathrm{range}}_i^{\left(1\right)}-\frac{1}{n_i}\underset{j=1}{\overset{n_i}{.Math.}}.Math.{\stackrel{\_}{z}}_{i,j}^{\left(1\right)}-{\underset{\_}{z}}_{i,j}^{\left(1\right)}.Math.s.t.{\sigma }_{\min }^{\left(1\right)}{\beta }_i\le \frac{1}{n_i}\underset{j=1}{\overset{n_i}{.Math.}}{\alpha }_{i,j}\le {\sigma }_{\max }^{\left(1\right)}{\beta }_i\frac{1}{n_i}\underset{j=1}{\overset{n_i}{.Math.}}{\lambda }_{i,j}^{\left(1\right)}\ge \left(1-\rho \right)\times 100\%& \left(4\right)\end{array}$ where range.sub.i.sup.(1) denotes the difference between the maximum and minimum value of data points in each optimization problem of 1.sup.st layer, i=1, 2, . . . , m; z.sub.i,j.sup.(1) and z.sub.i,j.sup.(1) respectively represent the upper and lower bounds of the constructed intervals in 1.sup.st layer; σ.sub.min.sup.(1) and σ.sub.max.sup.(1) are set to control α.sub.i,j for not being excessively far from β.sub.i; step 4: construction of long-term prediction intervals the granular computing based long-term prediction intervals construction method is to predict the fuzzy membership grades, that is, to form relationship û.sub.k=f(u.sub.k−n.sub.I, . . . , u.sub.k−1), where û.sub.k denotes the predicted fuzzy membership grades, u.sub.k−n.sub.I, . . . , u.sub.k−1 are the elements of the matrix of the fuzzy membership grades, n.sub.I refers to the number of inputs in the relationship among the fuzzy membership grades; for clarifying the process, the following definitions will be given in terms of numeric values; first, defining the probability of a prototype p(v.sub.i1, . . . , v.sub.in.sub.I), the probability of a segment p(z.sub.k−n.sub.I, . . . , z.sub.k−1) and the symbiosis matrix as follows: $\begin{array}{cc}p\left(v_{i1},.Math.,v_{{\mathrm{in}}_I}\right)\stackrel{.}{=}\frac{\underset{j=1}{\overset{N}{.Math.}}{𝕆}_j}{N}& \left(5\right)\end{array}$ $\begin{array}{cc}p\left(z_{k-n_I},.Math.,z_{k-1}\right)\stackrel{.}{=}\left\{p\left(v_{i1},.Math.,v_{{\mathrm{in}}_I}\right).Math.\left\{i1,.Math.,{\mathrm{in}}_I\right\}\in \left[1,c\right]\right\}& \left(6\right)\end{array}$ $\begin{array}{cc}\mathbb{P}\stackrel{.}{=}\left\{p\left(v_i.Math.v_{j1},.Math.,v_{{\mathrm{jn}}_I}\right).Math.i=1,.Math.,c;\left\{j1,.Math.,{\mathrm{jn}}_I\right\}\in \left[1,c\right]\right\}& \left(7\right)\end{array}$ where {v.sub.i1, . . . , v.sub.in.sub.I}, {i1, i2, . . . , in.sub.I}∈[1, c] are the elements of prototype matrix V; z.sub.k−n.sub.I, . . . , z.sub.k−1 are the elements of the data segments Z; .sub.j is a marker variable, which considers {u.sub.h1,j=max(u.sub.h1), . . . , u.sub.hn.sub.I.sub.,j=max(u.sub.hn.sub.I)}, where {u.sub.h1,j, u.sub.hd,j, . . . , u.sub.hn.sub.I.sub.,j} are the membership grades from u.sub.h1, u.sub.h2, . . . , u.sub.hn.sub.I, {h1, h2, . . . , hn.sub.I}∈[1, c]; .sub.j will be tagged as 1 if h1=i1, . . . , hn.sub.I=in.sub.I, otherwise 0; p(v.sub.i|v.sub.j) is the conditional probability, denoting the probability of the fact that the maximum membership grade of Z.sub.k is towards v.sub.i, given that the maximum membership grades of {z.sub.k−n.sub.I, . . . , z.sub.k−1} are towards {v.sub.j1, . . . , v.sub.jn.sub.I}; based on the above definitions, the probability of segment z.sub.k can be computed as {circumflex over (p)}(z.sub.k)=p(z.sub.k−n.sub.I, . . . , z.sub.k−2, z.sub.k−1).sup.T×, and the predicted {circumflex over (z)}.sub.k can be obtained by: $\begin{array}{cc}{\hat{z}}_k=\frac{\underset{i=1}{\overset{c}{.Math.}}{\hat{p}}_{i}V}{\underset{i=1}{\overset{c}{.Math.}}{\hat{p}}_i}& \left(8\right)\end{array}$ where {circumflex over (p)}.sub.i is the element of {circumflex over (p)}(z.sub.k); step 5: reinforcement learning for the structural parameters regarding the determination on the structure of the multi-layer granular computing model as a one-step Markov Decision Process, and then implementing Monte-Carlo method, including parameters m and n.sub.i, i=1, 2, . . . , m; defining state S, action A and reward R as follows: S—a determined structure for multi-layer granular computing model; A—to change the parameters m and n.sub.i, i=1, 2, . . . , m; R—the spec of the constructed prediction intervals; considering the large quantities of to-be-determined parameters, the gradient descend strategy is employed to learn the value function π.sub.ω(s, a); assuming π.sub.ω(s, a) is a multi-layer perceptron neural network as follows:
π.sub.ω(s,a)=f(ω.sup.T.Math.ϕ(s,a)+b)  (9) where ϕ(s, a) refers to the feature vector of state-action pair, defining as ϕ(s, a)=(m, n.sub.1, n.sub.2, . . . , n.sub.m).sup.T; b denotes the bias of the neural network; f is the activation function, which is sigmoid function; defining a derivative performance function as follows:
J(ω)≐q.sub.π.sub.ω(s.sub.0)  (10) where q.sub.π.sub.ωis the real-value function of π.sub.ω(s, a); s.sub.0 is the initial state; computing the gradient of J(ω) with regard to ω and apply policy gradient theorem, the weights ω can be updated by:
ω.sub.t+1=ω.sub.t+τγ∇.sub.ω.sub.tlnπ.sub.ω.sub.t(s,a)r.sub.t  (11) where τ denotes the step size, γ is the discounting factor; r.sub.t is the reward obtained at t, which can be formulated as: $\begin{array}{cc}{r}_t=\frac{1}{T}\underset{i=1}{\overset{T}{.Math.}}.Math.{\stackrel{\_}{z}}_i-\underset{\_}{z_i}.Math.& \left(12\right)\end{array}$ assuming the dimension of searching space for the structure parameters as L, the procedures can be summarized as follows: (1) initializing τ>0, γ>0, ω∈.sup.max(m), taking l samples from L for training; (2) given t=1˜l, computing Eq. (12) and then Eq. (11) in step 4, obtaining ω.sub.opt for the policy value function; considering the independence of the problems, the computation can be conducted by using parallel strategy for accelerating the process; (3) given t=1˜l, compute Eq. (9), the optimal structural parameters are then determined as the feature vector when the policy value function is at its maximum value:
ϕ.sub.opt(s,a)=arg max.sub.t=1,2, . . . , L(π.sub.ω.sub.opt(s.sub.t, a.sub.t))  (13) (4) using ϕ.sub.opt(s, a) to compute Eq. (3)-(8), the final long-term prediction intervals can be obtained; and generating and consuming the industrial energy system using the obtained final long-term prediction intervals.

Description

DESCRIPTION OF DRAWINGS

(1) FIG. 1—Structure of the secondary gas system in steel industry.

(2) FIG. 2—Flow-chart for application of the present invention.

(3) FIG. 3—Multi-layer structure for the assignment and optimization of the information granularity.

(4) FIG. 4(a)—Constructed long-term prediction intervals by MVE for #2 BFG generation.

(5) FIG. 4(b)—Constructed long-term prediction intervals by one-layer Granular Computing method for #2 BFG generation.

(6) FIG. 4(c)—Constructed long-term prediction intervals by the method of the present invention for #2 BFG generation.

(7) FIG. 5(a)—Constructed long-term prediction intervals by MVE for #1 COG consumption.

(8) FIG. 5(b)—Constructed long-term prediction intervals by one-layer granular computing method for #1 COG consumption.

(9) FIG. 5(c)—Constructed long-term prediction intervals by the method of the present invention for #1 COG consumption.

(10) The MVE denotes Mean-Variance Estimation method.

DETAILED DESCRIPTION

(11) In order to further clarify the technical solution and implementation of the present invention, a secondary energy system of Shanghai Baosteel co., which behaves superior industrial automatic level in China, is deployed as an illustrative example. It can be concluded from the structure as depicted in FIG. 1 that 4 furnaces, 6 ovens and 6 converters form as the generation units, and cold/hot rolling, sintering, etc., are deemed as the consumption units. Specifically, low pressure boiler and power station are typically regarded as adjustable units. Besides, the gas tanks are used for storage and buffering. And the transmission system including mixture and pressure stations is to deliver the gas from the generation units to the consumption units. In real production, it is important for the staffs to keep balance between generation and consumption so as to support the production as well as save the energy and reduce the emission. The secondary gas network covers many processes, such as steel-making, iron-making, steel rolling, etc., which exhibits considerable complexity. And its characteristics involving strong nonlinear and large delay lead to difficulties on accurately estimate the future trends of generation and consumption amounts. Aims at solving the above problem, the present invention develops research and application for the prediction on the secondary gas system.

(12) The detailed procedures for implementation are as follows:

(13) Step 1: Data Pre-Processing

(14) Data of generation and consumption units of the industrial energy system are collected from real-time relational database, and essential noise elimination, filtering and imputation are implemented.

(15) Step 2: FCM

(16) The data are divided into segments with equal length, i.e., Z={z.sub.1, z.sub.2, . . . , z.sub.N}, where z.sub.i ∈.sup.n, n denotes the number of data points in each segment, and N denotes the number of segments. FCM clustering algorithm is implemented so as to obtain the prototype matrix V={v.sub.1, v.sub.2, . . . , v.sub.c} and the corresponding fuzzy membership grades U={u.sub.1, u.sub.2, . . . , u.sub.N}, where v.sub.i ∈.sup.n, u.sub.i ∈.sup.c, c denotes the dimension of the prototype matrix.

(17) Step 3: Establishment of the Multi-Layer Granular Computing Model

(18) As shown in FIG. 1, the information granularity α.sub.i,j and β.sub.i are assigned from bottom to top on the prototype matrix V={v.sub.1, v.sub.2, . . . , v.sub.c}, where i=1, 2, . . . , m; j=1, 2, . . . , n.sub.i and n.sub.1≠n.sub.2≠ . . . ≠n.sub.m. As such, the numeric prototypes are successfully extended into the intervals. In order to optimize the above parameters, this method defines two indices, i.e., Coverage coy and Specificity spec, as follows:

(19) $\begin{array}{cc}\mathrm{cov}\stackrel{.}{=}\frac{1}{T}\underset{i=1}{\overset{T}{.Math.}}{\lambda }_i& \left(1\right)\end{array}$$\begin{array}{cc}\mathrm{spec}\stackrel{.}{=}\mathrm{range}-\frac{1}{T}\underset{i=1}{\overset{T}{.Math.}}.Math.{\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i.Math.& \left(2\right)\end{array}$
where T denotes the number of data points in a sample. λ.sub.i is a marker variable, which will be tagged as 1 if the constructed prediction interval covers the data point, otherwise it will be tagged as 0. range denotes the difference between the maximum and minimum value of the data points. z.sub.i and z.sub.i respectively represent the upper and lower bounds of the constructed prediction intervals.

(20) The objective for optimizing this model is to maxmize the coy and spec. In detail, the coy should be greater than or equal to the prescribed confidence level (1−ρ)×100%, where ρ∈[0,1] denotes the level of significance. In order to avoid the difficulty and cumbersomeness of solving the multi-objective problem, Eq. (1) is considered as the constraints, meaning that the col) should be greater than or equal to the objective confidenece interval. The direction for optimizing the information granularities is opposite with the one for assignment. The optimization models are established as follows:

(21) (1) 2.sup.nd Layer

(22) $\begin{array}{cc}\max {\mathrm{range}}^{\left(2\right)}-\frac{1}{m}\underset{i=1}{\overset{m}{.Math.}}.Math.{\stackrel{\_}{z}}_i^{\left(2\right)}-{\underset{\_}{z}}_i^{\left(2\right)}.Math.s.t.{\sigma }_{\min }^{\left(2\right)}\epsilon \le \frac{1}{m}\underset{i=1}{\overset{m}{.Math.}}{\beta }_i\le {\sigma }_{\max }^{\left(2\right)}\epsilon \frac{1}{m}\underset{i=1}{\overset{m}{.Math.}}{\lambda }_i^{\left(2\right)}\ge \left(1-\rho \right)\times 100\%& \left(3\right)\end{array}$
where range.sup.(2) denotes the difference between the maximum and minimum value of data points in 2.sup.nd layer. z.sub.i.sup.(2) and z.sub.i.sup.(2) respectively represent the upper and lower bounds of the intervals. ε is a hyper-parameter to control the overall information granulartiy. σ.sub.min.sup.(2) and σ.sub.max.sup.(2) are set to control β.sub.i for not excessively far from ε. λ.sub.i.sup.(2) is a marker variable which is similar with λ.sub.i, i.e., it will be tagged as 1 if the constructed prediction interval covers the data point in 2.sup.nd layer, otherwise it will be tagged as 0.

(23) (2) 1.sup.st Layer

(24) Being different with 2.sup.nd layer, the 1.sup.st layer is to solve a number of m optimization problems, each of which can be described as follows:

(25) $\begin{array}{cc}\max {\mathrm{range}}_i^{\left(1\right)}-\frac{1}{n_i}\underset{j=1}{\overset{n_i}{.Math.}}.Math.{\stackrel{\_}{z}}_{i,j}^{\left(1\right)}-{\underset{\_}{z}}_{i,j}^{\left(1\right)}.Math.s.t.{\sigma }_{\min }^{\left(1\right)}{\beta }_i\le \frac{1}{n_i}\underset{j=1}{\overset{n_i}{.Math.}}{\alpha }_{i,j}\le {\sigma }_{\max }^{\left(1\right)}{\beta }_i\frac{1}{n_i}\underset{j=1}{\overset{n_i}{.Math.}}{\lambda }_{i,j}^{\left(1\right)}\ge \left(1-\rho \right)\times 100\%& \left(4\right)\end{array}$
where range.sub.i.sup.(1) denotes the difference between the maximum and minimum value of data points in each optimization problem of 1.sup.st layer, i=1, 2, . . . , m. z.sub.i,j.sup.(1) and z.sub.i,j.sup.(1). respectively represent the upper and lower bounds of the constructed intervals in 1.sup.st layer. σ.sub.min.sup.(1) and σ.sub.max.sup.(1) are set to control α.sub.i,j for not being excessively far from β.sub.i.

(26) Considering the convergence rate and solving speed, Differential Evolution (DE) algorithm is deployed in the present invention to solve the optimization problems. It should be noted that based on the independency between the optimization problems in the 1.sup.st layer, the present invention utilizes parallel strategy to largely reduce the computing time, so that the practical demands on timeliness can be fully satisfied.

(27) Step 4: Construction of Long-Term Prediction Intervals

(28) The Granular Computing based long-term prediction intervals construction method is basically to predict the fuzzy membership grades, in other words, to form relationship û.sub.k=f (u.sub.k−n.sub.I, . . . , u.sub.k−1), where û.sub.k denotes the predicted fuzzy membership grades, u.sub.k−n.sub.I, . . . , u.sub.k−1 are the elements of the matrix of the fuzzy membership grades U, n.sub.I refers to the number of inputs in the relationship among the fuzzy membership grades. The relationship in the present invention is established in view of probability. For clarifying the process, the following definitions will be given in terms of numeric values. First, define the probability of a prototype p(v.sub.i1, . . . , v.sub.in.sub.I), the probability of a segment p(z.sub.k−n.sub.I, . . . , z.sub.k-−1) and the symbiosis matrix as follows:

(29) $\begin{array}{cc}p\left(v_{i1},.Math.,v_{{\mathrm{in}}_I}\right)\stackrel{.}{=}\frac{\underset{j=1}{\overset{N}{.Math.}}{𝕆}_j}{N}& \left(5\right)\end{array}$$\begin{array}{cc}p\left(z_{k-n_I},.Math.,z_{k-1}\right)\stackrel{.}{=}\left\{p\left(v_{i1},.Math.,v_{{\mathrm{in}}_I}\right).Math.\left\{i1,.Math.,{\mathrm{in}}_I\right\}\in \left[1,c\right]\right\}& \left(6\right)\end{array}$$\begin{array}{cc}\mathbb{P}\stackrel{.}{=}\left\{p\left(v_i.Math.v_{j1},.Math.,v_{{\mathrm{jn}}_I}\right).Math.i=1,.Math.,c;\left\{j1,.Math.,{\mathrm{jn}}_I\right\}\in \left[1,c\right]\right\}& \left(7\right)\end{array}$
where {v.sub.i1, . . . , v.sub.in.sub.I}, {i1, i2, . . . , in.sub.I}∈[1, c] are the elements of prototype matrix V. z.sub.k−n.sub.I, . . . , z.sub.k−1 are the elements of the data segments Z. .sub.j is a marker variable, which considers {u.sub.h1,j=max(u.sub.h1), . . . , u.sub.hn.sub.I.sub.,j=max(u.sub.hn.sub.I)}, where {u.sub.h1,j, u.sub.h2,j, . . . , u.sub.hn.sub.I.sub.,j} are the membership grades from u.sub.h1, u.sub.h2, . . . , u.sub.hn.sub.I, {h1, h2, . . . , hn.sub.I}∈[1, c]. .sub.j will be tagged as 1 if h1=i1, . . . , hn.sub.I=in.sub.I, otherwise 0. p(v.sub.i|v.sub.j) is the conditional probability, denoting the probability of the fact that the maximum membership grade of z.sub.k is towards v.sub.i, given that the maximum membership grades of {z.sub.k−n.sub.I, . . . , z.sub.k−1} are towards {v.sub.j1, . . . , v.sub.jn.sub.I}.

(30) Based on the above definitions, the probability of segment z.sub.k can be computed as {circumflex over (p)}(z.sub.k)=p(z.sub.k−n.sub.I, . . . , z.sub.k−2, z.sub.k−1).sup.T×, and the predicted {circumflex over (z)}.sub.k can be obtained by:

(31) $\begin{array}{cc}{\hat{z}}_k=\frac{\underset{i=1}{\overset{c}{.Math.}}{\hat{p}}_{i}V}{\underset{i=1}{\overset{c}{.Math.}}{\hat{p}}_i}& \left(8\right)\end{array}$
where {circumflex over (p)}.sub.i is the element of {circumflex over (p)}(z.sub.k).

(32) Step 5: Reinforcement Learning for the Structural Parameters

(33) The determination on the structure of the multi-layer granular computing model is regarded as a one-step Markov Decision Process, and then Monte-Carlo method is implemented including parameters m and n.sub.i, i=1, 2, . . . , m. State S, action A and reward R are defined as follows:

(34) S—a determined structure for multi-layer granular computing model.

(35) A—to change the parameters m and n.sub.i, i=1, 2, . . . , m.

(36) R—the spec of the constructed prediction intervals.

(37) Considering the large quantities of to-be-determined parameters, the gradient descend strategy is employed to learn the value function π.sub.ω(s, a). Assuming π.sub.ω(s, a) is a multi-layer perceptron neural network as follows:
π.sub.ω(s,a)=f(ω.sup.T.Math.ϕ(s,a)+b)  (9)
where ϕ(s, a) refers to the feature vector of state-action pair, defining as ϕ(s, a)=(m, n.sub.1, n.sub.2, . . . , n.sub.m).sup.T. b denotes the bias of the neural network. f is the activation function, which is sigmoid function in the present invention.

(38) Define a derivative performance function as follows:
J(ω)≐q.sub.π.sub.ω(s.sub.0)  (10)
where q.sub.π.sub.ω is the real-value function of π.sub.ω(s, a). s.sub.0 is the initial state. The gradient of J(ω) with regard to ω is computed and policy gradient theorem is applied, the weights ω can be updated by:
ω.sub.t+1=ω.sub.t+τγ∇.sub.ω.sub.tln π.sub.ω.sub.t(s,a)r.sub.t  (11)
where τ denotes the step size, γ is the discounting factor. r.sub.t is the reward obtained at t, which can be formulated as:

(39) $\begin{array}{cc}{r}_t=\frac{1}{T}\underset{i=1}{\overset{T}{.Math.}}.Math.{\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i.Math.& \left(12\right)\end{array}$

(40) Assuming the dimension of searching space for the structure parameters as L, the procedures can be summarized as follows:

(41) (1) Initializing τ>0, γ>0, ω∈.sup.max(m), taking l samples from L for training.

(42) (2) Given t=1˜l, Eq. (12) and then Eq. (11) in step 4 are computed, ω.sub.opt for the policy value function is obtained. Considering the independence of the problems, the computation can be conducted by using parallel strategy for accelerating the process.

(43) (3) Given t=1˜=l1, Eq. (9) is computed, the optimal structural parameters are then determined as the feature vector when the policy value function is at its maximum value:
ϕ.sub.opt(s,a)=arg max.sub.t=1,2, . . . ,L(π.sub.ω.sub.opt(s.sub.t,a.sub.t))  (13)

(44) (4) ϕ.sub.opt(s, a) is used to compute Eq. (3)-(8), the final long-term prediction intervals can be obtained.

(45) It can be concluded from the above process that the present invention assigns the information granularity layer by layer and optimize them with parallel strategy, so as to improve the computing efficiency and prediction accuracy on one hand. And it adaptively determines the multi-layer structure in terms of reinforcement learning on the other hand.

(46) The constructed long-term prediction intervals for #2 BFG generation and #1 COG generation are given as FIGS. 4 and 5, in which the prediction length is 480 points, i.e., 8 hours. (a) shows the results of Mean-Variance Estimation, (b) for one-layer granular computing model, and (c) for method of the present invention. The dotted line denotes the real values, and the gray zone represents the constructed prediction intervals. Statistics involving prediction accuracy and computing efficiency are given as Table 1, including Prediction Intervals Coverage Probability (PICP), Prediction Intervals Normalized Average Width (PINAW), Interval Score (IS) and Computing Time (CT). The PICP, PINAW and IS are defined as follows:

(47) $\begin{array}{cc}\mathrm{PICP}=\frac{1}{T_{\mathrm{test}}}\underset{i=1}{\overset{T_{\mathrm{test}}}{.Math.}}{\lambda }_i\times 100\%& \left(14\right)\end{array}$$\begin{array}{cc}\mathrm{PINAW}=\frac{1}{T_{\mathrm{test}}\left(d_{\max }-d_{\min }\right)}\underset{i=1}{\overset{T_{\mathrm{test}}}{.Math.}}.Math.{\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i.Math.& \left(15\right)\end{array}$$\begin{array}{cc}\mathrm{IS}\left({\stackrel{\_}{z}}_i,{\underset{\_}{z}}_i,t_i\right)=\frac{1}{T_{\mathrm{test}}}\underset{i=1}{\overset{T_{\mathrm{test}}}{.Math.}}{e}_i\times 100\%& \left(16\right)\end{array}$
where T.sub.test denotes the number of data points in testing set. λ.sub.i is marker variable which will be tagged as 1 if the constructed prediction interval covers the data points in training set, otherwise it will be 0. z.sub.i and z.sub.i respectively represents the upper and lower bounds of the prediction intervals. d.sub.max and d.sub.min respectively refers to the maximum and minimum value of the testing set. e.sub.i is defined as a piecewise variable as follows:

(48) $\begin{array}{cc}{e}_i=\{\begin{array}{c}-2\rho \left({\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i\right)-4\left({\stackrel{\_}{z}}_i-d_i\right),\mathrm{when}{d}_i<{\underset{\_}{z}}_i\\ -2\rho \left({\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i\right),\mathrm{when}{\underset{\_}{z}}_i\le d_i\le {\stackrel{\_}{z}}_i\\ -2\rho \left({\stackrel{\_}{z}}_i-{\underset{\_}{z}}_i\right)-4\left(d_i-{\stackrel{\_}{z}}_i\right),\mathrm{when}{d}_i>{\stackrel{\_}{z}}_i\end{array}& \left(17\right)\end{array}$
where d.sub.i is the data points in testing set, ρ=0.1. Obviously, the present invention exhibits superiority on both accuracy and efficiency comparing with other commonly deployed methods.

(49) TABLE-US-00001 TABLE 1 Comparison of accuracy and computing time on the three methods for construction of long-term prediction intervals Item Method PICP (%) PINAW IS CT (s) #2 BFG MVE 67.76 0.33 −63.08 11.34 generation One-layer Granular 65.71 0.30 −64.35 53.17 Computing The present invention 96.46 0.26 −18.33 9.81 #1 COG MVE 77.50 0.87 −1.63 10.54 generation One-layer Granular 88.54 0.88 −1.27 59.93 Computing The present invention 95.13 0.73 −0.92 7.98