💥1 概述
时间序列预测方法是科学、经济、工程等领域的研究重点之一。经典的时间序列预测方法在用于非线性系统预测时有一定的困难,而神经网络具有较好的非线性特性,为时间序列预测开辟了新的途径。但神经网络具有易陷入局部极小值以及全局搜索能力弱等缺点;而遗传算法具有较好的全局最优搜索能力,遗传神经网络将两者结合,既保留了遗传算法的全局寻优的特点,又兼有神经网络的非线性特性和收敛的快速性。Mackey-Glass(MG)混沌时间序列具有非线性特性,是时间序列预测问题中的基准问题之一,具有代表性。
时滞混沌系统即具有混沌运动的时滞系统。时滞系统是系统中一处或几处的信号传递有时间延迟的系统。所谓混沌是指具有以下特点的一类现象：由确定性产生；具有有界性；具有非周期性；初始条件具有极端敏感性。
📚2 运行结果
🎉3 参考文献
[1]邵海见,邓星.基于RBF神经网络结构选择方法的Mackey-Glass与Lorenz混沌时间序列预测建模[J].江苏科技大学学报(自然科学版),2018,32(05):701-706.
👨💻4 Matlab代码
%% mackeyglass % This script generates a Mackey-Glass time series using the 4th order % Runge-Kutta method. %% Input parameters a = 0.2; % value for a in eq (1) b = 0.1; % value for b in eq (1) tau = 17; % delay constant in eq (1) x0 = 1.2; % initial condition: x(t=0)=x0 deltat = 0.1; % time step size (which coincides with the integration step) sample_n = 5000; % total no. of samples, excluding the given initial condition interval = 1; % output is printed at every 'interval' time steps %% Main algorithm % * x_t : x at instant t , i.e. x(t) (current value of x) % * x_t_minus_tau : x at instant (t-tau) , i.e. x(t-tau) % * x_t_plus_deltat : x at instant (t+deltat), i.e. x(t+deltat) (next value of x) % * X : the (sample_n+1)-dimensional vector containing x0 plus all other computed values of x % * T : the (sample_n+1)-dimensional vector containing time samples % * x_history : a circular vector storing all computed samples within x(t-tau) and x(t) time = 0; index = 1; history_length = floor(tau/deltat); x_history = zeros(history_length, 1); % here we assume x(t)=0 for -tau <= t < 0 x_t = x0; X = zeros(sample_n, 1); % vector of all generated x samples T = zeros(sample_n, 1); % vector of time samples for i = 1:sample_n X(i) = x_t; if tau == 0 x_t_minus_tau = 0.0; else x_t_minus_tau = x_history(index); end x_t_plus_deltat = mackeyglass_rk4(x_t, x_t_minus_tau, deltat, a, b); if (tau ~= 0) x_history(index) = x_t_plus_deltat; index = mod(index, history_length)+1; end time = time + deltat; T(i) = time; x_t = x_t_plus_deltat; end % Save training and test data Data = X; save('Dataset\Data.mat',''); figure plot(T, X); set(gca,'xlim',[0, T(end)]); xlabel('t'); ylabel('x(t)'); title(sprintf('A Mackey-Glass time serie (tau=%d)', tau));
主函数部分代码：
clc clear all close all %% Load Mackey Glass Time series data load Dataset\Data.mat %% Training and Testing datasets % For training Tr=1:4000; % First 4000 samples for training Xr(Tr)=Data(Tr); % Selecting a chuck of series data x(t) % For testing Ts=4000:5000; % Last 1000 samples for testing Xs(Ts)=Data(Ts); % Selecting a chuck of series data x(t) %% LMS Parameters % We run the LMS algorithm for different learning rates etaValues = [5e-4 1e-3 5e-3 0.01]; % Learning rate M=5; % Order of LMS filter W_init=randn(M+1,1); % Initialize weights figure(2) plot(Tr(2*M:end-M),Xr(Tr(2*M:end-M))); % Actual values of mackey glass series figure(3) plot(Ts,Xs(Ts)); % Actual unseen data for eta = etaValues U=zeros(1,M+1); % Initialize values of taps W=W_init; % Initialize weights E=[]; % Initialize squared error vector %% Learning weights of LMS (Training) for i=Tr(1):Tr(end)-1 U(1:end-1)=U(2:end); % Shifting of tap window U(end)=Xr(i); % Input (past/current samples) Y(i)=W'*U'; % Predicted output e(i)=Xr(i+1)-Y(i); % Error in predicted output W=W+eta*e(i)*U'; % Weight update rule of LMS E(i)=e(i).^2; % Concatenate current squared error end %% Prediction of a next outcome of series using previous samples (Testing) for i=Ts(1):Ts(end) U(1:end-1)=U(2:end); % Shifting of tap window U(end)=Xs(i); % Input (past/current samples) Y(i)=W'*U'; % Calculating output (future value) e(i)=Xs(i)-Y(i); % Error in predicted output E(i)=e(i).^2; % Current mean squared error (MSE) end % Plot the squared error over the training sample iterations figure(1),hold on; plot(Tr(1:end-1),E(:,Tr(1:end-1))); % MSE curve hold off; % Plot the predicted training data figure(2), hold on; plot(Tr(2*M:end-M),Y(Tr(2*M:end-M))') % Predicted values during training hold off; % Comment out the following parts to plot prediction of the test data figure(3), hold on; plot(Ts(2*M:end),Y(Ts(2*M:end))'); % Predicted values of mackey glass series (testing) hold off; MSEtr= mean(E(Tr)); % MSE of training MSEts= mean(E(Ts)); % MSE of testing disp(['MSE for test samples (Learning Rate: ' num2str(eta) '):' num2str(MSEts)]); end
