目录
引言
应用领域
VMD代码实现
智能优化算法优化VMD
引言
VMD(变分模态分解)是一种新的非线性自适应信号分解方法,它通过变分原理将复杂信号分解为若干个具有不同频率中心和带宽的本征模态函数(Intrinsic Mode Functions, IMFs)。
变分原理:VMD的分解过程即变分问题的求解过程。在该算法中,IMF被定义为一个有带宽限制的调幅-调频函数。
目标:VMD算法的目标是找到若干模态函数,使得每个模态函数的带宽最小
自适应性强:VMD可以自适应地选择模态数量和频率中心。
高分辨率:VMD具有较高的时频分辨率,能够 取信号的时频特征。
抗噪性强:VMD对噪声具有较强的鲁棒性,能够有效分离信号和噪声。
避免问题:VMD将信号分量的获取过程转移到变分框架内,采用非递归的处理策略,能有效避免模态混叠、过包络、欠包络、边界效应等问题。
具体推导参考:
对变分模态分解(VMD)的理解_vmd的自适应形态-CSDN博客
变分模态分解(VMD)学习-CSDN博客
应用领域
信号处理:VMD广泛应用于各种信号处理任务,如语音信号处理、地震信号分析等。
机械故障诊断:VMD可以用于机械设备的故障诊断,通过分析振动信号,识别设备的故障状态。
生物医学信号分析:VMD在生物医学信号分析中具有重要应用,如心电信号分解、脑电信号分析等。
VMD代码实现
matalb中VMD函数,参照如下:
function [u, u_hat, omega] = VMD(signal, alpha, tau, K, DC, init, tol)% Period and sampling frequency of input signal
save_T = length(signal);
fs = 1/save_T;% extend the signal by mirroring
T = save_T;
f_mirror(1:T/2) = signal(T/2:-1:1);
f_mirror(T/2+1:3*T/2) = signal;
f_mirror(3*T/2+1:2*T) = signal(T:-1:T/2+1);
f = f_mirror;% Time Domain 0 to T (of mirrored signal)
T = length(f);
t = (1:T)/T;% Spectral Domain discretization
freqs = t-0.5-1/T;% Maximum number of iterations (if not converged yet, then it won't anyway)
N = 100;% For future generalizations: individual alpha for each mode
Alpha = alpha*ones(1,K);% Construct and center f_hat
f_hat = fftshift((fft(f)));
f_hat_plus = f_hat;
f_hat_plus(1:T/2) = 0;% matrix keeping track of every iterant // could be discarded for mem
u_hat_plus = zeros(N, length(freqs), K);% Initialization of omega_k
omega_plus = zeros(N, K);
switch initcase 1for i = 1:Komega_plus(1,i) = (0.5/K)*(i-1);endcase 2omega_plus(1,:) = sort(exp(log(fs) + (log(0.5)-log(fs))*rand(1,K)));otherwiseomega_plus(1,:) = 0;
end% if DC mode imposed, set its omega to 0
if DComega_plus(1,1) = 0;
end% start with empty dual variables
lambda_hat = zeros(N, length(freqs));% other inits
uDiff = tol+eps; % update step
n = 1; % loop counter
sum_uk = 0; % accumulator% ----------- Main loop for iterative updateswhile ( uDiff > tol && n < N ) % not converged and below iterations limit% update first mode accumulatork = 1;sum_uk = u_hat_plus(n,:,K) + sum_uk - u_hat_plus(n,:,1);% update spectrum of first mode through Wiener filter of residualsu_hat_plus(n+1,:,k) = (f_hat_plus - sum_uk - lambda_hat(n,:)/2)./(1+Alpha(1,k)*(freqs - omega_plus(n,k)).^2);% update first omega if not held at 0if ~DComega_plus(n+1,k) = (freqs(T/2+1:T)*(abs(u_hat_plus(n+1, T/2+1:T, k)).^2)')/sum(abs(u_hat_plus(n+1,T/2+1:T,k)).^2);end% update of any other modefor k=2:K% accumulatorsum_uk = u_hat_plus(n+1,:,k-1) + sum_uk - u_hat_plus(n,:,k);% mode spectrumu_hat_plus(n+1,:,k) = (f_hat_plus - sum_uk - lambda_hat(n,:)/2)./(1+Alpha(1,k)*(freqs - omega_plus(n,k)).^2);% center frequenciesomega_plus(n+1,k) = (freqs(T/2+1:T)*(abs(u_hat_plus(n+1, T/2+1:T, k)).^2)')/sum(abs(u_hat_plus(n+1,T/2+1:T,k)).^2);end% Dual ascentlambda_hat(n+1,:) = lambda_hat(n,:) + tau*(sum(u_hat_plus(n+1,:,:),3) - f_hat_plus);% loop countern = n+1;% converged yet?uDiff = eps;for i=1:KuDiff = uDiff + 1/T*(u_hat_plus(n,:,i)-u_hat_plus(n-1,:,i))*conj((u_hat_plus(n,:,i)-u_hat_plus(n-1,:,i)))';enduDiff = abs(uDiff);end%------ Postprocessing and cleanup% discard empty space if converged early
N = min(N,n);
omega = omega_plus(1:N,:);% Signal reconstruction
u_hat = zeros(T, K);
u_hat((T/2+1):T,:) = squeeze(u_hat_plus(N,(T/2+1):T,:));
u_hat((T/2+1):-1:2,:) = squeeze(conj(u_hat_plus(N,(T/2+1):T,:)));
u_hat(1,:) = conj(u_hat(end,:));u = zeros(K,length(t));for k = 1:Ku(k,:)=real(ifft(ifftshift(u_hat(:,k))));
end% remove mirror part
u = u(:,T/4+1:3*T/4);% recompute spectrum
clear u_hat;
for k = 1:Ku_hat(:,k)=fftshift(fft(u(k,:)))';
endend
智能优化算法优化VMD
VMD(变分模态分解)在解决EMD(经验模态分解)的模态混叠问题上表现出色,但其性能高度依赖于参数的选择,特别是惩罚因子alpha
和模态分解数k
。手动调整这些参数既繁琐又不经济,因此研究者们已经开始探索利用群智能优化算法来自动选择这些参数。
-
定义优化目标函数:根据具体的应用需求,选择一个或多个评估指标来定义优化目标函数。
-
初始化算法参数:设置群智能优化算法的参数,如粒子数量、迭代次数、搜索空间范围等。
-
初始化候选解:在搜索空间内随机初始化一组候选解,每个候选解代表一组VMD参数(
alpha
和k
)。 -
评估候选解:使用VMD对每个候选解进行信号分解,并计算优化目标函数的值。
-
更新候选解:根据优化目标函数的值,利用群智能优化算法的更新机制来更新候选解。
-
终止条件判断:检查是否满足终止条件(如达到最大迭代次数、优化目标函数值收敛等)。如果满足,则输出最优解;否则,返回步骤4继续迭代。
clc; %% 加载数据
load('data.mat');
signal = signal_data; % 假设数据变量名为 signal_data %% 设定优化算法参数
Max_iter = 10; % 迭代次数
sizepop = 20; % 种群规模
lb = [1000, 3]; % 变量下限(VMD 参数 alpha 和 K 的下限)
ub = [3000, 10]; % 变量上限(VMD 参数 alpha 和 K 的上限)
dim = length(lb); % 优化参数个数(VMD 的 alpha 和 K) fobj = @(x) -objfun(x, signal); %% 调用蜣螂优化算法进行VMD参数寻优
[best_fitness, best_params, Convergence_curve] = DBO(sizepop, Max_iter, lb, ub, dim, fobj); %% 最优参数
alpha_opt = best_params(1);
K_opt = round(best_params(2));
tau = 0;
DC = 0;
init = 1;
tol = 1e-6; %% VMD分解
[u_opt, u_hat_opt, omega_opt] = VMD(signal, alpha_opt, tau, K_opt, DC, init, tol); %% 绘制VMD分解曲线图
figure;
for k = 1:length(u_opt) subplot(length(u_opt), 1, k); plot(u_opt{k}); title(['模态 ', num2str(k)]); xlabel('时间'); ylabel('振幅');
end %% 绘制优化算法的迭代曲线图
figure;
plot(1:length(Convergence_curve), Convergence_curve);
title('优化算法的迭代曲线');
xlabel('迭代次数');
ylabel('适应度值');