for i = 2:n % 预测 x(:, i) = A * x(:, i-1); P = A * P0 * A' + Q; % 更新 K = P * C' / (C * P * C' + R); x(:, i) = x(:, i) + K * (y(i) - C * x(:, i)); P = (eye(2) - K * C) * P; P0 = P;

这是一个卡尔曼滤波器的实现，用于估计一个系统的状态。其中，A是状态转移矩阵，C是观测矩阵，Q是系统噪声的协方差矩阵，R是观测噪声的协方差矩阵，x是状态向量，y是观测向量，P是状态估计误差的协方差矩阵，K是卡尔曼增益。具体实现过程如下：

1. 初始化状态向量和状态估计误差的协方差矩阵P0。
2. 对于每个时间步i，先进行预测，即利用状态转移矩阵A和上一时刻的状态向量x(:,i-1)来估计当前时刻的状态向量x(:,i)。
3. 根据预测得到的状态向量x(:,i)和协方差矩阵P，计算卡尔曼增益K。
4. 利用卡尔曼增益K，将观测向量y(i)和预测得到的状态向量x(:,i)融合，得到更准确的状态估计。
5. 更新状态估计误差的协方差矩阵P，用于下一时刻的预测。
6. 将更新后的协方差矩阵P赋值给P0，用于下一时刻的初始化。

通过卡尔曼滤波器，可以对系统状态进行实时估计和预测，具有广泛的应用，如导航、控制、信号处理等领域。

相关问题

DD=xlsread('residual.xlsx') P=DD(1:621,1)' N=length(P) n=486 F =P(1:n+2) Yt=[0,diff(P,1)] L=diff(P,2) Y=L(1:n) a=length(L)-length(Y) aa=a Ux=sum(Y)/n yt=Y-Ux b=0 for i=1:n b=yt(i)^2/n+b end v=sqrt(b) Y=zscore(Y) f=F(1:n) t=1:n R0=0 for i=1:n R0=Y(i)^2/n+R0 end for k=1:20 R(k)=0 for i=k+1:n R(k)=Y(i)*Y(i-k)/n+R(k) end end x=R/R0 X1=x(1);xx(1,1)=1;X(1,1)=x(1);B(1,1)=x(1); K=0;T=X1 for t=2:n at=Y(t)-T(1)*Y(t-1) K=(at)^2+K end U(1)=K/(n-1) for i =1:19 B(i+1,1)=x(i+1); xx(1,i+1)=x(i); A=toeplitz(xx); XX=A\B XXX=XX(i+1); X(1,i+1)=XXX; K=0;T=XX; for t=i+2:n r=0 for j=1:i+1 r=T(j)*Y(t-j)+r end at= Y(t)-r K=(at)^2+K end U(i+1)=K/(n-i+1) end q=20 S(1,1)=R0; for i = 1:q-1 S(1,i+1)=R(i); end G=toeplitz(S) W=inv(G)*[R(1:q)]' U=20*U for i=1:20 AIC2(i)=n*log(U(i))+2*(i) end q=20 C=0;K=0 for t=q+2:n at=Y(t)+Y(q+1); for i=1:q at=-W(i)*Y(t-i)-W(i)*Y(q-i+1)+at; end at1=Y(t-1); for i=1:q at1=-W(i)*Y(t-i-1)+at1 end C=at*at1+C K=(at)^2+K end p=C/K XT=[L(n-q+1:n+a)] for t=q+1:q+a m(t)=0 for i=1:q m(t)=W(i)*XT(t-i)+m(t) end end m=m(q+1:q+a) for i =1:a m(i)=Yt(n+i+1)+m(i) z1(i)=P(n+i+1)+m(i); end for t=q+1:n r=0 for i=1:q r=W(i)*Y(t-i)+r end at= Y(t)-r end figure for t=q+1:n y(t)=0 for i=1:q y(t)=W(i)*Y(t-i)+y(t) end y(t)=y(t)+at y(t)=Yt(t+1)-y(t) y(t)=P(t+1)-y(t) end D_a=P(n+2:end-1); for i=1:a e6_a(i)=D_a(i)-z1(i) PE6_a(i)= (e6_a(i)/D_a(i))*100 end e6_a PE6_a 1-abs(PE6_a) mae6_a=sum(abs(e6_a)) /6 MAPE6_a=sum(abs(PE6_a))/6 Z(1)=0;Xt=0 for i =1:q Xt(1,i)=Y(n-q+i) end for i =1:q Z(1)=W(i)*Xt(q-i+1)+Z(1) end for l=2:q K(l)=0 for i=1:l-1 K(l)=W(i)*Z(l-i)+K(l) end G(l)=0 for j=l:q G(l)=W(j)*Xt(q+l-j)+G(l) end Z(l)=K(l)+G(l) end for l=q+1:aa K(l)=0 for i=1:q K(l)=W(i)*Z(l-i)+K(l) end Z(l)=K(l) end r=Z*v+Ux r(1)=Yt(n+2)+r(1) z(1)=P(n+2)+r(1) for i=2:aa r(i)=r(i-1)+r(i) z(i)=z(i-1)+r(i) end D=P(n+2:end-1) for i=1:aa e6(i)=D(i)-z(i) PE6(i)= (e6(i)/D(i))*100 end e6 PE6 1-abs(PE6) mae6=sum(abs(e6)) /6 MAPE6=sum(abs(PE6))/6把单步预测的完整代码单独摘出来

以下是该段代码的注释：

% 读取数据
DD = xlsread('residual.xlsx');
P = DD(1:621, 1)';
N = length(P);
n = 486;

% 获取一阶差分、二阶差分、原始序列的部分
F = P(1:n+2);
Yt = [0, diff(P, 1)];
L = diff(P, 2);
Y = L(1:n);

% 计算Ux、v、Y的z-score
Ux = sum(Y) / n;
yt = Y - Ux;
v = sqrt(sum(yt.^2) / n);
Y = zscore(Y);

% 计算R、X、U、AIC2、C、K、m、y、e6、PE6、mae6、MAPE6等
R0 = sum(Y.^2) / n;
R = zeros(1, 20);
for k = 1:20
    for i = k+1:n
        R(k) = R(k) + Y(i) * Y(i-k) / n;
    end
end

X1 = R(1);
xx(1, 1) = 1;
X(1, 1) = X1;
B(1, 1) = X1;
K = 0;
T = X1;
for t = 2:n
    at = Y(t) - T * Y(t-1);
    K = at^2 + K;
end
U(1) = K / (n-1);

for i = 1:19
    B(i+1, 1) = R(i+1);
    xx(1, i+1) = R(i);
    A = toeplitz(xx);
    XX = A \ B;
    XXX = XX(i+1);
    X(1, i+1) = XXX;
    K = 0;
    T = X(1, 1:i+1);
    for t = i+2:n
        r = 0;
        for j = 1:i+1
            r = T(j) * Y(t-j) + r;
        end
        at = Y(t) - r;
        K = at^2 + K;
    end
    U(i+1) = K / (n-i+1);
end

q = 20;
S(1,1) = R0;
for i = 1:q-1
    S(1, i+1) = R(i);
end
G = toeplitz(S);
W = inv(G) * [R(1:q)]';
U = 20 * U;

for i = 1:20
    AIC2(i) = n*log(U(i)) + 2*(i);
end

C = 0;
K = 0;
for t = q+2:n
    at = Y(t) + Y(q+1);
    for i = 1:q
        at = -W(i) * Y(t-i) - W(i) * Y(q-i+1) + at;
    end
    at1 = Y(t-1);
    for i = 1:q
        at1 = -W(i) * Y(t-i-1) + at1;
    end
    C = at * at1 + C;
    K = at^2 + K;
end

p = C / K;
XT = [L(n-q+1:n+a)];
for t = q+1:q+a
    m(t) = 0;
    for i = 1:q
        m(t) = W(i) * XT(t-i) + m(t);
    end
end
m = m(q+1:q+a);

for t = q+1:n
    y(t) = 0;
    for i = 1:q
        y(t) = W(i) * Y(t-i) + y(t);
    end
    y(t) = y(t) + Y(t) - Yt(t+1);
    y(t) = P(t+1) - y(t);
end

D_a = P(n+2:end-1);
for i = 1:a
    e6_a(i) = D_a(i) - (P(n+i+1) + m(i));
    PE6_a(i) = (e6_a(i) / D_a(i)) * 100;
end
mae6_a = sum(abs(e6_a)) / 6;
MAPE6_a = sum(abs(PE6_a)) / 6;

Z(1) = 0;
Xt = 0;
for i = 1:q
    Xt(1, i) = Y(n-q+i);
end
for i = 1:q
    Z(1) = W(i) * Xt(q-i+1) + Z(1);
end
for l = 2:q
    K(l) = 0;
    for i = 1:l-1
        K(l) = W(i) * Z(l-i) + K(l);
    end
    G(l) = 0;
    for j = l:q
        G(l) = W(j) * Xt(q+l-j) + G(l);
    end
    Z(l) = K(l) + G(l);
end
for l = q+1:aa
    K(l) = 0;
    for i = 1:q
        K(l) = W(i) * Z(l-i) + K(l);
    end
    Z(l) = K(l);
end

r = Z * v + Ux;
r(1) = Yt(n+2) + r(1);
z(1) = P(n+2) + r(1);
for i = 2:aa
    r(i) = r(i-1) + r(i);
    z(i) = z(i-1) + r(i);
end

D = P(n+2:end-1);
for i = 1:aa
    e6(i) = D(i) - z(i);
    PE6(i) = (e6(i) / D(i)) * 100;
end
mae6 = sum(abs(e6)) / 6;
MAPE6 = sum(abs(PE6)) / 6;

该代码主要实现了单步预测的过程，其中主要的计算包括：

1. 计算一阶差分、二阶差分和原始序列的部分；
2. 计算Ux、v、Y的z-score；
3. 计算R、X、U、AIC2、C、K、m、y、e6、PE6、mae6、MAPE6等；
4. 实现单步预测过程，计算得到预测值z1和z；
5. 计算e6、PE6、mae6和MAPE6等评价指标。

%% % ----------------------- 卡尔曼滤波 ----------------------------- % -------说明 %X(k^l)=Ak*X(k)+W(k); %Y(k)=Ck*X(k)+V(k) %% clear;clc; %基本参数值 Ak=exp(-0.02); Ck=1; Qk=1-exp(-0.04); Rk=1; %初始时刻的卡尔曼最优值设置 X0=0; P0=1; %观测值y(k) Y=[-3.2 -0.8 -14 -16 -17 -18 -3.3 -2.4 -18 -0.3 -0.4 -0.8 -19 -2.0 -1.2 ... -11 -14 -0.9 0.8 10 0.2 0.5 2.4 -0.5 0.5 -13 0.5 10 -12 0.5 -0.6 -15 -0.7 15 ... 0.5 -0.7 -2.0 -19 -17 -11 -14]; %数据长度 N=length(Y); for k=1:N if k==1 %k=l时由初值开始计算 %预测 X_pre(k)= P_pre(k)= K(k)= %卡尔曼增益 X_kalman(k)= I=eye(size(K(k)));%生成单位矩阵 P_kalman(k)=; else %k>l时开始递推 %预测 X_pre(k)= X_pre(k)= %更新 K(k)= X_kalman(k)= I=eye(size(K(k))); %卡尔曼增益 P_kalman(k) = end end M=1:N; T=0.02*M %作图,画出x(t)的波形 figure () plot(T,Y,'r','LineWidth',1); hold on; plot(T,X,'b','LineWidth',1); legend('测量信号y(t)','Kalman估计信号x(t)') 请补全上述代码

补全代码如下：

clear;clc;

%基本参数值
Ak=exp(-0.02);
Ck=1;
Qk=1-exp(-0.04);
Rk=1;

%初始时刻的卡尔曼最优值设置
X0=0;
P0=1;

%观测值y(k)
Y=[-3.2 -0.8 -14 -16 -17 -18 -3.3 -2.4 -18 -0.3 -0.4 -0.8 -19 -2.0 -1.2 ...
    -11 -14 -0.9 0.8 10 0.2 0.5 2.4 -0.5 0.5 -13 0.5 10 -12 0.5 -0.6 -15 -0.7 15 ...
    0.5 -0.7 -2.0 -19 -17 -11 -14];

%数据长度
N=length(Y);

for k=1:N
    if k==1 %k=l时由初值开始计算
        
        %预测
        X_pre(k)=Ak*X0;
        P_pre(k)=Ak*P0*Ak'+Qk;
        
        K(k)=P_pre(k)*Ck'/(Ck*P_pre(k)*Ck'+Rk); %卡尔曼增益
        
        X_kalman(k)=X_pre(k)+K(k)*(Y(k)-Ck*X_pre(k)); %更新
        I=eye(size(K(k)));%生成单位矩阵
        P_kalman(k)=(I-K(k)*Ck)*P_pre(k); 
        
    else %k>l时开始递推
        
        %预测
        X_pre(k)=Ak*X_kalman(k-1);
        P_pre(k)=Ak*P_kalman(k-1)*Ak'+Qk;
        
        K(k)=P_pre(k)*Ck'/(Ck*P_pre(k)*Ck'+Rk); %卡尔曼增益
        
        X_kalman(k)=X_pre(k)+K(k)*(Y(k)-Ck*X_pre(k)); %更新
        I=eye(size(K(k))); %卡尔曼增益
        P_kalman(k)=(I-K(k)*Ck)*P_pre(k);
        
    end
end

M=1:N;
T=0.02*M;

%作图,画出x(t)的波形
figure()
plot(T,Y,'r','LineWidth',1); 
hold on; 
plot(T,X_kalman,'b','LineWidth',1); 
legend('测量信号y(t)','Kalman估计信号x(t)')