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应同学邀请，运行程序生成多变量灰色预测结果。因为很久没用过MATLAB，对它不熟，遂花了点时间，把该程序转成python版的。
原版MATLAB程序（省略部分注释）：
clc,clear all;
format long g;
fprintf(2, 'OBGM(1,N)模型: \n');
X0=input('请输入因变量序列(第一个分号前面部分)与自变量序列(多个自变量序列之间用分号隔开)，如：[1 2 3;4 5 6]\n:')
X00=input('请输入预测时的自变量序列(多个自变量序列之间用分号隔开)，如：[911 12 13]\n:')
% 书上的数据
%X0 = [287.6 312.8 350.4 401.9 480.9 498.3 520 543.7;19978 21989 24725 26738 29073 32903 36469 40321;3371.5 3966.6 3748.5 4858.4 5493.5 5910.6 6462.8 7032.2;1848.5 2053.3 2131.7 2303.1 2764 3048.8 3294.3 3566.4];
%X00 = [43910; 7562.3; 3746.8];
% epsilon:
epsilon=0.5;
[n,m]=size(X0);
F = zeros(n, m);
for u=1:n
ago=0;
for v=1:m
ago=ago+X0(u,v);
F(u,v)=ago;
end
end
% Calculate mode parameters
Y=X0';
Y=Y(:,1);
Y(1,:)=[];
B=F';
B(1,:)=[];B(:,1)=[];
Bm=[];Li=[];Co=[];
for k=1:m-1
Bm=[Bm,(epsilon-1)*F(1,k)-epsilon*F(1,k+1)];
Li=[Li,k+1];
Co=[Co,1];
end
B=[B,Bm'];
B=[B,Li'];
B=[B,Co'];
disp('B:');
disp(B);
% Calculate b2,b3,...
P=(B'*B)^(-1)*B'*Y;
disp('B.T*B:');
disp(B'*B);
disp('B.T*B**-1:');
disp((B'*B)^(-1));
disp('_*B.T:');
disp((B'*B)^(-1)*B');
disp('P:');
disp(P);
in_var_number=n-1;
a = P(in_var_number+1);
c = P(in_var_number+2);
d = P(in_var_number+3);
theta_1 = 1/(1+epsilon*a);
theta_2 = 1-a/(1+epsilon*a);
theta_3 = c/(1+epsilon*a);
theta_4 = d/(1+epsilon*a);
% Calcute Simu1
Simu1 = [];
x01=X0(1,1);
Simu1=[Simu1,x01];
for k=2:m
s1=0;s2=0;
for u=1:k-1
for i=2:n
s1=s1+theta_1*power(theta_2,u-1)*P(i-1)*F(i,k-u+1);
end
end
for v=0:k-2
s2=s2+power(theta_2,v)*(k-v)*theta_3+theta_4;
end
s3=power(theta_2,k-1)*x01;
Simu1=[Simu1,s1+s2+s3];
end
% Calculate Fore0
Xf=X0;
Xf(1,:)=[];
X0_F=[Xf,X00];
[g, h] =size(X0_F);
for u=1:g
ago=0;
for v=1:h
ago=ago+X0_F(u,v);
X0_F_1(u,v)=ago;
end
end
Fore1=[];
for k=m+1:h
s1=0;s2=0;
for u=1:k-1
for i=2:n
s1=s1+theta_1*power(theta_2,u-1)*P(i-1)*X0_F_1(i-1,k-u+1);
end
end
for v=0:k-2
s2=s2+power(theta_2,v)*((k-v)*theta_3+theta_4);
end
s3=power(theta_2,k-1)*x01;
Fore1=[Fore1,s1+s2+s3];
end
Fore1=[F(1,m),Fore1];
Fore0=[];
for fo=1:length(Fore1)-1
ss=Fore1(fo+1)-Fore1(fo);
Fore0=[Fore0,roundn(ss,-3)];
end
% A;
A=zeros(n,5);
A(1,1)=1;
A(1,2)=X0(1,1);
A(1,3)=X0(1,1);
% mp:
mp=0;
disp('length X0');
disp(length(X0));
disp(' X0 size:');
disp(size(X0));
for k=2:length(X0)
A(k,1)=k;
A(k,2)=X0(1,k);
A(k,3)=roundn(Simu1(k)-Simu1(k-1),-3);
A(k,4)=roundn(A(k,3)-A(k,2),-3);
A(k,5)=roundn(100*abs(A(k,4))/A(k,2),-3);
mp=mp+100*abs(A(k,4))/A(k,2);
end
mp=roundn(mp/(length(X0)-1),-4);
disp('---------------------------------------');
disp('(1)模型参数b2,b3,..,bn及a,c,d分别为:');
disp(roundn(P,-6));
disp('(2)误差检验表为:');
disp(' 序号 实际数据 模拟数据 残差 相对误差(%)');
disp(A);
disp('(3)平均相对模拟百分误差为(%)');
disp(mp);
str1='(4)未来[';
str2=int2str(length(Fore0));
str3=']步的预测值分别为:';
str=[str1, str2, str3];
disp(str);
disp(Fore0);直接转化为python版（需要有numpy库，可通过pip install numpy安装）：
#!/usr/bin/env python3
import math
import numpy as np
print("OBGM(1, N)模型:")
X0 = input('请输入因变量序列:')
X00 = input('请输入预测时的自变量序列:')
# 书上的数据
#X0 = [[287.6,312.8, 350.4, 401.9, 480.9, 498.3, 520, 543.7],[19978,21989,24725, 26738, 29073, 32903, 36469, 40321],[3371.5, 3966.6, 3748.5, 4858.4, 5493.5, 5910.6, 6462.8, 7032.2],[1848.5, 2053.3, 2131.7, 2303.1, 2764, 3048.8, 3294.3, 3566.4]]
#X00 = [[43910],[7562.3],[3746.8]]
X0 = np.array(X0)
X00 = np.array(X00)
epsilon = 0.5
n, m = X0.shape
F = np.zeros((n, m))
for u in range(n):
ago = 0
for v in range(m):
ago += X0[u, v]
F[u, v] = ago
# Calculate mode parameters
Y = np.transpose(X0[0][1:])
B = np.delete(F.T[1:], 0, axis=1)
Bm = []
Li = []
Co = []
for k in range(m - 1):
Bm.append((epsilon - 1) * F[0, k] - epsilon * F[0, k + 1])
Li.append(k + 2)
Co.append(1)
for L in (Bm, Li, Co):
B = np.c_[B, np.transpose(np.array(L))]
# Calculate b2,b3,...
P = np.linalg.inv(B.T.dot(B)).dot(B.T).dot(Y)
in_var_number = n - 1
a = P[in_var_number]
c = P[in_var_number + 1]
d = P[in_var_number + 2]
_t = 1 + epsilon * a
theta_1 = 1 / _t
theta_2 = 1 - a / _t
theta_3 = c / _t
theta_4 = d / _t
# Calculate Simu
Simu1 = [X0[0, 0]]
for k in range(1, m):
s1 = s2 = s3 = 0
for u in range(k - 1):
for i in range(1, n):
s1 += theta_1 * math.pow(theta_2, u) * P[i - 1] * F[i, k - u]
for v in range(k - 1):
s2 += math.pow(theta_2, v) * ((k - v + 1) * theta_3 + theta_4)
s3 = math.pow(theta_2, k) * Simu1[0]
Simu1.append(s1 + s2 + s3)
# % Calcute g
Xf = X0[1:]
X0_F = np.c_[Xf, np.array(X00)]
X0_F_1 = np.zeros(X0_F.shape)
g, h = X0_F.shape
for u in range(g):
ago = 0
for v in range(h):
ago += X0_F[u, v]
X0_F_1[u, v] = ago
Fore1 = [F[0, m - 1]]
for k in range(m, h):
s1 = s2 = s3 = 0
for u in range(k - 1):
for i in range(1, n):
s1 += (
theta_1
* math.pow(theta_2, u)
* P[i - 1]
* X0_F_1[i - 1, k - u]
)
for v in range(k - 1):
s2 += math.pow(theta_2, v) * ((k - v + 1) * theta_3 + theta_4)
s3 = math.pow(theta_2, k) * X0[0, 0]
Fore1.append(s1 + s2 + s3)
Fore0 = [round(a - b, 3) for a, b in zip(Fore1[1:], Fore1[:-1])]
# % A
A = np.zeros((n, 5))
A[0, 0] = 1
A[0, 1] = A[0, 2] = X0[0, 0]
# % mp
mp = 0
for k in range(1, len(X0)):
A[k, 0] = k + 1
A[k, 1] = X0[0, k]
A[k, 2] = round(Simu1[k] - Simu1[k - 1], 3)
A[k, 3] = round(A[k, 2] - A[k, 1], 3)
A[k, 4] = round(100 * abs(A[k, 3]) / A[k, 1], 3)
mp += 100 * abs(A[k, 3]) / A[k, 1]
mp = round(mp / (len(X0) - 1), 4)
print("-" * 20)
print("(1)模型参数b2,b3,..,bn及a,c,d分别为:")
print(np.around(P, 6))
print("(2)误差检验表为:")
print(" 序号 实际数据 模拟数据 残差 相对误差(%)")
print(A)
print("(3)平均相对模拟百分误差为(%)")
print(mp)
print(f"(4)未来[{len(Fore0)]步的预测值分别为:")
print(Fore0)
优化后：
#!/usr/bin/env python
# coding=utf-8
import re
import numpy as np
def summed_array(arr):
a = np.zeros(arr.shape)
for i in range(arr.shape[0]):
s = 0
for j in range(arr.shape[1]):
s += arr[i, j]
a[i, j] = s
return a
def gen_list(first, start, end, arr, index_delta, P, X0, *thetas):
a = [first]
t1, t2, t3, t4 = thetas
for k in range(start, end):
s1 = s2 = 0
for u in range(k - 1):
for i in range(1, X0.shape[0]):
s1 += t1 * t2 ** u * P[i - 1] * arr[i - index_delta, k - u]
s2 = sum(t2 ** v * ((k - v + 1) * t3 + t4) for v in range(k - 1))
s3 = t2 ** k * X0[0, 0]
a.append(s1 + s2 + s3)
return a
def matlab_string_to_array(s):
ss = s.strip().strip('[]').split(';')
a = [[float(i) for i in re.findall(r'[-0-9.]+', j)] for j in ss]
return np.array(a)
def main():
print("OBGM(1, N)模型:")
tip0 = "请输入因变量序列(第一个分号前面部分)与自变量序列(多个自变量序列之间用分号隔开)，如：[1 2 3;4 5 6]\n:"
tip00 = "请输入预测时的自变量序列(多个自变量序列之间用分号隔开)，如：[911 12 13]\n:"
try:
X0, X00 = [matlab_string_to_array(raw_input(t)) for t in (tip0, tip00)]
except NameError:
X0, X00 = [matlab_string_to_array(input(t)) for t in (tip0, tip00)]
"""
# 书上的数据
X0 = [
[287.6, 312.8, 350.4, 401.9, 480.9, 498.3, 520, 543.7],
[19978, 21989, 24725, 26738, 29073, 32903, 36469, 40321],
[3371.5, 3966.6, 3748.5, 4858.4, 5493.5, 5910.6, 6462.8, 7032.2],
[1848.5, 2053.3, 2131.7, 2303.1, 2764, 3048.8, 3294.3, 3566.4],
]
X00 = [[43910], [7562.3], [3746.8]]
X0 = np.array(X0)
X00 = np.array(X00)
"""
epsilon = 0.5
n, m = X0.shape
F = summed_array(X0)
# Calculate mode parameters
Y = X0[0][1:].T
B = np.delete(F.T[1:], 0, axis=1)
Bm = [
(epsilon - 1) * F[0, k] - epsilon * F[0, k + 1] for k in range(m - 1)
]
Li = [k + 2 for k in range(m - 1)]
Co = [1] * (m - 1)
for L in (Bm, Li, Co):
B = np.c_[B, np.array(L).T]
# Calculate b2,b3,...
P = np.linalg.inv(B.T.dot(B)).dot(B.T).dot(Y)
in_var_number = n - 1
a, c, d = P[in_var_number: in_var_number + 3]
t = 1 + epsilon * a
thetas = (1 / t, 1 - a / t, c / t, d / t)
# Calculate Simu
Simu1 = gen_list(X0[0, 0], 1, m, F, 0, P, X0, *thetas)
# Calcute g
Xf = X0[1:]
X0_F = np.c_[Xf, X00]
X0_F_1 = summed_array(X0_F)
Fore1 = gen_list(F[0, m - 1], m, X0_F.shape[1], X0_F_1, 1, P, X0, *thetas)
Fore0 = [round(a - b, 3) for a, b in zip(Fore1[1:], Fore1[:-1])]
# A
A = np.zeros((n, 5))
A[0, 0] = 1
A[0, 1] = A[0, 2] = X0[0, 0]
# mp
mp = 0
for k in range(1, len(X0)):
A[k, 0] = k + 1
A[k, 1] = X0[0, k]
A[k, 2] = round(Simu1[k] - Simu1[k - 1], 3)
A[k, 3] = round(A[k, 2] - A[k, 1], 3)
A[k, 4] = round(100 * abs(A[k, 3]) / A[k, 1], 3)
mp += 100 * abs(A[k, 3]) / A[k, 1]
mp = round(mp / (len(X0) - 1), 4)
print("-" * 20)
print("(1)模型参数b2,b3,..,bn及a,c,d分别为:")
print(np.around(P, 6))
print("\n(2)误差检验表为:")
print(" 序号 实际数据 模拟数据 残差 相对误差(%)")
print(A)
print("\n(3)平均相对模拟百分误差为(%)")
print(mp)
print("\n(4)未来[{}]步的预测值分别为:".format(len(Fore0)))
print(Fore0)
if __name__ == "__main__":
main()