GRA算法步骤

step1：确定参考序列 $x_0$ 和比较序列 $x_i$；

step2：对原始数据变换，将第 i 个属性的第 k 个数据 $x_i(k)$ 转换为 $y_i(k)$ 。方法有：

初值变换： $y_i(k) = \frac{x_i(k)}{x_i(1)}$.

均值变换： $y_i(k) = \frac{x_i(k)}{\overline{x}_i}$.

百分比变换(变换成小于1的数)：$y_i(k) = \frac{x_i(k)}{\max_k {x}_i(k)}$ .

倍数变换(变换成大于1的数)：$y_i(k) = \frac{x_i(k)}{\min_k {x}_i(k)}$ .

归一化变换：$y_i(k) = \frac{x_i(k)}{∑_k {x}_i(k)}$ .

区间变换：$y_i(k) = \frac{x_i(k) - \min_k {x}_i(k)}{\max_k {x}_i(k) - \min_k {x}_i(k)}$ .

step3：求绝对差序列，即比较序列与参考序列的差值 $△_{0i}(k) =|x_0(k)-x_i(k)| $ .

step4：使用下面公式计算灰关联系数，$γ(x_0(k),x_i(k))=\frac{△_{min}+ρ△_{max}}{△_{0i}(k)+ρ△_{max}}$，一般取分辨系数ρ=0.5。

step5：使用下面公式计算灰关联度，$γ(x_0,x_i)=\frac{1}{N}∑_{k=1}^N w_kγ(x_0(k),x_i(k)) $ ，wk为第k条数据权重。

step6：得到关键属性

如果参考序列 ${x_0}$ 为最优值数据列，那么灰关联度 $γ(x_0,x_i)$ 越大，则第 i 个属性越好；

如果参考序列 ${x_0}$ 为最劣值数据列，那么灰关联度 $γ(x_0,x_i)$ 越大，则第 i 个属性越不好；

GRA算法的Python实现

import numpy as np

import pandas as pd

import seaborn as sns

import matplotlib.pyplot as plt

from scipy.sparse import issparse

## 矩阵检查，必须是数据型，必须是二维的

def check_array(array, dtype="numeric"):

if issparse(array):

raise TypeError(‘PCA does not support sparse input.‘) # 不接受稀疏矩阵

if array.ndim != 2:

raise ValueError(‘Expected 2D array.‘) # 只接受二维矩阵

array = np.array(array, dtype=np.float64)

return array

class GRA():

def __init__(self, k=0, norm_method=‘norm‘, rho=0.5):

self.k = k

self.norm_method = norm_method

self.rho = rho

def fit(self, X, k=None):

‘‘‘ 与单个参考序列比较 ‘‘‘

if not k==None: self.k = k

X = check_array(X)

Y = self.__normalization(X) #归一化

self.r = self.__calculation_relevancy(Y)

return self.r

def fit_all(self, X):

‘‘‘ 所有序列依次为参考序列，互相比较 ‘‘‘

X = check_array(X)

self.data = np.zeros([X.shape[1], X.shape[1]])

for k in range(X.shape[1]):

self.k = k

Y = self.__normalization(X) #归一化

r = self.__calculation_relevancy(Y)

self.data[:,k] = r

return self.data

def __normalization(self, X):

if self.norm_method == ‘mean‘:

Y = self.__mean(X)

elif self.norm_method == ‘initial‘:

Y = self.__initial(X)

elif self.norm_method == ‘norm‘:

Y = self.__norm(X)

elif self.norm_method == ‘section‘:

Y = self.__section(X)

elif self.norm_method == ‘max‘:

Y = self.__max(X)

elif self.norm_method == ‘min‘:

Y = self.__min(X)

else:

raise ValueError("Unrecognized norm_method=‘{0}‘".format(self.norm_method))

print(Y)

return Y

def __norm(self, X):

Xsum = np.sum(X, axis=0)

for i in range(X.shape[1]):

X[:, i] = X[:, i]/Xsum[i]

return X

def __mean(self, X):

‘‘‘ 平均值归一化 ‘‘‘

Xmean = np.mean(X, axis=0, keepdims=True) #每一列的平均

for i in range(X.shape[1]):

X[:, i] = X[:, i]/Xmean[0][i]

return X

def __initial(self, X):

‘‘‘ 初值归一化 ‘‘‘

X0 = X[0,:]

for i in range(X.shape[1]):

X[:, i] = X[:, i]/X0[i]

return X

def __max(self, X):

‘‘‘ 百分比归一化 ‘‘‘

Xmax = np.max(X, axis=0)

for i in range(X.shape[1]):

X[:, i] = X[:, i]/Xmax[i]

return X

def __min(self, X):

‘‘‘ 倍数归一化 ‘‘‘

Xmin = np.min(X, axis=0)

for i in range(X.shape[1]):

X[:, i] = X[:, i]/(Xmin[i]+0.000001) #避免除数为零

return X

def __section(self, X):

Xmax = np.max(X, axis=0)

Xmin = np.min(X, axis=0)

for i in range(X.shape[1]):

X[:, i] = (X[:, i]-Xmin[i])/(Xmax[i]-Xmin[i])

return X

def __calculation_relevancy(self, X):

‘‘‘ 计算关联度 ‘‘‘

# 计算参考序列与比较序列差值

Delta = np.zeros((X.shape))

for i in range(X.shape[1]):

Delta[:, i] = np.fabs(X[:, i]-X[:, self.k])

# 计算关联系数

t = np.delete(Delta, self.k, axis=1)

mmax=t.max().max()

mmin=t.min().min()

ksi=((mmin+self.rho*mmax)/(Delta+self.rho*mmax))

# 计算关联度

r = ksi.sum(axis=0) / ksi.shape[0]

return r

def sort_comparison(self):

idxs = np.argsort(-self.r)

data = []

for idx in idxs:

if idx == self.k: continue

data.append([‘第{}个特征‘.format(idx), self.r[idx]])

df = pd.DataFrame(

data=np.array(data),

columns=[‘特征‘,‘相关度‘],

index=[f"{i+1}" for i in range(len(data))],

)

print(‘\n与第{}个特征相关度的从大到小排序：‘.format(self.k))

print(df)

return df

def ShowGRAHeatMap(self):

‘‘‘ 灰色关联结果矩阵可视化 ‘‘‘

df = pd.DataFrame(

data=self.data,

columns=[f"{i}" for i in range(self.data.shape[1])],

index=[f"{i}" for i in range(self.data.shape[0])],

)

colormap = plt.cm.RdBu

plt.figure()

plt.title(‘Pearson Correlation of Features‘)

sns.heatmap(df.astype(float), linewidths=0.1, vmax=1.0, square=True, cmap=colormap, linecolor=‘white‘, annot=True)

plt.show()

接口调用

test 1

import numpy as np

X = np.array([[0.732, 0.646, 0.636, 0.598, 0.627],

[0.038, 0.031, 0.042, 0.036, 0.043],

[0.507, 0.451, 0.448, 0.411, 0.122],

[0.048, 0.034, 0.030, 0.030, 0.031],

[183.25, 207.28, 240.98, 290.80, 370.00],

[24.03, 44.98, 62.79, 83.44, 127.22],

[85508, 74313, 85966, 100554, 109804],

[175.87, 175.72, 183.69, 277.11, 521.26],

[10, 13, 13, 1, 1],])

X=X.T # 每一行为一条记录，每一列为一个特征数据

print(X.shape)

print(X)

gra = GRA(k=0, norm_method=‘min‘)

r = gra.fit(X)

print(r)

gra.sort_comparison()

tese 2

from GRA import GRA

import numpy as np

X = np.array([[0.732, 0.646, 0.636, 0.598, 0.627],

[0.038, 0.031, 0.042, 0.036, 0.043],

[0.507, 0.451, 0.448, 0.411, 0.122],

[0.048, 0.034, 0.030, 0.030, 0.031],

[183.25, 207.28, 240.98, 290.80, 370.00],

[24.03, 44.98, 62.79, 83.44, 127.22],

[85508, 74313, 85966, 100554, 109804],

[175.87, 175.72, 183.69, 277.11, 521.26],

[10, 13, 13, 1, 1],])

gra = GRA(norm_method=‘initial‘)

data = gra.fit_all(X)

print(data)

gra.ShowGRAHeatMap()

参考：

原文：https://www.cnblogs.com/yejifeng/p/13152728.html