经过一个多月的努力，终于完成了BP网络，参考的资料为：
1、Training feed-forward networks with the Marquardt algorithm
2、The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems
3、Neural Network Design
4、http://deeplearning.stanford.edu/wiki/index.php/UFLDL%E6%95%99%E7%A8%8B 中介绍的神经网络部分
以下给出Python脚本：
import numpy as np
from math import exp, pow
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import sys
import copy
from scipy.linalg import norm, pinv
class Layer:
def __init__(self,w, b, neure_number, transfer_function, layer_index):
self.transfer_function = transfer_function
self.neure_number = neure_number
self.layer_index = layer_index
self.w = w
self.b = b
class NetStruct:
def __init__(self, x, y, hidden_layers, activ_fun_list, performance_function = 'mse'):
if len(hidden_layers) == len(activ_fun_list):
activ_fun_list.append('line')
self.active_fun_list = activ_fun_list
self.performance_function = performance_function
x = np.array(x)
y = np.array(y)
if(x.shape[1] != y.shape[1]):
print 'The dimension of x and y are not same.'
sys.exit()
self.x = x
self.y = y
input_eles = self.x.shape[0]
output_eles = self.y.shape[0]
tmp = []
tmp.append(input_eles)
tmp.extend(hidden_layers)
tmp.append(output_eles)
self.hidden_layers = np.array(tmp)
self.layer_num = len(self.hidden_layers)
self.layers = []
for i in range(0, len(self.hidden_layers)):
if i == 0:
self.layers.append(Layer([],[],\
self.hidden_layers[i], 'none', i))
continue
f = self.hidden_layers[i - 1]
s = self.hidden_layers[i]
self.layers.append(Layer(np.random.randn(s, f),np.random.randn(s, 1),\
self.hidden_layers[i], activ_fun_list[i-1], i))
class Train:
def __init__(self, net_struct, mu = 1e-3, beta = 10, iteration = 100, tol = 0.1):
self.net_struct = net_struct
self.mu = mu
self.beta = beta
self.iteration = iteration
self.tol = tol
def train(self, method = 'lm'):
if(method == 'lm'):
self.lm()
def sim(self, x):
self.net_struct.x = x
self.forward()
layer_num = len(self.net_struct.layers)
predict = self.net_struct.layers[layer_num - 1].output_val
return predict
def actFun(self, z, activ_type = 'sigm'):
if activ_type == 'sigm':
f = 1.0 / (1.0 + np.exp(-z))
elif activ_type == 'tanh':
f = (np.exp(z) + np.exp(-z)) / (np.exp(z) + np.exp(-z))
elif activ_type == 'radb':
f = np.exp(-z * z)
elif activ_type == 'line':
f = z
return f
def actFunGrad(self, z, activ_type = 'sigm'):
if activ_type == 'sigm':
grad = self.actFun(z, activ_type) * (1.0 - self.actFun(z, activ_type))
elif activ_type == 'tanh':
grad = 1.0 - self.actFun(z, activ_type) * self.actFun(z, activ_type)
elif activ_type == 'radb':
grad = -2.0 * z * self.actFun(z, activ_type)
elif activ_type == 'line':
m = z.shape[0]
n = z.shape[1]
grad = np.ones((m, n))
return grad
def forward(self):
layer_num = len(self.net_struct.layers)
for i in range(0, layer_num):
if i == 0:
curr_layer = self.net_struct.layers[i]
curr_layer.input_val = self.net_struct.x
curr_layer.output_val = self.net_struct.x
continue
before_layer = self.net_struct.layers[i - 1]
curr_layer = self.net_struct.layers[i]
curr_layer.input_val = curr_layer.w.dot(before_layer.output_val) + curr_layer.b
curr_layer.output_val = self.actFun(curr_layer.input_val,
self.net_struct.active_fun_list[i - 1])
def backward(self):
layer_num = len(self.net_struct.layers)
last_layer = self.net_struct.layers[layer_num - 1]
last_layer.error = -self.actFunGrad(last_layer.input_val,
self.net_struct.active_fun_list[layer_num - 2])
layer_index = range(1, layer_num - 1)
layer_index.reverse()
for i in layer_index:
curr_layer = self.net_struct.layers[i]
curr_layer.error = (last_layer.w.transpose().dot(last_layer.error)) \
* self.actFunGrad(curr_layer.input_val,self.net_struct.active_fun_list[i - 1])
last_layer = curr_layer
def parDeriv(self):
layer_num = len(self.net_struct.layers)
for i in range(1, layer_num):
befor_layer = self.net_struct.layers[i - 1]
befor_input_val = befor_layer.output_val.transpose()
curr_layer = self.net_struct.layers[i]
curr_error = curr_layer.error
curr_error = curr_error.reshape(curr_error.shape[0]*curr_error.shape[1], 1, order='F')
row = curr_error.shape[0]
col = befor_input_val.shape[1]
a = np.zeros((row, col))
num = befor_input_val.shape[0]
neure_number = curr_layer.neure_number
for i in range(0, num):
a[neure_number*i:neure_number*i + neure_number,:] = \
np.repeat([befor_input_val[i,:]],neure_number,axis = 0)
tmp_w_par_deriv = curr_error * a
curr_layer.w_par_deriv = np.zeros((num, befor_layer.neure_number * curr_layer.neure_number))
for i in range(0, num):
tmp = tmp_w_par_deriv[neure_number*i:neure_number*i + neure_number,:]
tmp = tmp.reshape(tmp.shape[0] * tmp.shape[1], order='C')
curr_layer.w_par_deriv[i, :] = tmp
curr_layer.b_par_deriv = curr_layer.error.transpose()
def jacobian(self):
layers = self.net_struct.hidden_layers
row = self.net_struct.x.shape[1]
col = 0
for i in range(0, len(layers) - 1):
col = col + layers[i] * layers[i + 1] + layers[i + 1]
j = np.zeros((row, col))
layer_num = len(self.net_struct.layers)
index = 0
for i in range(1, layer_num):
curr_layer = self.net_struct.layers[i]
w_col = curr_layer.w_par_deriv.shape[1]
b_col = curr_layer.b_par_deriv.shape[1]
j[:, index : index + w_col] = curr_layer.w_par_deriv
index = index + w_col
j[:, index : index + b_col] = curr_layer.b_par_deriv
index = index + b_col
return j
def gradCheck(self):
W1 = self.net_struct.layers[1].w
b1 = self.net_struct.layers[1].b
n = self.net_struct.layers[1].neure_number
W2 = self.net_struct.layers[2].w
b2 = self.net_struct.layers[2].b
x = self.net_struct.x
p = []
p.extend(W1.reshape(1,W1.shape[0]*W1.shape[1],order = 'C')[0])
p.extend(b1.reshape(1,b1.shape[0]*b1.shape[1],order = 'C')[0])
p.extend(W2.reshape(1,W2.shape[0]*W2.shape[1],order = 'C')[0])
p.extend(b2.reshape(1,b2.shape[0]*b2.shape[1],order = 'C')[0])
old_p = p
jac = []
for i in range(0, x.shape[1]):
xi = np.array([x[:,i]])
xi = xi.transpose()
ji = []
for j in range(0, len(p)):
W1 = np.array(p[0:2*n]).reshape(n,2,order='C')
b1 = np.array(p[2*n:2*n+n]).reshape(n,1,order='C')
W2 = np.array(p[3*n:4*n]).reshape(1,n,order='C')
b2 = np.array(p[4*n:4*n+1]).reshape(1,1,order='C')
z2 = W1.dot(xi) + b1
a2 = self.actFun(z2)
z3 = W2.dot(a2) + b2
h1 = self.actFun(z3)
p[j] = p[j] + 0.00001
W1 = np.array(p[0:2*n]).reshape(n,2,order='C')
b1 = np.array(p[2*n:2*n+n]).reshape(n,1,order='C')
W2 = np.array(p[3*n:4*n]).reshape(1,n,order='C')
b2 = np.array(p[4*n:4*n+1]).reshape(1,1,order='C')
z2 = W1.dot(xi) + b1
a2 = self.actFun(z2)
z3 = W2.dot(a2) + b2
h = self.actFun(z3)
g = (h[0][0]-h1[0][0])/0.00001
ji.append(g)
jac.append(ji)
p = old_p
return jac
def jjje(self):
layer_number = self.net_struct.layer_num
e = self.net_struct.y - \
self.net_struct.layers[layer_number - 1].output_val
e = e.transpose()
j = self.jacobian()
#check gradient
#j1 = -np.array(self.gradCheck())
#jk = j.reshape(1,j.shape[0]*j.shape[1])
#jk1 = j1.reshape(1,j1.shape[0]*j1.shape[1])
#plt.plot(jk[0])
#plt.plot(jk1[0],'.')
#plt.show()
jj = j.transpose().dot(j)
je = -j.transpose().dot(e)
return[jj, je]
def lm(self):
mu = self.mu
beta = self.beta
iteration = self.iteration
tol = self.tol
y = self.net_struct.y
self.forward()
pred = self.net_struct.layers[self.net_struct.layer_num - 1].output_val
pref = self.perfermance(y, pred)
for i in range(0, iteration):
print 'iter:',i, 'error:', pref
#1) step 1:
if(pref < tol):
break
#2) step 2:
self.backward()
self.parDeriv()
[jj, je] = self.jjje()
while(1):
#3) step 3:
A = jj + mu * np.diag(np.ones(jj.shape[0]))
delta_w_b = pinv(A).dot(je)
#4) step 4:
old_net_struct = copy.deepcopy(self.net_struct)
self.updataNetStruct(delta_w_b)
self.forward()
pred1 = self.net_struct.layers[self.net_struct.layer_num - 1].output_val
pref1 = self.perfermance(y, pred1)
if (pref1 < pref):
mu = mu / beta
pref = pref1
break
mu = mu * beta
self.net_struct = copy.deepcopy(old_net_struct)
def updataNetStruct(self, delta_w_b):
layer_number = self.net_struct.layer_num
index = 0
for i in range(1, layer_number):
before_layer = self.net_struct.layers[i - 1]
curr_layer = self.net_struct.layers[i]
w_num = before_layer.neure_number * curr_layer.neure_number
b_num = curr_layer.neure_number
w = delta_w_b[index : index + w_num]
w = w.reshape(curr_layer.neure_number, before_layer.neure_number, order='C')
index = index + w_num
b = delta_w_b[index : index + b_num]
index = index + b_num
curr_layer.w += w
curr_layer.b += b
def perfermance(self, y, pred):
error = y - pred
return norm(error) / len(y)
def plotSamples(self, n = 40):
x = np.array([np.linspace(0, 3, n)])
x = x.repeat(n, axis = 0)
y = x.transpose()
z = np.zeros((n, n))
for i in range(0, x.shape[0]):
for j in range(0, x.shape[1]):
z[i][j] = self.sampleFun(x[i][j], y[i][j])
fig = plt.figure()
ax = fig.gca(projection='3d')
surf = ax.plot_surface(x, y, z, cmap='autumn', cstride=2, rstride=2)
ax.set_xlabel("X-Label")
ax.set_ylabel("Y-Label")
ax.set_zlabel("Z-Label")
plt.show()
def sinSamples(n):
x = np.array([np.linspace(-0.5, 0.5, n)])
#x = x.repeat(n, axis = 0)
y = x + 0.2
z = np.zeros((n, 1))
for i in range(0, x.shape[1]):
z[i] = np.sin(x[0][i] * y[0][i])
X = np.zeros((n, 2))
n = 0
for xi, yi in zip(x.transpose(), y.transpose()):
X[n][0] = xi
X[n][1] = yi
n = n + 1
return X,z
def peaksSamples(n):
x = np.array([np.linspace(-3, 3, n)])
x = x.repeat(n, axis = 0)
y = x.transpose()
z = np.zeros((n, n))
for i in range(0, x.shape[0]):
for j in range(0, x.shape[1]):
z[i][j] = sampleFun(x[i][j], y[i][j])
X = np.zeros((n*n, 2))
x_list = x.reshape(n*n,1 )
y_list = y.reshape(n*n,1)
z_list = z.reshape(n*n,1)
n = 0
for xi, yi in zip(x_list, y_list):
X[n][0] = xi
X[n][1] = yi
n = n + 1
return X,z_list.transpose()
def sampleFun(x, y):
z = 3*pow((1-x),2) * exp(-(pow(x,2)) - pow((y+1),2)) \
- 10*(x/5 - pow(x, 3) - pow(y, 5)) * exp(-pow(x, 2) - pow(y, 2)) \
- 1/3*exp(-pow((x+1), 2) - pow(y, 2))
return z
if __name__ == '__main__':
hidden_layers = [10,10] #设置网络层数，共两层，每层10个神经元
activ_fun_list = ['sigm','sigm']#设置隐层的激活函数类型，可以设置为tanh,radb，tanh,line类型，如果不显式的设置最后一层为line
[X, z] = peaksSamples(20) #产生训练数据点
X = X.transpose()
bp = NetStruct(X, z, hidden_layers, activ_fun_list) #初始化网络信息
tr = Train(bp) #初始化训练网络的类
tr.train() #训练
[XX, z0] = peaksSamples(40) #产生测试数据
XX = XX.transpose()
z1 = tr.sim(XX) #用训练好的神经网络预测数据，z1为预测结果
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z0[0]) #真值
ax.plot(z1[0],'r.') #预测值
plt.legend((r'real data', r'predict data'))
plt.show()
以上代码计算的结果如下图，由于初始值等原因的影响偶尔收敛效果会变差，不过大多数时候都可以收敛到下图的结果，以后再改进，欢迎指正。
