python版本选择
这里选的python版本是2.7，因为我之前用python3试了几次，发现在画3d图的时候会报错，所以改用了2.7。
数据集选择
数据集我选了一个包含两个变量，三个参数的数据集，这样可以画出3d图形对结果进行验证。
部分函数总结
symbols()函数：首先要安装sympy库才可以使用。用法：
> x1 = symbols('x2')
> x1 + 1
x2 + 1
在这个例子中，x1和x2是不一样的，x2代表的是一个函数的变量，而x1代表的是python中的一个变量，它可以表示函数的变量，也可以表示其他的任何量，它替代x2进行函数的计算。实际使用的时候我们可以将x1,x2都命名为x，但是我们要知道他们俩的区别。
再看看这个例子：
> x = symbols('x')
> expr = x + 1
> x = 2
> print(expr)
x + 1
作为python变量的x被2这个数值覆盖了，所以它现在不再表示函数变量x，而expr依然是函数变量x+1的别名，所以结果依然是x+1。
subs()函数：既然普通的方法无法为函数变量赋值，那就肯定有函数来实现这个功能，用法：
> (1 + x*y).subs(x, pi)#一个参数时的用法
pi*y + 1
> (1 + x*y).subs({x:pi, y:2})#多个参数时的用法
1 + 2*pi
diff()函数：求偏导数，用法：result=diff(fun,x)，这个就是求fun函数对x变量的偏导数，结果result也是一个变量，需要赋值才能得到准确结果。
代码实现：
from __future__ import division
from sympy import symbols, diff, expand
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
data = {'x1': [100, 50, 100, 100, 50, 80, 75, 65, 90, 90],
'x2': [4, 3, 4, 2, 2, 2, 3, 4, 3, 2],
'y': [9.3, 4.8, 8.9, 6.5, 4.2, 6.2, 7.4, 6.0, 7.6, 6.1]}#初始化数据集
theta0, theta1, theta2 = symbols('theta0 theta1 theta2', real=True) # y=theta0+theta1*x1+theta2*x2,定义参数
costfuc = 0 * theta0
for i in range(10):
costfuc += (theta0 + theta1 * data['x1'][i] + theta2 * data['x2'][i] - data['y'][i]) ** 2
costfuc /= 20#初始化代价函数
dtheta0 = diff(costfuc, theta0)
dtheta1 = diff(costfuc, theta1)
dtheta2 = diff(costfuc, theta2)
rtheta0 = 1
rtheta1 = 1
rtheta2 = 1#为参数赋初始值
costvalue = costfuc.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})
newcostvalue = 0#用cost的值的变化程度来判断是否已经到最小值了
count = 0
alpha = 0.0001#设置学习率，一定要设置的比较小，否则无法到达最小值
while (costvalue - newcostvalue > 0.00001 or newcostvalue - costvalue > 0.00001) and count < 1000:
count += 1
costvalue = newcostvalue
rtheta0 = rtheta0 - alpha * dtheta0.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})
rtheta1 = rtheta1 - alpha * dtheta1.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})
rtheta2 = rtheta2 - alpha * dtheta2.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})
newcostvalue = costfuc.subs({theta0: rtheta0, theta1: rtheta1, theta2: rtheta2})
rtheta0 = round(rtheta0, 4)
rtheta1 = round(rtheta1, 4)
rtheta2 = round(rtheta2, 4)#给结果保留4位小数，防止数值溢出
print(rtheta0, rtheta1, rtheta2)
fig = plt.figure()
ax = Axes3D(fig)
ax.scatter(data['x1'], data['x2'], data['y']) # 绘制散点图
xx = np.arange(20, 100, 1)
yy = np.arange(1, 5, 0.05)
X, Y = np.meshgrid(xx, yy)
Z = X * rtheta1 + Y * rtheta2 + rtheta0
ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=plt.get_cmap('rainbow'))
plt.show()#绘制3d图进行验证
结果：
实例扩展：
'''
梯度下降算法
Batch Gradient Descent
Stochastic Gradient Descent SGD
'''
__author__ = 'epleone'
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import sys
# 使用随机数种子， 让每次的随机数生成相同，方便调试
# np.random.seed(111111111)
class GradientDescent(object):
eps = 1.0e-8
max_iter = 1000000 # 暂时不需要
dim = 1
func_args = [2.1, 2.7] # [w_0, .., w_dim, b]
def __init__(self, func_arg=None, N=1000):
self.data_num = N
if func_arg is not None:
self.FuncArgs = func_arg
self._getData()
def _getData(self):
x = 20 * (np.random.rand(self.data_num, self.dim) - 0.5)
b_1 = np.ones((self.data_num, 1), dtype=np.float)
# x = np.concatenate((x, b_1), axis=1)
self.x = np.concatenate((x, b_1), axis=1)
def func(self, x):
# noise太大的话， 梯度下降法失去作用
noise = 0.01 * np.random.randn(self.data_num) + 0
w = np.array(self.func_args)
# y1 = w * self.x[0, ] # 直接相乘
y = np.dot(self.x, w) # 矩阵乘法
y += noise
return y
@property
def FuncArgs(self):
return self.func_args
@FuncArgs.setter
def FuncArgs(self, args):
if not isinstance(args, list):
raise Exception(
'args is not list, it should be like [w_0, ..., w_dim, b]')
if len(args) == 0:
raise Exception('args is empty list!!')
if len(args) == 1:
args.append(0.0)
self.func_args = args
self.dim = len(args) - 1
self._getData()
@property
def EPS(self):
return self.eps
@EPS.setter
def EPS(self, value):
if not isinstance(value, float) and not isinstance(value, int):
raise Exception("The type of eps should be an float number")
self.eps = value
def plotFunc(self):
# 一维画图
if self.dim == 1:
# x = np.sort(self.x, axis=0)
x = self.x
y = self.func(x)
fig, ax = plt.subplots()
ax.plot(x, y, 'o')
ax.set(xlabel='x ', ylabel='y', title='Loss Curve')
ax.grid()
plt.show()
# 二维画图
if self.dim == 2:
# x = np.sort(self.x, axis=0)
x = self.x
y = self.func(x)
xs = x[:, 0]
ys = x[:, 1]
zs = y
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(xs, ys, zs, c='r', marker='o')
ax.set_xlabel('X Label')
ax.set_ylabel('Y Label')
ax.set_zlabel('Z Label')
plt.show()
else:
# plt.axis('off')
plt.text(
0.5,
0.5,
"The dimension(x.dim > 2) \n is too high to draw",
size=17,
rotation=0.,
ha="center",
va="center",
bbox=dict(
boxstyle="round",
ec=(1., 0.5, 0.5),
fc=(1., 0.8, 0.8), ))
plt.draw()
plt.show()
# print('The dimension(x.dim > 2) is too high to draw')
# 梯度下降法只能求解凸函数
def _gradient_descent(self, bs, lr, epoch):
x = self.x
# shuffle数据集没有必要
# np.random.shuffle(x)
y = self.func(x)
w = np.ones((self.dim + 1, 1), dtype=float)
for e in range(epoch):
print('epoch:' + str(e), end=',')
# 批量梯度下降，bs为1时 等价单样本梯度下降
for i in range(0, self.data_num, bs):
y_ = np.dot(x[i:i + bs], w)
loss = y_ - y[i:i + bs].reshape(-1, 1)
d = loss * x[i:i + bs]
d = d.sum(axis=0) / bs
d = lr * d
d.shape = (-1, 1)
w = w - d
y_ = np.dot(self.x, w)
loss_ = abs((y_ - y).sum())
print('\tLoss = ' + str(loss_))
print('拟合的结果为:', end=',')
print(sum(w.tolist(), []))
print()
if loss_ < self.eps:
print('The Gradient Descent algorithm has converged!!\n')
break
pass
def __call__(self, bs=1, lr=0.1, epoch=10):
if sys.version_info < (3, 4):
raise RuntimeError('At least Python 3.4 is required')
if not isinstance(bs, int) or not isinstance(epoch, int):
raise Exception(
"The type of BatchSize/Epoch should be an integer number")
self._gradient_descent(bs, lr, epoch)
pass
pass
if __name__ == "__main__":
if sys.version_info < (3, 4):
raise RuntimeError('At least Python 3.4 is required')
gd = GradientDescent([1.2, 1.4, 2.1, 4.5, 2.1])
# gd = GradientDescent([1.2, 1.4, 2.1])
print("要拟合的参数结果是: ")
print(gd.FuncArgs)
print("===================\n\n")
# gd.EPS = 0.0
gd.plotFunc()
gd(10, 0.01)
print("Finished!")