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：
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>>> x1 = symbols('x2')>>> x1 + 1x2 + 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：
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>>> 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：
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>>> (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：
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from __future__ import divisionfrom sympy import symbols, diff, expandimport numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Ddata = {'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 * theta0for i in range(10): costfuc += (theta0 + theta1 * data['x1'][i] + theta2 * data['x2'][i] - data['y'][i]) ** 2costfuc /= 20#初始化代价函数dtheta0 = diff(costfuc, theta0)dtheta1 = diff(costfuc, theta1)dtheta2 = diff(costfuc, theta2)rtheta0 = 1rtheta1 = 1rtheta2 = 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 = 0alpha = 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 + rtheta0ax.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：
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'''梯度下降算法Batch Gradient DescentStochastic Gradient Descent SGD'''__author__ = 'epleone'import numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3Dimport 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 passif __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!") |
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