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拟合
欠拟合
过拟合
正确的拟合
解决过拟合的方法#xff1a;正则化 线性回归模型和逻辑回归模型都存在欠拟合和过拟合的情况。
拟合
来自百度的解释#xff1a;
数据拟合又称曲线拟合#xff0c;俗称拉曲线#xff0c;是一种把现有数据透过数学方法来代入一条…目录
拟合
欠拟合
过拟合
正确的拟合
解决过拟合的方法正则化 线性回归模型和逻辑回归模型都存在欠拟合和过拟合的情况。
拟合
来自百度的解释
数据拟合又称曲线拟合俗称拉曲线是一种把现有数据透过数学方法来代入一条数式的表示方式。科学和工程问题可以通过诸如采样、实验等方法获得若干离散的数据根据这些数据我们往往希望得到一个连续的函数也就是曲线或者更加密集的离散方程与已知数据相吻合这过程就叫做拟合(fitting)。
个人理解拟合就是根据已有数据来建立的一个数学模型这个数据模型能最大限度的包含现有的数据。这样预测的数据就能最大程度的符合现有情况。
欠拟合
所建立的模型与现有数据匹配度较低如下图的分类模型决策边界并不能很好的区分目前的数据
当训练数据的特征值较少的时候会出现欠拟合 过拟合
模型过于匹配现有数据导致模型不能推广应用到更多数据中去。当训练数据的特征值太多的时候会出现这种情况。 正确的拟合
介于欠拟合和过拟合之间 解决过拟合的方法正则化 解决过拟合的方法是将模型正则化就是说把不是主要特征的w_j调整为无限接近于0然后训练模型这样来寻找最优的模型。这样存在一个问题怎么分辨特征是不是主要特征呢这个是不好分辨的因此是把所有的特征都正则化正则化的公式为:
线性回归cost function: 逻辑回归cost function: 适用于线性回归和逻辑回归的梯度下降函数 实现代码
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
from plt_overfit import overfit_example, outputnp.set_printoptions(precision8)def sigmoid(z):Compute the sigmoid of zArgs:z (ndarray): A scalar, numpy array of any size.Returns:g (ndarray): sigmoid(z), with the same shape as zg 1/(1np.exp(-z))return gdef compute_cost_linear_reg(X, y, w, b, lambda_ 1):Computes the cost over all examplesArgs:X (ndarray (m,n): Data, m examples with n featuresy (ndarray (m,)): target valuesw (ndarray (n,)): model parameters b (scalar) : model parameterlambda_ (scalar): Controls amount of regularizationReturns:total_cost (scalar): cost m X.shape[0]n len(w)cost 0.for i in range(m):f_wb_i np.dot(X[i], w) b #(n,)(n,)scalar, see np.dotcost cost (f_wb_i - y[i])**2 #scalar cost cost / (2 * m) #scalar reg_cost 0for j in range(n):reg_cost (w[j]**2) #scalarreg_cost (lambda_/(2*m)) * reg_cost #scalartotal_cost cost reg_cost #scalarreturn total_cost #scalarnp.random.seed(1)
X_tmp np.random.rand(5,6)
y_tmp np.array([0,1,0,1,0])
w_tmp np.random.rand(X_tmp.shape[1]).reshape(-1,)-0.5
b_tmp 0.5
lambda_tmp 0.7
cost_tmp compute_cost_linear_reg(X_tmp, y_tmp, w_tmp, b_tmp, lambda_tmp)print(Regularized cost:, cost_tmp)def compute_cost_logistic_reg(X, y, w, b, lambda_ 1):Computes the cost over all examplesArgs:Args:X (ndarray (m,n): Data, m examples with n featuresy (ndarray (m,)): target valuesw (ndarray (n,)): model parameters b (scalar) : model parameterlambda_ (scalar): Controls amount of regularizationReturns:total_cost (scalar): cost m,n X.shapecost 0.for i in range(m):z_i np.dot(X[i], w) b #(n,)(n,)scalar, see np.dotf_wb_i sigmoid(z_i) #scalarcost -y[i]*np.log(f_wb_i) - (1-y[i])*np.log(1-f_wb_i) #scalarcost cost/m #scalarreg_cost 0for j in range(n):reg_cost (w[j]**2) #scalarreg_cost (lambda_/(2*m)) * reg_cost #scalartotal_cost cost reg_cost #scalarreturn total_cost #scalarnp.random.seed(1)
X_tmp np.random.rand(5,6)
y_tmp np.array([0,1,0,1,0])
w_tmp np.random.rand(X_tmp.shape[1]).reshape(-1,)-0.5
b_tmp 0.5
lambda_tmp 0.7
cost_tmp compute_cost_logistic_reg(X_tmp, y_tmp, w_tmp, b_tmp, lambda_tmp)print(Regularized cost:, cost_tmp)def compute_gradient_linear_reg(X, y, w, b, lambda_): Computes the gradient for linear regression Args:X (ndarray (m,n): Data, m examples with n featuresy (ndarray (m,)): target valuesw (ndarray (n,)): model parameters b (scalar) : model parameterlambda_ (scalar): Controls amount of regularizationReturns:dj_dw (ndarray (n,)): The gradient of the cost w.r.t. the parameters w. dj_db (scalar): The gradient of the cost w.r.t. the parameter b. m,n X.shape #(number of examples, number of features)dj_dw np.zeros((n,))dj_db 0.for i in range(m): err (np.dot(X[i], w) b) - y[i] for j in range(n): dj_dw[j] dj_dw[j] err * X[i, j] dj_db dj_db err dj_dw dj_dw / m dj_db dj_db / m for j in range(n):dj_dw[j] dj_dw[j] (lambda_/m) * w[j]return dj_db, dj_dwnp.random.seed(1)
X_tmp np.random.rand(5,3)
y_tmp np.array([0,1,0,1,0])
w_tmp np.random.rand(X_tmp.shape[1])
b_tmp 0.5
lambda_tmp 0.7
dj_db_tmp, dj_dw_tmp compute_gradient_linear_reg(X_tmp, y_tmp, w_tmp, b_tmp, lambda_tmp)print(fdj_db: {dj_db_tmp}, )
print(fRegularized dj_dw:\n {dj_dw_tmp.tolist()}, )def compute_gradient_logistic_reg(X, y, w, b, lambda_): Computes the gradient for linear regression Args:X (ndarray (m,n): Data, m examples with n featuresy (ndarray (m,)): target valuesw (ndarray (n,)): model parameters b (scalar) : model parameterlambda_ (scalar): Controls amount of regularizationReturnsdj_dw (ndarray Shape (n,)): The gradient of the cost w.r.t. the parameters w. dj_db (scalar) : The gradient of the cost w.r.t. the parameter b. m,n X.shapedj_dw np.zeros((n,)) #(n,)dj_db 0.0 #scalarfor i in range(m):f_wb_i sigmoid(np.dot(X[i],w) b) #(n,)(n,)scalarerr_i f_wb_i - y[i] #scalarfor j in range(n):dj_dw[j] dj_dw[j] err_i * X[i,j] #scalardj_db dj_db err_idj_dw dj_dw/m #(n,)dj_db dj_db/m #scalarfor j in range(n):dj_dw[j] dj_dw[j] (lambda_/m) * w[j]return dj_db, dj_dw np.random.seed(1)
X_tmp np.random.rand(5,3)
y_tmp np.array([0,1,0,1,0])
w_tmp np.random.rand(X_tmp.shape[1])
b_tmp 0.5
lambda_tmp 0.7
dj_db_tmp, dj_dw_tmp compute_gradient_logistic_reg(X_tmp, y_tmp, w_tmp, b_tmp, lambda_tmp)print(fdj_db: {dj_db_tmp}, )
print(fRegularized dj_dw:\n {dj_dw_tmp.tolist()}, )plt.close(all)
display(output)
ofit overfit_example(True) 逻辑回归输出为
