probability = sigmoid(X * w.T)

这个公式表示的是 logistic 回归模型中的预测概率值。其中，X 是输入样本的特征向量，w 是模型的参数向量，T 表示矩阵转置操作。sigmoid 函数将 X 和 w 的内积映射到 [0,1] 区间，表示该样本属于正例的概率。在训练过程中，我们会根据实际标签值和预测概率值计算损失函数，并使用梯度下降等优化算法来更新参数，使得模型能够更准确地预测样本标签。

相关问题

import numpy as np def sigmoid(x): # the sigmoid function return 1/(1+np.exp(-x)) class LogisticReg(object): def __init__(self, indim=1): # initialize the parameters with all zeros # w: shape of [d+1, 1] self.w = np.zeros((indim + 1, 1)) def set_param(self, weights, bias): # helper function to set the parameters # NOTE: you need to implement this to pass the autograde. # weights: vector of shape [d, ] # bias: scaler def get_param(self): # helper function to return the parameters # NOTE: you need to implement this to pass the autograde. # returns: # weights: vector of shape [d, ] # bias: scaler def compute_loss(self, X, t): # compute the loss # X: feature matrix of shape [N, d] # t: input label of shape [N, ] # NOTE: return the average of the log-likelihood, NOT the sum. # extend the input matrix # compute the loss and return the loss X_ext = np.concatenate((X, np.ones((X.shape[0], 1))), axis=1) # compute the log-likelihood def compute_grad(self, X, t): # X: feature matrix of shape [N, d] # grad: shape of [d, 1] # NOTE: return the average gradient, NOT the sum. def update(self, grad, lr=0.001): # update the weights # by the gradient descent rule def fit(self, X, t, lr=0.001, max_iters=1000, eps=1e-7): # implement the .fit() using the gradient descent method. # args: # X: input feature matrix of shape [N, d] # t: input label of shape [N, ] # lr: learning rate # max_iters: maximum number of iterations # eps: tolerance of the loss difference # TO NOTE: # extend the input features before fitting to it. # return the weight matrix of shape [indim+1, 1] def predict_prob(self, X): # implement the .predict_prob() using the parameters learned by .fit() # X: input feature matrix of shape [N, d] # NOTE: make sure you extend the feature matrix first, # the same way as what you did in .fit() method. # returns the prediction (likelihood) of shape [N, ] def predict(self, X, threshold=0.5): # implement the .predict() using the .predict_prob() method # X: input feature matrix of shape [N, d] # returns the prediction of shape [N, ], where each element is -1 or 1. # if the probability p>threshold, we determine t=1, otherwise t=-1

以下是您提供的代码的完整实现，包括_param()和get_param()方法的实现：

import numpy as np


def sigmoid(x):
    # the sigmoid function
    return 1 / (1 + np.exp(-x))


class LogisticReg(object):
    def __init__(self, indim=1):
        # initialize the parameters with all zeros
        # w: shape of [d+1, 1]
        self.w = np.zeros((indim + 1, 1))

    def set_param(self, weights, bias):
        # helper function to set the parameters
        # weights: vector of shape [d, ]
        # bias: scalar
        self.w[:-1] = weights.reshape(-1, 1)
        self.w[-1] = bias

    def get_param(self):
        # helper function to return the parameters
        # returns:
        # weights: vector of shape [d, ]
        # bias: scalar
        weights = self.w[:-1].reshape(-1)
        bias = self.w[-1]
        return weights, bias

    def compute_loss(self, X, t):
        # compute the loss
        # X: feature matrix of shape [N, d]
        # t: input label of shape [N, ]
        # NOTE: return the average of the log-likelihood, NOT the sum.

        # extend the input matrix
        X_ext = np.concatenate((X, np.ones((X.shape[0], 1))), axis=1)

        # compute the log-likelihood
        z = X_ext @ self.w
        y = sigmoid(z)
        loss = -np.mean(t * np.log(y) + (1 - t) * np.log(1 - y))
        return loss

    def compute_grad(self, X, t):
        # X: feature matrix of shape [N, d]
        # grad: shape of [d, 1]
        # NOTE: return the average gradient, NOT the sum.
        X_ext = np.concatenate((X, np.ones((X.shape[0], 1))), axis=1)
        z = X_ext @ self.w
        y = sigmoid(z)
        grad = np.mean(X_ext.T @ (y - t), axis=1)
        return grad

    def update(self, grad, lr=0.001):
        # update the weights
        # by the gradient descent rule
        self.w -= lr * grad.reshape(-1, 1)

    def fit(self, X, t, lr=0.001, max_iters=1000, eps=1e-7):
        # implement the .fit() using the gradient descent method.
        # args:
        #   X: input feature matrix of shape [N, d]
        #   t: input label of shape [N, ]
        #   lr: learning rate
        #   max_iters: maximum number of iterations
        #   eps: tolerance of the loss difference
        # TO NOTE:
        #   extend the input features before fitting to it.
        #   return the weight matrix of shape [indim+1, 1]
        X_ext = np.concatenate((X, np.ones((X.shape[0], 1))), axis=1)
        prev_loss = np.inf
        for i in range(max_iters):
            loss = self.compute_loss(X, t)
            if np.abs(loss - prev_loss) < eps:
                break
            grad = self.compute_grad(X, t)
            self.update(grad, lr)
            prev_loss = loss
        return self.w

    def predict_prob(self, X):
        # implement the .predict_prob() using the parameters learned by .fit()
        # X: input feature matrix of shape [N, d]
        #   NOTE: make sure you extend the feature matrix first,
        #   the same way as what you did in .fit() method.
        # returns the prediction (likelihood) of shape [N, ]
        X_ext = np.concatenate((X, np.ones((X.shape[0], 1))), axis=1)
        z = X_ext @ self.w
        y = sigmoid(z)
        return y

    def predict(self, X, threshold=0.5):
        # implement the .predict() using the .predict_prob() method
        # X: input feature matrix of shape [N, d]
        # returns the prediction of shape [N, ], where each element is -1 or 1.
        # if the probability p > threshold, we determine t = 1, otherwise t = -1
        prob = self.predict_prob(X)
        predictions = np.where(prob > threshold, 1, -1)
        return predictions

这个类实现了逻辑回归的基本功能，包括参数设置、计算损失、计算梯度、更新权重、拟合数据、预测概率和预测类别等方法。您可以使用这个类来拟合二分类问题的数据，并进行预测。