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为什么数据增强能防止过拟合:以曲线拟合为例

发布于2020-10-23 22:08     阅读(831)     评论(0)     点赞(24)     收藏(4)


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多项式+最小二乘法做曲线拟合的数学基础

记n次多项式的表达式为:
y = a 0 + a 1 x 1 + a 2 x 2 + . . . + a n x n y = a_0 + a_1x^1 + a_2x^2 + ... + a_nx^n y=a0+a1x1+a2x2+...+anxn
拟合的目的就是利用已知点求取所有的系数 a i a_i ai

假设我们有m个点用来做拟合,m个点的坐标为: ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x m , y m ) (x_1, y_1), (x_2, y_2), ... , (x_m, y_m) (x1,y1),(x2,y2),...,(xm,ym)

上述点的坐标均为已知内容,将其代入多项式可得:
a 0 + a 1 x 1 + a 2 x 1 2 + . . . + a n x 1 n = y 1 a 0 + a 1 x 2 + a 2 x 2 2 + . . . + a n x 2 n = y 2 . . . a 0 + a 1 x m + a 2 x m 2 + . . . + a n x m n = y m a_0 + a_1{x_1} + a_2{x_1}^2 + ... + a_n{x_1}^n = y_1 \\[2ex] a_0 + a_1{x_2} + a_2{x_2}^2 + ... + a_n{x_2}^n = y_2 \\[2ex] ... \\[2ex] a_0 + a_1{x_m} + a_2{x_m}^2 + ... + a_n{x_m}^n = y_m a0+a1x1+a2x12+...+anx1n=y1a0+a1x2+a2x22+...+anx2n=y2...a0+a1xm+a2xm2+...+anxmn=ym
将上述方程组写为线性方程形式:
[ 1 x 1 1 x 1 2 . . . x 1 n 1 x 2 1 x 2 2 . . . x 2 n . . . 1 x m 1 x m 2 . . . x m n ] [ a 0 a 1 . . . a n ] = [ y 1 y 2 . . . y m ] [1x11x12...x1n1x21x22...x2n...1xm1xm2...xmn]

111x11x21xm1x12x22...xm2.........x1nx2nxmn
[a0a1...an]
a0a1...an
= [y1y2...ym]
y1y2...ym
111x11x21xm1x12x22...xm2.........x1nx2nxmna0a1...an=y1y2...ym
记:
X = [ 1 x 1 1 x 1 2 . . . x 1 n 1 x 2 1 x 2 2 . . . x 2 n . . . 1 x m 1 x m 2 . . . x m n ] X = [1x11x12...x1n1x21x22...x2n...1xm1xm2...xmn]
111x11x21xm1x12x22...xm2.........x1nx2nxmn
X=111x11x21xm1x12x22...xm2.........x1nx2nxmn

A = [ a 0 a 1 . . . a n ] A = [a0a1...an]
a0a1...an
A=a0a1...an

Y = [ y 1 y 2 . . . y m ] Y = [y1y2...ym]
y1y2...ym
Y=y1y2...ym

那么有如下推导:
X A = Y X T X A = X T Y ( X T X ) − 1 X T X A = ( X T X ) − 1 X T Y A = ( X T X ) − 1 X T Y XA=Y \\[2ex] X^TXA=X^TY \\[2ex] (X^TX)^{-1}X^TXA=(X^TX)^{-1}X^TY \\[2ex] A=(X^TX)^{-1}X^TY XA=YXTXA=XTY(XTX)1XTXA=(XTX)1XTYA=(XTX)1XTY
最后的表达式就是A的最小二乘解,将求解出的A代入到多项式中,就可以得到多项式关于已知点的最小二乘拟合曲线。
注意,只有当 m > n m>n m>n 时,得到的是最小二乘意义下的解;当 m = n m=n m=n 时得到的是精确解,此时拟合曲线可以精准穿过所有的已知点;当 m < n m<n m<n 时,就不能称之为最小二乘解了(该叫什么我也忘了,好像是最小方差解?),此时尽管多项式本身有更加富足的能力穿过所有的已知点,但是通过上述线性代数求解方法则得不到这样的解。

程序

在代码中,将系数 a i a_i ai 称为模型 model,将待拟合的已知点称为训练集train data,将用来测试的点称为测试集test data

代码中拟合是fit,fit的目的就是通过训练集求出模型(我们这里只是借用了训练的说法,实际上也没啥可训的,求解过程是一步到位。。。)。做预测是predict,predict的目的是针对任意输入,使用训练得到的模型求得输出。

代码构造了x_train, y_train, x_test, y_test用于测试,并根据这些数据画了一些图片以帮助理解。

# -*- coding: utf-8 -*-
import numpy as np
import matplotlib.pyplot as plt

DEGREE = [2, 4, 6, 8]
DEGREE_AUG = [2, 4, 6, 8, 10, 12]


class PolynomialModel(object):
    def __init__(self, degree):
        self.model = None
        self.degree = degree

    def _generate_x_matrix(self, X):
        """
        generate X matrix

        Parameters:
        -----------
        X: list or 1-dim numpy array
            x coordinates of points

        Returns:
        --------
        x_matrix: 2-dim numpy array
            X matrix
        """
        x_matrix = np.ones([len(X), self.degree + 1])
        for i in range(1, self.degree + 1):
            x_matrix[:, i] = np.power(X, i)
        return x_matrix

    def fit(self, X, Y):
        """
        compute model by least square fitting

        Parameters:
        -----------
        X, Y: list or 1-dim numpy array
            coordinates of points (x, y)
        """
        Y = np.array(Y)
        x_mat = self._generate_x_matrix(X)
        Y = np.reshape(Y, [Y.shape[0], 1])
        self.model = np.linalg.inv(x_mat.T @ x_mat) @ x_mat.T @ Y

    def predict(self, X):
        """
        predict output of input X, using the computed model

        Parameters:
        -----------
        X: list or 1-dim numpy array
            x coordinates of points
        """
        x_mat = self._generate_x_matrix(X)
        Y = x_mat @ self.model
        return Y


def compute_mse(y1, y2):
    """
    compute mean square error

    Parameters:
    -----------
    y1, y2: list or numpy array, with same shape

    Returns:
    --------
    mse: mean square error
    """
    y1 = np.array(y1)
    y2 = np.array(y2)
    assert np.product(y1.shape) == np.product(y2.shape)
    y2 = np.reshape(y2, y1.shape)
    mse = np.mean(np.power(y1 - y2, 2))
    return mse


def gauss_augmentation(X, Y, aug_num, std):
    """
    augment data using gauss distributed random number

    Parameters:
    -----------
    X, Y: list or 1-dim numpy array
        coordinates of points (x, y)
    aug_num: augmentation number for each point
    std: std of random number

    Returns:
    --------
    x_aug, y_aug: augmented points
    """
    assert len(X) == len(Y)
    length = len(X)

    x_aug = []
    y_aug = []
    for i in range(length):
        x = X[i] + np.random.randn(aug_num) * std
        y = Y[i] + np.random.randn(aug_num) * std
        x_aug.append(x)
        y_aug.append(y)
    x_aug = np.array(x_aug).reshape([-1])
    y_aug = np.array(y_aug).reshape([-1])
    return x_aug, y_aug


def get_x_range(x, increment):
    """
    get evenly distributed coordinates by input x, the low limit and high
    limit of coordinates are min(x) and max(x), separately.

    Parameters:
    -----------
    x: list or 1-dim numpy array
        x coordinates of points
    increment: float number
        interval of adjacent coordinates

    Returns:
    --------
    range: evenly distributed coordinates
    """
    x = np.array(x)
    xmin = np.min(x)
    xmax = np.max(x) + increment
    range = np.arange(xmin, xmax, increment)
    return range


def plot_original_dataset(x_train, y_train, x_test, y_test):
    plt.figure(1, figsize=(9, 6))
    plt.plot(x_train, y_train,
             color='red', marker='o', markersize=6,
             linewidth=0, label='train_data')
    plt.plot(x_test, y_test,
             color='green', marker='o', markersize=6,
             linewidth=0, label='test_data')
    plt.legend()
    plt.savefig('original_dataset.png')


def plot_augmented_dataset(x_train, y_train, x_test, y_test, x_aug, y_aug):
    plt.figure(2, figsize=(9, 6))
    plt.plot(x_train, y_train,
             color='red', marker='o', markersize=6,
             linewidth=0, label='train_data')
    plt.plot(x_test, y_test,
             color='green', marker='o', markersize=6,
             linewidth=0, label='test_data')
    plt.plot(x_aug, y_aug,
             color='blue', marker='o', markersize=3,
             linewidth=0, label='augmented_data')
    plt.legend()
    plt.savefig('augmented_dataset.png')


def plot_no_augmentation_model(x_train, y_train, x_test, y_test):
    x_range = get_x_range(x_train, 0.1)
    plt.figure(3, figsize=(15, 9))
    for i, degree in enumerate(DEGREE):
        plt.subplot(2, 2, i + 1)
        poly_model = PolynomialModel(degree)
        poly_model.fit(x_train, y_train)
        y_predict = poly_model.predict(x_range)
        plt.plot(x_train, y_train,
                 color='red', marker='o', markersize=6,
                 linewidth=0, label='train_data')
        plt.plot(x_test, y_test,
                 color='green', marker='o', markersize=6,
                 linewidth=0, label='test_data')
        plt.plot(x_range, y_predict, color='black', label='fitted curve')
        plt.title('degree = %d' % degree)
        plt.legend()
    plt.savefig('no_augmentation_model.png')


def plot_loss_for_overfitting(x_train, y_train, x_test, y_test):
    plt.figure(4, figsize=(9, 6))
    degrees = get_x_range(DEGREE, 1)
    train_loss = []
    test_loss = []
    for _, degree in enumerate(degrees):
        poly_model = PolynomialModel(degree)
        poly_model.fit(x_train, y_train)
        train_predict = poly_model.predict(x_train)
        test_predict = poly_model.predict(x_test)
        train_mse = compute_mse(y_train, train_predict)
        test_mse = compute_mse(y_test, test_predict)
        train_loss.append(train_mse)
        test_loss.append(test_mse)
    plt.plot(degrees, train_loss, color='red', label='train_loss')
    plt.plot(degrees, test_loss, color='green', label='test_loss')
    plt.xlabel('degree')
    plt.ylabel('mean square error')
    plt.legend()
    plt.savefig('loss_for_overfitting.png')


def plot_augmentation_model(x_train, y_train, x_aug, y_aug):
    x_range = get_x_range(x_train, 0.1)
    plt.figure(5, figsize=(15, 15))
    for i, degree in enumerate(DEGREE_AUG):
        plt.subplot(3, 2, i + 1)
        poly_model = PolynomialModel(degree)
        poly_model.fit(x_aug, y_aug)
        y_predict = poly_model.predict(x_range)
        plt.plot(x_train, y_train,
                 color='red', marker='o', markersize=6,
                 linewidth=0, label='train_data')
        plt.plot(x_test, y_test,
                 color='green', marker='o', markersize=6,
                 linewidth=0, label='test_data')
        plt.plot(x_range, y_predict, color='black', label='fitted curve')
        plt.title('degree = %d' % degree)
        plt.legend()
    plt.savefig('augmentation_model.png')


def plot_loss_for_augmentation(x_train, y_train, x_test, y_test, x_aug, y_aug):
    plt.figure(6, figsize=(9, 6))
    degrees = get_x_range(DEGREE_AUG, 1)
    train_loss = []
    test_loss = []
    for _, degree in enumerate(degrees):
        poly_model = PolynomialModel(degree)
        poly_model.fit(x_aug, y_aug)
        train_predict = poly_model.predict(x_train)
        test_predict = poly_model.predict(x_test)
        train_mse = compute_mse(y_train, train_predict)
        test_mse = compute_mse(y_test, test_predict)
        train_loss.append(train_mse)
        test_loss.append(test_mse)
    plt.plot(degrees, train_loss, color='red', label='train_loss')
    plt.plot(degrees, test_loss, color='green', label='test_loss')
    plt.xlabel('degree')
    plt.ylabel('mean square error')
    plt.legend()
    plt.savefig('loss_for_augmentation.png')


if __name__ == '__main__':
    np.random.seed(1337)
    x_train = [-3.0, -2.1, -0.9, 0.1, 1.2, 2.0, 3]
    y_train = [2.5, 1.2, 1.1, -2.9, -0.7, -3.2, 1.3]

    x_test = [-3.0, -2.7, -2.3, -2.0, -1.8, -1.6, -1.3, -1.0, -0.9, -0.6, -0.2,
              0.1, 0.4, 0.7, 1.0, 1.2, 1.5, 1.8, 2.0, 2.3, 2.5, 2.7, 3.0]
    y_test = [2.4, 2.1, 1.6, 1.1, 1.3, 1.0, 1.2, 1.0, 0.8, -0.2, -1.3, -2.3,
              -2.7, -2.3, -1.5, -1.2, -1.5, -2.9, -2.5, -1.3, -1.1, -0.4, 1.1]

    x_aug, y_aug = gauss_augmentation(x_train, y_train, 20, 0.4)

    plot_original_dataset(x_train, y_train, x_test, y_test)
    plot_augmented_dataset(x_train, y_train, x_test, y_test, x_aug, y_aug)
    plot_no_augmentation_model(x_train, y_train, x_test, y_test)
    plot_loss_for_overfitting(x_train, y_train, x_test, y_test)
    plot_augmentation_model(x_train, y_train, x_aug, y_aug)
    plot_loss_for_augmentation(x_train, y_train, x_test, y_test, x_aug, y_aug)

解释

训练和测试数据

如图
在这里插入图片描述

使用原始训练数据做曲线拟合

如图,随着多项式的次数(degree)从2变到6,拟合曲线对训练数据(红色点)的逼近程度越来越高,但是对测试数据的逼近程度却不是如此,特别是degree从4变到6以后,拟合曲线与测试数据的逼近程度反而增大,这就是过拟合现象。

当degree等于8时曲线反而不能完全穿过训练数据,关于此我们前面已有解释。
在这里插入图片描述

过拟合现象的loss表现

随着模型变大,测试集loss反而开始升高,这就是典型的过拟合现象在loss上的表现。
在这里插入图片描述

数据增强

数据增强方式是以训练集中的各个点为基准加上高斯分布的随机数,在其周围构造出一些随机点,这些随机点就是增强后的数据。此例中针对每个训练集中的点构造了20个增强的数据点。
在这里插入图片描述

使用增强后的数据做曲线拟合

从图中可以看到,随着模型增大,拟合曲线不仅对训练集的逼近程度越来越高,对测试集的逼近程度也越来越高。这就是数据增强的威力:防止过拟合,提升在测试集上的效果;另外关键是,我们并没有额外地付出数据采集成本。
在这里插入图片描述

数据增强后的loss表现

从loss上也能看出数据增强带来明显的效果改善,随着模型增大,不仅训练集loss在降低,测试集loss也在不断降低。
在这里插入图片描述

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作者:9384vfnv

链接: https://www.pythonheidong.com/blog/article/607783/f1b276d08719c9572bca/

来源: python黑洞网

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