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  • 支持向量机回归和支持向量机Support Vector Machine(SVM) is a supervised machine learning algorithm that is usually used in solving binary classification problems. It can also be applied in multi-class ...

    支持向量机回归和支持向量机

    Support Vector Machine(SVM) is a supervised machine learning algorithm that is usually used in solving binary classification problems. It can also be applied in multi-class classification problems and regression problems. This article represents the mathematics behind the binary-class linear Support Vector Machines. Understanding mathematics helps implement and tune the models in practice. Moreover, you can build your own support vector machine model from scratch, and compare it with the one from Scikit-Learn. For details, you can read this article along with another article of mine.

    支持向量机(SVM)是一种监督型机器学习算法,通常用于解决二进制分类问题。 它也可以应用于多类分类问题和回归问题。 本文介绍了二元类线性支持向量机背后的数学原理。 了解数学有助于在实践中实现和调整模型。 此外,您可以从头开始构建自己的支持向量机模型,并将其与Scikit-Learn的模型进行比较。 有关详细信息,您可以阅读本文以及我的另一篇文章

    Specifically, this report explains the key concepts of linear support vector machine, including the primal form and its dual form for both hard margin and soft margin case; the concept of support vectors, max-margin, and the generalization process.

    具体而言,本报告解释了线性支持向量机的关键概念,包括硬边际和软边际情况的原始形式及其对偶形式; 支持向量,最大边距和归纳过程的概念。

    Key Concepts of SVM

    SVM的关键概念

    Assume we have n training points, each observation i has p features (i.e. x_i has p dimensions), and is in two classes y_i=-1 or y_i = 1. Suppose we have two classes of observations that are linearly separable. That means we can draw a hyperplane through our feature space such that all instances of one class are on one side of the hyperplane, and all instances of the other class are on the opposite side. (A hyperplane in p dimensions is a p-1 dimensional subspace. In the two-dimensional example that follows, a hyperplane is just a line.) We define a hyperplane as:

    假设我们有n个训练点,每个观测值i具有p个特征(即x_i具有p个维),并且处于两类y_i = -1或y_i =1。假设我们有两类线性可分离的观测值。 这意味着我们可以在特征空间中绘制一个超平面,这样一类的所有实例都在超平面的一侧,而另一类的所有实例都在另一侧。 (p维上的超平面是p-1维子空间。在下面的二维示例中,超平面只是一条线。)我们将超平面定义为:

    where ˜w is a p-vector and ˜b is a real number. For convenience, we require that ˜w = 1, so the quantity x * ˜w + ˜b is the distance from point x to the hyperplane.

    其中〜w是p向量,〜b是实数。 为了方便起见,我们要求〜w = 1,所以数量x *〜w +〜b是从点x到超平面的距离。

    Image for post
    image from Wikipedia
    图片来自维基百科

    Thus we can label our classes with y = +1/-1, and the requirement that the hyperplane divides the classes becomes:

    因此,我们可以用y = + 1 / -1标记我们的类,超平面划分这些类的要求变为:

    Image for post

    How should we choose the best hyperplane?

    我们应该如何选择最佳的超平面?

    The approach to answering this question is to choose the plane that results in the largest margin M between the two classes, which is called the Maximal Margin Classifier.

    回答此问题的方法是选择在两个类别之间产生最大边距M的平面,这称为最大边距分类器。

    Image for post
    image from Wikipedia
    图片来自维基百科

    From the previous graph, we can see that H1 doesn’t separate the two classes; for H2 and H3, we will choose H3 because H3 has a larger margin. Mathematically, we choose ˜b and ˜w to maximize M, given the constraints:

    从上一张图中,我们可以看到H1并没有将这两个类别分开。 对于H2和H3,我们将选择H3,因为H3具有较大的边距。 在数学上,给定约束,我们选择〜b和〜w以最大化M:

    Image for post

    Defining w =˜ w / M and b =˜b / M, we can rewrite this as:

    定义w = 〜w / Mb =〜b / M ,我们可以重写为:

    Image for post

    and

    Image for post

    The support vectors

    支持向量

    The support vectors are the data points that lie closest to the separating hyperplane. They are the most difficult data points to classify. Moreover, support vectors are the elements of the training set that would change the position of the dividing hyperplane if removed. The optimization algorithm to generate the weights proceeds in such a way that only the support vectors determine the weights and thus the boundary. Mathematically support vectors are defined as:

    支持向量是最靠近分离超平面的数据点。 它们是最难分类的数据点。 此外,支持向量是训练集的元素,如果被移除,它们将改变划分超平面的位置。 生成权重的优化算法以这样的方式进行:只有支持向量才能确定权重,从而确定边界。 数学上的支持向量定义为:

    Image for post

    Hard-margin SVM

    硬利润支持向量机

    The hard-margin SVM is very strict with the support vectors crossing the hyperplane. It doesn’t allow any support vectors to be classified in the wrong class. To maximize the margin of the hyperplane, the hard-margin support vector machine is facing the optimization problem:

    硬边支持向量机非常严格,支持向量跨过超平面。 不允许将任何支持向量分类为错误的类。 为了最大化超平面的余量,硬余量支持向量机面临着优化问题:

    Image for post

    Soft-margin SVM and the hyper-parameter C

    软边距SVM和超参数C

    In general, classes are not linearly separable. This may be because the class boundary is not linear, but often there is no clear boundary. To deal with this case, the support vector machine adds a set of “slack variables”, which forgive excursions of a few points into, or even across, the margin, like showing in the graph below:

    通常,类不是线性可分离的。 这可能是因为类边界不是线性的,但是通常没有明确的边界。 为了解决这种情况,支持向量机添加了一组“松弛变量”,这些宽恕原谅了到边缘甚至跨越边缘的几个点的偏移,如下图所示:

    Image for post
    image by Author
    图片作者

    We want to minimize the total amount of slacks while maximizing the width of the margin, which is called soft-margin support vector machine. This is more widely used, and the objective function becomes:

    我们要在使边距的宽度最大的同时,使松弛的总量最小化,这称为软边距支持向量机。 这被更广泛地使用,目标函数变为:

    Image for post

    for some constant C. This optimization problem is called the primal problem. The constant C represents the “cost” of the slack. When C is small, it is efficient to allow more points into the margin to achieve a larger margin. Larger C will produce boundaries with fewer support vectors. By increasing the number of support vectors, SVM reduces its variance, since it depends less on any individual observation. Reducing variance makes the model more generalized. Thus, decreasing C will increase the number of support vectors and reduce over-fitting.

    对于一些常数C。 此优化问题称为原始问题。 常数C代表松弛的“成本”。 当C较小时,允许有更多点进入边距以实现更大的边距是有效的。 C越大,边界越少,支持向量也越少。 通过增加支持向量的数量,SVM减少了方差,因为它较少依赖于任何单独的观察。 减少方差使模型更通用。 因此,降低C将增加支持向量的数量并减少过度拟合。

    With Lagrange multipliers:

    使用拉格朗日乘数:

    Image for post
    two constraints
    两个约束

    we can rewrite the constrained optimization problem as the primal Lagrangian function :

    我们可以将约束优化问题重写为原始拉格朗日函数:

    Image for post

    Instead of minimizing over w, b, subject to constraints, we can maximize over the multipliers subject to the relations obtained previously for w, b. This is called the dual Lagrangian formulation:

    代替将wb最小化(受约束),我们可以使乘数最大化,该乘数受先前为wb获得的关系的约束。 这称为双重拉格朗日公式:

    Image for post

    This is now a reasonably straightforward quadratic programming problem, solved with Sequential Minimization Optimization. There are a lot of programming tools you can use to solve the optimizing problem. You can use the CVX tool in Matlab to solve this question. Or if you are familiar with python, you can use the CVXOPT package to solve it. I have another article at Medium that discusses the use of the CVXOPT package, and how to apply it to solve SVM in the dual formulation. Once we have solved this problem for \alpha, we can easily work out the coefficients:

    现在这是一个相当简单的二次规划问题,可通过顺序最小化优化解决。 您可以使用许多编程工具来解决优化问题。 您可以使用Matlab中的CVX工具来解决此问题。 或者,如果您熟悉python,则可以使用CVXOPT软件包来解决它。 我在Medium上还有另一篇文章,讨论了CVXOPT软件包的用法,以及如何将其应用于对偶公式中的SVM解决方案。 解决\ alpha的问题后,我们可以轻松计算出系数:

    Image for post

    Waling through the math behind the Support Vector Machines algorithm definitely helps understand the implementation of the model. It gives insights on choosing the right model for the right questions and choosing the right value for the hyper-parameters.

    仔细阅读支持向量机算法背后的数学知识,无疑有助于理解模型的实现。 它为选择正确的问题模型和为超参数选择正确的值提供了见解。

    Hope this helps. Thank you all for reading!

    希望这可以帮助。 谢谢大家的阅读!

    翻译自: https://towardsdatascience.com/explain-support-vector-machines-in-mathematic-details-c7cc1be9f3b9

    支持向量机回归和支持向量机

    展开全文
  • 一维支持向量机SVM代码(MATLAB),包括支持向量机分类和支持向量机回归SVC&SVR;。另外还包括与BP神经网络的比较结果。
  • 支持向量机 回归分析It is a common misconception that support vector machines are only useful when solving classification problems. 常见的误解是,支持向量机仅在解决分类问题时才有用。 The purpose of ...

    支持向量机 回归分析

    It is a common misconception that support vector machines are only useful when solving classification problems.

    常见的误解是,支持向量机仅在解决分类问题时才有用。

    The purpose of using SVMs for regression problems is to define a hyperplane as in the image above, and fit as many instances as is feasible within this hyperplane while at the same time limiting margin violations.

    使用SVM解决回归问题的目的是定义一个超平面,如上图所示,并在此超平面内尽可能多地容纳实例,同时限制了违反边界的情况。

    In this way, SVMs used in this manner differ from classification tasks, where the objective is to fit the largest possible hyperplane between two separate classes (while also limiting margin violations).

    通过这种方式,以这种方式使用的SVM与分类任务不同,分类任务的目的是在两个单独的类之间容纳最大可能的超平面(同时还限制违反边界的行为)。

    As a matter of fact, SVMs can handle regression modelling quite effectively. Let’s take hotel bookings as an example.

    实际上,SVM可以非常有效地处理回归建模。 让我们以酒店预订为例。

    预测酒店客户的平均每日房价 (Predicting Average Daily Rates Across Hotel Customers)

    Suppose that we are building a regression model to predict the average daily rate (or rate that a customer pays on average per day) for a hotel booking. A model is constructed with the following features:

    假设我们正在建立回归模型,以预测酒店预订的平均每日房价(或客户平均每天所支付的房价)。 构建具有以下特征的模型:

    • Cancellation (whether a customer cancels their booking or not)

      取消(客户是否取消预订)
    • Country of Origin

      出生国家
    • Market Segment

      细分市场
    • Deposit Type

      存款类型
    • Customer Type

      客户类型
    • Required Car Parking Spaces

      所需停车位
    • Week of Arrival

      抵达周

    Note that the ADR values are also populated for customers that cancelled — the response variable in this case reflects the ADR that would have been paid had the customer proceeded with the booking.

    请注意,还会为已取消的客户填充ADR值-在这种情况下,响应变量反映的是如果客户继续进行预订将要支付的ADR。

    The original study by Antonio, Almeida and Nunes (2016) can be accessed from the References section below.

    Antonio,Almeida和Nunes(2016)的原始研究可从下面的“参考”部分获得。

    建筑模型 (Model Building)

    Using the features as outlined above, the SVM model is trained and validated on the training set (H1), with the predictions compared to the actual ADR values across the test set (H2).

    使用上述功能,可以在训练集(H1)上对SVM模型进行训练和验证,并将预测结果与整个测试集(H2)的实际ADR值进行比较。

    The model is trained as follows:

    该模型的训练如下:

    >>> from sklearn.svm import LinearSVR
    >>> svm_reg = LinearSVR(epsilon=1.5)
    >>> svm_reg.fit(X_train, y_train)LinearSVR(C=1.0, dual=True, epsilon=1.5, fit_intercept=True,
    intercept_scaling=1.0, loss='epsilon_insensitive', max_iter=1000,
    random_state=None, tol=0.0001, verbose=0)>>> predictions = svm_reg.predict(X_val)
    >>> predictionsarray([100.75090575, 109.08222631, 79.81544167, ..., 94.50700112,
    55.65495607, 65.5248653 ])

    Now, the same model is used on the features in the test set to generate predicted ADR values:

    现在,对测试集中的特征使用相同的模型来生成预测的ADR值:

    bpred = svm_reg.predict(atest)
    bpred

    Let’s compare the predicted ADR to actual ADR on a mean absolute error (MAE) and root mean squared error (RMSE) basis.

    让我们在平均绝对误差(MAE)和均方根误差(RMSE)的基础上将预测的ADR与实际ADR进行比较。

    >>> mean_absolute_error(btest, bpred)
    29.50931462735928>>> print('mse (sklearn): ', mean_squared_error(btest,bpred))
    >>> math.sqrt(mean_squared_error(btest, bpred))
    44.60420935095296

    Note that the sensitivity of the SVM to additional training instances is set by the epsilon (ϵ) parameter, i.e. the higher the parameter, the more of an impact additional training instances has on the model results.

    请注意,SVM对其他训练实例的敏感性由epsilon(ϵ)参数设置,即参数越高,其他训练实例对模型结果的影响就越大。

    In this instance, a large margin of 1.5 was used. Here is the model performance when a margin of 0.5 is used.

    在这种情况下,使用了1.5的大余量。 这是使用0.5的裕度时的模型性能。

    >>> mean_absolute_error(btest, bpred)29.622491512816826>>> print('mse (sklearn): ', mean_squared_error(btest,bpred))
    >>> math.sqrt(mean_squared_error(btest, bpred))44.7963000500928

    We can see that there has been virtually no change in the MAE or RMSE parameters through modifying the ϵ parameter.

    我们可以看到,通过修改ϵ参数,MAE或RMSE参数几乎没有变化。

    That said, we want to ensure that the SVM model is not overfitting. Specifically, if we find that the best fit is achieved when ϵ = 0, then this might be a sign that the model is overfitting.

    也就是说,我们要确保SVM模型不会过拟合。 具体来说,如果我们发现当ϵ = 0时达到了最佳拟合则可能表明该模型过度拟合。

    Here are the results when we set ϵ = 0.

    当我们设置ϵ = 0时,结果如下。

    • MAE: 31.86

      MAE: 31.86

    • RMSE: 47.65

      RMSE: 47.65

    Given that we are not seeing higher accuracy when ϵ = 0, there does not seem to be any evidence that overfitting is an issue in our model — at least not from this standpoint.

    考虑到当ϵ = 0时我们看不到更高的精度,因此似乎没有任何证据表明过度拟合是我们模型中的一个问题-至少从这个角度来看不是这样。

    SVM性能与神经网络相比如何? (How Does SVM Performance Compare To A Neural Network?)

    When using the same features, how does the SVM performance accuracy compare to that of a neural network?

    当使用相同的功能时,SVM的性能精度与神经网络的性能相比如何?

    Consider the following neural network configuration:

    考虑以下神经网络配置:

    >>> model = Sequential()
    >>> model.add(Dense(8, input_dim=8, kernel_initializer='normal', activation='elu'))
    >>> model.add(Dense(1669, activation='elu'))
    >>> model.add(Dense(1, activation='linear'))
    >>> model.summary()Model: "sequential"
    _________________________________________________________________
    Layer (type) Output Shape Param #
    =================================================================
    dense (Dense) (None, 8) 72
    _________________________________________________________________
    dense_1 (Dense) (None, 1669) 15021
    _________________________________________________________________
    dense_2 (Dense) (None, 1) 1670
    =================================================================
    Total params: 16,763
    Trainable params: 16,763
    Non-trainable params: 0
    _________________________________________________________________

    The model is trained across 30 epochs with a batch size of 150:

    该模型经过30个时期训练,批量大小为150

    >>> model.compile(loss='mse', optimizer='adam', metrics=['mse','mae'])
    >>> history=model.fit(X_train, y_train, epochs=30, batch_size=150, verbose=1, validation_split=0.2)
    >>> predictions = model.predict(X_test)

    The following MAE and RMSE are obtained on the test set:

    在测试集上获得以下MAE和RMSE:

    • MAE: 29.89

      MAE: 29.89

    • RMSE: 43.91

      RMSE: 43.91

    We observed that when ϵ was set to 1.5 for the SVM model, the MAE and RMSE came in at 29.5 and 44.6 respectively. In this regard, the SVM has matched the neural network in prediction accuracy on the test set.

    我们观察到 SVM模型设置为1.5 ,MAE和RMSE分别为29.5和44.6。 在这方面,SVM在测试集的预测准确性方面已与神经网络相匹配。

    结论 (Conclusion)

    It is a common misconception that SVMs are only suitable for working with classification data.

    常见的误解是SVM仅适用于分类数据。

    However, we have seen in this example that the SVM model has been quite effective at predicting ADR values for the neural network.

    但是,我们在此示例中看到,SVM模型在预测神经网络的ADR值方面非常有效。

    Many thanks for reading, and any questions or feedback appreciated.

    非常感谢您的阅读,并感谢任何问题或反馈。

    The GitHub repository for this example, as well as other relevant references are available below.

    下面提供了此示例的GitHub存储库以及其他相关参考。

    Disclaimer: This article is written on an “as is” basis and without warranty. It was written with the intention of providing an overview of data science concepts, and should not be interpreted as professional advice in any way.

    免责声明:本文按“原样”撰写,不作任何担保。 它旨在提供数据科学概念的概述,并且不应以任何方式解释为专业建议。

    翻译自: https://towardsdatascience.com/support-vector-machines-and-regression-analysis-ad5d94ac857f

    支持向量机 回归分析

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  • 主要介绍了libsvm支持向量机回归示例,需要的朋友可以参考下
  • 支持向量机回归

    千次阅读 2014-07-23 16:33:05
    支持向量机回归教程: http://wenku.baidu.com/view/2977157da26925c52cc5bfcc.html http://wenku.baidu.com/view/91b59bddad51f01dc281f1dd.html?re=view 最小二乘支持向量回归机: ...

    支持向量机回归教程:

    http://wenku.baidu.com/view/2977157da26925c52cc5bfcc.html

    http://wenku.baidu.com/view/91b59bddad51f01dc281f1dd.html?re=view


    最小二乘支持向量回归机:

    http://www.doc88.com/p-2748734476827.html

    matlab2012b安装教程:

    http://wenku.baidu.com/view/05fd8a3b83c4bb4cf7ecd159.html

    展开全文
  • 支持向量机回归在矿区GPS高程转换中的应用,张健,郝蒙蒙,基于统计学习理论和支持向量机原理,提出了支持向量机回归应用于矿区GPS高程转换的方法用以精化矿区似大地水准面,研究了支持向量
  • 目录 支持向量机基本思想 标准的线性支持向量回归模型 非线性支持向量机 支持向量机用于分类:硬...支持向量机回归: 希望\(f(x)\)与​\(y\)尽可能的接近。 支持向量机基本思想 英文名:support vector...

    支持向量机用于分类:硬间隔和软件间隔支持向量机。尽可能分对

    支持向量机回归: 希望\(f(x)\)与​\(y\)尽可能的接近。

    支持向量机基本思想

    英文名:support vector regression

    简记:SVR

    标准的线性支持向量回归模型

    学习的模型:

    \[f(x)=w^Tx+b\]

    假设能容忍\(f(x)\)\(y\)之间差别绝对值\(\xi\),这就以\(f(x)=w^Tx+b\)形成了一个\(2\xi\)的间隔带,因此模型
    \[ \min \frac{1}{2}w^Tw\\ s.t -\xi<=f(x_i)-y_i<=\xi \]
    但是上述条件太过严苛,因此增加惩罚项,
    \[ \min \frac{1}{2}w^Tw+C\sum(\epsilon_i+\hat{\epsilon}_i)\\ s.t. \begin{cases}f(x_i)-y_i<=\xi+\epsilon_i\\ y_i-f(x_i)<=\xi+\hat{\epsilon}_i\\ \hat{\epsilon}_i>=0,\epsilon_i>=0 \end{cases} \]
    构造Lagrange函数
    \[ \begin{aligned} L :=\frac{1}{2}\|\omega\|^{2} &+C \sum\left(\xi_i+\xi^{\prime}_i\right)-\sum_{i=1}^{N}\left(\eta_{i} \xi_{i}+\eta_{i}^{'} \xi_{i}6{'}\right) \\ &+\sum \alpha_{i}\left(y_{i}-\omega^{T} x_{i}-b-\varepsilon-\xi_{i}\right) \\ &+\sum \alpha_{i}^{'}\left(\omega^{T} x_{i}+b-y_{i}-\varepsilon-\xi_{i}^{\prime}\right) \end{aligned}\tag{1} \]
    求偏导
    \[ \frac{\partial L}{\partial \omega}=\omega-\sum\left(\alpha_{i}-\alpha_{i}\right) x_{i}=0 \Rightarrow \omega=\sum\left(\alpha_{i}-\alpha_{i}^{\prime}\right) x_{i}\tag{2} \]

    \[ \frac{\partial L}{\partial b}=\sum_{i=1}^{N}\left(\alpha_{i}-\alpha_{i}^{\prime}\right)=0 \tag{3} \]

    \[ \frac{\partial L}{\partial \xi_{i}^{\prime}}=C-\alpha_{i}^{'}-\eta_{i}^{\prime}=0 \tag{4} \]

    \[ \frac{\partial L}{\partial \xi_{i}}=C-\alpha_{i}-\eta_{i}=0 \tag{5} \]

    将(2)-(4)带回(1),可得对偶问题
    \[ \begin{aligned} \min L(\boldsymbol{\alpha})=& \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N}\left(\alpha_{i}-\alpha_{i}^{*}\right)\left(\alpha_{j}-\alpha_{j}^{*}\right)\left\langle x_{i}, x_{j}\right\rangle \\ &+\varepsilon \sum_{i=1}^{N}\left(\alpha_{i}+\alpha_{i}^{*}\right)-\sum_{i=1}^{N} y_{i}\left(\alpha_{i}-\alpha_{i}^{*}\right) \\ \text { s.t. } & \sum_{n=1}^{N}\left(\alpha_{n}-\alpha_{n}^{*}\right)=0 \end{aligned} \]
    再将(2)带回\(Y=w^Tx+b\),可得线性回归模型
    \[ y(x)=\sum_{i=1}^{N}\left(\alpha_{i}-\alpha_{i}^{*}\right) x_{i}^{T} x+b \]

    非线性支持向量机

    考虑模型
    \[ y=f(x)+b \]
    \(f(x)\)是非线性函数,存在一个由\(X\)所在空间到希尔伯特空间的映射,使得
    \[ f(x)=w^T\varphi(x) \]
    因此,建立如下的优化问题
    \[ \min \frac{1}{2}\|\omega\|^{T}+C \sum_{i}\left(\xi_{i}+\xi_{i}^{\prime}\right)\\ \begin{cases} y\left(x_{i}\right)-\omega^{T} \varphi\left(x_{i}\right)-b \leq \xi_{i} \\ \omega^{T} \varphi\left(x_{i}\right)+b-y\left(x_{i}\right) & \leq \xi_{i} \\ \xi_{i} & \geq 0 \\ \xi_{i} & \geq 0 \end{cases} \]
    构造lagrange函数
    \[ \begin{aligned} L :=\frac{1}{2}\|\omega\|^{2} &+C \sum\left(\xi+\xi^{\prime}\right)-\sum\left(\eta_{i} \xi_{i}+\eta_{i} \xi_{i}^{\prime}\right) \\ &+\sum \alpha_{i}\left(y_{i}-w^{T} \varphi\left(x_{i}\right)-b-\varepsilon_{i}-\xi_{i}\right) \\ &+\sum \alpha_{\mathrm{i}}^{\prime}\left(w^{T} \varphi\left(x_{i}\right)+b-y_{i}-\varepsilon_{i}^{'}-\xi_{i}^{\prime}\right) \end{aligned} \]
    求偏导
    \[ \begin{cases}\frac{\partial L}{\partial w}=w-\sum\left(\alpha_{i}-\alpha_{i}\right) \varphi\left(x_{i}\right)=0\\ \frac{\partial L}{\partial b} =\sum\left(\alpha_{i}-\alpha_{i}^{\prime}\right)=0 \\ \frac{\partial L}{\partial \xi_{i}^{\prime}} =C-\alpha_{i}^{'}-\eta_{i}^{\prime}=0 \\ \frac{\partial L}{\partial \xi_{i}} =C-\alpha_{i}-\eta_{i}=0 \end{cases} \]
    再带回优化问题可得

    \[\min _{t}-\frac{1}{2} \sum\left(\alpha_{i}-\alpha_{i}^{\prime}\right)\left(\alpha_{j}-\alpha_{j}^{\prime}\right) \varphi\left(x_{i}\right)^{T} \varphi\left(x_{j}\right)-\varepsilon \sum\left(\alpha_{i}+\alpha_{i}^{\prime}\right)+\sum y_{i}\left(\alpha_{i}-\alpha_{i}^{'}\right)\\s t . \sum\left(\alpha_{i}-\alpha_{i}^{\prime}\right)=0\]

    再次将\(w\)带回模型
    \[ y=\sum\left(\alpha_{i}-\alpha_{i}^{'}\right) \varphi\left(x_{i}\right)^{T} \varphi(x)+b \]

    转载于:https://www.cnblogs.com/xiemaycherry/p/10560877.html

    展开全文
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