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31 simple questions to evaluate your Python basic syntax knowledge! Lần cập nhật gần đây nhất 17 thg 11, 2020 An toàn dữ liệuNhà phát triển có thể cung cấp thông tin tại đây về cách họ thu thập và sử dụng dữ liệu của bạn. Tìm hiểu thêm về mục an toàn dữ liệu Không có thông tin Gắn cờ là không thích hợp Trang web http://jpglomot.com Email Địa chỉ Jean-Philippe Glomot 17, rue de Strasbourg 94500 Champigny sur Marne France Bass Chords & Scales Jean-Philippe Glomot 4,3 Piano Chords & Scales Jean-Philippe Glomot 4,4
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Chord Progression Composer Jean-Philippe Glomot 4,2 Guitar Chord Finder Jean-Philippe Glomot Gắn cờ là không thích hợp Trụ sở chính: Văn phòng: Số 27-3RD, Sunrise D, The Manor Central Park, đường Nguyễn Xiển, phường Đại Kim, quận Hoàng Mai, TP. Hà Nội. Liên hệ truyền thông: 0929.536.185 Email: [email protected] Chịu trách nhiệm nội dung: Ông Trần Anh Tú TEK4.VN giữ bản quyền nội dung trên website này. Cấm sao chép dưới mọi hình thức nếu không có sự chấp thuận bằng văn bản. Support vector machines (SVMs) are a set of supervised learning methods used for classification, regression and outliers detection. The advantages of support vector machines are:
The disadvantages of support vector machines include:
The support vector
machines in scikit-learn support both dense ( 1.4.1. Classification¶
As other classifiers, >>> from sklearn import svm >>> X = [[0, 0], [1, 1]] >>> y = [0, 1] >>> clf = svm.SVC() >>> clf.fit(X, y) SVC() After being fitted, the model can then be used to predict new values: >>> clf.predict([[2., 2.]]) array([1]) SVMs decision function (detailed in the Mathematical formulation) depends on some subset of the training data, called the support vectors. Some properties of these support vectors can be found in attributes >>> # get support vectors >>> clf.support_vectors_ array([[0., 0.], [1., 1.]]) >>> # get indices of support vectors >>> clf.support_ array([0, 1]...) >>> # get number of support vectors for each class >>> clf.n_support_ array([1, 1]...) 1.4.1.1. Multi-class classification¶
>>> X = [[0], [1], [2], [3]] >>> Y = [0, 1, 2, 3] >>> clf = svm.SVC(decision_function_shape='ovo') >>> clf.fit(X, Y) SVC(decision_function_shape='ovo') >>> dec = clf.decision_function([[1]]) >>> dec.shape[1] # 4 classes: 4*3/2 = 6 6 >>> clf.decision_function_shape = "ovr" >>> dec = clf.decision_function([[1]]) >>> dec.shape[1] # 4 classes 4 On the other hand, >>> lin_clf = svm.LinearSVC() >>> lin_clf.fit(X, Y) LinearSVC() >>> dec = lin_clf.decision_function([[1]]) >>> dec.shape[1] 4 See Mathematical formulation for a complete description of the decision function. Note that the For “one-vs-rest”
In the case of “one-vs-one”
The shape of This might be clearer with an example: consider a three class problem with class 0 having three support vectors \(v^{0}_0,
v^{1}_0, v^{2}_0\) and class 1 and 2 having two support vectors \(v^{0}_1, v^{1}_1\) and \(v^{0}_2, v^{1}_2\) respectively. For each support vector \(v^{j}_i\), there are two dual coefficients. Let’s call the coefficient of support vector \(v^{j}_i\) in the classifier between classes \(i\) and \(k\) \(\alpha^{j}_{i,k}\). Then
1.4.1.2. Scores and probabilities¶The The cross-validation involved in Platt scaling is an expensive operation for large datasets. In addition, the probability estimates may be inconsistent with the scores:
Platt’s method is also
known to have theoretical issues. If confidence scores are required, but these do not have to be probabilities, then it is advisable to set Please note that when 1.4.1.3. Unbalanced problems¶In problems
where it is desired to give more importance to certain classes or certain individual samples, the parameters
1.4.2. Regression¶The method of Support Vector Classification can be extended to solve regression problems. This method is called Support Vector Regression. The model produced by support vector classification (as described above) depends only on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin. Analogously, the model produced by Support Vector Regression depends only on a subset of the training data, because the cost function ignores samples whose prediction is close to their target. There are three different implementations of Support Vector Regression: As with classification classes, the fit method will take as argument vectors X, y, only that in this case y is expected to have floating point values instead of integer values: >>> from sklearn import svm >>> X = [[0, 0], [2, 2]] >>> y = [0.5, 2.5] >>> regr = svm.SVR() >>> regr.fit(X, y) SVR() >>> regr.predict([[1, 1]]) array([1.5]) 1.4.3. Density estimation, novelty detection¶The class See Novelty and Outlier Detection for the description and usage of OneClassSVM. 1.4.4. Complexity¶Support Vector Machines are powerful tools, but their compute and storage requirements increase rapidly with the number of training vectors. The core of an SVM is a quadratic programming problem (QP), separating support vectors from the rest of the training data. The QP solver used by the libsvm-based implementation scales between \(O(n_{features} \times n_{samples}^2)\) and \(O(n_{features} \times n_{samples}^3)\) depending on how efficiently the libsvm cache is used in practice (dataset dependent). If the data is very sparse \(n_{features}\) should be replaced by the average number of non-zero features in a sample vector. For the linear case, the algorithm used in 1.4.5. Tips on Practical Use¶
1.4.6. Kernel functions¶The kernel function can be any of the following:
Different kernels are specified by the >>> linear_svc = svm.SVC(kernel='linear') >>> linear_svc.kernel 'linear' >>> rbf_svc = svm.SVC(kernel='rbf') >>> rbf_svc.kernel 'rbf' See also Kernel Approximation for a solution to use RBF kernels that is much faster and more scalable. 1.4.6.1. Parameters of the RBF Kernel¶When training an SVM with the Radial Basis Function (RBF) kernel, two parameters must be considered: Proper choice of 1.4.6.2. Custom Kernels¶You can define your own kernels by either giving the kernel as a python function or by precomputing the Gram matrix. Classifiers with custom kernels behave the same way as any other classifiers, except that:
1.4.6.2.1. Using Python functions as kernels¶You can use
your own defined kernels by passing a function to the Your kernel must take as arguments two matrices of shape The following code defines a linear kernel and creates a classifier instance that will use that kernel: >>> import numpy as np >>> from sklearn import svm >>> def my_kernel(X, Y): ... return np.dot(X, Y.T) ... >>> clf = svm.SVC(kernel=my_kernel) 1.4.6.2.2. Using the Gram matrix¶You can pass pre-computed kernels by using the >>> import numpy as np >>> from sklearn.datasets import make_classification >>> from sklearn.model_selection import train_test_split >>> from sklearn import svm >>> X, y = make_classification(n_samples=10, random_state=0) >>> X_train , X_test , y_train, y_test = train_test_split(X, y, random_state=0) >>> clf = svm.SVC(kernel='precomputed') >>> # linear kernel computation >>> gram_train = np.dot(X_train, X_train.T) >>> clf.fit(gram_train, y_train) SVC(kernel='precomputed') >>> # predict on training examples >>> gram_test = np.dot(X_test, X_train.T) >>> clf.predict(gram_test) array([0, 1, 0]) 1.4.7. Mathematical formulation¶A support vector machine constructs a hyper-plane or set of hyper-planes in a high or infinite dimensional space, which can be used for classification, regression or other tasks. Intuitively, a good separation is achieved by the hyper-plane that has the largest distance to the nearest training data points of any class (so-called functional margin), since in general the larger the margin the lower the generalization error of the classifier. The figure below shows the decision function for a linearly separable problem, with three samples on the margin boundaries, called “support vectors”: In general, when the problem isn’t linearly separable, the support vectors are the samples within the margin boundaries. We recommend [13] and [14] as good references for the theory and practicalities of SVMs. 1.4.7.1. SVC¶Given training vectors \(x_i \in \mathbb{R}^p\), i=1,…, n, in two classes, and a vector \(y \in \{1, -1\}^n\), our goal is to find \(w \in \mathbb{R}^p\) and \(b \in \mathbb{R}\) such that the prediction given by \(\text{sign} (w^T\phi(x) + b)\) is correct for most samples. SVC solves the following primal problem: \[ \begin{align}\begin{aligned}\min_ {w, b, \zeta} \frac{1}{2} w^T w + C \sum_{i=1}^{n} \zeta_i\\\begin{split}\textrm {subject to } & y_i (w^T \phi (x_i) + b) \geq 1 - \zeta_i,\\ & \zeta_i \geq 0, i=1, ..., n\end{split}\end{aligned}\end{align} \] Intuitively, we’re trying to maximize the margin (by minimizing \(||w||^2 = w^Tw\)), while incurring a penalty when a sample is misclassified or within the margin boundary. Ideally, the value
\(y_i (w^T \phi (x_i) + b)\) would be \(\geq 1\) for all samples, which indicates a perfect prediction. But problems are usually not always perfectly separable with a hyperplane, so we allow some samples to be at a distance \(\zeta_i\) from their correct margin boundary. The penalty term The dual problem to the primal is \[ \begin{align}\begin{aligned}\min_{\alpha} \frac{1}{2} \alpha^T Q \alpha - e^T \alpha\\\begin{split} \textrm {subject to } & y^T \alpha = 0\\ & 0 \leq \alpha_i \leq C, i=1, ..., n\end{split}\end{aligned}\end{align} \] where \(e\) is the vector of all ones, and \(Q\) is an \(n\) by \(n\) positive semidefinite matrix, \(Q_{ij} \equiv y_i y_j K(x_i, x_j)\), where \(K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\) is the kernel. The terms \(\alpha_i\) are called the dual coefficients, and they are upper-bounded by \(C\). This dual representation highlights the fact that training vectors are implicitly mapped into a higher (maybe infinite) dimensional space by the function \(\phi\): see kernel trick. Once the optimization problem is solved, the output of decision_function for a given sample \(x\) becomes: \[\sum_{i\in SV} y_i \alpha_i K(x_i, x) + b,\] and the predicted class correspond to its sign. We only need to sum over the support vectors (i.e. the samples that lie within the margin) because the dual coefficients \(\alpha_i\) are zero for the other samples. These parameters can be accessed through the
attributes Note While SVM models derived from libsvm and liblinear use 1.4.7.2. LinearSVC¶The primal problem can be equivalently formulated as \[\min_ {w, b} \frac{1}{2} w^T w + C \sum_{i=1}^{n}\max(0, 1 - y_i (w^T \phi(x_i) + b)),\] where we make use of the hinge loss. This is the form that is directly optimized by
1.4.7.3. NuSVC¶The \(\nu\)-SVC formulation [15] is a reparameterization of the \(C\)-SVC and therefore mathematically equivalent. We introduce a new parameter \(\nu\) (instead of \(C\)) which controls the number of support vectors and margin errors: \(\nu \in (0, 1]\) is an upper bound on the fraction of margin errors and a lower bound of the fraction of support vectors. A margin error corresponds to a sample that lies on the wrong side of its margin boundary: it is either misclassified, or it is correctly classified but does not lie beyond the margin. 1.4.7.4. SVR¶Given training vectors \(x_i \in \mathbb{R}^p\), i=1,…, n, and a vector \(y \in \mathbb{R}^n\) \(\varepsilon\)-SVR solves the following primal problem: \[ \begin{align}\begin{aligned}\min_ {w, b, \zeta, \zeta^*} \frac{1}{2} w^T w + C \sum_{i=1}^{n} (\zeta_i + \zeta_i^*)\\\begin{split}\textrm {subject to } & y_i - w^T \phi (x_i) - b \leq \varepsilon + \zeta_i,\\ & w^T \phi (x_i) + b - y_i \leq \varepsilon + \zeta_i^*,\\ & \zeta_i, \zeta_i^* \geq 0, i=1, ..., n\end{split}\end{aligned}\end{align} \] Here, we are penalizing samples whose prediction is at least \(\varepsilon\) away from their true target. These samples penalize the objective by \(\zeta_i\) or \(\zeta_i^*\), depending on whether their predictions lie above or below the \(\varepsilon\) tube. The dual problem is \[ \begin{align}\begin{aligned}\min_{\alpha, \alpha^*} \frac{1}{2} (\alpha - \alpha^*)^T Q (\alpha - \alpha^*) + \varepsilon e^T (\alpha + \alpha^*) - y^T (\alpha - \alpha^*)\\\begin{split} \textrm {subject to } & e^T (\alpha - \alpha^*) = 0\\ & 0 \leq \alpha_i, \alpha_i^* \leq C, i=1, ..., n\end{split}\end{aligned}\end{align} \] where \(e\) is the vector of all ones, \(Q\) is an \(n\) by \(n\) positive semidefinite matrix, \(Q_{ij} \equiv K(x_i, x_j) = \phi (x_i)^T \phi (x_j)\) is the kernel. Here training vectors are implicitly mapped into a higher (maybe infinite) dimensional space by the function \(\phi\). The prediction is: \[\sum_{i \in SV}(\alpha_i - \alpha_i^*) K(x_i, x) + b\] These parameters can be accessed through the attributes 1.4.7.5. LinearSVR¶The primal problem can be equivalently formulated as \[\min_ {w, b} \frac{1}{2} w^T w + C \sum_{i=1}\max(0, |y_i - (w^T \phi(x_i) + b)| - \varepsilon),\] where we make use of the epsilon-insensitive loss, i.e. errors of less than \(\varepsilon\) are ignored. This is the form that is directly optimized by
1.4.8. Implementation details¶Internally, we use libsvm [12] and liblinear [11] to handle all computations. These libraries are wrapped using C and Cython. For a description of the implementation and details of the algorithms used, please refer to their respective papers. |