萌新向Python数据分析及数据挖掘 第三章 机器学习常用算法 第四节 PCA与梯度上升 （下）实操篇

import numpy as np
from sklearn import datasets# 载入数据包

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digits = datasets.load_digits()#读取数据
X = digits.data#定义X
y = digits.target#定义y

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from sklearn.model_selection import train_test_split #载入数据切分工具
X_train, X_test, y_train, y_test = train_test_split(X,y,test_size = 0.2)#数据切分

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%time
from sklearn.neighbors import KNeighborsClassifier #载入KNN分类器
knn_clf = KNeighborsClassifier()# 设置分类器
knn_clf.fit(X_train,y_train)

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knn_clf.score(X_test,y_test)

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from sklearn.decomposition import PCA
pca = PCA(n_components=2)
pca.fit(X_train)
X_train_reduction = pca.transform(X_train)
X_test_reduction = pca.transform(X_test)

 

Init signature: PCA(n_components=None, copy=True, whiten=False, svd_solver='auto', tol=0.0, iterated_power='auto', random_state=None) Docstring:
Principal component analysis (PCA)

Linear dimensionality reduction using Singular Value Decomposition of the data to project it to a lower dimensional space.

It uses the LAPACK implementation of the full SVD or a randomized truncated SVD by the method of Halko et al. 2009, depending on the shape of the input data and the number of components to extract.

It can also use the scipy.sparse.linalg ARPACK implementation of the truncated SVD.

Notice that this class does not support sparse input. See :class:TruncatedSVD for an alternative with sparse data.

Read more in the :ref:User Guide <PCA>.

Parameters

n_components : int, float, None or string Number of components to keep. if n_components is not set all components are kept::

    n_components == min(n_samples, n_features)

if n_components == 'mle' and svd_solver == 'full', Minka's MLE is used
to guess the dimension
if ``0 < n_components < 1`` and svd_solver == 'full', select the number
of components such that the amount of variance that needs to be
explained is greater than the percentage specified by n_components
n_components cannot be equal to n_features for svd_solver == 'arpack'.

copy : bool (default True) If False, data passed to fit are overwritten and running fit(X).transform(X) will not yield the expected results, use fit_transform(X) instead.

whiten : bool, optional (default False) When True (False by default) the components_ vectors are multiplied by the square root of n_samples and then divided by the singular values to ensure uncorrelated outputs with unit component-wise variances.

Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometime
improve the predictive accuracy of the downstream estimators by
making their data respect some hard-wired assumptions.

svd_solver : string {'auto', 'full', 'arpack', 'randomized'} auto : the solver is selected by a default policy based on X.shape and n_components: if the input data is larger than 500x500 and the number of components to extract is lower than 80% of the smallest dimension of the data, then the more efficient 'randomized' method is enabled. Otherwise the exact full SVD is computed and optionally truncated afterwards. full : run exact full SVD calling the standard LAPACK solver via scipy.linalg.svd and select the components by postprocessing arpack : run SVD truncated to n_components calling ARPACK solver viascipy.sparse.linalg.svds. It requires strictly 0 < n_components < X.shape[1] randomized : run randomized SVD by the method of Halko et al.

.. versionadded:: 0.18.0

tol : float >= 0, optional (default .0) Tolerance for singular values computed by svd_solver == 'arpack'.

.. versionadded:: 0.18.0

iterated_power : int >= 0, or 'auto', (default 'auto') Number of iterations for the power method computed by svd_solver == 'randomized'.

.. versionadded:: 0.18.0

random_state : int, RandomState instance or None, optional (default None) If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random. Used when svd_solver == 'arpack' or 'randomized'.

.. versionadded:: 0.18.0

Attributes

components_ : array, shape (n_components, n_features) Principal axes in feature space, representing the directions of maximum variance in the data. The components are sorted by explained_variance_.

explainedvariance : array, shape (n_components,) The amount of variance explained by each of the selected components.

Equal to n_components largest eigenvalues
of the covariance matrix of X.

.. versionadded:: 0.18

explained_varianceratio : array, shape (n_components,) Percentage of variance explained by each of the selected components.

If ``n_components`` is not set then all components are stored and the
sum of explained variances is equal to 1.0.

singularvalues : array, shape (n_components,) The singular values corresponding to each of the selected components. The singular values are equal to the 2-norms of the n_components variables in the lower-dimensional space.

mean_ : array, shape (n_features,) Per-feature empirical mean, estimated from the training set.

Equal to `X.mean(axis=0)`.

ncomponents : int The estimated number of components. When n_components is set to 'mle' or a number between 0 and 1 (with svd_solver == 'full') this number is estimated from input data. Otherwise it equals the parameter n_components, or n_features if n_components is None.

noisevariance : float The estimated noise covariance following the Probabilistic PCA model from Tipping and Bishop 1999. See "Pattern Recognition and Machine Learning" by C. Bishop, 12.2.1 p. 574 orhttp://www.miketipping.com/papers/met-mppca.pdf. It is required to computed the estimated data covariance and score samples.

Equal to the average of (min(n_features, n_samples) - n_components)
smallest eigenvalues of the covariance matrix of X.

References

For n_components == 'mle', this class uses the method of Thomas P. Minka: Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604

Implements the probabilistic PCA model from: M. Tipping and C. Bishop, Probabilistic Principal Component Analysis, Journal of the Royal Statistical Society, Series B, 61, Part 3, pp. 611-622 via the score and score_samples methods. See http://www.miketipping.com/papers/met-mppca.pdf

For svd_solver == 'arpack', refer to scipy.sparse.linalg.svds.

For svd_solver == 'randomized', see: Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions Halko, et al., 2009 (arXiv:909) A randomized algorithm for the decomposition of matrices Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert

Examples

import numpy as np from sklearn.decomposition import PCA X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]]) pca = PCA(n_components=2) pca.fit(X) PCA(copy=True, iterated_power='auto', n_components=2, random_state=None, svd_solver='auto', tol=0.0, whiten=False) print(pca.explained_varianceratio) # doctest: +ELLIPSIS [ 0.99244... 0.00755...] print(pca.singularvalues) # doctest: +ELLIPSIS [ 6.30061... 0.54980...]

pca = PCA(n_components=2, svd_solver='full') pca.fit(X) # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE PCA(copy=True, iterated_power='auto', n_components=2, random_state=None, svd_solver='full', tol=0.0, whiten=False) print(pca.explained_varianceratio) # doctest: +ELLIPSIS [ 0.99244... 0.00755...] print(pca.singularvalues) # doctest: +ELLIPSIS [ 6.30061... 0.54980...]

pca = PCA(n_components=1, svd_solver='arpack') pca.fit(X) PCA(copy=True, iterated_power='auto', n_components=1, random_state=None, svd_solver='arpack', tol=0.0, whiten=False) print(pca.explained_varianceratio) # doctest: +ELLIPSIS [ 0.99244...] print(pca.singularvalues) # doctest: +ELLIPSIS [ 6.30061...]

See also

KernelPCA SparsePCA TruncatedSVD IncrementalPCA

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knn_clf = KNeighborsClassifier()# 设置分类器
knn_clf.fit(X_train_reduction,y_train)
knn_clf.score(X_test_reduction,y_test)

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pca.explained_variance_ratio_

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%time

knn_clf = KNeighborsClassifier()# 设置分类器
knn_clf.fit(X_train_reduction,y_train)

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 pca = PCA(n_components=X_train.shape[1])
 pca.fit(X_train) 
pca.explained_variance_ratio_
    

 

Signature: pca.fit(X, y=None) Docstring: Fit the model with X.

Parameters

X : array-like, shape (n_samples, n_features) Training data, where n_samples in the number of samples and n_features is the number of features.

y : Ignored.

Returns

self : object Returns the instance itself.

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pca = PCA(0.95)
pca.fit(X_train)

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pca.n_components_

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X_train_reduction = pca.transform(X_train)
X_test_reduction = pca.transform(X_test)

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knn_clf = KNeighborsClassifier()# 设置分类器
knn_clf.fit(X_train_reduction,y_train)

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knn_clf.score(X_test_reduction,y_test)

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pca = PCA(n_components=2)
pca.fit(X)
X_reduction = pca.transform(X)
import matplotlib.pyplot as plt

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for i  in range(10):
    plt.scatter(X_reduction[y==i,0],X_reduction[y==i,1],alpha=0.8)
plt.show()

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import numpy as np
from sklearn.datasets import fetch_mldata

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mnist = fetch_mldata('MNIST original')

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mnist

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X, y = mnist['data'],  mnist['target']

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X_train = np.array(X[:60000],dtype=float)
Y_train = np.array(y[:60000 ],dtype=float)
X_test = np.array(X[60000:],dtype=float)
Y_test = np.array(y[60000:],dtype=float)

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from sklearn.neighbors import KNeighborsClassifier #载入KNN分类器
knn_clf = KNeighborsClassifier()# 设置分类器 
%time knn_clf.fit(X_train,Y_train)

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 %time knn_clf.score(X_test,Y_test)

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from sklearn.decomposition import PCA
pca = PCA(0.9)
pca.fit(X_train)
X_train_reduction = pca.transform(X_train)
X_test_reduction = pca.transform(X_test)

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X_train_reduction.shape

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knn_clf = KNeighborsClassifier()# 设置分类器
knn_clf.fit(X_train_reduction,Y_train)

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knn_clf.score(X_test_reduction,Y_test)

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from sklearn.datasets import fetch_lfw_people #人脸数据库

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faces = fetch_lfw_people()

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faces

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faces.keys()

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faces.data.shape

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faces.images.shape

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random_indexes = np.random.permutation(len(faces.data))
X = faces.data[random_indexes]

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example_faces = X[:36,:]
example_faces.shape

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def plot_faces(faces):
    fig,axes = plt.subplots(6,6,figsize=(10,10),
                            subplot_kw={'xticks':[],'yticks':[]},
                            gridspec_kw=dict(hspace=0.1,wspace=0.1))
    for i, ax in enumerate(axes.flat):
        ax.imshow(faces[i].reshape(62,47),cmap='bone')
    plt.show()

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 plot_faces(example_faces)

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faces.target_names

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len(faces.target_names)

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%%time
from sklearn.decomposition import PCA
pca = PCA(svd_solver='randomized')
pca.fit(X)

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pca.components_.shape

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plot_faces(pca.components_[:36])#绘制主成分排前36的特征脸

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faces2 = fetch_lfw_people(min_faces_per_person=60)

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faces2.data.shape