Python-sklearn-降维

Sklearn 降维与流形学习

sklearn.decomposition和sklearn.manifold提供 PCA、SVD、NMF、t-SNE 等降维工具。

🎯 矩阵分解

1.PCA— 主成分分析 ⭐

fromsklearn.decompositionimportPCA model=PCA(n_components=None,# 保留的成分数# int: 成分数# float (0~1): 保留的方差比例# 'mle': 自动选择（Minka's MLE）# None: 保留所有 min(n_samples, n_features)copy=True,whiten=False,# 白化处理svd_solver='auto',# 'auto','full','arpack','randomized'tol=0.0,# arpack 的容差iterated_power='auto',# randomized 的幂迭代次数n_oversamples=10,# randomized 的过采样数random_state=None,)model.fit(X)# 核心属性print(model.components_)# 主成分（特征向量）(n_components, n_features)print(model.explained_variance_)# 各成分的方差 ⭐print(model.explained_variance_ratio_)# 各成分的方差比例 ⭐print(model.singular_values_)# 奇异值print(model.mean_)# 训练数据的均值print(model.n_components_)# 实际成分数print(model.n_features_)# 特征数print(model.n_samples_)# 样本数print(model.noise_variance_)# 噪声方差# 变换X_pca=model.transform(X)X_pca=model.fit_transform(X)# 逆变换（近似重建）X_reconstructed=model.inverse_transform(X_pca)# 增量 PCA（大数据集）pca.partial_fit(X_batch)# 不适用于普通 PCA

2.IncrementalPCA— 增量 PCA

fromsklearn.decompositionimportIncrementalPCA model=IncrementalPCA(n_components=None,whiten=False,copy=True,batch_size=None# 每批样本数)# 分批拟合forbatchinnp.array_split(X,10):model.partial_fit(batch)X_pca=model.transform(X)

3.KernelPCA— 核 PCA

fromsklearn.decompositionimportKernelPCA model=KernelPCA(n_components=None,kernel='linear',# 'linear','poly','rbf','sigmoid','cosine','precomputed'gamma=None,# rbf/poly/sigmoid 的参数degree=3,# poly 次数coef0=1,# poly/sigmoid 的独立项kernel_params=None,alpha=1.0,# 学习到的逆变换正则化参数fit_inverse_transform=False,eigen_solver='auto',# 'auto','dense','arpack','randomized'tol=0,max_iter=None,iterated_power='auto',remove_zero_eig=False,random_state=None,copy_X=True,n_jobs=None)model.fit(X)X_kpca=model.transform(X)# 逆变换（需 fit_inverse_transform=True）X_reconstructed=model.inverse_transform(X_kpca)

4.SparsePCA/MiniBatchSparsePCA— 稀疏 PCA

fromsklearn.decompositionimportSparsePCA,MiniBatchSparsePCA model=SparsePCA(n_components=None,alpha=1,# 稀疏控制参数ridge_alpha=0.01,# 岭惩罚max_iter=1000,tol=1e-8,method='lars',# 'lars' 或 'cd'（坐标下降）n_jobs=None,random_state=None)model.fit(X)print(model.components_)# MiniBatch 版本（大数据）model=MiniBatchSparsePCA(n_components=None,alpha=1,batch_size=100,random_state=42)model.fit(X)

5.TruncatedSVD— 截断 SVD ⭐

不中心化数据，直接进行 SVD 分解（适合稀疏矩阵，如 TF-IDF）。

fromsklearn.decompositionimportTruncatedSVD model=TruncatedSVD(n_components=2,algorithm='randomized',# 'arpack' 或 'randomized'n_iter=5,# 幂迭代次数（randomized）n_oversamples=10,random_state=None,tol=0.0)model.fit(X)print(model.components_)print(model.explained_variance_)print(model.explained_variance_ratio_)print(model.singular_values_)X_svd=model.transform(X)X_approx=model.inverse_transform(X_svd)

6.NMF— 非负矩阵分解

fromsklearn.decompositionimportNMF model=NMF(n_components=None,init=None,# 'random','nndsvd','nndsvda','nndsvdar','custom'solver='cd',# 'cd'（坐标下降）或 'mu'（乘法更新）beta_loss='frobenius',# 'frobenius' 或 'kullback-leibler' 或 floattol=1e-4,max_iter=200,random_state=None,alpha_W=0.0,# W 正则化alpha_H='same',# H 正则化（'same' 或 float）l1_ratio=0.0,# L1/L2 比率（0=仅L2, 1=仅L1）shuffle=False,# 按样本划分时的打乱顺序verbose=0)model.fit(X)print(model.components_)# H 矩阵 (n_components, n_features)print(model.n_components_)print(model.reconstruction_err_)print(model.n_iter_)# W 矩阵（样本在成分上的表示）W=model.transform(X)X_approx=model.inverse_transform(W)

7.DictionaryLearning— 字典学习

fromsklearn.decompositionimportDictionaryLearning model=DictionaryLearning(n_components=None,alpha=1,# 稀疏控制max_iter=1000,tol=1e-8,fit_algorithm='lars',# 'lars' 或 'cd'transform_algorithm='omp',# 'lars','lasso_lars','lasso_cd','omp','threshold'transform_n_nonzero_coefs=None,transform_alpha=None,n_jobs=None,code_init=None,dict_init=None,callback=None,verbose=False,random_state=None)model.fit(X)print(model.components_)# 字典

8.FactorAnalysis— 因子分析

fromsklearn.decompositionimportFactorAnalysis model=FactorAnalysis(n_components=None,tol=1e-2,copy=True,max_iter=1000,noise_variance_init=None,# 噪声方差初始值svd_method='randomized',iterated_power=3,rotation='varimax',# None 或 'varimax'（正交旋转）random_state=0)model.fit(X)X_transformed=model.transform(X)print(model.components_)print(model.noise_variance_)print(model.loglike_)

9.FastICA— 独立成分分析（ICA）

fromsklearn.decompositionimportFastICA model=FastICA(n_components=None,algorithm='parallel',# 'parallel' 或 'deflation'whiten='unit-variance',# 'unit-variance' 或 True/Falsefun='logcosh',# 'logcosh','exp','cube'fun_args=None,# 函数参数max_iter=200,tol=1e-4,w_init=None,whiten_solver='svd',random_state=None)model.fit(X)print(model.components_)# 独立成分print(model.mixing_)# 混合矩阵print(model.mean_)# 各特征的均值print(model.n_iter_)# 迭代次数X_ica=model.transform(X)X_reconstructed=model.inverse_transform(X_ica)

10.LatentDirichletAllocation— LDA 主题模型

fromsklearn.decompositionimportLatentDirichletAllocation model=LatentDirichletAllocation(n_components=10,# 主题数doc_topic_prior=None,# Dirichlet 先验 αtopic_word_prior=None,# Dirichlet 先验 βlearning_method='batch',# 'batch' 或 'online'learning_decay=0.7,# online 学习率衰减learning_offset=10.0,max_iter=10,batch_size=128,evaluate_every=-1,# 每多少次迭代评估困惑度total_samples=1e6,perp_tol=1e-1,mean_change_tol=1e-3,max_doc_update_iter=100,n_jobs=None,verbose=0,random_state=None)model.fit(X)print(model.components_)# 主题-词矩阵 (n_topics, n_words)print(model.n_batch_iter_)# batch 方法的迭代次数print(model.bound_)# 变分下界# 文档-主题分布doc_topics=model.transform(X)# (n_docs, n_topics)

🌀 流形学习（Manifold Learning）

1.TSNE— t-SNE ⭐

fromsklearn.manifoldimportTSNE model=TSNE(n_components=2,# 嵌入维度（通常 2 或 3）perplexity=30.0,# 有效邻居数 (5-50)early_exaggeration=12.0,# 早期放大因子learning_rate='auto',# 学习率 (10-1000)，'auto'=max(N/12, 200)n_iter=1000,# 迭代优化次数n_iter_without_progress=300,# 无改善则提前停止min_grad_norm=1e-7,metric='euclidean',# 距离度量metric_params=None,init='pca',# 'random','pca' 或 ndarrayverbose=0,random_state=None,method='barnes_hut',# 'barnes_hut' 或 'exact'angle=0.5,# barnes_hut 的角度-速度权衡n_jobs=None,perplexity_max_iter=100# 1.3+: perplexity 校准最大迭代)X_embedded=model.fit_transform(X)# 关键属性print(model.n_iter_)# 实际迭代数print(model.kl_divergence_)# 最终的 KL 散度print(model.embedding_)# 嵌入坐标print(model.learning_rate_)# 实际使用的学习率

调参指导:

# perplexity: 5-50，数据集越大值越大# learning_rate: 10-1000# n_iter: 至少 250，通常 1000-5000# 建议多次运行取最佳结果# 大规模数据集（>5000 样本）先 PCA 降维再 t-SNEfromsklearn.pipelineimportmake_pipelinefromsklearn.decompositionimportPCA pipeline=make_pipeline(PCA(n_components=50,random_state=42),TSNE(n_components=2,random_state=42))X_embedded=pipeline.fit_transform(X)

2.MDS— 多维缩放

fromsklearn.manifoldimportMDS model=MDS(n_components=2,metric=True,# True=度量 MDS, False=非度量 MDSn_init=4,# SMACOF 运行次数max_iter=300,eps=1e-3,# 收敛容差n_jobs=None,random_state=None,dissimilarity='euclidean',# 'euclidean' 或 'precomputed'normalized_stress='auto'# 是否返回标准化应力)X_embedded=model.fit_transform(X)print(model.stress_)# 应力值（越低越好）print(model.n_iter_)print(model.embedding_)

3.Isomap— 等度量映射

fromsklearn.manifoldimportIsomap model=Isomap(n_neighbors=5,# 邻居数n_components=2,eigen_solver='auto',# 'auto','arpack','dense'tol=0,max_iter=None,path_method='auto',# 'auto','FW'(Floyd-Warshall),'D'(Dijkstra)neighbors_algorithm='auto',# 'auto','brute','kd_tree','ball_tree'n_jobs=None,metric='minkowski',p=2,metric_params=None)X_embedded=model.fit_transform(X)print(model.embedding_)print(model.dist_matrix_)# 训练数据的全对最短路径距离print(model.nbrs_)# NearestNeighbors 实例print(model.kernel_pca_)# KernelPCA 实例print(model.n_features_in_)

4.LocallyLinearEmbedding— 局部线性嵌入（LLE）

fromsklearn.manifoldimportLocallyLinearEmbedding model=LocallyLinearEmbedding(n_neighbors=5,n_components=2,reg=0.001,# 正则化常数eigen_solver='auto',tol=1e-6,max_iter=100,method='standard',# 'standard','hessian','modified','ltsa'hessian_tol=0.0001,# Hessian LLE 的容差modified_tol=1e-12,# Modified LLE 的容差neighbors_algorithm='auto',random_state=None,n_jobs=None)X_embedded=model.fit_transform(X)print(model.embedding_)print(model.reconstruction_error_)# 与 reconstruction error_ 相关联print(model.nbrs_)

5.SpectralEmbedding— 谱嵌入

fromsklearn.manifoldimportSpectralEmbedding model=SpectralEmbedding(n_components=2,affinity='nearest_neighbors',# 'nearest_neighbors','rbf','precomputed',callablegamma=None,# RBF 核参数random_state=None,eigen_solver=None,# None,'arpack','lobpcg','amg'eigen_tol='auto',n_neighbors=None,# nearest_neighbors 的邻居数n_jobs=None)X_embedded=model.fit_transform(X)print(model.embedding_)print(model.affinity_matrix_)

📊 PCA 实用技巧

确定保留的方差

importnumpyasnpimportmatplotlib.pyplotaspltfromsklearn.decompositionimportPCA pca=PCA().fit(X)# 累积方差比例cumsum=np.cumsum(pca.explained_variance_ratio_)# 找到 95% 方差所需的成分数n_95=np.argmax(cumsum>=0.95)+1print(f"Components for 95% variance:{n_95}")# 可视化fig,(ax1,ax2)=plt.subplots(1,2,figsize=(14,5))# 碎石图ax1.bar(range(1,len(pca.explained_variance_ratio_)+1),pca.explained_variance_ratio_)ax1.set_xlabel('Principal Component')ax1.set_ylabel('Explained Variance Ratio')# 累积方差ax2.plot(range(1,len(cumsum)+1),cumsum,'bo-')ax2.axhline(y=0.95,color='r',linestyle='--',label='95%')ax2.axvline(x=n_95,color='r',linestyle='--')ax2.set_xlabel('Number of Components')ax2.set_ylabel('Cumulative Explained Variance')ax2.legend()plt.tight_layout()plt.show()

PCA 逆变换（重建）

pca=PCA(n_components=0.95)# 只保留 95% 方差的成分X_reduced=pca.fit_transform(X)X_reconstructed=pca.inverse_transform(X_reduced)# 计算重建误差reconstruction_error=np.mean((X-X_reconstructed)**2)

📝 算法选择指南

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