[机器学习读书笔记] - PCA降维

1. 原理

blog.csdn.net/u010376788/…

Python 代码：

def compute_pca(X, n_components=2):
    """
    Input:
        X: of dimension (m,n) where each row corresponds to an observation, and each column is a feature (variable)
        n_components: Number of components you want to keep.
    Output:
        X_reduced: data transformed in reduced dimension
    """

    # mean center the data
    X_demeaned = X - np.mean(X,axis=0)
    
    # calculate the covariance matrix
    covariance_matrix = np.cov(X_demeaned, rowvar=False)

    # calculate eigenvectors & eigenvalues of the covariance matrix
    eigen_vals, eigen_vecs = np.linalg.eigh(covariance_matrix, UPLO='L')
    
    # sort eigenvalue in increasing order (get the indices from the sort)
    idx_sorted = np.argsort(eigen_vals)
    
    # reverse the order so that it's from highest to lowest.
    idx_sorted_decreasing = idx_sorted[::-1]
    
    # sort the eigen values by idx_sorted_decreasing
    eigen_vals_sorted = eigen_vals[idx_sorted_decreasing]

    # sort eigenvectors using the idx_sorted_decreasing indices
    eigen_vecs_sorted = eigen_vecs[:,idx_sorted_decreasing]

    # select the first n eigenvectors (n is desired dimension
    # of rescaled data array, or dims_rescaled_data)
    eigen_vecs_subset = eigen_vecs_sorted[:,0:n_components]

    # transform the data by multiplying the transpose of the eigenvectors 
    # with the transpose of the de-meaned data
    # Then take the transpose of that product.
    X_reduced = np.dot(eigen_vecs_subset.transpose(),X_demeaned.transpose()).transpose()

    return X_reduced
    
X = np.random.rand(3, 10)
print(X)
X_reduced = compute_pca(X, n_components=2)
print("Your original matrix was " + str(X.shape) + " and it became:")
print(X_reduced)