주성분 분석

introduction

• The principal components of matrix are linear transformations of the original columns into uncorrelated columns arranged in order of decreasing variance

memo

• https://math.stackexchange.com/questions/3869/what-is-the-intuitive-relationship-between-svd-and-pca
• https://mathematica.stackexchange.com/questions/50987/principal-components-how-to-obtain-linear-transformations
• https://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues

computational resource

• https://drive.google.com/file/d/0B8XXo8Tve1cxT0hBUmdPLUd1VHM/view
• https://jakevdp.github.io/PythonDataScienceHandbook/05.09-principal-component-analysis.html

관련된 항목들

• 특이값 분해

노트

• The first step in PCA is to draw a new axis representing the direction of maximum variation through the data.
• This is because a significant feature is one which exhibits differences between groups, and PCA captures differences between groups.
• Therefore, using significant features for the PCA will always see some sort of grouping.
• This is simply because PCA captures the variation that exists in the feature data and you have chosen all features.
• Principal Component Analysis and Factor Analysis are data reduction methods to re-express multivariate data with fewer dimensions.
• PCA is closely related to the Karhunen-Loève (KL) expansion.
• PCA, the eigenvectors \(\vec{\varphi}_i\) of the covariance matrix \(\Sigma\) are usually referred to as principal components or eigenmodes.
• Please note that PCA is sensitive to the relative scaling of the original attributes.
• In this chapter, we describe the basic idea of PCA and, demonstrate how to compute and visualize PCA using R software.
• Basics Understanding the details of PCA requires knowledge of linear algebra.
• PCA assumes that the directions with the largest variances are the most “important” (i.e, the most principal).
• Note that, the PCA method is particularly useful when the variables within the data set are highly correlated.
• XLSTAT provides a complete and flexible PCA feature to explore your data directly in Excel.
• PCA dimensions are also called axes or Factors.
• PCA can thus be considered as a Data Mining method as it allows to easily extract information from large datasets.
• XLSTAT lets you add variables (qualitative or quantitative) or observations to the PCA after it has been computed.
• The first edition of this book was the first comprehensive text written solely on principal component analysis.
• In order to achieve this, principal component analysis (PCA) was conducted on joint moment waveform data from the hip, knee and ankle.
• PCA was also performed comparing all data from each individual across CMJnas and CMJas conditions.
• PCA was used in this study to extract common patterns of moment production during the vertical jump under two task constraints.
• In biomechanics, PCA has sometimes been used to compare time-normalized waveforms.
• Hence, PCA allows us to find the direction along which our data varies the most.
• Applying PCA to N-dimensional data set yields N N-dimensional eigenvectors, N eigenvalues and 1 N-dimensional center point.
• A simple example is provided by comparing the singular spectrum from a singular value decomposition (SVD) with that of a traditional PCA.
• Note the robustness of PCA.
• Components are then grouped into subspaces preserving the order determined by the maximum variance property of PCA.
• λ N represent the eigenvalues from a PCA of the data.
• Principal Component Analysis is an appropriate tool for removing the collinearity.
• Right-click on the tab of PCA Plot Data1 and select Duplicate.
• The new sheet is named as PCA Plot Data2.
• Because of the versatility and interpretability of PCA, it has been shown to be effective in a wide variety of contexts and disciplines.
• PCA's main weakness is that it tends to be highly affected by outliers in the data.
• In the following sections, we will look at other unsupervised learning methods that build on some of the ideas of PCA.
• Find the principal components for one data set and apply the PCA to another data set.
• For example, you can preprocess the training data set by using PCA and then train a model.
• Use coeff (principal component coefficients) and mu (estimated means of XTrain ) to apply the PCA to a test data set.
• To use the trained model for the test set, you need to transform the test data set by using the PCA obtained from the training data set.
• The estimated noise covariance following the Probabilistic PCA model from Tipping and Bishop 1999.
• Implements the probabilistic PCA model from: Tipping, M. E., and Bishop, C. M. (1999).
• Some of the important differences and similarities between PCA and MLPCA are summarized in Table 2 and are briefly discussed here.
• One of the most convenient features of PCA that is lost in the transition to MLPCA is the simultaneous estimation of all subspace models.
• Of course, some properties of PCA remain the same for MLPCA.
• In addition, the columns of U and V remain orthonormal for both PCA and MLPCA.
• The PCA score plot of the first two PCs of a data set about food consumption profiles.
• Principal Component Analysis is a dimension-reduction tool that can be used advantageously in such situations.
• The main idea behind principal component analysis is to derive a linear function \({\bf y}\) for each of the vector variables \({\bf z}_i\).
• But if we want to tease out variation, PCA finds a new coordinate system in which every point has a new (x,y) value.
• PCA is useful for eliminating dimensions.
• 3D example With three dimensions, PCA is more useful, because it's hard to see through a cloud of data.
• To see the "official" PCA transformation, click the "Show PCA" button.
• In this section we will start by visualizing the data as well as consider a simplified, geometric view of what a PCA model look like.
• The PCA method starts with the "Road" class and computes the mean value for each attribute for that class.
• The PCA method computes class scores based on the training samples you select.
• Intensive Principal Component Analysis Classical PCA takes a set of data examples and infers features which are linearly uncorrelated.
• The features to be analyzed with PCA are compared via their Euclidean distance.
• This arises because both InPCA and PCA/MDS rely on mean shifing the input data before finding an eigenbasis.
• Thus, we view InPCA as a natural generalization of PCA to probability distributions and MDS to non-Euclidean embeddings.
• As an added benefit, each of the “new” variables after PCA are all independent of one another.
• If you answered “yes” to all three questions, then PCA is a good method to use.
• Our original data transformed by PCA.
• Here, I walk through an algorithm for conducting PCA.
• PCA is used in exploratory data analysis and for making predictive models.
• PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis.
• PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component.
• PCA essentially rotates the set of points around their mean in order to align with the principal components.
• This article will begin by introducing the basic ideas of PCA, discussing what it can and cannot do.
• Many techniques have been developed for this purpose, but principal component analysis (PCA) is one of the oldest and most widely used.
• PCA can be based on either the covariance matrix or the correlation matrix.
• Section 3c discusses one of the extensions of PCA that has been most active in recent years, namely robust PCA (RPCA).