# 주성분 분석

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## 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

## 노트

- The first step in PCA is to draw a new axis representing the direction of maximum variation through the data.
^{[1]} - This is because a significant feature is one which exhibits differences between groups, and PCA captures differences between groups.
^{[1]} - Therefore, using significant features for the PCA will always see some sort of grouping.
^{[1]} - This is simply because PCA captures the variation that exists in the feature data and you have chosen all features.
^{[1]} - Principal Component Analysis and Factor Analysis are data reduction methods to re-express multivariate data with fewer dimensions.
^{[2]} - PCA is closely related to the Karhunen-Loève (KL) expansion.
^{[3]} - PCA, the eigenvectors \(\vec{\varphi}_i\) of the covariance matrix \(\Sigma\) are usually referred to as principal components or eigenmodes.
^{[3]} - Please note that PCA is sensitive to the relative scaling of the original attributes.
^{[4]} - In this chapter, we describe the basic idea of PCA and, demonstrate how to compute and visualize PCA using R software.
^{[5]} - Basics Understanding the details of PCA requires knowledge of linear algebra.
^{[5]} - PCA assumes that the directions with the largest variances are the most “important” (i.e, the most principal).
^{[5]} - Note that, the PCA method is particularly useful when the variables within the data set are highly correlated.
^{[5]} - XLSTAT provides a complete and flexible PCA feature to explore your data directly in Excel.
^{[6]} - PCA dimensions are also called axes or Factors.
^{[6]} - PCA can thus be considered as a Data Mining method as it allows to easily extract information from large datasets.
^{[6]} - XLSTAT lets you add variables (qualitative or quantitative) or observations to the PCA after it has been computed.
^{[6]} - The first edition of this book was the first comprehensive text written solely on principal component analysis.
^{[7]} - In order to achieve this, principal component analysis (PCA) was conducted on joint moment waveform data from the hip, knee and ankle.
^{[8]} - PCA was also performed comparing all data from each individual across CMJnas and CMJas conditions.
^{[8]} - PCA was used in this study to extract common patterns of moment production during the vertical jump under two task constraints.
^{[8]} - In biomechanics, PCA has sometimes been used to compare time-normalized waveforms.
^{[8]} - Hence, PCA allows us to find the direction along which our data varies the most.
^{[9]} - Applying PCA to N-dimensional data set yields N N-dimensional eigenvectors, N eigenvalues and 1 N-dimensional center point.
^{[9]} - A simple example is provided by comparing the singular spectrum from a singular value decomposition (SVD) with that of a traditional PCA.
^{[10]} - Note the robustness of PCA.
^{[10]} - Components are then grouped into subspaces preserving the order determined by the maximum variance property of PCA.
^{[10]} - λ N represent the eigenvalues from a PCA of the data.
^{[10]} - Principal Component Analysis is an appropriate tool for removing the collinearity.
^{[11]} - Right-click on the tab of PCA Plot Data1 and select Duplicate.
^{[11]} - The new sheet is named as PCA Plot Data2.
^{[11]} - Because of the versatility and interpretability of PCA, it has been shown to be effective in a wide variety of contexts and disciplines.
^{[12]} - PCA's main weakness is that it tends to be highly affected by outliers in the data.
^{[12]} - In the following sections, we will look at other unsupervised learning methods that build on some of the ideas of PCA.
^{[12]} - Find the principal components for one data set and apply the PCA to another data set.
^{[13]} - For example, you can preprocess the training data set by using PCA and then train a model.
^{[13]} - Use coeff (principal component coefficients) and mu (estimated means of XTrain ) to apply the PCA to a test data set.
^{[13]} - 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.
^{[13]} - The estimated noise covariance following the Probabilistic PCA model from Tipping and Bishop 1999.
^{[14]} - Implements the probabilistic PCA model from: Tipping, M. E., and Bishop, C. M. (1999).
^{[14]} - Some of the important differences and similarities between PCA and MLPCA are summarized in Table 2 and are briefly discussed here.
^{[15]} - One of the most convenient features of PCA that is lost in the transition to MLPCA is the simultaneous estimation of all subspace models.
^{[15]} - Of course, some properties of PCA remain the same for MLPCA.
^{[15]} - In addition, the columns of U and V remain orthonormal for both PCA and MLPCA.
^{[15]} - The PCA score plot of the first two PCs of a data set about food consumption profiles.
^{[16]} - Principal Component Analysis is a dimension-reduction tool that can be used advantageously in such situations.
^{[17]} - The main idea behind principal component analysis is to derive a linear function \({\bf y}\) for each of the vector variables \({\bf z}_i\).
^{[17]} - But if we want to tease out variation, PCA finds a new coordinate system in which every point has a new (x,y) value.
^{[18]} - PCA is useful for eliminating dimensions.
^{[18]} - 3D example With three dimensions, PCA is more useful, because it's hard to see through a cloud of data.
^{[18]} - To see the "official" PCA transformation, click the "Show PCA" button.
^{[18]} - 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.
^{[19]} - The PCA method starts with the "Road" class and computes the mean value for each attribute for that class.
^{[20]} - The PCA method computes class scores based on the training samples you select.
^{[20]} - Intensive Principal Component Analysis Classical PCA takes a set of data examples and infers features which are linearly uncorrelated.
^{[21]} - The features to be analyzed with PCA are compared via their Euclidean distance.
^{[21]} - This arises because both InPCA and PCA/MDS rely on mean shifing the input data before finding an eigenbasis.
^{[21]} - Thus, we view InPCA as a natural generalization of PCA to probability distributions and MDS to non-Euclidean embeddings.
^{[21]} - As an added benefit, each of the “new” variables after PCA are all independent of one another.
^{[22]} - If you answered “yes” to all three questions, then PCA is a good method to use.
^{[22]} - Our original data transformed by PCA.
^{[22]} - Here, I walk through an algorithm for conducting PCA.
^{[22]} - PCA is used in exploratory data analysis and for making predictive models.
^{[23]} - PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis.
^{[23]} - PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component.
^{[23]} - PCA essentially rotates the set of points around their mean in order to align with the principal components.
^{[23]} - This article will begin by introducing the basic ideas of PCA, discussing what it can and cannot do.
^{[24]} - Many techniques have been developed for this purpose, but principal component analysis (PCA) is one of the oldest and most widely used.
^{[24]} - PCA can be based on either the covariance matrix or the correlation matrix.
^{[24]} - Section 3c discusses one of the extensions of PCA that has been most active in recent years, namely robust PCA (RPCA).
^{[24]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}What does Principal Component Analysis (PCA) show? - ↑ Principal Component Analysis
- ↑
^{3.0}^{3.1}Principal Component Analysis - ↑ Principal Component Analysis
- ↑
^{5.0}^{5.1}^{5.2}^{5.3}Principal Component Analysis Essentials - ↑
^{6.0}^{6.1}^{6.2}^{6.3}Principal Component Analysis (PCA) - ↑ Principal Component Analysis
- ↑
^{8.0}^{8.1}^{8.2}^{8.3}Principal Component Analysis Reveals the Proximal to Distal Pattern in Vertical Jumping Is Governed by Two Functional Degrees of Freedom - ↑
^{9.0}^{9.1}OpenCV: Introduction to Principal Component Analysis (PCA) - ↑
^{10.0}^{10.1}^{10.2}^{10.3}Component retention in principal component analysis with application to cDNA microarray data - ↑
^{11.0}^{11.1}^{11.2}Principal Component Analysis - ↑
^{12.0}^{12.1}^{12.2}In Depth: Principal Component Analysis - ↑
^{13.0}^{13.1}^{13.2}^{13.3}Principal component analysis of raw data - ↑
^{14.0}^{14.1}sklearn.decomposition.PCA — scikit-learn 0.23.2 documentation - ↑
^{15.0}^{15.1}^{15.2}^{15.3}Principal Component Analysis - an overview - ↑ What is principal component analysis (PCA) and how it is used?
- ↑
^{17.0}^{17.1}6.5.5. Principal Components - ↑
^{18.0}^{18.1}^{18.2}^{18.3}Principal Component Analysis explained visually - ↑ 6.5. Principal Component Analysis (PCA) — Process Improvement using Data
- ↑
^{20.0}^{20.1}Principal Components Analysis Background - ↑
^{21.0}^{21.1}^{21.2}^{21.3}Visualizing probabilistic models and data with Intensive Principal Component Analysis - ↑
^{22.0}^{22.1}^{22.2}^{22.3}A One-Stop Shop for Principal Component Analysis - ↑
^{23.0}^{23.1}^{23.2}^{23.3}Principal component analysis - ↑
^{24.0}^{24.1}^{24.2}^{24.3}Principal component analysis: a review and recent developments