The purpose of this post is to give the reader detailed understanding of Principal Component Analysis with the necessary mathematical proofs. In real world data analysis tasks we analyze complex data i.e. multi dimensional data. We plot the data and find various patterns in it or use it to train some machine learning models. One way to think about dimensions is that suppose you have an data point x, if we consider this data point as a physical object then dimensions are merely a basis of view, like where is the data located when it is observed from horizontal axis or vertical axis. As the dimensions of data increases, the difficulty to visualize it and perform computations on it also increases. Variance: It is a measure of the variability or it simply measures how spread the data set is. Mathematically, it is the average squared deviation from the mean score.

Below is an example combining PCA plots with code similar to the above. These plots illustrate something further with regard to the relationships among modalities. In property words, the different modalities spread out more clearly than they do in concept words. This makes sense because in language, properties define concepts . An example of this code is use is available here (with data here).

We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non-iterative, so is not vulnerable to a bad choice of initialisation. Our theory provides great detail on the statistical and computational trade-off in our procedure, revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate. Numerical studies provide further insight into the procedure and confirm its highly competitive finite-sample performance.

In this post, we will learn about Principal Component Analysis (PCA) -- a popular dimensionality reduction technique in Machine Learning. Our goal is to form an intuitive understanding of PCA without going into all the mathematical details. At the time of writing this post, the population of the United States is roughly 325 million. You may think millions of people will have a million different ideas, opinions, and thoughts, after all, every person is unique. Let's say you select 20 top political questions in the United States and ask millions of people to answer these questions using a yes or a no.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, MONTH 2016 1 Polar n -Complex and n -Bicomplex Singular V alue Decomposition and Principal Component Pursuit Tak-Shing T. Chan, Member, IEEE and Yi-Hsuan Y ang, Member, IEEE Abstract--Informed by recent work on tensor singular value decomposition and circulant algebra matrices, this paper presents a new theoretical bridge that unifies the hypercomplex and tensor-based approaches to singular value decomposition and robust principal component analysis. We begin our work by extending the principal component pursuit to Olariu's polar n - complex numbers as well as their bicomplex counterparts. In so doing, we have derived the polar n -complex and n -bicomplex proximity operators for both the 1-and trace-norm regularizers, which can be used by proximal optimization methods such as the alternating direction method of multipliers. Experimental results on two sets of audio data show that our algebraically-informed formulation outperforms tensor robust principal component analysis. We conclude with the message that an informed definition of the trace norm can bridge the gap between the hypercomplex and tensor-based approaches. Our approach can be seen as a general methodology for generating other principal component pursuit algorithms with proper algebraic structures. I NTRODUCTION T HE robust principal component analysis (RPCA) [1] has received a lot of attention lately in many application areas of signal processing [2]-[5]. Owing to the NPhardness of the above formulation, the principal component pursuit (PCP) [1] has been proposed to solve this relaxed problem instead [6]: min L, S ‖L ‖ λ‖S ‖ 1 s.t. X L S, (2) where ‖·‖ is the trace norm (sum of the singular values),‖·‖ 1 is the entrywise 1-norm, andλ can be set toc/ max(l,m) where c is a positive parameter [1], [2]. The trace norm and the 1-norm are the tightest convex relaxations of the rank and Manuscript received August 26, 2015; revised May 26, 2016 and July 16, 2016; accepted September 3, 2016. This work was supported by a grant from the Ministry of Science and Technology under the contract MOST102-2221-E-001-004-MY3 and the Academia Sinica Career Development Program. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Masahiro Y ukawa. The authors are with the Research Center for Information Technology Innovation, Academia Sinica, Taipei 11564, Taiwan (email: taksh-ingchan@citi.sinica.edu.tw;

In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our three-phase analysis further provides a finite-sample error bound for the running estimate, which matches the minimax information lower bound for principal component analysis under the additional assumption of bounded samples.

Big Data Analytics is a buzzword nowadays. Everyone is talking about it. Big data Analytics has found application in many sectors like medicine, politics, dating. Though big data analytics is used in bettering many aspects of human life, it comes with its own problems. One of them is'Curse of dimensionality'.

Principal component analysis (PCA) is largely adopted for chemical process monitoring and numerous PCA-based systems have been developed to solve various fault detection and diagnosis problems. Since PCA-based methods assume that the monitored process is linear, nonlinear PCA models, such as autoencoder models and kernel principal component analysis (KPCA), has been proposed and applied to nonlinear process monitoring. However, KPCA-based methods need to perform eigen-decomposition (ED) on the kernel Gram matrix whose dimensions depend on the number of training data. Moreover, prefixed kernel parameters cannot be most effective for different faults which may need different parameters to maximize their respective detection performances. Autoencoder models lack the consideration of orthogonal constraints which is crucial for PCA-based algorithms. To address these problems, this paper proposes a novel nonlinear method, called neural component analysis (NCA), which intends to train a feedforward neural work with orthogonal constraints such as those used in PCA. NCA can adaptively learn its parameters through backpropagation and the dimensionality of the nonlinear features has no relationship with the number of training samples. Extensive experimental results on the Tennessee Eastman (TE) benchmark process show the superiority of NCA in terms of missed detection rate (MDR) and false alarm rate (FAR). The source code of NCA can be found in https://github.com/haitaozhao/Neural-Component-Analysis.git.

We present a new technique called contrastive principal component analysis (cPCA) that is designed to discover low-dimensional structure that is unique to a dataset, or enriched in one dataset relative to other data. The technique is a generalization of standard PCA, for the setting where multiple datasets are available -- e.g. a treatment and a control group, or a mixed versus a homogeneous population -- and the goal is to explore patterns that are specific to one of the datasets. We conduct a wide variety of experiments in which cPCA identifies important dataset-specific patterns that are missed by PCA, demonstrating that it is useful for many applications: subgroup discovery, visualizing trends, feature selection, denoising, and data-dependent standardization. We provide geometrical interpretations of cPCA and show that it satisfies desirable theoretical guarantees. We also extend cPCA to nonlinear settings in the form of kernel cPCA. We have released our code as a python package and documentation is on Github.

Stochastic principal component analysis (SPCA) has become a popular dimensionality reduction strategy for large, high-dimensional datasets. We derive a simplified algorithm, called Lazy SPCA, which has reduced computational complexity and is better suited for large-scale distributed computation. We prove that SPCA and Lazy SPCA find the same approximations to the principal subspace, and that the pairwise distances between samples in the lower-dimensional space is invariant to whether SPCA is executed lazily or not. Empirical studies find downstream predictive performance to be identical for both methods, and superior to random projections, across a range of predictive models (linear regression, logistic lasso, and random forests). In our largest experiment with 4.6 million samples, Lazy SPCA reduced 43.7 hours of computation to 9.9 hours. Overall, Lazy SPCA relies exclusively on matrix multiplications, besides an operation on a small square matrix whose size depends only on the target dimensionality.