Matrix decompositions | Dimension reduction
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. The principal components of a collection of points in a real coordinate space are a sequence of unit vectors, where the -th vector is the direction of a line that best fits the data while being orthogonal to the first vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Principal component analysis (PCA) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest. In data analysis, the first principal component of a set of variables, presumed to be jointly normally distributed, is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. PCA is used in exploratory data analysis and for making predictive models. It is commonly used for dimensionality reduction by projecting each data point onto only the first few principal components to obtain lower-dimensional data while preserving as much of the data's variation as possible. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The -th principal component can be taken as a direction orthogonal to the first principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. (Wikipedia).
(8.1.1) Systems of Autonomous Nonlinear Differential Equations and Phase Plane Analysis
This video defines autonomous systems of differential equations, how to analyze phase portraits and determine the equilibrium solutions. https://mathispower4u.com
From playlist Differential Equations: Complete Set of Course Videos
Solve a multi step equation with two variables and distributive property ex 19, –7=3(t–5)–t
👉 Learn how to solve multi-step equations with parenthesis. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-step equation with parenthes
From playlist How to Solve Multi Step Equations with Parenthesis
A first example problem solving a linear, second-order, homogeneous, ODE with variable coefficients around a regular singular point.
From playlist Differential Equations
Learn how to solve a multi step equation with multiple fractions
👉 Learn how to solve multi-step equations with parenthesis. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-step equation with parenthes
From playlist How to Solve Multi Step Equations with Parenthesis
👉 Learn how to solve multi-step equations with parenthesis. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-step equation with parenthes
From playlist How to Solve Multi Step Equations with Parenthesis
Solve a multi step equation with variables on the same side ex 15, 4(3y–1)–5y=–11
👉 Learn how to solve multi-step equations with parenthesis. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-step equation with parenthes
From playlist How to Solve Multi Step Equations with Parenthesis
How to Determine if Functions are Linearly Independent or Dependent using the Definition
How to Determine if Functions are Linearly Independent or Dependent using the Definition If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member https://www.youtube.com/channel/UCr7lmzIk63PZnBw3bezl-Mg/join Th
From playlist Zill DE 4.1 Preliminary Theory - Linear Equations
B01 An introduction to separable variables
In this first lecture I explain the concept of using the separation of variables to solve a differential equation.
From playlist Differential Equations
Solving a multi step equation using distributive property
👉 Learn how to solve multi-step equations with parenthesis and variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To
From playlist How to Solve Multi Step Equations with Parenthesis on Both Sides
08b Machine Learning: Principal Component Analysis
Lecture of principal component analysis for dimensionality reduction and general inference, learning about the structures in our subsurface data. Follow along with the demonstration workflow in Python's scikit-learn package: https://github.com/GeostatsGuy/PythonNumericalDemos/blob/master/
From playlist Machine Learning
Deep Learning Lecture 6.3 - PCA part 2
Principal Component Analysis - PCA Algorithm - Properties of PCA - Equivalence between maximum projection variance and minimal reconstruction error - Applications to images
From playlist Deep Learning Lecture
19 Data Analytics: Principal Component Analysis
Lecture on unsupervised machine learning with principal component analysis for dimensional reduction, inference and prediction.
From playlist Data Analytics and Geostatistics
Robust Principal Component Analysis (RPCA)
Robust statistics is essential for handling data with corruption or missing entries. This robust variant of principal component analysis (PCA) is now a workhorse algorithm in several fields, including fluid mechanics, the Netflix prize, and image processing. Book Website: http://databoo
From playlist Data-Driven Science and Engineering
Eigendecomposition is a technique that finds "special" vectors associated with square matrices. Eigendecomposition is the basis for many important techniques in data analysis, including principal components analyses, blind-source-separation, and other spatial filters. You'll also see a com
From playlist OLD ANTS #9) Matrix analysis
Lecture 15A : From Principal Components Analysis to Autoencoders
Neural Networks for Machine Learning by Geoffrey Hinton [Coursera 2013] Lecture 15A : From Principal Components Analysis to Autoencoders
From playlist Neural Networks for Machine Learning by Professor Geoffrey Hinton [Complete]
Data Analysis 6: Principal Component Analysis (PCA) - Computerphile
PCA - Principle Component Analysis - finally explained in an accessible way, thanks to Dr Mike Pound. This is part 6 of the Data Analysis Learning Playlist: https://www.youtube.com/playlist?list=PLzH6n4zXuckpfMu_4Ff8E7Z1behQks5ba This Learning Playlist was designed by Dr Mercedes Torres
From playlist Data Analysis with Dr Mike Pound
Lecture 15.1 — From PCA to autoencoders [Neural Networks for Machine Learning]
Lecture from the course Neural Networks for Machine Learning, as taught by Geoffrey Hinton (University of Toronto) on Coursera in 2012. Link to the course (login required): https://class.coursera.org/neuralnets-2012-001
From playlist [Coursera] Neural Networks for Machine Learning — Geoffrey Hinton
Applying distributive property with a negative one to solve the multi step equation
👉 Learn how to solve multi-step equations with parenthesis. An equation is a statement stating that two values are equal. A multi-step equation is an equation which can be solved by applying multiple steps of operations to get to the solution. To solve a multi-step equation with parenthes
From playlist How to Solve Multi Step Equations with Parenthesis
PCA In Machine Learning | Principal Component Analysis | Machine Learning Tutorial | Simplilearn
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