This document discusses issues related to linear regression analysis, including suppression and enhancement effects. It begins with an introduction and provides graphical representations of correlations between variables. The document then discusses how adding an uncorrelated predictor variable can increase the significance of other predictors, and how correlated predictors can enhance the overall model fit. It provides examples and equations to explain these "befuddling issues" in more detail.
Stochastic differential equations (SDEs) describe systems with random components. Common methods to solve SDEs include spectral and perturbation methods. The spectral method represents variables and parameters as mean values plus fluctuations. Taking the expected value of the SDE yields equations for the mean and fluctuations that can be solved. The perturbation method expresses variables and parameters as power series expansions. Introducing these into the SDE allows analytical or numerical solution. SDEs are used to model systems with uncertain parameters like groundwater flow with random hydraulic conductivity.
- The document discusses representation of stochastic processes in real and spectral domains and Monte Carlo sampling.
- Stochastic processes can be represented in the real (time or space) domain using autocorrelation and variogram functions, and in the spectral domain using power spectral density functions.
- Monte Carlo sampling uses techniques to generate random numbers from a probability density function for random sampling.
Double Clamped and Cantilever Beam Theoretical Solution and Numerical Solutio...Tasos Lazaridis
This document presents a theoretical and numerical analysis of the displacement of double clamped and cantilever beams under a load. It derives the theoretical solutions using calculus of variations and defines the boundary conditions. It also models the beams in ANSYS and SolidWorks to calculate the displacement numerically using finite element analysis. The theoretical and numerical results are then compared to calculate the relative differences.
Cs8092 computer graphics and multimedia unit 3SIMONTHOMAS S
The document discusses various methods for representing 3D objects in computer graphics, including polygon meshes, curved surfaces defined by equations or splines, and sweep representations. It also covers 3D transformations like translation, rotation, and scaling. Key representation methods discussed are polygonal meshes, NURBS curves and surfaces, and extruded and revolved shapes. Transformation operations covered are translation using addition of a offset vector, and rotation using a rotation matrix.
This document discusses stochastic models for site characterization. It describes several continuous models for generating random fields including the multivariate normal method, LU decomposition method, and turning bands method. The multivariate normal method models a random vector as having a multivariate normal distribution defined by a mean vector and covariance matrix. The LU decomposition method generates a random field with a given covariance structure by decomposing the covariance matrix into lower and upper triangular matrices. It provides numerical examples of applying the LU decomposition method to generate correlated random variables at two points.
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.Igor Moiseev
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.
The geometry of strokes arises in the control problems of Reeds–Shepp car, Dubins’ car, modeling of vision and some others. The main problem is to characterize the shortest paths and minimal distances on the plane, equipped with the structure of geometry of strokes.
This problem is formulated as an optimal control problem in 3-space with 2 dimensional control and a quadratic integral cost. Here is studied the symmetries of the sub-Riemannian structure, extremals of the optimal control problem, the Maxwell stratum, conjugate points and boundary value problem for the corresponding Hamiltonian system.
The 3D Smith Chart program is a new Java tool for visualizing and designing active and passive microwave circuits. It generalizes the traditional 2D Smith chart onto the surface of a sphere, addressing limitations of the 2D chart. Key features include reading measurement files, plotting reflection coefficients for complex impedances, and aiding oscillator and amplifier design by visualizing infinite mismatch and stability circles. The tool aims to provide a complete graphical solution for microwave circuit measurement and design.
Stochastic differential equations (SDEs) describe systems with random components. Common methods to solve SDEs include spectral and perturbation methods. The spectral method represents variables and parameters as mean values plus fluctuations. Taking the expected value of the SDE yields equations for the mean and fluctuations that can be solved. The perturbation method expresses variables and parameters as power series expansions. Introducing these into the SDE allows analytical or numerical solution. SDEs are used to model systems with uncertain parameters like groundwater flow with random hydraulic conductivity.
- The document discusses representation of stochastic processes in real and spectral domains and Monte Carlo sampling.
- Stochastic processes can be represented in the real (time or space) domain using autocorrelation and variogram functions, and in the spectral domain using power spectral density functions.
- Monte Carlo sampling uses techniques to generate random numbers from a probability density function for random sampling.
Double Clamped and Cantilever Beam Theoretical Solution and Numerical Solutio...Tasos Lazaridis
This document presents a theoretical and numerical analysis of the displacement of double clamped and cantilever beams under a load. It derives the theoretical solutions using calculus of variations and defines the boundary conditions. It also models the beams in ANSYS and SolidWorks to calculate the displacement numerically using finite element analysis. The theoretical and numerical results are then compared to calculate the relative differences.
Cs8092 computer graphics and multimedia unit 3SIMONTHOMAS S
The document discusses various methods for representing 3D objects in computer graphics, including polygon meshes, curved surfaces defined by equations or splines, and sweep representations. It also covers 3D transformations like translation, rotation, and scaling. Key representation methods discussed are polygonal meshes, NURBS curves and surfaces, and extruded and revolved shapes. Transformation operations covered are translation using addition of a offset vector, and rotation using a rotation matrix.
This document discusses stochastic models for site characterization. It describes several continuous models for generating random fields including the multivariate normal method, LU decomposition method, and turning bands method. The multivariate normal method models a random vector as having a multivariate normal distribution defined by a mean vector and covariance matrix. The LU decomposition method generates a random field with a given covariance structure by decomposing the covariance matrix into lower and upper triangular matrices. It provides numerical examples of applying the LU decomposition method to generate correlated random variables at two points.
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.Igor Moiseev
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.
The geometry of strokes arises in the control problems of Reeds–Shepp car, Dubins’ car, modeling of vision and some others. The main problem is to characterize the shortest paths and minimal distances on the plane, equipped with the structure of geometry of strokes.
This problem is formulated as an optimal control problem in 3-space with 2 dimensional control and a quadratic integral cost. Here is studied the symmetries of the sub-Riemannian structure, extremals of the optimal control problem, the Maxwell stratum, conjugate points and boundary value problem for the corresponding Hamiltonian system.
The 3D Smith Chart program is a new Java tool for visualizing and designing active and passive microwave circuits. It generalizes the traditional 2D Smith chart onto the surface of a sphere, addressing limitations of the 2D chart. Key features include reading measurement files, plotting reflection coefficients for complex impedances, and aiding oscillator and amplifier design by visualizing infinite mismatch and stability circles. The tool aims to provide a complete graphical solution for microwave circuit measurement and design.
Cs8092 computer graphics and multimedia unit 2SIMONTHOMAS S
This document discusses two-dimensional graphics transformations and matrix representations. It covers topics such as translation, rotation, scaling, reflections, shearing, and representing composite transformations using matrix multiplication. Homogeneous coordinates are also introduced as a way to represent 2D points using 3-dimensional vectors and matrices for transformations.
Smith chart:A graphical representation.amitmeghanani
The document discusses the Smith chart, which is a graphical tool used to solve transmission line problems. Some key points:
- The Smith chart was developed in 1939 and allows tedious transmission line calculations to be done graphically.
- It provides a mapping between the normalized impedance plane and the reflection coefficient plane. Circles of constant resistance and reactance are plotted, along with the reflection coefficient.
- Parameters like impedance, admittance, reflection coefficient, VSWR can all be plotted and derived from locations on the chart.
- Examples are given of using the Smith chart to determine input impedance, reflection coefficient, and stub matching of transmission lines with various termination impedances.
Vector mechanics for engineers statics 7th chapter 5 Nahla Hazem
This problem involves locating the centroid of a plane area shown in multiple problems. The solution provides the area (A) of each section, the x and y coordinates, the moment of area about the x-axis (xA), and the moment of area about the y-axis (yA). It then calculates the x and y coordinates of the centroid by taking the sum of the xA and yA values, respectively, and dividing each by the total area.
- Semi Regular Meshes can be subdivided using regular 1:4 subdivision or represented as Spherical Geometry Images mapped to the unit sphere.
- Subdivision Surfaces are generated by applying local interpolators repeatedly to refine a coarse control mesh. Common subdivision schemes include Linear, Butterfly, and Loop which are demonstrated in examples.
- Biorthogonal Wavelets can be constructed on meshes using a Lifting Scheme to create wavelet coefficients with vanishing moments, allowing for compression of mesh signals. Invariant neighborhoods are used to analyze the refinement of meshes across scales.
This document provides examples and explanations for writing and graphing linear equations. It defines key vocabulary like linear equation, slope, y-intercept, slope-intercept form, and point-slope form. Examples demonstrate finding the slope and y-intercept of lines given in various forms and writing the equation of a line given the slope and a point. The examples are worked through step-by-step with explanations of the process.
This document provides examples and explanations for writing and graphing linear equations. It defines key vocabulary like slope, y-intercept, slope-intercept form, and point-slope form. Examples demonstrate finding the slope and y-intercept of lines given in various forms and writing the equation of a line given the slope and a point. The examples are worked through step-by-step to model how to set up and solve for the line equation in different forms.
This document contains a question bank with multiple choice and short answer questions covering various topics in digital electronics across 5 units. Some of the key topics covered include binary, hexadecimal, octal number systems; logic gates and families (TTL, CMOS, ECL); flip-flops, counters, registers; basics of memory units (RAM, ROM); programmable logic devices (PLA, PLD); and asynchronous sequential circuits including hazards. The document is divided into parts for each unit with concepts ranging from introductory to advanced level questions.
Euler's equation can be used to find curves or trajectories that extremize (minimize or maximize) functionals which are defined as integrals along curves. It does this by requiring the first variation of the integral, with respect to variations of the curve, to vanish. This results in an equation involving derivatives of the integrand which must be satisfied by any extremizing curve. Examples include finding geodesics as curves of shortest distance, surfaces of minimal area obtained by rotating curves, and trajectories that extremize the action in physics.
This document outlines different ways to parametrize surfaces, including:
- Graphs of functions can be parametrized using the function.
- Planes can be parametrized using a point on the plane and two vectors.
- Coordinate surfaces like cylinders can be parametrized using the coordinate conversions.
- Surfaces of revolution can be parametrized using a radius function and angle.
The document provides examples of parametrizing planes, cylinders, and surfaces of revolution and outlines other topics like implicit vs explicit descriptions and more complex parametrizations.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
This document presents a novel method called the Eigenfunction Expansion Method (EFEM) for analytically solving transient heat conduction problems with phase change in cylindrical coordinates. The method involves formulating the governing equations and associated boundary conditions, introducing coefficients, solving the eigenvalue problems, and representing the solution as a series expansion of the eigenfunctions. Dimensionless parameters are introduced to simplify the problem. The EFEM is then applied to solve a one-dimensional phase change problem. Results show that increasing the number of terms in the series expansion decreases the truncation error and that the Stefan number affects the melting fraction evolution over time.
The document discusses using measures of risk and dependence to analyze the risk of an aggregation function g(X) of multiple risks X1, ..., Xd represented as a random vector X. Specifically, it covers how to model the risks X, measure correlations between risks, compare risks under dependence versus independence, and determine the contribution of each risk Xi to the overall risk. Examples of applications to finance, environmental risks, and credit risk are provided.
This document discusses transformations of functions including translations, stretches, reflections, and combinations of functions. It begins by explaining how translating a function's graph vertically or horizontally shifts the graph up/down or left/right by a given amount. Stretching or shrinking a graph vertically stretches or shrinks the y-values, while reflecting graphs flips them over an axis. Functions can also be combined by addition, subtraction, multiplication, composition. Composition applies one function to the output of another. Several examples demonstrate applying transformations to graphs and combining simple functions.
The document discusses different computer graphics display systems and algorithms for drawing lines. It describes raster scan and random scan display systems. Raster scan systems sweep an electron beam across the screen row by row to draw the image based on values stored in a frame buffer. It also covers the digital differential analyzer (DDA) and Bresenham's algorithms for drawing lines on a digital display. DDA calculates increments to move the line incrementally pixel by pixel, while Bresenham's uses a decision parameter to efficiently draw lines on a raster display. An example demonstrates applying each algorithm to draw a line between two points.
This document contains 25 multiple choice questions from a GATE exam on electrical engineering. Each question is one mark and includes the question, possible answers, and solution. The questions cover topics including calculus, signals and systems, electromagnetics, circuits, power systems, and machines. The summary provides an overview of the document content and structure without reproducing the full text.
Prévision de consommation électrique avec adaptive GAMCdiscount
The document discusses generalized additive models (GAM) for short-term electricity load forecasting. GAMs are smooth additive models that decompose a response variable into additive components like trends, cyclic patterns, and nonlinear effects. They summarize how GAMs can model various drivers of electricity consumption, including temperature effects, day-of-week patterns, and lagged load values. Big additive models (BAM) allow applying GAMs to large electricity load datasets. BAMs use QR decomposition and online updating to efficiently estimate high-dimensional additive models.
6161103 10.5 moments of inertia for composite areasetcenterrbru
1) Moments of inertia for composite areas can be determined by dividing the area into its composite parts, finding the moment of inertia of each part about its centroidal axis and the reference axis using the parallel axis theorem, and taking the algebraic sum.
2) The procedure was demonstrated by calculating the moment of inertia of a composite area made of a rectangle and circle, and another made of three rectangles.
3) For the second example, the cross-sectional area was divided into three rectangles, the moment of inertia of each was found about the x and y axes using the parallel axis theorem, and summed to find the total moment of inertia.
Low Complexity Regularization of Inverse ProblemsGabriel Peyré
This document discusses regularization techniques for inverse problems. It begins with an overview of compressed sensing and inverse problems, as well as convex regularization using gauges. It then discusses performance guarantees for regularization methods using dual certificates and L2 stability. Specific examples of regularization gauges are given for various models including sparsity, structured sparsity, low-rank, and anti-sparsity. Conditions for exact recovery using random measurements are provided for sparse vectors and low-rank matrices. The discussion concludes with the concept of a minimal-norm certificate for the dual problem.
6161103 10.4 moments of inertia for an area by integrationetcenterrbru
This document discusses calculating moments of inertia for planar areas using integration. It describes:
1) Choosing a differential element for integration that has size in only one direction to simplify the calculation.
2) The procedure involves specifying a rectangular differential element and orienting it parallel or perpendicular to the axis of rotation.
3) Moments of inertia are calculated through single or double integration, depending on whether the element has thickness in one or two directions.
The document discusses how the first derivative can be used to analyze curves geometrically. It indicates that the first derivative measures the slope of the tangent line to a curve. If the derivative is positive, the curve is increasing; if negative, decreasing; and if zero, the curve is stationary. As an example, it finds the stationary points of the curve y = 3x^2 - x^3 and determines that they are a minimum at (0,0) and an inflection point at (2,4).
This document discusses two issues in linear regression modeling: suppression and enhancement. Suppression occurs when the sign of a standardized regression coefficient is opposite to the zero-order correlation between the predictor and dependent variable. This can happen when another predictor accounts for variability in the first predictor. Enhancement occurs when the R-squared value from a multiple regression model is higher than the sum of the individual zero-order correlations, indicating correlated predictors provide additional explanatory power together than alone. An example from a 1941 study by Horst is provided to illustrate suppression.
Applications Of One Type Of Euler-Lagrange Fractional Differential EquationIRJET Journal
This document presents applications of one type of Euler-Lagrange fractional differential equation involving the composition of left Riemann-Liouville and right Caputo fractional derivatives of order α, where 0 < α < 1. First, some examples of ordinary harmonic oscillators described by second-order differential equations are transformed into this fractional differential equation form. Next, the expanded form of the fractional differential equation is obtained using finite differences and the definitions of the fractional derivatives. This is also expressed in matrix notation. Finally, the document describes using Matlab script to numerically solve this type of equation and graphically represent the approximate solutions for various values of α.
Cs8092 computer graphics and multimedia unit 2SIMONTHOMAS S
This document discusses two-dimensional graphics transformations and matrix representations. It covers topics such as translation, rotation, scaling, reflections, shearing, and representing composite transformations using matrix multiplication. Homogeneous coordinates are also introduced as a way to represent 2D points using 3-dimensional vectors and matrices for transformations.
Smith chart:A graphical representation.amitmeghanani
The document discusses the Smith chart, which is a graphical tool used to solve transmission line problems. Some key points:
- The Smith chart was developed in 1939 and allows tedious transmission line calculations to be done graphically.
- It provides a mapping between the normalized impedance plane and the reflection coefficient plane. Circles of constant resistance and reactance are plotted, along with the reflection coefficient.
- Parameters like impedance, admittance, reflection coefficient, VSWR can all be plotted and derived from locations on the chart.
- Examples are given of using the Smith chart to determine input impedance, reflection coefficient, and stub matching of transmission lines with various termination impedances.
Vector mechanics for engineers statics 7th chapter 5 Nahla Hazem
This problem involves locating the centroid of a plane area shown in multiple problems. The solution provides the area (A) of each section, the x and y coordinates, the moment of area about the x-axis (xA), and the moment of area about the y-axis (yA). It then calculates the x and y coordinates of the centroid by taking the sum of the xA and yA values, respectively, and dividing each by the total area.
- Semi Regular Meshes can be subdivided using regular 1:4 subdivision or represented as Spherical Geometry Images mapped to the unit sphere.
- Subdivision Surfaces are generated by applying local interpolators repeatedly to refine a coarse control mesh. Common subdivision schemes include Linear, Butterfly, and Loop which are demonstrated in examples.
- Biorthogonal Wavelets can be constructed on meshes using a Lifting Scheme to create wavelet coefficients with vanishing moments, allowing for compression of mesh signals. Invariant neighborhoods are used to analyze the refinement of meshes across scales.
This document provides examples and explanations for writing and graphing linear equations. It defines key vocabulary like linear equation, slope, y-intercept, slope-intercept form, and point-slope form. Examples demonstrate finding the slope and y-intercept of lines given in various forms and writing the equation of a line given the slope and a point. The examples are worked through step-by-step with explanations of the process.
This document provides examples and explanations for writing and graphing linear equations. It defines key vocabulary like slope, y-intercept, slope-intercept form, and point-slope form. Examples demonstrate finding the slope and y-intercept of lines given in various forms and writing the equation of a line given the slope and a point. The examples are worked through step-by-step to model how to set up and solve for the line equation in different forms.
This document contains a question bank with multiple choice and short answer questions covering various topics in digital electronics across 5 units. Some of the key topics covered include binary, hexadecimal, octal number systems; logic gates and families (TTL, CMOS, ECL); flip-flops, counters, registers; basics of memory units (RAM, ROM); programmable logic devices (PLA, PLD); and asynchronous sequential circuits including hazards. The document is divided into parts for each unit with concepts ranging from introductory to advanced level questions.
Euler's equation can be used to find curves or trajectories that extremize (minimize or maximize) functionals which are defined as integrals along curves. It does this by requiring the first variation of the integral, with respect to variations of the curve, to vanish. This results in an equation involving derivatives of the integrand which must be satisfied by any extremizing curve. Examples include finding geodesics as curves of shortest distance, surfaces of minimal area obtained by rotating curves, and trajectories that extremize the action in physics.
This document outlines different ways to parametrize surfaces, including:
- Graphs of functions can be parametrized using the function.
- Planes can be parametrized using a point on the plane and two vectors.
- Coordinate surfaces like cylinders can be parametrized using the coordinate conversions.
- Surfaces of revolution can be parametrized using a radius function and angle.
The document provides examples of parametrizing planes, cylinders, and surfaces of revolution and outlines other topics like implicit vs explicit descriptions and more complex parametrizations.
This document provides formulas and methods for solving ordinary differential equations and vector calculus problems that are covered in an Engineering Mathematics course. It includes:
1. Seven methods for finding the complementary function for ODEs with constant coefficients depending on the nature of the roots.
2. Methods for finding the particular integral for ODEs with constant coefficients, including four types of functions the right side could be.
3. An overview of key concepts in vector calculus including vector differential operators, gradient, divergence, curl, and theorems like Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
This document presents a novel method called the Eigenfunction Expansion Method (EFEM) for analytically solving transient heat conduction problems with phase change in cylindrical coordinates. The method involves formulating the governing equations and associated boundary conditions, introducing coefficients, solving the eigenvalue problems, and representing the solution as a series expansion of the eigenfunctions. Dimensionless parameters are introduced to simplify the problem. The EFEM is then applied to solve a one-dimensional phase change problem. Results show that increasing the number of terms in the series expansion decreases the truncation error and that the Stefan number affects the melting fraction evolution over time.
The document discusses using measures of risk and dependence to analyze the risk of an aggregation function g(X) of multiple risks X1, ..., Xd represented as a random vector X. Specifically, it covers how to model the risks X, measure correlations between risks, compare risks under dependence versus independence, and determine the contribution of each risk Xi to the overall risk. Examples of applications to finance, environmental risks, and credit risk are provided.
This document discusses transformations of functions including translations, stretches, reflections, and combinations of functions. It begins by explaining how translating a function's graph vertically or horizontally shifts the graph up/down or left/right by a given amount. Stretching or shrinking a graph vertically stretches or shrinks the y-values, while reflecting graphs flips them over an axis. Functions can also be combined by addition, subtraction, multiplication, composition. Composition applies one function to the output of another. Several examples demonstrate applying transformations to graphs and combining simple functions.
The document discusses different computer graphics display systems and algorithms for drawing lines. It describes raster scan and random scan display systems. Raster scan systems sweep an electron beam across the screen row by row to draw the image based on values stored in a frame buffer. It also covers the digital differential analyzer (DDA) and Bresenham's algorithms for drawing lines on a digital display. DDA calculates increments to move the line incrementally pixel by pixel, while Bresenham's uses a decision parameter to efficiently draw lines on a raster display. An example demonstrates applying each algorithm to draw a line between two points.
This document contains 25 multiple choice questions from a GATE exam on electrical engineering. Each question is one mark and includes the question, possible answers, and solution. The questions cover topics including calculus, signals and systems, electromagnetics, circuits, power systems, and machines. The summary provides an overview of the document content and structure without reproducing the full text.
Prévision de consommation électrique avec adaptive GAMCdiscount
The document discusses generalized additive models (GAM) for short-term electricity load forecasting. GAMs are smooth additive models that decompose a response variable into additive components like trends, cyclic patterns, and nonlinear effects. They summarize how GAMs can model various drivers of electricity consumption, including temperature effects, day-of-week patterns, and lagged load values. Big additive models (BAM) allow applying GAMs to large electricity load datasets. BAMs use QR decomposition and online updating to efficiently estimate high-dimensional additive models.
6161103 10.5 moments of inertia for composite areasetcenterrbru
1) Moments of inertia for composite areas can be determined by dividing the area into its composite parts, finding the moment of inertia of each part about its centroidal axis and the reference axis using the parallel axis theorem, and taking the algebraic sum.
2) The procedure was demonstrated by calculating the moment of inertia of a composite area made of a rectangle and circle, and another made of three rectangles.
3) For the second example, the cross-sectional area was divided into three rectangles, the moment of inertia of each was found about the x and y axes using the parallel axis theorem, and summed to find the total moment of inertia.
Low Complexity Regularization of Inverse ProblemsGabriel Peyré
This document discusses regularization techniques for inverse problems. It begins with an overview of compressed sensing and inverse problems, as well as convex regularization using gauges. It then discusses performance guarantees for regularization methods using dual certificates and L2 stability. Specific examples of regularization gauges are given for various models including sparsity, structured sparsity, low-rank, and anti-sparsity. Conditions for exact recovery using random measurements are provided for sparse vectors and low-rank matrices. The discussion concludes with the concept of a minimal-norm certificate for the dual problem.
6161103 10.4 moments of inertia for an area by integrationetcenterrbru
This document discusses calculating moments of inertia for planar areas using integration. It describes:
1) Choosing a differential element for integration that has size in only one direction to simplify the calculation.
2) The procedure involves specifying a rectangular differential element and orienting it parallel or perpendicular to the axis of rotation.
3) Moments of inertia are calculated through single or double integration, depending on whether the element has thickness in one or two directions.
The document discusses how the first derivative can be used to analyze curves geometrically. It indicates that the first derivative measures the slope of the tangent line to a curve. If the derivative is positive, the curve is increasing; if negative, decreasing; and if zero, the curve is stationary. As an example, it finds the stationary points of the curve y = 3x^2 - x^3 and determines that they are a minimum at (0,0) and an inflection point at (2,4).
This document discusses two issues in linear regression modeling: suppression and enhancement. Suppression occurs when the sign of a standardized regression coefficient is opposite to the zero-order correlation between the predictor and dependent variable. This can happen when another predictor accounts for variability in the first predictor. Enhancement occurs when the R-squared value from a multiple regression model is higher than the sum of the individual zero-order correlations, indicating correlated predictors provide additional explanatory power together than alone. An example from a 1941 study by Horst is provided to illustrate suppression.
Applications Of One Type Of Euler-Lagrange Fractional Differential EquationIRJET Journal
This document presents applications of one type of Euler-Lagrange fractional differential equation involving the composition of left Riemann-Liouville and right Caputo fractional derivatives of order α, where 0 < α < 1. First, some examples of ordinary harmonic oscillators described by second-order differential equations are transformed into this fractional differential equation form. Next, the expanded form of the fractional differential equation is obtained using finite differences and the definitions of the fractional derivatives. This is also expressed in matrix notation. Finally, the document describes using Matlab script to numerically solve this type of equation and graphically represent the approximate solutions for various values of α.
Formulas for Surface Weighted Numbers on Graphijtsrd
The boundary value problem differential operator on the graph of a specific structure is discussed in this article. The graph has degree 1 vertices and edges that are linked at one common vertex. The differential operator expression with real valued potentials, the Dirichlet boundary conditions, and the conventional matching requirements define the boundary value issue. There are a finite number of eig nv lu s in this problem.The residues of the diagonal elements of the Weyl matrix in the eigenvalues are referred to as weight numbers. The ig nv lu s are monomorphic functions with simple poles.The weight numbers under consideration generalize the weight numbers of differential operators on a finite interval, which are equal to the reciprocals of the squared norms of eigenfunctions. These numbers, along with the eig nv lu s, serve as spectral data for unique operator reconstruction. The contour integration is used to obtain formulas for surfacethe weight numbers, as well as formulas for the sums in the case of superficial near ig nv lu s. On the graphs, the formulas can be utilized to analyze inverse spectral problems. Ghulam Hazrat Aimal Rasa "Formulas for Surface Weighted Numbers on Graph" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-6 | Issue-3 , April 2022, URL: https://www.ijtsrd.com/papers/ijtsrd49573.pdf Paper URL: https://www.ijtsrd.com/mathemetics/calculus/49573/formulas-for-surface-weighted-numbers-on-graph/ghulam-hazrat-aimal-rasa
Dr Omar Presrntation of (on the solution of Multiobjective (1).ppteyadabdallah
This document presents a solution algorithm for solving a multiobjective cutting stock problem in the aluminum industry where scrap is considered a fuzzy parameter. The problem involves casting molten aluminum into rods and cutting them into logs to meet customer demands while minimizing costs from inventory and scrap. The algorithm formulates the problem using fuzzy set concepts and models scrap as a fuzzy number. It then finds α-Pareto optimal solutions for different α-levels using a weighted objective function and nonlinear programming solved with branch-and-bound methods. An example demonstrates implementing the method.
A common unique random fixed point theorem in hilbert space using integral ty...Alexander Decker
This document presents a common unique random fixed point theorem for two continuous random operators defined on a non-empty closed subset of a Hilbert space.
The theorem proves that if two continuous random operators S and T satisfy a certain integral type condition (Condition A), then S and T have a unique common random fixed point.
The proof constructs a sequence of measurable functions {ng} and shows that it converges to the common unique random fixed point of S and T. It utilizes a rational inequality and the parallelogram law to show {ng} is a Cauchy sequence that converges, and its limit is the random fixed point.
Observations on Ternary Quadratic Equation z2 = 82x2 +y2IRJET Journal
This document analyzes the ternary quadratic equation z2 = 82x2 + y2 for its non-zero distinct integer solutions. Four different patterns of solutions are presented. Notable observations include some solutions being related to polygonal numbers, octahedral numbers, and four dimensional figurate numbers. Specific relations are shown between the solutions and expressions representing "nasty numbers". The analysis finds multiple parametric representations of solutions to the given equation.
This document provides an overview of the Smith chart. It discusses the history and construction of the Smith chart. The Smith chart is a graphical tool developed in 1939 for solving impedance matching and transformation problems in transmission lines. It allows users to determine values such as reflection coefficients, standing wave ratios, input impedances, and locations of voltage maxima and minima. The document provides examples of how to use the Smith chart to calculate these values for different impedances and transmission line lengths. It also discusses scales, circles and other features of the Smith chart.
This document provides an overview of Dirac delta functions and their properties. It defines the Dirac delta function as equal to 0 for all values of x except when x is 0, where it is equal to infinity. The key properties listed are that the Dirac delta function can be used to sample or shift other functions, replicate functions, and behave like the derivative of the Heaviside step function. Integrating a function multiplied by the Dirac delta returns the value of the function at 0.
The Smith chart is a graphical tool used to analyze high frequency circuits. It represents all possible complex impedances in terms of the reflection coefficient. Circles of constant resistance and arcs of constant reactance intersect on the chart to indicate impedance values. The chart allows users to determine impedances, reflection coefficients, voltage standing wave ratios and other transmission line parameters through graphical techniques. It remains a popular tool decades after its original conception due to providing a clever way to visualize complex impedance functions.
The paper examines the problem of systems redesign within the context of passive electrical networks and through analogies provides also the means of addressing issues of re-design of mechanical networks. The problem addressed here are special cases of the more general network redesign problem. Redesigning autonomous passive electric networks involves changing the network natural dynamics by modification of the types of elements, possibly their values, interconnection topology and possibly addition, or elimination of parts of the network. We investigate the modelling of systems, whose structure is not fixed but evolves during the system lifecycle. As such, this is a problem that differs considerably from a standard control problem, since it involves changing the system itself without control and aims to achieve the desirable system properties, as these may be expressed by the natural frequencies by system re-engineering. In fact, this problem involves the selection of alternative values for dynamic elements and non-dynamic elements within a fixed interconnection topology and/or alteration of the network interconnection topology and possible evolution of the cardinality of physical elements (increase of elements, branches). The aim of the paper is to define an appropriate representation framework that allows the deployment of control theoretic tools for the re-engineering of properties of a given network. We use impedance and admittance modelling for passive electrical networks and develop a systems framework that is capable of addressing “life-cycle design issues” of networks where the problems of alteration of existing topology and values of the elements, as well as issues of growth, or death of parts of the network are addressed.
We use the Natural Impedance/ Admittance (NI-A) models and we establish a representation of the different types of transformations on such models. This representation provides the means for an appropriate formulation of natural frequencies assignment using the Determinantal Assignment Problem framework defined on appropriate structured transformations. The developed natural representation of transformations are expressed as additive structured transformations. For the simpler case of RL or RC networks it is shown that the single parameter variation problem (dynamic or non-dynamic) is equivalent to Root Locus problems.
follow IEEE NTUA SB on facebook:
https://www.facebook.com/IeeeNtuaSB
An introduction to discrete wavelet transformsLily Rose
This document provides an overview of wavelet transforms and their applications. It introduces continuous and discrete wavelet transforms, including multiresolution analysis and the fast wavelet transform. It discusses how wavelet transforms can be used for image compression, edge detection, and digital watermarking due to properties like decomposing images into different frequency subbands. The fast wavelet transform allows efficient computation of wavelet coefficients by exploiting relationships between scales.
The document discusses various methods for describing and analyzing human joint motion, including Euler angles, joint coordinate systems, screw axes, and dual Euler angles. It presents a model of the golf swing as a 5-segment kinematic chain and calculates the joint-link transformation matrices using dual Euler angles. Individual joint velocities are determined and the total clubhead velocity is calculated as the sum of individual joint contributions. An experiment is described to verify the dual Euler angle analysis of golf swing motion.
A modeling approach for integrating durability engineering and robustness in ...Phuong Dx
This document presents mathematical models to integrate robustness, durability, and tolerances into the design of mechanical products like springs. The models optimize design parameters like wire diameter and coil diameter to minimize mass and stress while ensuring the spring provides the required deflection over its lifetime considering degradation. Three solutions are provided that incorporate different aspects: 1) ensures high reliability, 2) minimizes loss and quality, and 3) accounts for degradation over the spring's lifetime to better meet customer expectations. The models act as a decision support system for designers to evaluate tradeoffs between robustness, durability and manufacturing tolerances early in the design process.
This paper discusses the modeling and vibration Analyses of a rotor having multiple disk supported
by a continuous shaft for the first three modes. Normal modes of constrained structures method is used to
develop the equations. First three modes of the beam-disk system are considered.
This document discusses the computation of definite integrals involving certain polynomials expressed as hypergeometric functions. It defines several types of polynomials including Lucas polynomials, generalized harmonic numbers, Bernoulli polynomials, Gegenbauer polynomials, Laguerre polynomials, Hermite polynomials, Legendre polynomials, Chebyshev polynomials, Euler polynomials, and the generalized Riemann zeta function. It provides the explicit formulas and generating functions for each polynomial. The document contains new results for definite integrals expressed in terms of these polynomials and the hypergeometric function.
On Optimization of Manufacturing of Field-Effect Heterotransistors Frame-work...antjjournal
We consider an approach for increasing density of field-effect heterotransistors in a single-stage multi-path operational amplifier. At the same time one can obtain decreasing of dimensions of the above transistors. Dimensions of the elements could be decreased by manufacturing of these elements in a heterostructure with specific structure. The manufacturing is doing by doping of required areas of the heterostructure by diffusion or ion implantation with future optimization of annealing of dopant and/or radiation defects.
techDynamic characteristics and stability of cylindrical textured journal bea...ijmech
This document summarizes a study on the dynamic characteristics and stability of cylindrical textured journal bearings. The researchers numerically solved the Reynolds equation to analyze the effect of texture depth and density on the stiffness, damping coefficients, stability parameter, and whirl ratio of the bearing. They found that stability is enhanced with increasing texture depth, while there is an optimum texture density that results in maximum stability. Direct and cross stiffness/damping coefficients, stability margin, and whirl ratio were presented for different texture parameters.
Design of a novel controller to increase the frequency response of an aerospaceIAEME Publication
This document discusses the design of a novel controller called a Piecewise Predictive Estimator (PPE) to increase the frequency response of an aerospace electro-mechanical actuator. The PPE technique is combined with existing controllers like PID and LQR to reduce phase lag and increase bandwidth without increasing noise. Simulation results show the bandwidth increased from 22Hz to 25Hz with PPE. The PPE works by making piecewise predictions of the system state to reduce phase lag in a finite prediction horizon.
The document summarizes key concepts from Chapter 8 of the textbook "Fundamentals of Multimedia" on lossy compression algorithms. It introduces lossy compression and discusses distortion measures, rate-distortion theory, quantization techniques including uniform, non-uniform, and vector quantization. It also covers transform coding techniques such as the discrete cosine transform and its use in image compression standards to remove spatial redundancies by transforming pixel values into frequency coefficients.
The document discusses different methods for visualizing and interpreting machine learning models, including univariate, bivariate, and multivariate interpretations. Univariate interpretations using partial dependence plots (PDPs) show the effect of varying individual variables while holding others constant. PDPs vary across models, with logistic regression PDPs closely matching univariate projections but gradient boosting and decision tree PDPs being flatter. Bivariate and multivariate interpretations are needed to understand contextual effects and avoid overestimating variable importance from univariate analyses alone. Residual analysis also supports generally equivalent interpretations across models.
Gradient boosting is an ensemble technique that combines weak learners, such as decision trees, into a single strong learner. It works by iteratively fitting a prediction model to the negative gradient of the loss function, which represents the residuals or errors from the previous iteration. This process of residual fitting helps reduce bias and variance in the model. Gradient boosting has advantages over other techniques like bagging in that it can reduce both bias and variance through its iterative process, while avoiding overfitting by using weak learners and a regularization parameter.
This document discusses ensemble models and gradient boosting. It covers topics such as the bias-variance tradeoff in modeling, ensemble techniques like bagging and stacking, random forests, gradient boosting, and partial dependency plots. The document provides explanations of these techniques with examples and references bias-variance decomposition, bagging, random forests, and how random forests work by randomly selecting subsets of features and samples to grow multiple decision trees that are then combined through voting.
This document provides a summary of visual tools for interpreting machine learning models based on partial dependency plots and their variants. It introduces novel visual concepts such as overall, collapsed, and marginal partial dependency plots and shows how they can help with model interpretation. An example is provided using a simple dataset with 6 predictors and a binary target variable to classify fraudulent vs. valid insurance claims. Model interpretation focuses on identifying important variables and their effects rather than explaining individual predictions.
This document discusses various machine learning models for predicting insurance fraud, including gradient boosting, bagging, random forests, and logistic regression. It provides details on 16 models tested on a healthcare fraud dataset, including variable importance measures and partial dependence plots. While random forests appeared to fit the training data best, it was found to be overfitting, and gradient boosting allocated more importance to predictors other than the most important variable.
This document describes a study comparing different ensemble models and gradient boosting techniques on imbalanced datasets with varying percentages of fraud events. Three studies were conducted using datasets with 5%, 20%, and 50% fraud events. A variety of models were tested including logistic regression, bagging, gradient boosting, random forests and decision trees. The results showed that gradient boosting performed best overall, and was relatively unaffected by dataset imbalance compared to other techniques. Extreme imbalance significantly impacted the performance of decision trees. Re-sampling datasets to 50/50 had little effect on most model performances compared to using the original imbalanced data.
The document provides information on interpreting statistical and machine learning models. It discusses how interpretation depends on the intended audience and context. Various types of model interpretation are categorized, including univariate, bivariate, and multivariate interpretation. Interpretation methods like partial dependency plots are presented on a fraud detection example to show their use and limitations.
This document discusses various ensemble machine learning models and their tree representations. It describes two studies comparing gradient boosting to other methods, with and without data resampling. It also covers partial dependency plots and their modifications for interpreting "black box" models like gradient boosting, bagging, and random forests. Several analytical fraud detection problems are presented to compare different models on. Tree representations of bagging, gradient boosting, and random forests are shown for various models.
This document discusses ensemble models and gradient boosting. It covers topics such as the bias-variance tradeoff, bagging, stacking, random forests, gradient boosting, and interpreting models using partial dependency plots. The document provides an overview of these techniques and why ensemble methods are commonly used, noting that combining multiple models can help reduce variance and improve predictions compared to a single model. Examples are provided to help illustrate concepts like bagging and random forests.
This document discusses decision tree methodology and algorithms. It covers varieties of tree methods, the basic CART algorithm for binary and regression trees, splitting criteria, stopping rules, pruning, interpretation of fitted trees, and variable selection and importance. Examples of tree output and SAS code for finding splits are provided. Benefits and drawbacks of trees are outlined.
- The document discusses logistic regression models for binary classification problems. It covers interpreting coefficients in logistic regression models as odds ratios. An odds ratio above 1 indicates the variable increases the odds of the event, while an odds ratio below 1 decreases the odds.
- It also provides an example of how dummy variables are interpreted, where the exponentiated coefficient represents the odds ratio of the event occurring for that category versus the reference category. This allows easy comparison of probabilities between groups defined by the dummy variable.
This document provides an overview of linear regression modeling techniques. It discusses types of regression models for different dependent variable types, assumptions of linear regression, ordinary least squares estimation techniques, and issues that can arise like multicollinearity. Examples are provided to illustrate concepts like how beta coefficients can have different signs than bivariate correlations due to other predictor variables. Homework and interview questions are mentioned but not detailed. Non-linear regression models are briefly introduced.
The document discusses the curse of dimensionality and principal component analysis (PCA). It explains that as more features are added to multivariate studies, the sample becomes less representative of the actual data due to the exponential increase in sampling needed with higher dimensions. PCA is introduced as a technique to reduce the dimensionality of data while preserving as much information as possible. It works by transforming the data to a new coordinate system where the greatest variance by each component is achieved. Examples are provided to illustrate PCA and how it identifies the components that capture the most information.
The document discusses foundational concepts in probability theory, including different definitions of probability and axioms that define the probability of events. It notes classical and frequency definitions of probability have flaws and discusses the subjectivist definition. The axioms of probability are outlined, including that probabilities must be between 0 and 1 and the probability of the union of disjoint events equals the sum of individual probabilities. Key theorems on probability spaces are also summarized.
Analysts search for relationships between pairs of variables to build models and understand phenomena. Correlation measures the linear relationship between two variables, ranging from -1 to 1. Bivariate relationships do not necessarily extend to multiple variables. Correlations can differ based on the ranges of variables studied, as ranges impact regression results. Correlation does not imply causation, as demonstrated through an example of spurious correlations between bread consumption and negative health outcomes.
This document provides an introduction to exploratory data analysis (EDA). It defines EDA as the initial step of viewing and comprehending a data set which typically has observations as rows and variables as columns. The document outlines 5 steps for EDA: 1) defining the data set, 2) univariate analysis of individual variables, 3) bivariate analysis of variable pairs, 4) multivariate analysis including missing data and dimension reduction, and 5) outlier detection and variable transformations. Examples of data sets are also provided for demonstrating EDA techniques.
This document outlines the topics and approach for a statistics and data mining methods course. The course will cover 4 main topics over 4 lectures: exploratory data analysis, linear and logistic regression, classification trees and ensembles. Class participation and computer work will be graded, with originality and creativity rewarded. The seminars are intended to provide a short overview to spur further self-directed learning, as it is not possible to cover all material in depth. Students are encouraged to review concepts, think critically about problems, and consider alternatives.
This document discusses the results of three studies comparing gradient boosting models built with different class imbalance ratios. Study 1 used a 5% event rate, Study 2 used the original 20% rate, and Study 3 used a 50% rate. Overall, gradient boosting performed best across the studies and was relatively unaffected by the resampling. Extreme imbalance seriously impacted some models like random forests and decision trees for Study 1. While resampling had little effect on performance, it did result in different variable selection for some models. The document concludes that gradient boosting is generally robust to class imbalance issues compared to other methods.
This document discusses various machine learning models for predicting insurance fraud, including gradient boosting, bagging, random forests, and logistic regression. It provides details on 16 models tested on a dataset of insurance claims, including variable importance plots and partial dependency plots. Tree representations of some models are shown and compared. While random forests appeared to perform best based on the first level of the ensemble tree models, further analysis of validation results and variable importance suggested it was overfitting the data.
ViewShift: Hassle-free Dynamic Policy Enforcement for Every Data LakeWalaa Eldin Moustafa
Dynamic policy enforcement is becoming an increasingly important topic in today’s world where data privacy and compliance is a top priority for companies, individuals, and regulators alike. In these slides, we discuss how LinkedIn implements a powerful dynamic policy enforcement engine, called ViewShift, and integrates it within its data lake. We show the query engine architecture and how catalog implementations can automatically route table resolutions to compliance-enforcing SQL views. Such views have a set of very interesting properties: (1) They are auto-generated from declarative data annotations. (2) They respect user-level consent and preferences (3) They are context-aware, encoding a different set of transformations for different use cases (4) They are portable; while the SQL logic is only implemented in one SQL dialect, it is accessible in all engines.
#SQL #Views #Privacy #Compliance #DataLake
The Ipsos - AI - Monitor 2024 Report.pdfSocial Samosa
According to Ipsos AI Monitor's 2024 report, 65% Indians said that products and services using AI have profoundly changed their daily life in the past 3-5 years.
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...sameer shah
"Join us for STATATHON, a dynamic 2-day event dedicated to exploring statistical knowledge and its real-world applications. From theory to practice, participants engage in intensive learning sessions, workshops, and challenges, fostering a deeper understanding of statistical methodologies and their significance in various fields."
Global Situational Awareness of A.I. and where its headedvikram sood
You can see the future first in San Francisco.
Over the past year, the talk of the town has shifted from $10 billion compute clusters to $100 billion clusters to trillion-dollar clusters. Every six months another zero is added to the boardroom plans. Behind the scenes, there’s a fierce scramble to secure every power contract still available for the rest of the decade, every voltage transformer that can possibly be procured. American big business is gearing up to pour trillions of dollars into a long-unseen mobilization of American industrial might. By the end of the decade, American electricity production will have grown tens of percent; from the shale fields of Pennsylvania to the solar farms of Nevada, hundreds of millions of GPUs will hum.
The AGI race has begun. We are building machines that can think and reason. By 2025/26, these machines will outpace college graduates. By the end of the decade, they will be smarter than you or I; we will have superintelligence, in the true sense of the word. Along the way, national security forces not seen in half a century will be un-leashed, and before long, The Project will be on. If we’re lucky, we’ll be in an all-out race with the CCP; if we’re unlucky, an all-out war.
Everyone is now talking about AI, but few have the faintest glimmer of what is about to hit them. Nvidia analysts still think 2024 might be close to the peak. Mainstream pundits are stuck on the wilful blindness of “it’s just predicting the next word”. They see only hype and business-as-usual; at most they entertain another internet-scale technological change.
Before long, the world will wake up. But right now, there are perhaps a few hundred people, most of them in San Francisco and the AI labs, that have situational awareness. Through whatever peculiar forces of fate, I have found myself amongst them. A few years ago, these people were derided as crazy—but they trusted the trendlines, which allowed them to correctly predict the AI advances of the past few years. Whether these people are also right about the next few years remains to be seen. But these are very smart people—the smartest people I have ever met—and they are the ones building this technology. Perhaps they will be an odd footnote in history, or perhaps they will go down in history like Szilard and Oppenheimer and Teller. If they are seeing the future even close to correctly, we are in for a wild ride.
Let me tell you what we see.
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...Social Samosa
The Modern Marketing Reckoner (MMR) is a comprehensive resource packed with POVs from 60+ industry leaders on how AI is transforming the 4 key pillars of marketing – product, place, price and promotions.
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
State of Artificial intelligence Report 2023kuntobimo2016
Artificial intelligence (AI) is a multidisciplinary field of science and engineering whose goal is to create intelligent machines.
We believe that AI will be a force multiplier on technological progress in our increasingly digital, data-driven world. This is because everything around us today, ranging from culture to consumer products, is a product of intelligence.
The State of AI Report is now in its sixth year. Consider this report as a compilation of the most interesting things we’ve seen with a goal of triggering an informed conversation about the state of AI and its implication for the future.
We consider the following key dimensions in our report:
Research: Technology breakthroughs and their capabilities.
Industry: Areas of commercial application for AI and its business impact.
Politics: Regulation of AI, its economic implications and the evolving geopolitics of AI.
Safety: Identifying and mitigating catastrophic risks that highly-capable future AI systems could pose to us.
Predictions: What we believe will happen in the next 12 months and a 2022 performance review to keep us honest.
06-04-2024 - NYC Tech Week - Discussion on Vector Databases, Unstructured Data and AI
Round table discussion of vector databases, unstructured data, ai, big data, real-time, robots and Milvus.
A lively discussion with NJ Gen AI Meetup Lead, Prasad and Procure.FYI's Co-Found
Analysis insight about a Flyball dog competition team's performanceroli9797
Insight of my analysis about a Flyball dog competition team's last year performance. Find more: https://github.com/rolandnagy-ds/flyball_race_analysis/tree/main
2. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Introduction and data description
Overall Graphical View
Befuddlers: Analytics of suppression, redundancy and
enhancement
Befuddlers: Graphical presentation
Coefficient interpretation in multivariate setting
Befuddlers and co-linearity
3. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Befuddling issues in linear regression context due to
misunderstand of ‘conditioning’. Will show that:
- Uncorrelated predictor to dependent variable may increase
significance and fit of other predictors.
-Correlated predictors may enhance model fit.
-Extreme corr (x, z) does not always co-linearity.
Coefficient effect interpretation can be faulty when distinction
between zero-order and partial correlation is disregarded.
5. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Correlations, conditional correlations and Redundancy.
Linear model Y = a0 + a X + b Z + , with usual assumptions; circles below are
unit-variance circles (a + b + d + e = 1).
Y
X
Z
b d
e
a
r2
yx = b + d r2
yz = d + e R2 = b + d + e
pr2
yx = b / (a + b) pr2
yz = e/ (a + e),
sr2
yx = b sr2
yz = e
r2: zero order corr2 SSR(X, Z) / SST.
pr2: partial corr2 r2
yx.z= SSR(X/Z) / [SST – SSR(Z)] .
sr2 : semi-partial corr2 r2
y(x.z) = SSR(X/Z) / SST.
SST = 1 = a + b + d + e.
SSR(X) = b + d.
SSR(X / Z) = b.
SSR(X, Z) = b + d + e.
SSR(Z) = d + e.
But, SSR(X) + SSR(Z) > SSR(X,Z)
not always true.
8/24/2018
6. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Note that in previous slide:
R2 = b + d + e
R2 ≤ r2
yx + r2
yz = b + 2d + e
r2
yx = b + d r2
yz = d + e
‘d’ appears once in R2 while the sum of the marginal correlations
implies 2d. From previous slide, ‘d’ cannot be obtained via partial
or semi partial corrs alone. Instead, via marginal correlations:
R2 = (r2
yx + r2
yz – 2 ryx ryz rxz) / (1 – r2
xz) , and
“d” = r2
yx - sr2
yx = r2
yz - sr2
yz
Y
X
Z
b d
e
a
8/24/2018
7. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
2
xz
n
i i
i 1
xy
n n
2 2
i i
i 1 i 1
xy xz yz
yx y(x.z) 2
xz
(1 - r : det. of corr. matrix):
(X X)(Y Y)
Zero order r
(X X) (Y Y)
Semi
Correlations
partial
( r -
of different
r r )
sr r
(1 - r
orders.
)
8/24/2018
8. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
yx yx.z
yx xz yz
2 2
xz yz
yx.z yw.z xw.z
yx yx.zw 2 2
yw.z xw.z
Partial : pr r
( r - r r )
(1 - r ) (1 - r )
partial 2nd :
( r -
Correlations
r r )
pr r
(1
of different o
- r ) (1
rders.
- r )
8/24/2018
9. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Important relationships:
8/24/2018
2
.
2
.
2 2
. .
2
.
( / ) ( , ) ( )
( ) ( )
( / , )
( , )
( , ) ( , )
( , )
1
yx z
yx zw
y xwz y wz
y wz
SSR x z SSR x z SSR z
pr
SST SSR z SST SSR z
SSR x z w
pr
SST SSR z w
SSR x z SSR z w
SST SSR z w
R R
R
10. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Regr R69 = R72 R78
Corr Partial Semi
Y R Square Indep. Var
-0.01836688 0.00034119 0.00034102R69 0.0008622394 R72
R78 -0.02283033 0.00052507 0.00052490
pr2
R69.R72 = r2
R69.R72(R78) = (R2 – r2
R69.R72) / (1 - r2
R69.R72), also equal to
partial corr calculated from all zero order correlations.
sr2
R69.R72 = r2
R69(R72.R78) = (R2 – r2
R69.R72), equal to semi-partial
calculated from all zero order correlations. It is proportion of Var
(R69) fitted by R72 over and above what R78 has already fitted.
Let’s partition R2 ……………… (Note: This is
IMPORTANT!!!).
8/24/2018
11. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
2 2 2 2
0.123... 01 0(2.1) 0(3.12)
2
0( .123... 1)
that is, addition of non-redundant
X information.
...
for p independent vars. Note:
sum of correlations,semi-partial
p
p p
R r r r
r
8/24/2018
12. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Similarly for SSR:
Extra sum of Squares decomposition for SSR
(Type I)
1 2 1 2 1
3 2 1 1 2 1
2 3 1 1 1 2 3
2 2 1 2 1 2
1.2
2 2
( , ,,, ) ( ) ( / )
( / , ) ... ( / ,,, , )
( , / ) ( ) ( , , )
( ) ( , ) ( / )
( ) ( )
p
p p
Y
SSR X X X SSR X SSR X X
SSR X X X SSR X X X X
and SSR X X X SSE X SSE X X X
SSE X SSE X X SSR X X
R
SSE X SSE X
13. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Correlation and angles.
,
.....( )
:
( , )
( ) ( )
( )( )
( ) ( )
std
n-1
1
2 2
1 1
remember |X |=
1
X standardized
1
corr (X,Y) =
std
X
std std
xy
n
i i
i
xy
n n
i i
i i
X X
X
s
X YCov X Y
nVar X Var Y
r
X X Y Y
r
X X Y Y
Need to find length of standardized Variable to finish corr (X, Y)
and use the concept of inner product.
14. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
,
, 2
,
2
,2
, 2
,
( )
( 1)
( 1)( )
( )
( )
( 1)
i j i
i j
i j i
i j i
i i j
i j i
X X
z
X X
n
n X X
length z z
X X
n
Length of a standardized Variable
15. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
( ) ( ) cos
(
os
)
c
, | || | cos
From (1) before,
and by inner product definit
1 1
1
ion:
1
1 1
std std std std
n n
n
X Y X Y
n n
1) Corr (X, Y) = Corr standardized (X, Y).
2) All standardized variables have same length √(n-1).
3) Corr is always inner product of corresponding
standardized variables divided by n – 1 average
weighted sum of standardized X with weights given by
standardized Y, and vice-versa.
16. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Standardized and non-standardized regression coefficients
x
2
x 2 2 2
'
x
'
' xx
x x
y
'
x
Regular Coefficient b
(X and Z case, lower case: mean removed)
z xy zx zy
b
z x ( xz)
Standardized Coefficient b
b can be obtaineds
General case b b
s from straight regression.
X and Z case b
yx xz yz
2
xz
'
x yx
r (r * r ) Similarity with straight
(1 r ) regr coeff estimation.
Equality with
X case b r
corr coeff.
17. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Example.
Verifying length stdized
variable Result
Statistic
32.28Length X ~ N (-3, 1)
Length Y ~ N ( 5, 4) 51.89
Length Z ~ B (.3, 100) 5.83
Length STD(X) 9.95
Length STD(Y) 9.95
Length STD(Z) 9.95
# Obs 100.00
Sqrt (# Obs - 1) 9.95
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum
X 100 -3.07297 0.99339 -307.29669 -5.89441 -0.94043
Y 100 4.71923 2.16757 471.92321 -0.52986 9.43811
Z 100 0.34000 0.47610 34.00000 0 1.00000
STDX 100 0 1.00000 0 -2.84023 2.14673
STDY 100 0 1.00000 0 -2.42165 2.17703
STDZ 100 0 1.00000 0 -0.71414 1.38628
Correlations X Y Z STDX STDY STDZ
Statistic Computation
-3.073 4.719 0.340 0.000 0.000 0.000MEAN
STD 0.993 2.168 0.476 1.000 1.000 1.000
N 100.000 100.000 100.000 100.000 100.000 100.000
CORR X 1.000 -0.009 0.048 1.000 -0.009 0.048
Y 1.000 0.057 -0.009 1.000 0.057
Z 1.000 0.048 0.057 1.000
STDX 1.000 -0.009 0.048
STDY 1.000 0.057
STDZ 1.000
19. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Two befuddling issues.
1) Why or when is the sign of a standardized coefficient
opposite to sign of zero order correlation of
predictor with dependent variable?
(Suppression).
2) Why or when does addition of correlated predictor to
set of predictors cause R2 to be higher than sum of
individual zero order correlations?
(Enhancement).
8/24/2018
21. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Classical Suppression Example (Horst 1941).
Study of pilot performance (Y) from measures of
mechanical test (X) & verbal abilities test (Z).
When verbal ability (Z) was added to mechanical (X)
ability in equation, effect of X increased.
Happened because Z fitted variability in X, i.e., test of mechanical
ability also required verbal skills to read test directions. But Z did
not affect Y.
In fact, we have simultaneous equation system (SES)
with two dependent variables, X and Y.
Y = f (X, Z), X = g (Z)
But, specification of SES is far more difficult.
22. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Horst 1941
Z
Mechanical Ability
Verbal
Ability x
Y pilot
performance
Horst found corr (Y, X) > 0 (pilot performance related to
mechanical ability), corr (Z, X) > 0 (test performance for test
taking), and corr (Z, Y) = 0 (test taking did not assist in airplane
piloting).
23. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Correlations and Redundancy in linear models.
R2 decomposition. Let Bi be beta coefficients for equation with standardized
variables.
R2 = Bi
2 + 2 Bi Bj rij ( i j)
Formula does not decompose Var (Y) because some Bi Bj rij may be
negative. However, when all cross-terms are zero, R2 = Bi
2 and also
R2 = r2
yi ………………………………….(1)
(setting cross-correlations to zero): Case of Independence.
Y
X
Z
Independence No
Redundancy.
8/24/2018
24. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Correlations and redundancy in linear models.
Stepwises highly correlated variables with dependent variable, “hoping”
formula (1) to apply (but seldom happens).
Rather than independence, more commonly: Redundancy.
It occurs whenever (in absolute terms)
ryx > ryz rxz and ryz > ryx rxz rxz ≠ 0
sryx < ryx, and Bx < ryx.
RedundancyY
X
Z
d
8/24/2018
25. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Redundant info for Y =
a + bX + cZ
Corr
(Y,X)
Std
Beta
Corr
(Y,Z)
Corr
(X,Z)
Corr(Y,Z)
*Corr(X,Z)
Corr(Y,X)
*Corr(X,
Z)
Semi
(Y,X
/Z)
Y X Z Redundant
?
0.288 0.313 0.346 0.135 0.047 0.039 0.244
R
1
R
2
R3 N
R59 Y 0.288 -0.029 -0.050 -0.073 0.004 -0.021 0.286
R69 N 0.288 -0.070 -0.067 0.012 -0.001 0.004 0.289
R72 Y 0.288 0.024 0.025 0.006 0.000 0.002 0.288
R78 Y 0.288 -0.175 -0.209 -0.129 0.027 -0.037 0.264
Pairwise Redundancy: Surendra Data.
8/24/2018
sryx < ryx, and Bx < ryx.
26. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Suppression Effects.
Areas ‘a’, ‘b’ and ‘e’ can be understood as proportions
of “Y” variance.
Area ‘d’ does not have same interpretation, because
can take negative value relationship of suppression
or enhancement.
Y
X
Z
b d
e
a
8/24/2018
27. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
In this case, r2
yz ~ 0, and Z does not
directly affect Y, except in so far as
reducing unfitted variance of X.
bz=0, |bx.z|>|bx|, bx.zbx>0
a) bx.z > bx> 0 or
b) bx.z < bx < 0
R2 = 1 (two predictor case)
r2
xz = 1 - r2
xy , i.e., Z fits remaining X
variance.
It can be verified that:
R2 = pr2
yx = sr2
yx
Y
X
Classical
Suppression
Suppression Effects – Classical (some graphics).
Cohen and Cohen (1975).
Conger (1974) calls it “Traditional”. In later parlance, also case of
Enhancement and of Confounding (confounding used in logistic regression).
Y
Z
8/24/2018
28. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
In this case, ryz / ryx < rxz; Z
primarily suppresses unfitted
variance of X, and vice-versa.
Y
X
Z
Suppression Effects – Net (graphics).
Cohen and Cohen (1975)’s notation. Conger (1974) calls
it “negative”. Suppressor variable receives negative
coefficient, and other coefficient is larger than
correlation with dependent variable. Coefficient of
suppressor opposite to sign of zero order correlation
with dependent variable.
8/24/2018
29. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Suppression Effects – Cooperative Suppression.
(no graph, Cohen and Cohen (1975), Conger (1974) calls it “reciprocal”):
Positive correlation with dependent variable, but negative
correlation among pairs of independent variables. Thus,
when variable is partialled out from another, all measures of
fit are enhanced.
In later parlance, case of Enhancement and Confounding.
In this case, suppressor coefficient exceeds correlation with dependent
variable. In terms of correlations and regression coefficients:
8/24/2018
yx yz xz
x.z x x.z x
z.x z z.x z
r 0 r 0 r 0
|b | |b |, b b 0
|b | |b |, b b 0
30. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Pearson Correlation Coefficients, N = 10000
Prob > |r| under H0: Rho=0
Y X Z
Y 1.00000 0.24484
<.0001
0.12213
<.0001
X 0.24484
<.0001
1.00000 -0.93240
<.0001
Z 0.12213
<.0001
-0.9324
<.0001
1.00000
Cooperative
Suppression
Fit Measure 1 Fit Measure 2
Root
MSE
Depende
nt Mean
Coeff
Var
R-
Square
Adj R-
Sq
Value Value Value Value Value Value
Model
1.95 3.03 64.34 0.06 0.06 0.00X_ALONE
X_AND_Z 0.00 3.03 0.00 1.00 1.00 0.00
Cooperative
Suppression
Parameter
Estimate Pr > |t| VIF
Model Variable
3.79 0.00 .X_ALON
E
Intercept
X 0.12 0.00 .
X_AND_Z Intercept 0.00 1.00 0.00
X 1.33 0.00 7.66
Z 0.67 0.00 7.66
Simulated
Data
8/24/2018
Cooperative Example.
32. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Suppression Effects.
Detection:
Std. coeff (semi-partial correlation) > | ri | suppression.
If ri is zero or close to it classical suppression.
If sg (std coeff) = -sg (correlation) net suppression.
If std coeff > ri and sg(std coeff) = sg (ri ) cooperative
suppression.
8/24/2018
35. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Correlations and Redundancy in linear models.
Misconception (Hamilton, 1987).
Since R2 = r2
i (orthogonal case) R2 ri
2 (general case)? NO … (I)
Y = a + b X + c Z + , (with SSR (X, Z)) equivalent to:
1) Y = d + e X + 1 e1 = Y – est (Y), its SSR called SSR(X)
2) Z = f + g X + 2 e2 = Z – est(Z),
3) e1 = h e2 + 3 (no intercept model), SSR called SSR (Z/X) …… (II)
SSR(X, Z) = SSR(X) + SSR(Z / X) …….(1) (recall earlier slide)
R2 = SSR / SST, R2 > r2
i SSR(Z/X) > SSR(Z) ……. (III)
Deriving Working formulae in terms of simple correlations:
pr2
yz = r2
yz.x = SSR (Z / X) / [SST – SSR(X)], and with (1)
R2 = r2
x + r2
yz.x(1 - r2
x ) = zero order + semi-partial.
8/24/2018
36. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Correlations and Redundancy in linear models.
R2 rx
2 + rz.x
2 but R2 > rx
2 + rz
2 is possible.
Nec. and suff. condition for R2 > r2
i SSR(Z/X) > SSR(Z) is:
2
yx yzyx2
xz xz
yx 2
yz
yx yz
2 2 2
yx yx yz
2 2
yx yz
2 2xz Zx
yx yz
0
r 2r r
pr r (r ) 0
(1 r ) r r
2r r
r r
pr (1 r ).
r r "Enhancement"
Remember that: spr
Condition is:
8/24/2018
Currie and Korabinski (1984) call it ‘enhancement’. Hamilton (1987)
“synergism”.
37. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Correlations and Redundancy in linear models.
Since R2 > r2
i possible
1) X-Y scatter plots and correlation measures may be
inadequate for variable selection with correlated variables.
X-Y correlations can be near 0 while R2 could be extremely
high.
2) Variable Removal due to co-linearity suspicions may be
counterproductive.
3) Forward stepwise methods suffer most from co-linearity.
4) Note that Corr (Y, Z) ≈ 0 and Z may still be useful Effects
on Variable Selection? t-value of Z could be insignificant.
5) Enhancement counterintuitive: predictor contributes more
to regression in presence of other predictors than by itself.
8/24/2018
38. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Pearson Correlation Coefficients, N = 10000
Prob > |r| under H0: Rho=0
Y Z X
Y 1.00000 0.00214
0.8306
0.24484
<.0001
Z 0.00214
0.8306
1.00000 0.97008
<.0001
Simulated
Data, case
Of
Enhancement.
Fit Measure 1 Fit Measure 2
Root MSE
Dependent
Mean Coeff Var R-Square Adj R-Sq
Value Value Value Value Value Value
Model
1.95 3.03 64.34 0.06 0.06 0.00X_ALONE
X_AND_Z 0.00 3.03 0.00 1.00 1.00 0.00
Net Suppression
Estima
te SE
Pr >
|t| VIF
Model Variabl
e
3.79 0.04 0.00 .X_ALON
E
Interc.
X 0.12 0.00 0.00 .
X_AND_
Z
Interc. 0.00 0.00 1.00 0.00
X 1.33 0.00 0.00 7.66
Z 0.67 0.00 0.00 7.66
X on Z Interc. 1.87 0.04 0.00 0.00
Z -0.48 0.00 0.00 1.00
Y and Z hardly correlated,
X and Z highly correlated.
8/24/2018
40. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
x Y ,X z Y ,z
ˆ ˆ| | | r | and | | | r |, stdndrzd coeffs. (1)
8/24/2018
Unifying differing nomenclature and definitions.
Velicer (1978) changed focus from standardized coeffs
to R2 because in previous formulation, |corr| < 1 but
betas unconstrained.
He suggested:
called “enhancement” by Currie and Korabinski (1984).
Let’s call enhancement (1) and (2) together. Otherwise,
just suppression.
2 2 2
Y,X Y,ZR r r (2)
42. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Comparing Suppression and Enhancement
Effects.
Per Friedman & Wall (2005), standardized variables:
x Y , X
ˆ| | | r |
2 2 2
Y , X Y ,Zi y,i
Suppression :
ˆ| | | r | but R r r
2 2 2
Y , X Y ,Zi y,i
Re dundancy :
ˆ| | | r | and R r r
2 2 2
Y ,X Y ,Zi y,i
Enhancement :
ˆ| | | r | but R r r
8/24/2018
43. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Betas, suppression and enhancement Examples, Y = f (X,
Z). (Friedman-Wall 2005).
yx yz
2 2
yx yz yx yz xzY,X Y,Z X,Z 2
X 2 2
X,Z xz
xz xz yi2 2
yx yz
2 2
yx yz yx yz x
2r r
r r
r r 2r r rr r *r
ˆ , for std X, and R
1 r (1 r )
Enhancement
r or r 0, since r 0 by assumption.
Nec. and suff. conditions :
1) r r and 2) r r 1 r
z
yx yz xz
0 region of enhanc.
if r ,r >0 and r 0 enhancement.
8/24/2018
44. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Cooperative
Suppression+
Enhancement
Redundancy
Net
Suppression
Ne
Region Name 2
2 2
2 2
2 2
2 2
ˆ ˆ
( ,0)
ˆ ˆ(0, ) 0 0
2
( , ) 0
xz x z
yx yz yx yz
yz
x yx z yz yx yz
yx
yz yx yz
yx yx yz
yx yx yz
r std std R
I low r r r r
r
II r r r r
r
r r r
III r r r
r r r
IV
t
Suppresion +
Enhancement
2 2
2 2
2
( , ) 0yx yz
xz yx yx yz
yx yz
r r
r upper r r r
r r
8/24/2018
51. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Suppression and Enhancement Effects –
summary.
”Suppressor” variable: enhances predictive ability of
another variable by reducing irrelevant variance of
otherwise relevant variable. In case of standardized
coefficients, Z is suppressor variable for X if Bx > rYX.
(Note: not necessary that rYZ be strictly 0).
“Redundant” variables decrease weights of other
variables (Conger, 1974).
“Enhancer” variable: increases overall R2 beyond sum
of zero-order correlations.
8/24/2018
52. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
kY k
k k k
xy xz yz
yx y(x.z) 2
xz
k
x ˆyx x
yx ˆy(x .x )
Velicer Suppression 2 predictor case :
( r - r r )
sr r
(1 - r )
J predictor case (Smith,1992) :
ˆx prediction on remaining (J-1) predictors.
( r - r r
sr r
k k
k k
k k
k k
k k k k
k k
ˆx
2
ˆx x
2 2
yx yx
x y ˆyx
ˆ ˆx x x x2 2
yx ˆyx
)
(1 - r )
Velicer's criterion : r sr
2r r
r or r 0
r r
8/24/2018
54. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
2
1 2
2
/
Confusion on signs of coefficients
and interpretation for
( )
{
( )
} ( ) ( )yi
xy xy
xi
xy
Y X
sY Y
b r r
sX X
sg r sg b
. .
. 2
2
But in multivariate: ,
estimated equation (emphasizing "partial")
ˆ ,
1
( ) ( )
( ) ( ) and 1
YX Z YZ X
Y YX YZ XZ
YX Z
X XZ
YX
YX YZ XZ XZ
Y X Z
Y a b X c Z where
s r r r
b
s r
sg b sg r
abs r abs r r r
8/24/2018
55. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
If recall partial and semi-partial correlation formulae
. 2
.
1
*
( ) ( )
Y YX YZ XZ
YX Z
X XZ
YX Z
Y
yx
X
Y
X
s r r r
b
s r
b
s
sr
s
s
semi partial
s
sg sg semi partial
8/24/2018
Coefficient signs in multivariate setting cannot necessarily
connote expected effects derived from theoretical analysis.
57. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Some Definitions.
Setting: linear models, specifically regression.
Co-linearity: existence of (almost) perfect linear
relationships among predictors, such that estimated
coefficients are unstable in repeated samples. Notice
that pair-wise or any other correlation notion is NOT
part of definition; instead LINEAR DEPENDENCE or
INDEPENDENCE is at its core.
8/24/2018
58. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Full (Exact) Colinearity
Equivalent conditions:
( ) ( ' ) | ' | 0rank X rank X X p X X
One or more predictors can be exactly expressed in terms of the
others. Sampling variance of some β = ∞, non-unique coeffs.
2
1 ( ).iR for some ith predictor s
8/24/2018
59. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Linear Regression Near Co-linearity: more likely.
(X’X) “wobbly”, “almost singular”. Almost??. Detour:
2
2 2
2
2 2
i
1
( ) . ,
( 1) 1
var( ),
: regr X on other X's.
i
i i
i i
i
Var b
n s R
s X
R R
8/24/2018
60. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
Present Practice, derived from ‘small’ datasets experience.
2
1
1 iR
: Variance Inflation Factor of Xi. √VIF_Xi affects CI of βi
multiplicatively.
Rule of thumb: VIF > 10 strong possibility of colinearity.
(1 / VIF) also called Tolerance.
2 2
2 2 ' 2
2 2 2
i
2 -1
,
1 1
( ) . . ,
( 1) 1 1
var( ), : regr X on other X's.
If X standardized, corr = cov matrix
= ( ' )
i
i i i i i
i i i
i i i
Var b
n s R X X R
s X R R
R X X
8/24/2018
61. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
2
2 2
1
( ) . ,
( 1) 1i
i i
Var b
n s R For given ‘p’ model.
1) In data mining, p → ∞, and R2 does not decrease with p
2
2
lim
lim 1
p i
p
R
R
Naïve estimation with (almost) all variables for the sake of
prediction (data mining disregards interpretation and with
powerful hard- and soft-ware) at least colinearity.
Data Mining World.
8/24/2018
62. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
2
i 2
xz
2
x,z i
1ˆX, Z indep. standardized, Var( )
1
ˆ1 Var( ) ? Not necessarily.
8/24/2018
When corr (X, Z) is very large, for “given” sigma-sq, var of
beta coefficient grows to infinity. But sigma-sq does not
necessarily stay fixed.
63. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 2009
2
2 2
i xz2
xz
2 2
xz yx yz yx yz
2
xz i
1 R 1ˆVar( ) (1), and r 1 R 1.
n 3 1 r
r *r (1 r )(1 r ),
ˆR (r Var( ) 0.
...
extreme values: r
= extreme values)=1, and
8/24/2018
Different Formulation.
65. Leonardo Auslender M008 Ch. 3 – Copyright 2008Leonardo Auslender Copyright 20098/24/2018
Region II and III: se’s increase with increasing co-linearity, but decrease at
extremes. Fig 5 and 6 show that high correlation coexist with small se’s
under enhancement and even under Suppression.