This document discusses classical and fuzzy relations. It begins by introducing relations and their importance in fields like engineering, science, and mathematics. It then contrasts classical/crisp relations with fuzzy relations. Classical relations have binary relatedness between elements, while fuzzy relations have degrees of relatedness on a continuum between completely related and not related. The document provides examples and explanations of crisp relations, fuzzy relations, Cartesian products, compositions, and equivalence/tolerance relations. It demonstrates these concepts with examples involving sets of cities and bacteria strains.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Fuzzy logic is a form of multivalued logic that allows intermediate values between conventional evaluations like true/false, yes/no, or 0/1. It provides a mathematical framework for representing uncertainty and imprecision in measurement and human cognition. The document discusses the history of fuzzy logic, key concepts like membership functions and linguistic variables, common fuzzy logic operations, and applications in fields like control systems, home appliances, and cameras. It also notes some drawbacks like difficulty in tuning membership functions and potential confusion with probability theory.
Soft computing is an emerging approach to computing that aims to mimic human reasoning and learning in uncertain and imprecise environments. It includes neural networks, fuzzy logic, and genetic algorithms. The main goals of soft computing are to develop intelligent machines to solve real-world problems that are difficult to model mathematically, while exploiting tolerance for uncertainty like humans. Some applications of soft computing include consumer appliances, robotics, food preparation devices, and game playing. Soft computing is well-suited for problems not solvable by traditional computing due to its characteristics of tractability, low cost, and high machine intelligence.
This document discusses data compression techniques for digital images. It explains that compression reduces the amount of data needed to represent an image by removing redundant information. The compression process involves an encoder that transforms the input image, and a decoder that reconstructs the output image. The encoder uses three main stages: a mapper to reduce interpixel redundancy, a quantizer to reduce accuracy and psychovisual redundancy, and a symbol encoder to assign variable-length codes to the quantized values. The decoder performs the inverse operations of the encoder and mapper to reconstruct the original image, but does not perform the inverse of quantization which is a lossy process.
This document discusses classical sets and fuzzy sets. It defines classical sets as having distinct elements that are either fully included or excluded from the set. Fuzzy sets allow for gradual membership, with elements having degrees of membership between 0 and 1. Operations like union, intersection, and complement are defined for both classical and fuzzy sets, with fuzzy set operations accounting for degrees of membership. Properties of classical and fuzzy sets and relations are also covered, noting differences like fuzzy sets not following the law of excluded middle.
Soft computing is an emerging approach to computing that aims to solve computationally hard problems using inexact solutions that are tolerant of imprecision, uncertainty, partial truth, and approximation. It uses techniques like fuzzy logic, neural networks, evolutionary computation, and probabilistic reasoning to model human-like decision making. Unlike hard computing which requires precise modeling and solutions, soft computing is well-suited for real-world problems where ideal models are not available. The key constituents of soft computing are fuzzy logic, evolutionary computation, neural networks, and machine learning.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Fuzzy logic is a form of multivalued logic that allows intermediate values between conventional evaluations like true/false, yes/no, or 0/1. It provides a mathematical framework for representing uncertainty and imprecision in measurement and human cognition. The document discusses the history of fuzzy logic, key concepts like membership functions and linguistic variables, common fuzzy logic operations, and applications in fields like control systems, home appliances, and cameras. It also notes some drawbacks like difficulty in tuning membership functions and potential confusion with probability theory.
Soft computing is an emerging approach to computing that aims to mimic human reasoning and learning in uncertain and imprecise environments. It includes neural networks, fuzzy logic, and genetic algorithms. The main goals of soft computing are to develop intelligent machines to solve real-world problems that are difficult to model mathematically, while exploiting tolerance for uncertainty like humans. Some applications of soft computing include consumer appliances, robotics, food preparation devices, and game playing. Soft computing is well-suited for problems not solvable by traditional computing due to its characteristics of tractability, low cost, and high machine intelligence.
This document discusses data compression techniques for digital images. It explains that compression reduces the amount of data needed to represent an image by removing redundant information. The compression process involves an encoder that transforms the input image, and a decoder that reconstructs the output image. The encoder uses three main stages: a mapper to reduce interpixel redundancy, a quantizer to reduce accuracy and psychovisual redundancy, and a symbol encoder to assign variable-length codes to the quantized values. The decoder performs the inverse operations of the encoder and mapper to reconstruct the original image, but does not perform the inverse of quantization which is a lossy process.
This document discusses classical sets and fuzzy sets. It defines classical sets as having distinct elements that are either fully included or excluded from the set. Fuzzy sets allow for gradual membership, with elements having degrees of membership between 0 and 1. Operations like union, intersection, and complement are defined for both classical and fuzzy sets, with fuzzy set operations accounting for degrees of membership. Properties of classical and fuzzy sets and relations are also covered, noting differences like fuzzy sets not following the law of excluded middle.
Soft computing is an emerging approach to computing that aims to solve computationally hard problems using inexact solutions that are tolerant of imprecision, uncertainty, partial truth, and approximation. It uses techniques like fuzzy logic, neural networks, evolutionary computation, and probabilistic reasoning to model human-like decision making. Unlike hard computing which requires precise modeling and solutions, soft computing is well-suited for real-world problems where ideal models are not available. The key constituents of soft computing are fuzzy logic, evolutionary computation, neural networks, and machine learning.
The document discusses the Fast Fourier Transform (FFT) algorithm. It begins by explaining how the Discrete Fourier Transform (DFT) and its inverse can be computed on a digital computer, but require O(N2) operations for an N-point sequence. The FFT was discovered to reduce this complexity to O(NlogN) operations by exploiting redundancy in the DFT calculation. It achieves this through a recursive decomposition of the DFT into smaller DFT problems. The FFT provides a significant speedup and enables practical spectral analysis of long signals.
The document discusses artificial neural networks and backpropagation. It provides an overview of backpropagation algorithms, including how they were developed over time, the basic methodology of propagating errors backwards, and typical network architectures. It also gives examples of applying backpropagation to problems like robotics, space robots, handwritten digit recognition, and face recognition.
Image restoration and degradation modelAnupriyaDurai
This document discusses image restoration and degradation. It provides an overview of image restoration techniques which attempt to reverse degradation processes and restore lost image information. Several types of image degradation are described, including motion blur, noise, and misfocus. Common noise models are explained, such as Gaussian, salt and pepper, Erlang, exponential, and uniform noise. Methods for estimating degradation models from observed images are also summarized, including using image observations, experimental replication of degradation, and mathematical modeling.
This document discusses image compression techniques. It begins by defining image compression as reducing the data required to represent a digital image. It then discusses why image compression is needed for storage, transmission and other applications. The document outlines different types of redundancies that can be exploited in compression, including spatial, temporal and psychovisual redundancies. It categorizes compression techniques as lossless or lossy and describes several algorithms for each type, including Huffman coding, LZW coding, DPCM, DCT and others. Key aspects like prediction, quantization, fidelity criteria and compression models are also summarized.
Homomorphic filtering is a technique used to remove multiplicative noise from images by transforming the image into the logarithmic domain, where the multiplicative components become additive. This allows the use of linear filters to separate the illumination and reflectance components, with a high-pass filter used to remove low-frequency illumination variations while preserving high-frequency reflectance edges. The filtered image is then transformed back to restore the original domain. Homomorphic filtering is commonly used to correct non-uniform illumination and simultaneously enhance contrast in grayscale images.
Neuro-fuzzy systems combine neural networks and fuzzy logic to overcome the limitations of each. They were created to achieve the mapping precision of neural networks and the interpretability of fuzzy systems. There are different types of neuro-fuzzy systems depending on whether the inputs, outputs, and weights are crisp or fuzzy. Two common models are fuzzy systems providing input to neural networks, and neural networks providing input to fuzzy systems. Neuro-fuzzy systems have applications in domains like measuring water opacity, improving financial ratings, and automatically adjusting devices.
This document provides an introduction to fuzzy logic and fuzzy sets. It discusses key concepts such as fuzzy sets having degrees of membership between 0 and 1 rather than binary membership, and fuzzy logic allowing for varying degrees of truth. Examples are given of fuzzy sets representing partially full tumblers and desirable cities to live in. Characteristics of fuzzy sets such as support, crossover points, and logical operations like union and intersection are defined. Applications mentioned include vehicle control systems and appliance control using fuzzy logic to handle imprecise and ambiguous inputs.
This document discusses various applications of parallel processing. It describes how parallel processing is used in numeric weather prediction to forecast weather by processing large amounts of observational data. It is also used in oceanography and astrophysics to study oceans and conduct particle simulations. Other applications mentioned include socioeconomic modeling, finite element analysis, artificial intelligence, seismic exploration, genetic engineering, weapon research, medical imaging, remote sensing, energy exploration, and more. The document also discusses loosely coupled and tightly coupled multiprocessors and the differences between the two approaches.
This document discusses image segmentation techniques. It describes how segmentation partitions an image into meaningful regions based on discontinuities or similarities in pixel intensity. The key methods covered are thresholding, edge detection using gradient and Laplacian operators, and the Hough transform for global line detection. Adaptive thresholding is also introduced as a technique to handle uneven illumination.
Transform coding is used in image and video processing to exploit correlations between neighboring pixels. The discrete cosine transform (DCT) is commonly used as it provides good energy compaction and de-correlation. DCT represents data as a sum of variable frequency cosine waves in the frequency domain. It has properties like separability and orthogonality that make it efficient for computation. DCT exhibits excellent energy compaction and removal of redundancy between pixels, making it useful for image and video compression.
The presentation gives basic insight into Information Theory, Entropies, various binary channels, and error conditions. It explains principles, derivations and problems in very easy and detailed manner with examples.
The document describes multilayer neural networks and their use for classification problems. It discusses how neural networks can handle continuous-valued inputs and outputs unlike decision trees. Neural networks are inherently parallel and can be sped up through parallelization techniques. The document then provides details on the basic components of neural networks, including neurons, weights, biases, and activation functions. It also describes common network architectures like feedforward networks and discusses backpropagation for training networks.
Here in the ppt a detailed description of Image Enhancement Techniques is given which includes topics like Basic Gray level Transformations,Histogram Processing.
Enhancement using Arithmetic/Logic Operations.
image averaging and image averaging methods.
Piecewise-Linear Transformation Functions
Discussed different types of dynamic interconnection networks. Graphically demonstrated single and multiple bus interconnection networks. Discussed different types of switch based interconnection networks. Graphically shown the mechanisms of crossbar, single and multistage interconnection networks. Graphically explained the working principle of omega network, Benes network, and baseline networks.
This presentation provides an introduction to the artificial neural networks topic, its learning, network architecture, back propagation training algorithm, and its applications.
The document discusses noise models and methods for removing additive noise from digital images. It describes several types of noise that can affect images, such as Gaussian, impulse, uniform, Rayleigh, gamma and exponential noise. It also presents various noise filters that can be used to remove noise, including mean filters like arithmetic, geometric and harmonic filters, and order statistics filters such as median, max, min and midpoint filters. The filters aim to reduce noise while retaining image detail as much as possible.
The document discusses various neural network learning rules:
1. Error correction learning rule (delta rule) adapts weights based on the error between the actual and desired output.
2. Memory-based learning stores all training examples and classifies new inputs based on similarity to nearby examples (e.g. k-nearest neighbors).
3. Hebbian learning increases weights of simultaneously active neuron connections and decreases others, allowing patterns to emerge from correlations in inputs over time.
4. Competitive learning (winner-take-all) adapts the weights of the neuron most active for a given input, allowing unsupervised clustering of similar inputs across neurons.
The document discusses fuzzy relations and fuzzy Cartesian products. It defines fuzzy relations as mappings between elements of two universes through their Cartesian product, with membership functions representing the strength of association. It provides an example of a fuzzy relation matrix and discusses operations like union, intersection, and composition of fuzzy relations. As an example, it gives fuzzy relations between universes X, Y and Z and calculates their composition using max-min and max-product methods.
This document summarizes part of a lecture on factor analysis from an machine learning course. It introduces the factor analysis model, which posits that observed data is generated by an underlying latent variable that is mapped to the observed space with noise. It describes the factor analysis model mathematically as a joint Gaussian distribution between the latent and observed variables. It also derives the E-step and M-step updates for performing maximum likelihood estimation of the factor analysis model parameters using EM algorithm.
The document discusses the Fast Fourier Transform (FFT) algorithm. It begins by explaining how the Discrete Fourier Transform (DFT) and its inverse can be computed on a digital computer, but require O(N2) operations for an N-point sequence. The FFT was discovered to reduce this complexity to O(NlogN) operations by exploiting redundancy in the DFT calculation. It achieves this through a recursive decomposition of the DFT into smaller DFT problems. The FFT provides a significant speedup and enables practical spectral analysis of long signals.
The document discusses artificial neural networks and backpropagation. It provides an overview of backpropagation algorithms, including how they were developed over time, the basic methodology of propagating errors backwards, and typical network architectures. It also gives examples of applying backpropagation to problems like robotics, space robots, handwritten digit recognition, and face recognition.
Image restoration and degradation modelAnupriyaDurai
This document discusses image restoration and degradation. It provides an overview of image restoration techniques which attempt to reverse degradation processes and restore lost image information. Several types of image degradation are described, including motion blur, noise, and misfocus. Common noise models are explained, such as Gaussian, salt and pepper, Erlang, exponential, and uniform noise. Methods for estimating degradation models from observed images are also summarized, including using image observations, experimental replication of degradation, and mathematical modeling.
This document discusses image compression techniques. It begins by defining image compression as reducing the data required to represent a digital image. It then discusses why image compression is needed for storage, transmission and other applications. The document outlines different types of redundancies that can be exploited in compression, including spatial, temporal and psychovisual redundancies. It categorizes compression techniques as lossless or lossy and describes several algorithms for each type, including Huffman coding, LZW coding, DPCM, DCT and others. Key aspects like prediction, quantization, fidelity criteria and compression models are also summarized.
Homomorphic filtering is a technique used to remove multiplicative noise from images by transforming the image into the logarithmic domain, where the multiplicative components become additive. This allows the use of linear filters to separate the illumination and reflectance components, with a high-pass filter used to remove low-frequency illumination variations while preserving high-frequency reflectance edges. The filtered image is then transformed back to restore the original domain. Homomorphic filtering is commonly used to correct non-uniform illumination and simultaneously enhance contrast in grayscale images.
Neuro-fuzzy systems combine neural networks and fuzzy logic to overcome the limitations of each. They were created to achieve the mapping precision of neural networks and the interpretability of fuzzy systems. There are different types of neuro-fuzzy systems depending on whether the inputs, outputs, and weights are crisp or fuzzy. Two common models are fuzzy systems providing input to neural networks, and neural networks providing input to fuzzy systems. Neuro-fuzzy systems have applications in domains like measuring water opacity, improving financial ratings, and automatically adjusting devices.
This document provides an introduction to fuzzy logic and fuzzy sets. It discusses key concepts such as fuzzy sets having degrees of membership between 0 and 1 rather than binary membership, and fuzzy logic allowing for varying degrees of truth. Examples are given of fuzzy sets representing partially full tumblers and desirable cities to live in. Characteristics of fuzzy sets such as support, crossover points, and logical operations like union and intersection are defined. Applications mentioned include vehicle control systems and appliance control using fuzzy logic to handle imprecise and ambiguous inputs.
This document discusses various applications of parallel processing. It describes how parallel processing is used in numeric weather prediction to forecast weather by processing large amounts of observational data. It is also used in oceanography and astrophysics to study oceans and conduct particle simulations. Other applications mentioned include socioeconomic modeling, finite element analysis, artificial intelligence, seismic exploration, genetic engineering, weapon research, medical imaging, remote sensing, energy exploration, and more. The document also discusses loosely coupled and tightly coupled multiprocessors and the differences between the two approaches.
This document discusses image segmentation techniques. It describes how segmentation partitions an image into meaningful regions based on discontinuities or similarities in pixel intensity. The key methods covered are thresholding, edge detection using gradient and Laplacian operators, and the Hough transform for global line detection. Adaptive thresholding is also introduced as a technique to handle uneven illumination.
Transform coding is used in image and video processing to exploit correlations between neighboring pixels. The discrete cosine transform (DCT) is commonly used as it provides good energy compaction and de-correlation. DCT represents data as a sum of variable frequency cosine waves in the frequency domain. It has properties like separability and orthogonality that make it efficient for computation. DCT exhibits excellent energy compaction and removal of redundancy between pixels, making it useful for image and video compression.
The presentation gives basic insight into Information Theory, Entropies, various binary channels, and error conditions. It explains principles, derivations and problems in very easy and detailed manner with examples.
The document describes multilayer neural networks and their use for classification problems. It discusses how neural networks can handle continuous-valued inputs and outputs unlike decision trees. Neural networks are inherently parallel and can be sped up through parallelization techniques. The document then provides details on the basic components of neural networks, including neurons, weights, biases, and activation functions. It also describes common network architectures like feedforward networks and discusses backpropagation for training networks.
Here in the ppt a detailed description of Image Enhancement Techniques is given which includes topics like Basic Gray level Transformations,Histogram Processing.
Enhancement using Arithmetic/Logic Operations.
image averaging and image averaging methods.
Piecewise-Linear Transformation Functions
Discussed different types of dynamic interconnection networks. Graphically demonstrated single and multiple bus interconnection networks. Discussed different types of switch based interconnection networks. Graphically shown the mechanisms of crossbar, single and multistage interconnection networks. Graphically explained the working principle of omega network, Benes network, and baseline networks.
This presentation provides an introduction to the artificial neural networks topic, its learning, network architecture, back propagation training algorithm, and its applications.
The document discusses noise models and methods for removing additive noise from digital images. It describes several types of noise that can affect images, such as Gaussian, impulse, uniform, Rayleigh, gamma and exponential noise. It also presents various noise filters that can be used to remove noise, including mean filters like arithmetic, geometric and harmonic filters, and order statistics filters such as median, max, min and midpoint filters. The filters aim to reduce noise while retaining image detail as much as possible.
The document discusses various neural network learning rules:
1. Error correction learning rule (delta rule) adapts weights based on the error between the actual and desired output.
2. Memory-based learning stores all training examples and classifies new inputs based on similarity to nearby examples (e.g. k-nearest neighbors).
3. Hebbian learning increases weights of simultaneously active neuron connections and decreases others, allowing patterns to emerge from correlations in inputs over time.
4. Competitive learning (winner-take-all) adapts the weights of the neuron most active for a given input, allowing unsupervised clustering of similar inputs across neurons.
The document discusses fuzzy relations and fuzzy Cartesian products. It defines fuzzy relations as mappings between elements of two universes through their Cartesian product, with membership functions representing the strength of association. It provides an example of a fuzzy relation matrix and discusses operations like union, intersection, and composition of fuzzy relations. As an example, it gives fuzzy relations between universes X, Y and Z and calculates their composition using max-min and max-product methods.
This document summarizes part of a lecture on factor analysis from an machine learning course. It introduces the factor analysis model, which posits that observed data is generated by an underlying latent variable that is mapped to the observed space with noise. It describes the factor analysis model mathematically as a joint Gaussian distribution between the latent and observed variables. It also derives the E-step and M-step updates for performing maximum likelihood estimation of the factor analysis model parameters using EM algorithm.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
Comparison Results of Trapezoidal, Simpson’s 13 rule, Simpson’s 38 rule, and ...BRNSS Publication Hub
Numerical integration plays very important role in mathematics. In this paper, overviews on the most common one, namely, trapezoidal Simpson’s 13 rule, Simpson’s 38 rule and Weddle’s rule. Different procedures compared and tried to evaluate the value of some definite integrals. A combined approach of different integral rules has been proposed for a definite integral to get more accurate value for all case
This document compares numerical integration methods for approximating definite integrals, including trapezoidal, Simpson's 1/3 rule, Simpson's 3/8 rule, and Weddle's rule. It presents the formulas for each method and applies them to evaluate several definite integrals. The results show that Weddle's rule provides the most accurate approximations compared to the other methods.
This document compares numerical integration methods for approximating definite integrals, including trapezoidal, Simpson's 1/3 rule, Simpson's 3/8 rule, and Weddle's rule. It presents the definitions and formulas for each method. Various definite integrals are evaluated using each method and the results are compared to the actual values, with Weddle's rule found to be the most accurate. The document proposes a new composite rule for numerical integration and concludes that Weddle's rule gives greater accuracy than the other methods tested.
This document discusses different number systems including binary, hexadecimal, and octal. It begins by reviewing place value in the decimal system. It then explains how to convert between binary and decimal, including using addition tables to add binary numbers. Next, it covers the hexadecimal system and how to convert between hexadecimal and decimal. Finally, it introduces relations and equivalence relations, including properties of relations, partitions, and representing relations using matrices.
1. A bi-variate random variable has a joint probability distribution function (PDF) that defines the probability of two random variables occurring together. The marginal PDF defines the probability of each variable individually, while the conditional PDF defines the probability of one variable given the other.
2. A multi-variate random variable contains multiple random variables defined by a mean vector and covariance matrix. A linear transformation of a Gaussian multi-variate random variable remains Gaussian with a transformed mean vector and covariance matrix.
3. If two random vectors are jointly Gaussian and uncorrelated, they are also independent, as their joint PDF can be written as the product of their individual PDFs.
On the Family of Concept Forming Operators in Polyadic FCADmitrii Ignatov
Triadic Formal Concept Analysis (3FCA) was introduced by Lehman and Wille almost two decades ago. And many researchers work in Data Mining and Formal Concept Analysis using the notions of closed sets, Galois and closure operators, closure systems. However, up-to-date even though that different researchers actively work on mining triadic and n-ary relations, a proper closure operator for enumeration of triconcepts, i.e. maximal triadic cliques of tripartite hypergaphs, was not introduced. In this talk we show that the previously introduced operators for obtaining triconcepts are not always consistent, describe their family and study their properties. We also introduce the notion of maximal switching generator to explain why such concept-forming operators are not closure operators due to violation of monotonicity property.
Principal component analysis (PCA) is a technique used to reduce the dimensionality of data by transforming correlated variables into a smaller number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. PCA involves computing the covariance matrix of the data and then determining the eigenvectors with the highest eigenvalues, which become the principal components.
This document discusses fuzzy relations, reasoning, and linguistic variables. It defines fuzzy relations as membership functions between elements of Cartesian product spaces. It describes the extension principle for mapping fuzzy sets through functions. Max-min and max-product composition are defined for combining fuzzy relations. Linguistic variables allow information to be expressed using fuzzy linguistic terms rather than numerical values. Operations on linguistic variables like concentration and dilation are discussed. Fuzzy if-then rules are defined using implication functions to model "if A then B" statements where A and B are linguistic values. Fuzzy reasoning uses these rules and facts to derive conclusions.
The document discusses solving linear differential equations with variable coefficients using power series representations. It begins by introducing series solution methods and properties of infinite series. It then discusses the Method of Frobenius for solving equations with variable coefficients that arise in cylindrical and spherical coordinate systems. The method involves finding the indicial equation to determine the index c, and then using the recurrence relation to determine the series coefficients an. An example is provided to illustrate the method where the roots of the indicial equation are distinct and not differing by an integer.
This lecture notes were written as part of the course "Pattern Recognition and Machine Learning" taught by Prof. Dinesh Garg at IIT Gandhinagar. This lecture notes deals with Linear Regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
Correlation & Regression for Statistics Social Sciencessuser71ac73
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
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8+8+8 Rule Of Time Management For Better ProductivityRuchiRathor2
This is a great way to be more productive but a few things to
Keep in mind:
- The 8+8+8 rule offers a general guideline. You may need to adjust the schedule depending on your individual needs and commitments.
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2. Relations This chapter introduce the notion of relation. The notion of relation is the basic idea behind numerous operations on sets suchas Cartesian products, composition of relations , difference of relations and intersections of relations and equivalence properties In all engineering , science and mathematically based fields, relations is very important 2
3. Relations Similarities can be described with relations. In this sense, relations is a very important notion to many different technologies like graph theory, data manipulation. Graphtheory 3
5. Inclassicalrelations (crisprelations), Relationshipsbetweenelements of thesetsare only in twodegrees; “completelyrelated” and “not related”. Fuzzyrelationstake on an infinitivenumber of degrees of relationshipsbetweentheextremes of “ completelyrelated” and “ not related” 5
6.
7. Crisp system -Complex systems hardto model -incomplete information leads to inaccuracy -numerical Fuzzy logic system -No traditional modeling,inferences based on knowledge - can handle incomplete information to some degree -linguistic 7
8. CartesianProduct Example 3.1. The elements in two sets A and B are given as A ={0, 1} and B ={a,b, c}. Various Cartesian products of these two sets can bewritten as shown: A × B ={(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)} B × A ={(a, 0), (a, 1), (b, 0), (b, 1), (c, 0), (c, 1)} A × A = A2={(0, 0), (0, 1), (1, 0), (1, 1)} B × B = B2={(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)} 8
9. CrispRelations Cartesianproduct is denoted in form A1 x A2 x…..x Ar Themostcommoncase is for r=2 andrepresentwith A1 x A2 The Cartesian product of two universes X and Y is determined as X × Y = {(x, y) | x ∈ X,y ∈ Y} This form showsthatthere is a matchingbetween X and Y , this is a unconstrainedmatching. 9
10. CrispRelations Every element in universe X is related completely toevery element in universe Y Thisrelationship’sstrenght is measuredbythecharacteristicsfunctionχ χX×Y(x, y) = 1, (x,y) ∈ X × Y 0, (x,y) ∉ X × Y Completerelationship is showedwith 1 and no relationship is showedwith 0 10
11. When the universes, or sets, are finite the relation can be conveniently represented by a matrix, called a relation matrix. X ={1, 2, 3} and Y ={a, b, c} Sagittal diagram of an unconstrained relation 11
12. Specialcases of theconstrainedCartesianproductforsetswhere r=2 arecalledidentityrelationdenoted IA IA ={(0, 0), (1, 1), (2, 2)} Specialcases of theunconstrainedCartesianproductforsetswhere r=2 arecalleduniversalrelationdenoted UA UA ={(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)} 12
13. Cardinality Of CripsRelations TheCardinality of therelation r between X and Y is n X x Y = nx * ny Power set (P(X x Y)), nP(X×Y) = 2(nXnY) 13
14. Operations On CripsRelations Define R and S as two separate relations on the Cartesian universe X × Y Union: R ∪ S -> χR∪S(x, y) : χR∪S(x, y) = max[χR(x, y), χS(x, y)] Intersection: R ∩ S -> χR∩S(x, y) : χR∩S(x, y) = min[χR(x, y), χS(x, y)] Complement:R ->χR(x, y) : χR(x, y) = 1 − χR(x, y) Containment: R ⊂ S ->χR(x, y) : χR(x, y) ≤ χS(x, y) 14
20. A chain is only as strong as itsweakestlink 20
21. Example Using themax–min composition operation,relationmatrices for Rand S would be expressed as µT(x1, z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 21
22. Example Using themax–min composition operation,relationmatrices for Rand S would be expressed as µT(x1, z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 µT(x1, z2) = max[min(1, 1), min(0, 0), min(1, 1), min(0, 0)] = 1 22
23. FuzzyRelations A fuzzy relation R is a mapping from the Cartesianspace X x Y to the interval [0,1], where thestrength of the mapping is expressed by themembership function of the relation μR(x,y) μR : A × B -> [0, 1] R = {((x, y), μR(x, y))| μR(x, y) ≥ 0 , x ∈ A, y ∈ B} 23
26. Cardinality of FuzzyRelations Since the cardinality of fuzzy sets on any universe is infinity, the cardinality of a fuzzyrelation between two or more universes is also infinity. 26
27. Operations on FuzzyRelations Let R and S be fuzzy relations on the Cartesian space X × Y. Then the following operationsapply for the membership values for various set operations: 27 Union: µR∪S(x, y) = max(µR (x, y),µS(x, y)) Intersection: µR∩S (x, y) = min(µR (x, y),µS (x, y)) Complement:µR(x, y) = 1 − µR(x, y) Containment:R⊂ S ⇒ µR (x, y) ≤ µS (x, y)
28. Fuzzy Cartesian Product and Composition A fuzzy relation R is a mapping from the Cartesianspace X x Y to the interval [0,1], where thestrength of the mapping is expressed by themembership function of the relation μR(x,y) μR: A × B -> [0, 1] R = {((x, y), μR(x, y))| μR(x, y) ≥ 0 , x ∈ A, y ∈ B} 28
29. Max-minComposition Two fuzzy relations R and S are defined on sets A,B and C. That is, R ⊆ A × B, S ⊆ B × C. Thecomposition S•R = SR of two relations R and S isexpressed by the relation from A to C: For(x, y) ∈ A × B, (y, z) ∈ B × C, µS•R(x, z) = max [min(µR(x, y), µS(y, z))]= ∨ [μR(x, y) ∧ μS(y, z)] MS•R= MR•MS(matrixnotation) 29
31. Max-productComposition Two fuzzy relations R and S are defined on sets A,B and C. That is, R ⊆ A × B, S ⊆ B × C. Thecomposition S•R = SR of two relations R and S isexpressed by the relation from A to C: For(x, y) ∈ A × B, (y, z) ∈ B × C, μS•R(x, z) = maxy[μR(x, y) • μS(y, z)] = ∨y[μR(x, y) • μS(y, z) MS•R= MR• MS(matrixnotation) 31
33. Example Suppose we have two fuzzy sets, Adefined on a universe of three discretetemperatures, X = {x1, x2, x3}, and Bdefined on a universe of two discrete pressures, Y ={y1, y2}, and we want to find the fuzzy Cartesian product between them. Fuzzy set Acouldrepresent the ‘‘ambient’’ temperature and fuzzy setBthe ‘‘near optimum’’ pressure for a certainheat exchanger, and the Cartesian productmight represent the conditions (temperature–pressurepairs) of the exchanger that are associated with ‘‘efficient’’ operations. 33
34. FuzzyCartesianproduct, usingµS•R(x, z) = max [min (µR (x, y), µS (y, z))]results in a fuzzyrelation R (of size 3 × 2) representing ‘‘efficient’’ conditions, 34
35. Example X = {x1, x2}, Y = {y1, y2}, and Z = {z1, z2, z3} Consider the following fuzzy relations: 35 Thentheresultingrelation, T, which relates elements of universe X to elements of universe Z, μT(x1, z1) = max[min(0.7, 0.9), min(0.5, 0.1)] = 0.7
37. Example A simple fuzzy system is given, which models the brake behaviour of a car driver depending on the car speed. The inference machine should determine the brake force for a given car speed. The speed is specified by the two linguistic terms"low"and "medium", and the brake force by "moderate"and "strong". The rule base includes the two rules (1) IF the car speed is low THEN the brake force is moderate (2) IF the car speed is medium THEN the brake force is strong 37
39. CrispEquivalenceRelation Arelation R on a universeXcan also be thought of as a relation fromXtoX. The relation R is an equivalence relation and it has the following three properties: Reflexivity Symmetry Transitivity 39
40. Reflexivity (xi ,xi ) ∈ R or χR(xi ,xi ) = 1 When a relation is reflexiveevery vertex in the graph originates a single loop, as shown in 40
43. CrispToleranceRelation A tolerance relation R (also called a proximity relation) on a universe X is a relation that exhibits only the properties of reflexivity and symmetry. A tolerance relation, R, can be reformed into an equivalence relation by at most (n − 1) compositions with itself, where n is the cardinal number of the set defining R, in this case X 43
44. Example Suppose in an airline transportation system we have a universe composed offive elements: the cities Omaha, Chicago, Rome, London, and Detroit. The airline is studyinglocations of potential hubs in various countries and must consider air mileage between citiesand takeoff and landing policies in the various countries. 44
45. Example These cities can be enumerated as the elements of a set, i.e., X ={x1,x2,x3,x4,x5}={Omaha, Chicago, Rome, London, Detroit} Suppose we have a tolerance relation, R1, that expresses relationships among these cities: This relation is reflexive and symmetric. 45
46. Example The graph for this tolerance relation If(x1,x5) ∈ R1can become an equivalence relation 46
47. Example: Thismatrix is equivalence relation because it has (x1,x5) 47 Five-vertex graph of equivalence relation (reflexive, symmetric, transitive)
49. Example Suppose, in a biotechnology experiment, five potentially new strains of bacteriahave been detected in the area around an anaerobic corrosion pit on a new aluminum-lithiumalloy used in the fuel tanks of a new experimental aircraft. In order to propose methods toeliminate the biocorrosion caused by these bacteria, the five strains must first be categorized.One way to categorize them is to compare them to one another. In a pairwise comparison, thefollowing " similarity" relation,R1, is developed. For example, the first strain (column 1) hasa strength of similarity to the second strain of 0.8, to the third strain a strength of 0 (i.e., norelation), to the fourth strain a strength of 0.1, and so on. Because the relation is for pairwisesimilarity it will be reflexive and symmetric. Hence, 49
50. 50 is reflexive and symmetric. However, it is not transitive μR(x1, x2) = 0.8, μR(x2, x5) = 0.9 ≥ 0.8 but μR(x1, x5) = 0.2 ≤ min(0.8, 0.9)
51. 51 One composition results in the following relation: where transitivity still does not result; for example, μR2(x1, x2) = 0.8 ≥ 0.5 and μR2(x2, x4) = 0.5 but μR2(x1, x4) = 0.2 ≤ min(0.8, 0.5)
52. 52 Finally, after one or two more compositions, transitivity results: