Naive Bayes is a kind of classifier which uses the Bayes Theorem. It predicts membership probabilities for each class such as the probability that given record or data point belongs to a particular class.
This document contains lecture slides from an ENEM602 engineering mathematics course taught by Dr. Eng. Mohammad Tawfik in Spring 2007. It discusses Lagrange interpolation, which is a method for interpolating a polynomial that passes through a set of points. The document provides examples of using Lagrange interpolation for 2, 3 and 4 points. It derives the general Lagrange interpolation formula and shows an example of applying it to find a 3rd order polynomial to interpolate 4 given data points. Students are assigned homework problems applying Lagrange interpolation.
This document introduces mathematical induction. It defines the principle of mathematical induction as having two steps: (1) the basis step, which shows a statement P(1) is true, and (2) the inductive step, which assumes P(k) is true and shows P(k+1) is also true. It provides an example of climbing an infinite ladder to illustrate these steps. It also notes some important points about mathematical induction, such as that it is expressing a rule of inference and in proofs we show P(k) implies P(k+1) rather than assuming P(k) is true for all k.
A neural network maps a set of inputs to a set of outputs. It is composed of nodes or units connected by links with weights. A neural network can compute or approximate functions, perform pattern recognition, signal processing, and learn to do any of these. A perceptron is a basic type of neural network that uses a threshold activation function. It can be trained to learn functions using the perceptron learning rule, which adjusts the weights to minimize errors between the network's output and the target output.
This document summarizes key concepts from Chapter 5 of the book "Pattern Recognition and Machine Learning" regarding neural networks.
1. Neural networks can overcome the curse of dimensionality by using nonlinear activation functions between layers. Common activation functions include sigmoid, tanh, and ReLU.
2. A feedforward neural network consists of an input layer, hidden layers with nonlinear activations, and an output layer. The network learns by adjusting weights in a process called backpropagation.
3. Bayesian neural networks treat the network weights as distributions and integrate them out to make predictions, avoiding overfitting. However, the posterior distribution cannot be expressed in closed form due to the nonlinear nature of neural networks.
This document defines and explains key concepts in fuzzy set theory, including fuzzy complements, unions, and intersections. It begins with an introduction to fuzzy sets as a generalization of classical sets that allows for gradual membership rather than binary membership. Membership functions assign elements a value between 0 and 1 indicating their degree of belonging to a set. The document then provides definitions and properties of fuzzy complements, unions, intersections, and other related concepts. It concludes with examples of applications of fuzzy set theory such as traffic monitoring systems, appliance controls, and medical diagnosis.
Presentation on Numerical Method (Trapezoidal Method)Syed Ahmed Zaki
The document discusses the trapezoidal method, which is a technique for approximating definite integrals. It provides the general formula for the trapezoidal rule, explains how it works by approximating the area under a function as a trapezoid, and discusses its history, advantages of being easy to use and having powerful convergence properties. An example application of the trapezoidal rule is shown, along with pseudocode and a C code implementation. The document concludes the trapezoidal rule can accurately integrate non-periodic and periodic functions.
Section 9: Equivalence Relations & CosetsKevin Johnson
This document discusses equivalence relations and cosets from abstract algebra. It contains the following key points:
1) It defines equivalence relations as relations that satisfy reflexivity, symmetry, and transitivity. Modular arithmetic and group conjugacy are given as examples of equivalence relations.
2) It introduces the concept of equivalence classes, which are the subsets of elements related by an equivalence relation. It proves that the equivalence classes partition the set.
3) It defines right cosets as translations of a subgroup by group elements. Examples are given of finding the right cosets of subgroups of Z6 and S3.
Naive Bayes is a kind of classifier which uses the Bayes Theorem. It predicts membership probabilities for each class such as the probability that given record or data point belongs to a particular class.
This document contains lecture slides from an ENEM602 engineering mathematics course taught by Dr. Eng. Mohammad Tawfik in Spring 2007. It discusses Lagrange interpolation, which is a method for interpolating a polynomial that passes through a set of points. The document provides examples of using Lagrange interpolation for 2, 3 and 4 points. It derives the general Lagrange interpolation formula and shows an example of applying it to find a 3rd order polynomial to interpolate 4 given data points. Students are assigned homework problems applying Lagrange interpolation.
This document introduces mathematical induction. It defines the principle of mathematical induction as having two steps: (1) the basis step, which shows a statement P(1) is true, and (2) the inductive step, which assumes P(k) is true and shows P(k+1) is also true. It provides an example of climbing an infinite ladder to illustrate these steps. It also notes some important points about mathematical induction, such as that it is expressing a rule of inference and in proofs we show P(k) implies P(k+1) rather than assuming P(k) is true for all k.
A neural network maps a set of inputs to a set of outputs. It is composed of nodes or units connected by links with weights. A neural network can compute or approximate functions, perform pattern recognition, signal processing, and learn to do any of these. A perceptron is a basic type of neural network that uses a threshold activation function. It can be trained to learn functions using the perceptron learning rule, which adjusts the weights to minimize errors between the network's output and the target output.
This document summarizes key concepts from Chapter 5 of the book "Pattern Recognition and Machine Learning" regarding neural networks.
1. Neural networks can overcome the curse of dimensionality by using nonlinear activation functions between layers. Common activation functions include sigmoid, tanh, and ReLU.
2. A feedforward neural network consists of an input layer, hidden layers with nonlinear activations, and an output layer. The network learns by adjusting weights in a process called backpropagation.
3. Bayesian neural networks treat the network weights as distributions and integrate them out to make predictions, avoiding overfitting. However, the posterior distribution cannot be expressed in closed form due to the nonlinear nature of neural networks.
This document defines and explains key concepts in fuzzy set theory, including fuzzy complements, unions, and intersections. It begins with an introduction to fuzzy sets as a generalization of classical sets that allows for gradual membership rather than binary membership. Membership functions assign elements a value between 0 and 1 indicating their degree of belonging to a set. The document then provides definitions and properties of fuzzy complements, unions, intersections, and other related concepts. It concludes with examples of applications of fuzzy set theory such as traffic monitoring systems, appliance controls, and medical diagnosis.
Presentation on Numerical Method (Trapezoidal Method)Syed Ahmed Zaki
The document discusses the trapezoidal method, which is a technique for approximating definite integrals. It provides the general formula for the trapezoidal rule, explains how it works by approximating the area under a function as a trapezoid, and discusses its history, advantages of being easy to use and having powerful convergence properties. An example application of the trapezoidal rule is shown, along with pseudocode and a C code implementation. The document concludes the trapezoidal rule can accurately integrate non-periodic and periodic functions.
Section 9: Equivalence Relations & CosetsKevin Johnson
This document discusses equivalence relations and cosets from abstract algebra. It contains the following key points:
1) It defines equivalence relations as relations that satisfy reflexivity, symmetry, and transitivity. Modular arithmetic and group conjugacy are given as examples of equivalence relations.
2) It introduces the concept of equivalence classes, which are the subsets of elements related by an equivalence relation. It proves that the equivalence classes partition the set.
3) It defines right cosets as translations of a subgroup by group elements. Examples are given of finding the right cosets of subgroups of Z6 and S3.
Unit I of the syllabus covers propositional logic and counting theory. It introduces concepts such as propositions, logical connectives like conjunction, disjunction, negation, implication and biconditional. It discusses how to represent compound statements using these connectives and their truth tables. The unit also covers topics like predicate logic, methods of proof, mathematical induction and fundamental counting principles like permutations and combinations. It aims to provide the logical foundations for discrete mathematics concepts that will be useful in computer science and information technology.
The document provides information about multi-layer perceptrons (MLPs) and backpropagation. It begins with definitions of perceptrons and MLP architecture. It then describes backpropagation, including the backpropagation training algorithm and cycle. Examples are provided, such as using an MLP to solve the exclusive OR (XOR) problem. Applications of backpropagation neural networks and options like momentum, batch vs sequential training, and adaptive learning rates are also discussed.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
Pattern recognition and Machine Learning.Rohit Kumar
Machine learning involves using examples to generate a program or model that can classify new examples. It is useful for tasks like recognizing patterns, generating patterns, and predicting outcomes. Some common applications of machine learning include optical character recognition, biometrics, medical diagnosis, and information retrieval. The goal of machine learning is to build models that can recognize patterns in data and make predictions.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
This document is from IFET College of Engineering and presents information on solving second order linear differential equations with constant coefficients. It defines such an equation as one where the highest order derivative is of order 2 and all coefficients are constants. The general solution is described as the sum of the complementary function and particular integral. Various cases are discussed for the complementary function depending on whether the roots are real/complex and distinct or repeated. Methods like variation of parameters and Cauchy's and Legendre's equations are also mentioned for solving related problems.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
hands on machine learning Chapter 6&7 decision tree, ensemble and random forestJaey Jeong
This document discusses decision trees and ensemble methods like random forests. It covers decision tree training and visualization using iris datasets. Ensemble methods like bagging, boosting and stacking are introduced. Random forests are ensembles of decision trees that split on a random subset of features at each node. Boosting methods like AdaBoost and gradient boosting aim to boost weak learners into a strong learner by focusing on misclassified samples.
This document is a report on Taylor's Theorem from a mathematics class. It begins with an introduction and objectives. It then defines Taylor's Theorem as giving an approximation of a function around a point using a Taylor polynomial. An example is worked through to approximate e to three decimal places using Taylor's formula. Two activities are presented involving the remainder term in Taylor's formula and applying it to polynomials. The document concludes with an assignment on using Taylor's formula for specific functions and approximating 1/e.
This document provides an introduction and overview of numerical analysis. It begins by stating that numerical analysis aims to find approximate solutions to complex mathematical problems through repeated computational steps when analytical solutions are not available or practical. It then discusses that numerical analysis is important because it allows for the conversion of physical phenomena into mathematical models that can be solved through basic arithmetic operations. Finally, it explains that numerical analysis involves developing algorithms and numerical techniques to solve problems, implementing those techniques using computers, and analyzing errors in approximate solutions.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
The document discusses the bisection method for finding roots of equations. It begins by defining the bisection method as a root finding technique that repeatedly bisects an interval and selects a subinterval containing the root. It notes that while simple and robust, the bisection method converges slowly. The document then provides the step-by-step algorithm for implementing the bisection method and works through an example of finding the root of f(x) = x^2 - 2 between 1 and 2. It concludes by presenting the bisection method code in C++.
This document provides an overview of calculus of variations, which generalizes the method of finding extrema of functions to functionals. It discusses how functionals take on extreme values when their path or curve satisfies certain necessary conditions, analogous to single-variable calculus. These necessary conditions are derived by applying the calculus of variations methodology to functionals dependent on a path and finding the Euler-Lagrange equation. Several examples from physics are described where extremizing a functional corresponds to minimizing time, length, or other physical quantities.
This document discusses methods for solving systems of linear equations, including the traditional method, matrix method, row echelon method, Gauss elimination method, and Gauss Jordan method. It provides examples working through solving systems of equations using Gauss elimination and Gauss Jordan. The key steps of each method like constructing the augmented matrix, row operations, and back substitution are demonstrated. Related fields where linear algebra is applied are also listed.
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.
These slides are distributed under a Creative Commons license for educational purposes and may not be used or distributed for commercial purposes. DeepLearning.AI makes the slides available as long as proper attribution is given. The full details of the license can be found at the provided URL.
The document discusses sequences and their properties. A sequence is a function whose domain is the positive integers. Sequences are commonly represented using subscript notation rather than standard function notation. The nth term of a sequence is denoted an. [/SUMMARY]
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.
This document provides an overview of deep learning and neural networks. It begins with definitions of machine learning, artificial intelligence, and the different types of machine learning problems. It then introduces deep learning, explaining that it uses neural networks with multiple layers to learn representations of data. The document discusses why deep learning works better than traditional machine learning for complex problems. It covers key concepts like activation functions, gradient descent, backpropagation, and overfitting. It also provides examples of applications of deep learning and popular deep learning frameworks like TensorFlow. Overall, the document gives a high-level introduction to deep learning concepts and techniques.
How CEOs should consider all the scenarios that patents are important for: patents as assets, sales tools, offense-defense tools. 科创企业应该发明、攻防、销售、资产等场景看专利。
Unit I of the syllabus covers propositional logic and counting theory. It introduces concepts such as propositions, logical connectives like conjunction, disjunction, negation, implication and biconditional. It discusses how to represent compound statements using these connectives and their truth tables. The unit also covers topics like predicate logic, methods of proof, mathematical induction and fundamental counting principles like permutations and combinations. It aims to provide the logical foundations for discrete mathematics concepts that will be useful in computer science and information technology.
The document provides information about multi-layer perceptrons (MLPs) and backpropagation. It begins with definitions of perceptrons and MLP architecture. It then describes backpropagation, including the backpropagation training algorithm and cycle. Examples are provided, such as using an MLP to solve the exclusive OR (XOR) problem. Applications of backpropagation neural networks and options like momentum, batch vs sequential training, and adaptive learning rates are also discussed.
Numerical solution of a system of linear equations by
1) LU FACTORIZATION METHOD.
2) GAUSS ELIMINATION METHOD.
3) MATRIX INVERSION BY GAUSS ELIMINATION METHOD.
Pattern recognition and Machine Learning.Rohit Kumar
Machine learning involves using examples to generate a program or model that can classify new examples. It is useful for tasks like recognizing patterns, generating patterns, and predicting outcomes. Some common applications of machine learning include optical character recognition, biometrics, medical diagnosis, and information retrieval. The goal of machine learning is to build models that can recognize patterns in data and make predictions.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
This document is from IFET College of Engineering and presents information on solving second order linear differential equations with constant coefficients. It defines such an equation as one where the highest order derivative is of order 2 and all coefficients are constants. The general solution is described as the sum of the complementary function and particular integral. Various cases are discussed for the complementary function depending on whether the roots are real/complex and distinct or repeated. Methods like variation of parameters and Cauchy's and Legendre's equations are also mentioned for solving related problems.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
hands on machine learning Chapter 6&7 decision tree, ensemble and random forestJaey Jeong
This document discusses decision trees and ensemble methods like random forests. It covers decision tree training and visualization using iris datasets. Ensemble methods like bagging, boosting and stacking are introduced. Random forests are ensembles of decision trees that split on a random subset of features at each node. Boosting methods like AdaBoost and gradient boosting aim to boost weak learners into a strong learner by focusing on misclassified samples.
This document is a report on Taylor's Theorem from a mathematics class. It begins with an introduction and objectives. It then defines Taylor's Theorem as giving an approximation of a function around a point using a Taylor polynomial. An example is worked through to approximate e to three decimal places using Taylor's formula. Two activities are presented involving the remainder term in Taylor's formula and applying it to polynomials. The document concludes with an assignment on using Taylor's formula for specific functions and approximating 1/e.
This document provides an introduction and overview of numerical analysis. It begins by stating that numerical analysis aims to find approximate solutions to complex mathematical problems through repeated computational steps when analytical solutions are not available or practical. It then discusses that numerical analysis is important because it allows for the conversion of physical phenomena into mathematical models that can be solved through basic arithmetic operations. Finally, it explains that numerical analysis involves developing algorithms and numerical techniques to solve problems, implementing those techniques using computers, and analyzing errors in approximate solutions.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
The document discusses the bisection method for finding roots of equations. It begins by defining the bisection method as a root finding technique that repeatedly bisects an interval and selects a subinterval containing the root. It notes that while simple and robust, the bisection method converges slowly. The document then provides the step-by-step algorithm for implementing the bisection method and works through an example of finding the root of f(x) = x^2 - 2 between 1 and 2. It concludes by presenting the bisection method code in C++.
This document provides an overview of calculus of variations, which generalizes the method of finding extrema of functions to functionals. It discusses how functionals take on extreme values when their path or curve satisfies certain necessary conditions, analogous to single-variable calculus. These necessary conditions are derived by applying the calculus of variations methodology to functionals dependent on a path and finding the Euler-Lagrange equation. Several examples from physics are described where extremizing a functional corresponds to minimizing time, length, or other physical quantities.
This document discusses methods for solving systems of linear equations, including the traditional method, matrix method, row echelon method, Gauss elimination method, and Gauss Jordan method. It provides examples working through solving systems of equations using Gauss elimination and Gauss Jordan. The key steps of each method like constructing the augmented matrix, row operations, and back substitution are demonstrated. Related fields where linear algebra is applied are also listed.
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.
These slides are distributed under a Creative Commons license for educational purposes and may not be used or distributed for commercial purposes. DeepLearning.AI makes the slides available as long as proper attribution is given. The full details of the license can be found at the provided URL.
The document discusses sequences and their properties. A sequence is a function whose domain is the positive integers. Sequences are commonly represented using subscript notation rather than standard function notation. The nth term of a sequence is denoted an. [/SUMMARY]
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.
This document provides an overview of deep learning and neural networks. It begins with definitions of machine learning, artificial intelligence, and the different types of machine learning problems. It then introduces deep learning, explaining that it uses neural networks with multiple layers to learn representations of data. The document discusses why deep learning works better than traditional machine learning for complex problems. It covers key concepts like activation functions, gradient descent, backpropagation, and overfitting. It also provides examples of applications of deep learning and popular deep learning frameworks like TensorFlow. Overall, the document gives a high-level introduction to deep learning concepts and techniques.
How CEOs should consider all the scenarios that patents are important for: patents as assets, sales tools, offense-defense tools. 科创企业应该发明、攻防、销售、资产等场景看专利。