This document provides an overview of supervised learning and perceptrons. It discusses two types of learning - associative and non-associative. Examples of associative learning include classical and operant conditioning. It also discusses reflexive vs declarative learning. The goal of supervised learning is to approximate an unknown function based on a set of input-output examples. Unsupervised learning involves clustering inputs without output labels. Perceptrons are a type of supervised learning model that learns by adjusting its weights according to a learning rule to minimize errors between predicted and target outputs.
Two Types of Novel Discrete Time Chaotic Systemsijtsrd
In this paper, two types of one dimensional discrete time systems are firstly proposed and the chaos behaviors are numerically discussed. Based on the time domain approach, an invariant set and equilibrium points of such discrete time systems are presented. Besides, the stability of equilibrium points will be analyzed in detail. Finally, Lyapunov exponent plots as well as state response and Fourier amplitudes of the proposed discrete time systems are given to verify and demonstrate the chaos behaviors. Yeong-Jeu Sun ""Two Types of Novel Discrete-Time Chaotic Systems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29853.pdf
Paper Url : https://www.ijtsrd.com/engineering/electrical-engineering/29853/two-types-of-novel-discrete-time-chaotic-systems/yeong-jeu-sun
The document provides an overview of neural networks. It begins by discussing biological inspiration from the human brain, including key facts about neurons and synapses. It then defines artificial neurons and various components like dendrites, axons, and synapses. The document explores different types of neural networks including feedforward, recurrent, self-organizing maps and time delay neural networks. It also covers common neural network architectures, learning algorithms, activation functions, and applications of neural networks.
This document describes a backpropagation algorithm for training second-order feedforward neural networks. It defines the architecture of these networks, which include first and second-order connections between units. The backpropagation algorithm is extended from traditional first-order networks to compute gradients and update both first and second-order weights during training. These networks are theoretically capable of universal function approximation like first-order networks. The document outlines the real and complex versions of the backpropagation algorithm for training these second-order neural networks.
Essentials of Chemical Reaction Engineering 1st Edition Fogler Solutions ManualCookMedge
The document contains copyright information for a solutions manual for the textbook "Essentials of Chemical Reaction Engineering". It states that the authors and publisher make no guarantees regarding errors or omissions in the book. It also restricts dissemination and sale of the solutions manual to protect the integrity of the work for instructors' intended pedagogical purposes. The document provides a summary table reviewing problems from Chapter 2 of the solutions manual, including assigned problems, alternate problems, estimated difficulty, and time to complete. It then provides sample solutions to several of the Chapter 2 problems.
This document discusses dynamics of structures with uncertainties. It begins with an introduction to stochastic single degree of freedom systems and how natural frequency variability can be modeled using probability distributions. It then discusses how to extend this approach to stochastic multi degree of freedom systems using stochastic finite element formulations and modal projections. Key challenges with statistical overlap of eigenvalues are noted. The document provides mathematical models of equivalent damping in stochastic systems and examples of stochastic frequency response functions.
This document is a class note for a signals and systems course. It was prepared by Stanley Chan at UC San Diego in 2011 using notes from Prof. Paul H. Siegel. The note covers topics in signals and systems including fundamentals of signals, systems, Fourier series, continuous-time Fourier transform, discrete-time Fourier transform, sampling theorem, and z-transform. It contains examples and definitions for key concepts. The textbook used for the course is Signals and Systems by Oppenheim and Wilsky.
1) This document summarizes research on phase transitions and self-organized criticality in networks of stochastic spiking neurons. It presents a mean-field model of neurons that exhibits continuous and discontinuous phase transitions between silent and active states.
2) The model shows power law distributions of neuronal avalanche sizes and durations near the critical point, consistent with experimental data. Introducing dynamic neuronal gains instead of static synapses allows the system to self-organize to a slightly supercritical state.
3) Future work is proposed to better understand the effects of network topology, inhibitory neurons, and to apply the model to more realistic large-scale neuronal networks modeling different brain regions. The research contributes to understanding phase transitions and
Two Types of Novel Discrete Time Chaotic Systemsijtsrd
In this paper, two types of one dimensional discrete time systems are firstly proposed and the chaos behaviors are numerically discussed. Based on the time domain approach, an invariant set and equilibrium points of such discrete time systems are presented. Besides, the stability of equilibrium points will be analyzed in detail. Finally, Lyapunov exponent plots as well as state response and Fourier amplitudes of the proposed discrete time systems are given to verify and demonstrate the chaos behaviors. Yeong-Jeu Sun ""Two Types of Novel Discrete-Time Chaotic Systems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-2 , February 2020, URL: https://www.ijtsrd.com/papers/ijtsrd29853.pdf
Paper Url : https://www.ijtsrd.com/engineering/electrical-engineering/29853/two-types-of-novel-discrete-time-chaotic-systems/yeong-jeu-sun
The document provides an overview of neural networks. It begins by discussing biological inspiration from the human brain, including key facts about neurons and synapses. It then defines artificial neurons and various components like dendrites, axons, and synapses. The document explores different types of neural networks including feedforward, recurrent, self-organizing maps and time delay neural networks. It also covers common neural network architectures, learning algorithms, activation functions, and applications of neural networks.
This document describes a backpropagation algorithm for training second-order feedforward neural networks. It defines the architecture of these networks, which include first and second-order connections between units. The backpropagation algorithm is extended from traditional first-order networks to compute gradients and update both first and second-order weights during training. These networks are theoretically capable of universal function approximation like first-order networks. The document outlines the real and complex versions of the backpropagation algorithm for training these second-order neural networks.
Essentials of Chemical Reaction Engineering 1st Edition Fogler Solutions ManualCookMedge
The document contains copyright information for a solutions manual for the textbook "Essentials of Chemical Reaction Engineering". It states that the authors and publisher make no guarantees regarding errors or omissions in the book. It also restricts dissemination and sale of the solutions manual to protect the integrity of the work for instructors' intended pedagogical purposes. The document provides a summary table reviewing problems from Chapter 2 of the solutions manual, including assigned problems, alternate problems, estimated difficulty, and time to complete. It then provides sample solutions to several of the Chapter 2 problems.
This document discusses dynamics of structures with uncertainties. It begins with an introduction to stochastic single degree of freedom systems and how natural frequency variability can be modeled using probability distributions. It then discusses how to extend this approach to stochastic multi degree of freedom systems using stochastic finite element formulations and modal projections. Key challenges with statistical overlap of eigenvalues are noted. The document provides mathematical models of equivalent damping in stochastic systems and examples of stochastic frequency response functions.
This document is a class note for a signals and systems course. It was prepared by Stanley Chan at UC San Diego in 2011 using notes from Prof. Paul H. Siegel. The note covers topics in signals and systems including fundamentals of signals, systems, Fourier series, continuous-time Fourier transform, discrete-time Fourier transform, sampling theorem, and z-transform. It contains examples and definitions for key concepts. The textbook used for the course is Signals and Systems by Oppenheim and Wilsky.
1) This document summarizes research on phase transitions and self-organized criticality in networks of stochastic spiking neurons. It presents a mean-field model of neurons that exhibits continuous and discontinuous phase transitions between silent and active states.
2) The model shows power law distributions of neuronal avalanche sizes and durations near the critical point, consistent with experimental data. Introducing dynamic neuronal gains instead of static synapses allows the system to self-organize to a slightly supercritical state.
3) Future work is proposed to better understand the effects of network topology, inhibitory neurons, and to apply the model to more realistic large-scale neuronal networks modeling different brain regions. The research contributes to understanding phase transitions and
1. The document provides resources and information for determining protein crystal structures using x-ray crystallography. It discusses topics like crystal lattices, diffraction, the phase problem, and structure refinement.
2. Key resources mentioned include the Advanced Light Source for collecting diffraction data, and computing software for analyzing structures.
3. The goals of x-ray crystallography are outlined as determining how the technique works, understanding potential sources of error, and what information is contained in the Protein Data Bank structure database.
- Artificial neural networks are inspired by biological neural networks and try to mimic their learning mechanisms by modifying synaptic strengths through an optimization process.
- Learning in neural networks can be formulated as a function approximation task where the network learns to approximate a function by minimizing an error measure through optimization of synaptic weights.
- A single hidden layer neural network is capable of learning nonlinear function approximations if general optimization methods are applied to update the synaptic weights.
This document proposes an emotion recognition system using EEG signals and time domain analysis. Five different machine learning algorithms (RVM, MLP, DT, SVM, Bayesian) are used to classify three emotions (happy, relaxed, sad) felt by subjects viewing different videos. EEG data is collected from subjects using a headset and amplifier. Six features are extracted from the time domain signals of each EEG channel. The document describes the data collection process, feature extraction methodology, and compares the performance of different algorithms at classifying emotions based on the EEG data.
The document discusses neural networks based on competition. It describes three fixed-weight competitive neural networks: Maxnet, Mexican Hat, and Hamming Net. Maxnet uses winner-take-all competition where only the neuron with the largest activation remains active. The Mexican Hat network enhances the activation of neurons receiving a stronger external signal by applying positive weights to nearby neurons and negative weights to those further away. An example demonstrates how the Mexican Hat network increases contrast over iterations.
Artificial Neural Networks Lect7: Neural networks based on competitionMohammed Bennamoun
This document summarizes key concepts about neural networks based on competition. It discusses fixed weight competitive networks including Maxnet, Mexican Hat, and Hamming Net. Maxnet uses winner-take-all competition where only the neuron with the largest activation remains on. The Mexican Hat network enhances contrast through excitatory connections to nearby neurons and inhibitory connections to farther neurons. Iterating the activations over time steps increases the activation of neurons with initially larger signals and decreases others. Kohonen self-organizing maps and their training in Matlab are also mentioned.
This document provides resources and an overview of topics for a course on crystallography and protein structure determination using X-ray crystallography. The course will involve crystallizing a protein, collecting data at the Advanced Light Source, and determining the protein's atomic structure. Key topics covered include X-ray scattering, the phase problem, structure refinement, and sources of errors. Online resources and contacts are provided for computing, tutorials, and beamline information.
1. The document discusses x-ray crystallography techniques for determining atomic structures, including crystallizing a protein, collecting diffraction data at an advanced light source, and determining structures.
2. Key aspects covered are x-ray diffraction, where x-rays scatter off electrons and the resolution is typically 1-3 Angstroms. Determining phases is also discussed as a challenge.
3. Resources provided include computing tools, online courses, and contacts for further information on crystallography techniques and resources.
Self-organizing networks can perform unsupervised clustering by mapping high-dimensional input patterns into a smaller number of clusters in output space through competitive learning. Fixed weight competitive networks like Maxnet, Mexican Hat net, and Hamming net use competitive learning with fixed weights. Maxnet uses winner-take-all competition to select the neuron whose weights best match the input. Mexican Hat net has both excitatory and inhibitory connections between neurons to enhance contrast. Hamming net determines which exemplar vector most closely matches the input using the Hamming distance measure.
This document discusses artificial neural networks and their learning processes. It provides an overview of biological inspiration for neural networks from the nervous system. It then describes artificial neurons and how they are modeled, including the McCulloch-Pitts model. Neural networks are composed of interconnected artificial neurons. Learning in neural networks and biological systems involves changing synaptic strengths. The document outlines learning rules and processes for artificial neural networks, including minimizing an error function through optimization techniques like backpropagation.
This document provides an overview of convolution, Fourier series, and the Fourier transform. It defines convolution as a mathematical operator that computes the overlap between two functions. Fourier series expresses periodic functions as an infinite sum of sines and cosines. The Fourier transform allows converting signals between the time and frequency domains. It describes how the discrete Fourier transform (DFT) represents a sampled signal as a sum of complex exponentials, and how the fast Fourier transform (FFT) efficiently computes the DFT. The document also introduces Fourier transform pairs and defines the delta function.
This document summarizes key aspects of neural networks and training algorithms. It introduces threshold logic units (TLUs) as basic units for neural networks and different methods for training single TLUs, including gradient descent, the Widrow-Hoff rule, and the generalized delta procedure. It then discusses feedforward neural networks and the backpropagation method for training multiple layers of neurons. Finally, it covers topics like generalization, accuracy, overfitting, and additional readings in the field of neural networks.
Lecture 02: Machine Learning for Language Technology - Decision Trees and Nea...Marina Santini
In this lecture, we talk about two different discriminative machine learning methods: decision trees and k-nearest neighbors. Decision trees are hierarchical structures.k-nearest neighbors are based on two principles: recollection and resemblance.
NIPS2017 Few-shot Learning and Graph ConvolutionKazuki Fujikawa
The document discusses meta-learning and prototypical networks for few-shot learning. It introduces prototypical networks, which learn a metric space such that classification can be performed by finding the nearest class prototype to a query example in embedding space. The document summarizes results on few-shot image classification benchmarks like Omniglot and miniImageNet, finding that prototypical networks achieve state-of-the-art performance.
A baisc ideas of statistical physics.pptxabhilasha7888
This document provides an overview of basic statistical physics concepts. It discusses topics like probability, microstates and macrostates, the principle of equal a priori probability, constraints, and equilibrium states of dynamic systems. The key points are:
1) Statistical physics applies statistical concepts to physics by looking at microscopic states and their probabilities at a macroscopic level.
2) The probability of a macrostate is given by the number of microstates in that macrostate times the probability of one microstate.
3) Constraints reduce the number of accessible microstates. Equilibrium in a dynamic system is reached when the system spends most time in the macrostate with the highest probability.
The document summarizes key concepts about linear time-invariant (LTI) systems from Chapter 2. It discusses:
1) LTI systems can be modeled as the sum of their impulse responses weighted by the input signal. This is known as the convolution sum/integral for discrete/continuous-time systems.
2) Any signal can be represented as a linear combination of shifted unit impulses. The output of an LTI system is the convolution of the input signal with the system's impulse response.
3) The impulse response completely characterizes an LTI system. The output is found by taking the convolution integral or sum of the input signal with the impulse response.
1) Molecular dynamics (MD) simulations numerically solve Newton's equations of motion to simulate the physical movements of atoms and molecules over time.
2) The Verlet algorithm is commonly used to integrate the equations of motion in MD simulations. It calculates new positions and velocities at each time step based on the forces between particles.
3) MD simulations sample the ensemble of all possible configurations over time. If run long enough, time averages from the simulation converge to ensemble averages, in accordance with the ergodic hypothesis. This allows MD to connect microscopic dynamics to macroscopic thermodynamics.
Dominance-Based Pareto-Surrogate for Multi-Objective OptimizationIlya Loshchilov
This document proposes a dominance-based Pareto surrogate model for multi-objective optimization using support vector machines. The model learns primary and secondary dominance constraints to build a surrogate function that preserves the Pareto dominance relations of training points. Experimental results show that using the surrogate to guide multi-objective evolutionary algorithms leads to 1.5-5x speedups in converging to the Pareto front on test problems compared to the original algorithms. However, the surrogate may prematurely converge the diversity of solutions, as it only considers convergence and not diversity maintenance. The model can incorporate additional preferences beyond dominance to further improve optimization.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
This document provides an introduction to dynamical systems and their mathematical modeling using differential equations. It discusses modeling dynamical systems using inputs, states, and outputs. It also covers simulating dynamical systems, equilibria, linearization, and system interconnections. Key topics include modeling dynamical systems using differential equations, the concept of inputs and outputs, interpreting mathematical models of dynamical systems, and converting higher-order models to first-order models.
This document provides an introduction to machine learning, covering key topics such as what machine learning is, common learning algorithms and applications. It discusses linear models, kernel methods, neural networks, decision trees and more. It also addresses challenges in machine learning like balancing fit and robustness, and evaluating model performance using techniques like ROC curves. The goal of machine learning is to build models that can learn from data to make predictions or decisions.
Supermarket Management System Project Report.pdfKamal Acharya
Supermarket management is a stand-alone J2EE using Eclipse Juno program.
This project contains all the necessary required information about maintaining
the supermarket billing system.
The core idea of this project to minimize the paper work and centralize the
data. Here all the communication is taken in secure manner. That is, in this
application the information will be stored in client itself. For further security the
data base is stored in the back-end oracle and so no intruders can access it.
More Related Content
Similar to Supervised learning in artificial neural networks
1. The document provides resources and information for determining protein crystal structures using x-ray crystallography. It discusses topics like crystal lattices, diffraction, the phase problem, and structure refinement.
2. Key resources mentioned include the Advanced Light Source for collecting diffraction data, and computing software for analyzing structures.
3. The goals of x-ray crystallography are outlined as determining how the technique works, understanding potential sources of error, and what information is contained in the Protein Data Bank structure database.
- Artificial neural networks are inspired by biological neural networks and try to mimic their learning mechanisms by modifying synaptic strengths through an optimization process.
- Learning in neural networks can be formulated as a function approximation task where the network learns to approximate a function by minimizing an error measure through optimization of synaptic weights.
- A single hidden layer neural network is capable of learning nonlinear function approximations if general optimization methods are applied to update the synaptic weights.
This document proposes an emotion recognition system using EEG signals and time domain analysis. Five different machine learning algorithms (RVM, MLP, DT, SVM, Bayesian) are used to classify three emotions (happy, relaxed, sad) felt by subjects viewing different videos. EEG data is collected from subjects using a headset and amplifier. Six features are extracted from the time domain signals of each EEG channel. The document describes the data collection process, feature extraction methodology, and compares the performance of different algorithms at classifying emotions based on the EEG data.
The document discusses neural networks based on competition. It describes three fixed-weight competitive neural networks: Maxnet, Mexican Hat, and Hamming Net. Maxnet uses winner-take-all competition where only the neuron with the largest activation remains active. The Mexican Hat network enhances the activation of neurons receiving a stronger external signal by applying positive weights to nearby neurons and negative weights to those further away. An example demonstrates how the Mexican Hat network increases contrast over iterations.
Artificial Neural Networks Lect7: Neural networks based on competitionMohammed Bennamoun
This document summarizes key concepts about neural networks based on competition. It discusses fixed weight competitive networks including Maxnet, Mexican Hat, and Hamming Net. Maxnet uses winner-take-all competition where only the neuron with the largest activation remains on. The Mexican Hat network enhances contrast through excitatory connections to nearby neurons and inhibitory connections to farther neurons. Iterating the activations over time steps increases the activation of neurons with initially larger signals and decreases others. Kohonen self-organizing maps and their training in Matlab are also mentioned.
This document provides resources and an overview of topics for a course on crystallography and protein structure determination using X-ray crystallography. The course will involve crystallizing a protein, collecting data at the Advanced Light Source, and determining the protein's atomic structure. Key topics covered include X-ray scattering, the phase problem, structure refinement, and sources of errors. Online resources and contacts are provided for computing, tutorials, and beamline information.
1. The document discusses x-ray crystallography techniques for determining atomic structures, including crystallizing a protein, collecting diffraction data at an advanced light source, and determining structures.
2. Key aspects covered are x-ray diffraction, where x-rays scatter off electrons and the resolution is typically 1-3 Angstroms. Determining phases is also discussed as a challenge.
3. Resources provided include computing tools, online courses, and contacts for further information on crystallography techniques and resources.
Self-organizing networks can perform unsupervised clustering by mapping high-dimensional input patterns into a smaller number of clusters in output space through competitive learning. Fixed weight competitive networks like Maxnet, Mexican Hat net, and Hamming net use competitive learning with fixed weights. Maxnet uses winner-take-all competition to select the neuron whose weights best match the input. Mexican Hat net has both excitatory and inhibitory connections between neurons to enhance contrast. Hamming net determines which exemplar vector most closely matches the input using the Hamming distance measure.
This document discusses artificial neural networks and their learning processes. It provides an overview of biological inspiration for neural networks from the nervous system. It then describes artificial neurons and how they are modeled, including the McCulloch-Pitts model. Neural networks are composed of interconnected artificial neurons. Learning in neural networks and biological systems involves changing synaptic strengths. The document outlines learning rules and processes for artificial neural networks, including minimizing an error function through optimization techniques like backpropagation.
This document provides an overview of convolution, Fourier series, and the Fourier transform. It defines convolution as a mathematical operator that computes the overlap between two functions. Fourier series expresses periodic functions as an infinite sum of sines and cosines. The Fourier transform allows converting signals between the time and frequency domains. It describes how the discrete Fourier transform (DFT) represents a sampled signal as a sum of complex exponentials, and how the fast Fourier transform (FFT) efficiently computes the DFT. The document also introduces Fourier transform pairs and defines the delta function.
This document summarizes key aspects of neural networks and training algorithms. It introduces threshold logic units (TLUs) as basic units for neural networks and different methods for training single TLUs, including gradient descent, the Widrow-Hoff rule, and the generalized delta procedure. It then discusses feedforward neural networks and the backpropagation method for training multiple layers of neurons. Finally, it covers topics like generalization, accuracy, overfitting, and additional readings in the field of neural networks.
Lecture 02: Machine Learning for Language Technology - Decision Trees and Nea...Marina Santini
In this lecture, we talk about two different discriminative machine learning methods: decision trees and k-nearest neighbors. Decision trees are hierarchical structures.k-nearest neighbors are based on two principles: recollection and resemblance.
NIPS2017 Few-shot Learning and Graph ConvolutionKazuki Fujikawa
The document discusses meta-learning and prototypical networks for few-shot learning. It introduces prototypical networks, which learn a metric space such that classification can be performed by finding the nearest class prototype to a query example in embedding space. The document summarizes results on few-shot image classification benchmarks like Omniglot and miniImageNet, finding that prototypical networks achieve state-of-the-art performance.
A baisc ideas of statistical physics.pptxabhilasha7888
This document provides an overview of basic statistical physics concepts. It discusses topics like probability, microstates and macrostates, the principle of equal a priori probability, constraints, and equilibrium states of dynamic systems. The key points are:
1) Statistical physics applies statistical concepts to physics by looking at microscopic states and their probabilities at a macroscopic level.
2) The probability of a macrostate is given by the number of microstates in that macrostate times the probability of one microstate.
3) Constraints reduce the number of accessible microstates. Equilibrium in a dynamic system is reached when the system spends most time in the macrostate with the highest probability.
The document summarizes key concepts about linear time-invariant (LTI) systems from Chapter 2. It discusses:
1) LTI systems can be modeled as the sum of their impulse responses weighted by the input signal. This is known as the convolution sum/integral for discrete/continuous-time systems.
2) Any signal can be represented as a linear combination of shifted unit impulses. The output of an LTI system is the convolution of the input signal with the system's impulse response.
3) The impulse response completely characterizes an LTI system. The output is found by taking the convolution integral or sum of the input signal with the impulse response.
1) Molecular dynamics (MD) simulations numerically solve Newton's equations of motion to simulate the physical movements of atoms and molecules over time.
2) The Verlet algorithm is commonly used to integrate the equations of motion in MD simulations. It calculates new positions and velocities at each time step based on the forces between particles.
3) MD simulations sample the ensemble of all possible configurations over time. If run long enough, time averages from the simulation converge to ensemble averages, in accordance with the ergodic hypothesis. This allows MD to connect microscopic dynamics to macroscopic thermodynamics.
Dominance-Based Pareto-Surrogate for Multi-Objective OptimizationIlya Loshchilov
This document proposes a dominance-based Pareto surrogate model for multi-objective optimization using support vector machines. The model learns primary and secondary dominance constraints to build a surrogate function that preserves the Pareto dominance relations of training points. Experimental results show that using the surrogate to guide multi-objective evolutionary algorithms leads to 1.5-5x speedups in converging to the Pareto front on test problems compared to the original algorithms. However, the surrogate may prematurely converge the diversity of solutions, as it only considers convergence and not diversity maintenance. The model can incorporate additional preferences beyond dominance to further improve optimization.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
This document provides an introduction to dynamical systems and their mathematical modeling using differential equations. It discusses modeling dynamical systems using inputs, states, and outputs. It also covers simulating dynamical systems, equilibria, linearization, and system interconnections. Key topics include modeling dynamical systems using differential equations, the concept of inputs and outputs, interpreting mathematical models of dynamical systems, and converting higher-order models to first-order models.
This document provides an introduction to machine learning, covering key topics such as what machine learning is, common learning algorithms and applications. It discusses linear models, kernel methods, neural networks, decision trees and more. It also addresses challenges in machine learning like balancing fit and robustness, and evaluating model performance using techniques like ROC curves. The goal of machine learning is to build models that can learn from data to make predictions or decisions.
Similar to Supervised learning in artificial neural networks (20)
Supermarket Management System Project Report.pdfKamal Acharya
Supermarket management is a stand-alone J2EE using Eclipse Juno program.
This project contains all the necessary required information about maintaining
the supermarket billing system.
The core idea of this project to minimize the paper work and centralize the
data. Here all the communication is taken in secure manner. That is, in this
application the information will be stored in client itself. For further security the
data base is stored in the back-end oracle and so no intruders can access it.
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
Discover the latest insights on Data Driven Maintenance with our comprehensive webinar presentation. Learn about traditional maintenance challenges, the right approach to utilizing data, and the benefits of adopting a Data Driven Maintenance strategy. Explore real-world examples, industry best practices, and innovative solutions like FMECA and the D3M model. This presentation, led by expert Jules Oudmans, is essential for asset owners looking to optimize their maintenance processes and leverage digital technologies for improved efficiency and performance. Download now to stay ahead in the evolving maintenance landscape.
Generative AI Use cases applications solutions and implementation.pdfmahaffeycheryld
Generative AI solutions encompass a range of capabilities from content creation to complex problem-solving across industries. Implementing generative AI involves identifying specific business needs, developing tailored AI models using techniques like GANs and VAEs, and integrating these models into existing workflows. Data quality and continuous model refinement are crucial for effective implementation. Businesses must also consider ethical implications and ensure transparency in AI decision-making. Generative AI's implementation aims to enhance efficiency, creativity, and innovation by leveraging autonomous generation and sophisticated learning algorithms to meet diverse business challenges.
https://www.leewayhertz.com/generative-ai-use-cases-and-applications/
Blood finder application project report (1).pdfKamal Acharya
Blood Finder is an emergency time app where a user can search for the blood banks as
well as the registered blood donors around Mumbai. This application also provide an
opportunity for the user of this application to become a registered donor for this user have
to enroll for the donor request from the application itself. If the admin wish to make user
a registered donor, with some of the formalities with the organization it can be done.
Specialization of this application is that the user will not have to register on sign-in for
searching the blood banks and blood donors it can be just done by installing the
application to the mobile.
The purpose of making this application is to save the user’s time for searching blood of
needed blood group during the time of the emergency.
This is an android application developed in Java and XML with the connectivity of
SQLite database. This application will provide most of basic functionality required for an
emergency time application. All the details of Blood banks and Blood donors are stored
in the database i.e. SQLite.
This application allowed the user to get all the information regarding blood banks and
blood donors such as Name, Number, Address, Blood Group, rather than searching it on
the different websites and wasting the precious time. This application is effective and
user friendly.
Accident detection system project report.pdfKamal Acharya
The Rapid growth of technology and infrastructure has made our lives easier. The
advent of technology has also increased the traffic hazards and the road accidents take place
frequently which causes huge loss of life and property because of the poor emergency facilities.
Many lives could have been saved if emergency service could get accident information and
reach in time. Our project will provide an optimum solution to this draw back. A piezo electric
sensor can be used as a crash or rollover detector of the vehicle during and after a crash. With
signals from a piezo electric sensor, a severe accident can be recognized. According to this
project when a vehicle meets with an accident immediately piezo electric sensor will detect the
signal or if a car rolls over. Then with the help of GSM module and GPS module, the location
will be sent to the emergency contact. Then after conforming the location necessary action will
be taken. If the person meets with a small accident or if there is no serious threat to anyone’s
life, then the alert message can be terminated by the driver by a switch provided in order to
avoid wasting the valuable time of the medical rescue team.
Open Channel Flow: fluid flow with a free surfaceIndrajeet sahu
Open Channel Flow: This topic focuses on fluid flow with a free surface, such as in rivers, canals, and drainage ditches. Key concepts include the classification of flow types (steady vs. unsteady, uniform vs. non-uniform), hydraulic radius, flow resistance, Manning's equation, critical flow conditions, and energy and momentum principles. It also covers flow measurement techniques, gradually varied flow analysis, and the design of open channels. Understanding these principles is vital for effective water resource management and engineering applications.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Applications of artificial Intelligence in Mechanical Engineering.pdf
Supervised learning in artificial neural networks
1. 1/120
Supervised Learning I:
Perceptrons and LMS
Instructor: Tai-Yue (Jason) Wang
Department of Industrial and Information Management
Institute of Information Management
2. 2/120
Two Fundamental Learning
Paradigms
Non-associative
an organism acquires the properties of a single
repetitive stimulus.
Associative
an organism acquires knowledge about the
relationship of either one stimulus to another, or one
stimulus to the organism’s own behavioral response to
that stimulus.
3. 3/120
Examples of Associative
Learning(1/2)
Classical conditioning
Association of an unconditioned stimulus (US) with a
conditioned stimulus (CS).
CS’s such as a flash of light or a sound tone produce
weak responses.
US’s such as food or a shock to the leg produce a
strong response.
Repeated presentation of the CS followed by the US,
the CS begins to evoke the response of the US.
Example: If a flash of light is always followed by
serving of meat to a dog, after a number of learning
trials the light itself begins to produce salivation.
4. 4/120
Examples of Associative
Learning(2/2)
Operant conditioning
Formation of a predictive relationship between a
stimulus and a response.
Example:
Place a hungry rat in a cage which has a lever on
one of its walls. Measure the spontaneous rate at
which the rat presses the lever by virtue of its
random movements around the cage. If the rat is
promptly presented with food when the lever is
pressed, the spontaneous rate of lever pressing
increases!
5. 5/120
Reflexive and Declarative Learning
Reflexive learning
repetitive learning is involved and recall does not
involve any awareness or conscious evaluation.
Declarative learning
established by a single trial or experience and
involves conscious reflection and evaluation for its
recall.
Constant repitition of declarative knowledge
often manifest itself in reflexive form.
6. 6/120
Two distinct stages:
short-term memory (STM)
long-term memory (LTM)
Inputs to the brain are
processed into STMs
which last at the most for a
few minutes.
Important Aspects of Human
Memory(1/4)
Recall process of these memories
is distinct from the memories
themselves.
Memory
recalled
SHORT TERM MEMORY
(STM)
LONG TERM MEMORY
(LTM)
Download
Input
stimulus
Recall
Process
7. 7/120
Information is
downloaded into
LTMs for more
permanent storage:
days, months, and
years.
Capacity of LTMs is
very large.
Important Aspects of Human
Memory(2/4)
Recall process of these memories
is distinct from the memories
themselves.
Memory
recalled
SHORT TERM MEMORY
(STM)
LONG TERM MEMORY
(LTM)
Download
Input
stimulus
Recall
Process
8. 8/120
Important Aspects of Human
Memory(3/4)
Recall of recent memories is more easily
disrupted than that of older memories.
Memories are dynamic and undergo
continual change and with time all
memories fade.
STM results in
physical changes in sensory receptors.
simultaneous and cohesive reverberation of
neuron circuits.
9. 9/120
Important Aspects of Human
Memory(4/4)
Long-term memory involves
plastic changes in the brain which take the form of
strengthening or weakening of existing synapses
the formation of new synapses.
Learning mechanism distributes the memory
over different areas
Makes robust to damage
Permits the brain to work easily from partially
corrupted information.
Reflexive and declarative memories may
actually involve different neuronal circuits.
10. 10/120
Learning Algorithms(1/2)
Define an architecture-dependent procedure to
encode pattern information into weights
Learning proceeds by modifying connection
strengths.
Learning is data driven:
A set of input–output patterns derived from a
(possibly unknown) probability distribution.
Output pattern might specify a desired system response for
a given input pattern
Learning involves approximating the unknown function as
described by the given data.
11. 11/120
Learning Algorithms(2/2)
Learning is data driven:
Alternatively, the data might comprise patterns that
naturally cluster into some number of unknown
classes
Learning problem involves generating a suitable
classification of the samples.
12. 12/120
Supervised Learning(1/2)
Data comprises a set
of discrete samples
drawn from the
pattern space where
each sample relates
an input vector Xk ∈
Rn to an output vector
Dk ∈ Rp.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-20
-15
-10
-5
0
5
10
15
20
x
f(x)
3 x
5
-1.2 x
4
-12.27 x
3
+3.288 x
2
+7.182 x
Q
k
k
k D
X
T 1
,
An example function described
by a set of noisy data points
13. 13/120
Supervised Learning(2/2)
The set of samples
describe the behavior
of an unknown
function f : Rn → Rp
which is to be
characterized.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-20
-15
-10
-5
0
5
10
15
20
x
f(x)
3 x
5
-1.2 x
4
-12.27 x
3
+3.288 x
2
+7.182 x
An example function described
by a set of noisy data points
14. 14/120
The Supervised Learning Procedure
We want the system to generate an output Dk in
response to an input Xk, and we say that the system has
learnt the underlying map if a stimulus Xk’ close to Xk
elicits a response Sk’ which is sufficiently close to Dk.
The result is a continuous function estimate.
Error information fed back for network adaptation
Error
Sk
Neural Network
Dx
Xk
15. 15/120
Unsupervised Learning(1/2)
Unsupervised learning provides the system with
an input Xk, and allow it to self-organize its
weights to generate internal prototypes of sample
vectors. Note: There is no teaching input involved
here.
The system attempts to represent the entire data
set by employing a small number of prototypical
vectors—enough to allow the system to retain a
desired level of discrimination between samples.
16. 16/120
Unsupervised Learning(2/2)
As new samples continuously buffer the system,
the prototypes will be in a state of constant flux.
This kind of learning is often called adaptive
vector quantization
17. 17/120
Clustering and Classification(1/3)
Given a set of data
samples {Xi}, Xi ∈ Rn,
is it possible to identify
well defined “clusters”,
where each cluster
defines a class of
vectors which are
similar in some broad
sense?
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-0.5
0
0.5
1
1.5
2
2.5
Cluster centroids
Cluster 1
Cluster 2
18. 18/120
Clustering and Classification(2/3)
Clusters help establish a
classification structure
within a data set that has
no categories defined in
advance.
Classes are derived from
clusters by appropriate
labeling.
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-0.5
0
0.5
1
1.5
2
2.5
Cluster centroids
Cluster 1
Cluster 2
19. 19/120
Clustering and Classification(3/3)
The goal of pattern
classification is to
assign an input pattern
to one of a finite
number of classes.
Quantization vectors
are called codebook
vectors. -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-0.5
0
0.5
1
1.5
2
2.5
Cluster centroids
Cluster 1
Cluster 2
21. 21/120
General Philosophy of Learning:
Principle of Minimal Disturbance
Adapt to reduce the output error for the
current training pattern, with minimal
disturbance to responses already learned.
22. 22/120
Error Correction and Gradient
Descent Rules
Error correction rules alter the weights of a
network using a linear error measure to reduce
the error in the output generated in response to
the present input pattern.
Gradient rules alter the weights of a network
during each pattern presentation by employing
gradient information with the objective of
reducing the mean squared error (usually
averaged over all training patterns).
23. 23/120
Learning Objective for TLNs(1/4)
Augmented Input and
Weight vectors
Objective: To design the
weights of a TLN to
correctly classify a given
set of patterns.
S
+1
WO
X1
X2
Xn
W1
W2
Wn
1
1
0
1
1
0
,
,
,
,
,
,
n
k
T
k
n
k
k
k
n
k
T
k
n
k
k
R
W
w
w
w
W
R
X
x
x
x
X
24. 24/120
Learning Objective for TLNs(2/4)
Assumption: A training
set of following form is
given
Each pattern Xk is tagged
to one of two classes C0
or C1 denoted by the
desired output dk being 0
or 1 respectively.
S
+1
WO
X1
X2
Xn
W1
W2
Wn
1
,
0
,
, 1
1
k
n
K
Q
k
k
k d
R
X
D
X
T
25. 25/120
Learning Objective for TLNs(3/4)
Two classes identified by two possible signal
states of the TLN
C0 by a signal S(yk) = 0, C1 by a signal S(yk) = 1.
Given two sets of vectors X0 andX1
belonging to classes C0 and C1 respectively
the learning procedure searches a solution
weight vector WS that correctly classifies the
vectors into their respective classes.
26. 26/120
Learning Objective for TLNs(4/4)
Context: TLNs
Find a weight vectorWS such that for all Xk ∈ X1,
S(yk) = 1; and for all Xk ∈ X0, S(yk) = 0.
Positive inner products translate to a +1 signal
and negative inner products to a 0 signal
Translates to saying that for all Xk ∈ X1, Xk
TWS
> 0; and for all Xk ∈ X0, Xk
TWS < 0.
27. 27/120
Pattern Space(1/2)
Points that satisfy XTWS
= 0 define a separating
hyperplane in pattern
space.
Two dimensional case:
Pattern space points on
one side of this
hyperplane (with an
orientation indicated by
the arrow) yield positive
inner products with WS
and thus generate a +1
neuron signal.
Activation
28. 28/120
Pattern Space(2/2)
Two dimensional case:
Pattern space points on
the other side of the
hyperplane generate a
negative inner product
with WS and consequently
a neuron signal equal to 0.
Points in C0 and C1 are
thus correctly classified
by such a placement of
the hyperplane.
Activation
29. 29/120
A Different View: Weight Space
(1/2)
Weight vector is a variable
vector.
WTXk = 0 represents a
hyperplane in weight space
Always passes through the
origin since W = 0 is a
trivial solution of WTXk =
0.
30. 30/120
A Different View: Weight Space
(2/2)
Called the pattern
hyperplane of pattern
Xk.
Locus of all pointsW
such thatWTXk = 0.
Divides the weight
space into two parts:
one which generates a
positive inner product
WTXk > 0, and the
other a negative inner
product WTX <0.
31. 31/120
Identifying a Solution Region from
Orientated Pattern Hyperplanes(1/2)
For each pattern Xk in
pattern space there is a
corresponding
hyperplane in weight
space.
For every point in
weight space there is a
corresponding
hyperplane in pattern
space.
W2
X3 X2
X1
X4
W1
Solution region
32. 32/120
Identifying a Solution Region from
Orientated Pattern Hyperplanes(2/2)
A solution region in
weight space with
four pattern
hyperplanes
1 = {X1,X2}
0 = {X3,X4}
W2
X3 X2
X1
X4
W1
Solution region
33. 33/120
Requirements of the Learning
Procedure
Linear separability guarantees the existence of a
solution region.
Points to be kept in mind in the design of an
automated weight update procedure :
It must consider each pattern in turn to assess the correctness of
the present classification.
It must subsequently adjust the weight vector to eliminate a
classification error, if any.
Since the set of all solution vectors forms a convex cone, the
weight update procedure should terminate as soon as it penetrates
the boundary of this cone (solution region).
34. 34/120
Design in Weight Space(1/3)
Assume: Xk ∈ X1 and
Wk
TXk as erroneously
non-positive.
For correct
classification, shift the
weight vector to some
position Wk+1 where
the inner product is
positive.
Wk+1
Wk
Wk
T Xk>0
WT Xk<0
Xk
35. 35/120
Design in Weight Space(2/3)
The smallest
perturbation in Wk
that produces the
desired change is, the
perpendicular distance
from Wk onto the
pattern hyperplane.
Wk+1
Wk
Wk
T Xk>0
WT Xk<0
Xk
36. 36/120
Design in Weight Space(3/3)
In weight space, the
direction
perpendicular to the
pattern hyperplane is
none other than that
of Xk itself.
Wk+1
Wk
Wk
T Xk>0
WT Xk<0
Xk
37. 37/120
Simple Weight Change Rule:
Perceptron Learning Law
If Xk ∈ X1 and Wk
TXk < 0 add a fraction of the
pattern to the weight Wk if one wishes the inner
product Wk
TXk to increase.
Alternatively, if Xk ∈ X0, and Wk
TXk is erroneously
non-negative we will subtract a fraction of the
pattern from the weight Wk in order to reduce this
inner product.
38. 38/120
Weight Space Trajectory
The weight space
trajectory
corresponding to the
sequential presentation
of four patterns with
pattern hyperplanes as
indicated:
1 = {X1,X2} and
0 = {X3,X4}
X3 X2
X1
X4
W1
Solution region
W2
39. 39/120
Linear Containment
Consider the set X0’ in which each element X0 is
negated.
Given a weight vectorWk, for any Xk ∈ X1 ∪ X0’,
Xk
T Wk > 0 implies correct classification and
Xk
TWk < 0 implies incorrect classification.
X‘ = X1 ∪ X0’ is called the adjusted training set.
Assumption of linear separability guarantees the
existence of a solution weight vector WS, such
that Xk
TWS > 0 ∀ Xk ∈ X
We say X’ is a linearly contained set.
40. 40/120
Recast of Perceptron Learning with
Linearly Contained Data
Since Xk ∈ X’, a misclassification of Xk will add
kXk to Wk.
ForXk ∈ X0‘, Xk actually represents the negative of
the original vector.
Therefore addition of kXk to Wk actually
amounts to subtraction of the original vector from
Wk.
42. 42/120
Perceptron Convergence
Theorem
Given: A linearly contained training set X’ and
any initial weight vectorW1.
Let SW be the weight vector sequence generated
in response to presentation of a training
sequence SX upon application of Perceptron
learning law. Then for some finite index k0 we
have: Wk0 = Wk0+1 = Wk0+2 = ・ ・ ・ = WS
as a solution vector.
See the text for detailed proofs.
46. 46/120
MATLAB Simulation
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-0.5
0
0.5
1
1.5
2
2.5
3
x
1
x
2
(0,0) (1,0)
(0,1) (1,1)
R
2
(a) Hyperplane movement depicted during Perceptron Learning
k=5
k=15
k=25
k=35
-4
-3
-2
-1
0
0
0.5
1
1.5
2
2.5
3
0
1
2
3
w0
(b) Weight space trajectories: W
0
= (0,0,0), W
S
=(-4 3 2)
w
1
w
2
47. 47/120
Perceptron Learning and Non-
separable Sets
Theorem:
Given a finite set of training patterns X, there
exists a number M such that if we run the
Perceptron learning algorithm beginning with
any initial set of weights,W1, then any weight
vector Wk produced in the course of the
algorithm will satisfyWk ≤ W1 +M
48. 48/120
Two Corollaries(1/2)
If, in a finite set of training patterns X, each
pattern Xk has integer (or rational) components
xi
k, then the Perceptron learning algorithm will
visit a finite set of distinct weight vectors Wk.
49. 49/120
Two Corollaries(2/2)
For a finite set of training patterns X, with
individual patterns Xk having integer (or
rational) components xi
k the Perceptron learning
algorithm will, in finite time, produce a weight
vector that correctly classifies all training
patterns iff X is linearly separable, or leave and
re-visit a specific weight vector iff X is linearly
non-separable.
50. 50/120
Handling Linearly Non-separable
Sets: The Pocket Algorithm(1/2)
Philosophy: Incorporate positive reinforcement in
a way to reward weights that yield a low error
solution.
Pocket algorithm works by remembering the
weight vector that yields the largest number of
correct classifications on a consecutive run.
51. 51/120
Handling Linearly Non-separable
Sets: The Pocket Algorithm(2/2)
This weight vector is kept in the “pocket”, and
we denote it as Wpocket .
While updating the weights in accordance with
Perceptron learning, if a weight vector is
discovered that has a longer run of consecutively
correct classifications than the one in the pocket,
it replaces the weight vector in the pocket.
54. 54/120
Pocket Convergence Theorem
Given a finite set of training examples, X,
and a probabilityp < 1, there exists an
integer k0 such that after any k > k0
iterations of the pocket algorithm, the
probability that the pocket weight
vectorWpocket is optimal exceeds p.
55. 55/120
Linear Neurons and Linear Error
Consider a training set of the form T = {Xk,
dk}, Xk ∈ Rn+1, dk ∈ R.
To allow the desired output to vary smoothly or
continuously over some interval consider a
linear signal function: sk = yk = Xk
TWk
The linear error ek due to a presented training
pair (Xk, dk), is the difference between the
desired output dk andthe neuronal signal sk:
ek = dk − sk = dk − Xk
TWk
56. 56/120
Operational Details of –LMS
–LMS error correction is
proportional to the error
itself
Each iteration reduces the
error by a factor of η.
η controls the stability and
speed of convergence.
Stability ensured
if 0 < η < 2.
Sk=Xk
TWk
+1
WO
W1
k
W2
k
Wn
k
X1
k
X2
k
Xn
k
59. 59/120
MATLAB Simulation Example
Synthetic data set
shown in the figure is
generated by artificially
scattering points around
a straight line: y = 0.5x
+ 0.333 and generate a
scatter of 200 points in
a ±0.1 interval in the y
direction. -2 -1 0 1 2
-20
-15
-10
-5
0
5
10
x
f
=
3x(x-1)(x-1.9)(x+0.7)(x+1.8)
60. 60/120
MATLAB Simulation Example
This is achieved by first
generating a random
scatter in the interval
[0,1].
Then stretching it to the
interval [−1, 1], and
finally scaling it to ±0.1
-2 -1 0 1 2
-20
-15
-10
-5
0
5
10
x
f
=
3x(x-1)(x-1.9)(x+0.7)(x+1.8)
62. 62/120
A Stochastic Setting
Assumption that the training set T is well defined
in advance is incorrect when the setting is
stochastic.
In such a situation, instead of deterministic
patterns, we have a sequence of samples {(Xk,
dk)} assumed to be drawn from a statistically
stationary population or process.
For adjustment of the neuron weights in response
to some pattern-dependent error measure, the
error computation has to be based on the
expectation of the error over the ensemble.
63. 63/120
Definition of Mean Squared Error
(MSE)(1/2)
We introduce the square error on a pattern Xk as
Assumption: The weights are held fixed at Wk
while computing the expectation.
(2)
2
2
1
(1)
2
1
2
1
2
2
2
k
T
k
k
T
k
k
T
k
k
k
k
k
T
k
k
k
W
X
X
W
W
X
d
d
e
W
X
d
64. 64/120
Definition of Mean Squared Error
(MSE)(2/2)
The mean-squared error can now be computed by
taking the expectation on both sides of (2):
(3)
2
1
2
1 2
k
T
k
k
T
k
k
T
k
k
k
k W
X
X
E
W
W
X
d
E
d
E
E
65. 65/120
To find optimal weight vector that
minimizes the mean-square error.
Our problem…
66. 66/120
Cross Correlations(1/2)
For convenience of expression we define the
pattern vector P as the cross-correlation between
the desired scalar output, dk, and the input vector,
Xk
and the input correlation matrix, R, as
(4)
,
,
,
d
E 1
k
k
n
k
k
k
k
T
k
T
x
d
x
d
d
E
X
P
(5)
1
E
1
1
1
1
1
1
k
k
n
k
n
k
k
n
k
n
k
n
k
k
k
k
k
n
k
T
k
x
x
x
x
x
x
x
x
x
x
x
x
E
X
X
R
67. 67/120
Cross Correlations (2/2)
Using Eqns. (4)-(5), we rewrite the MSE
expression of Eqn. (3) succinctly as
Note that since the MSE is a quadratic
function of the weights it represents a bowl
shaped surface in the (n+1) x 1 weight—
MSE cross space.
(6)
2
1
2
1 2
k
T
k
k
T
k
k RW
W
W
P
d
E
E
68. 68/120
First compute the gradient by straightforward
differentiation which is a linear function of
weights
To find the optimal set of weights, , simply set
which yields
Finding the Minimum Error(1/2)
(7)
,
,
0
RW
P
w
w
T
n
Ŵ
0
(8)
ˆ P
W
R
69. 69/120
This system of equations (8) is called the Weiner-
Hopf system and its solution is the Weiner solution
or the Weiner filter
is the point in weight space that represents the
minimum mean-square error
Finding the Minimum Error (2/2)
(9)
ˆ 1
P
R
W
min
70. 70/120
Computing the Optimal Filter(1/2)
First compute and . Substituting from Eqn.
(9) into Eqn. (6) yields:
1
R P
(12)
ˆ
2
1
2
1
(11)
2
1
2
1
(10)
ˆ
ˆ
ˆ
2
1
2
1
2
1
1
1
2
2
min
W
P
d
E
P
R
P
P
R
R
P
R
d
E
W
P
W
R
W
d
E
T
k
T
T
k
T
T
k
71. 71/120
Computing the Optimal Filter(2/2)
For the treatment of weight update procedures we
reformulate the expression for mean-square error
in terms of the deviation , of the weight
vector from the Weiner solution.
W
W
V ˆ
72. 72/120
Substituting into Eqn. (6)
Computing R(1/2)
W
V
W ˆ
(16)
ˆ
ˆ
ˆ
2
1
(15)
2
1
(14)
ˆ
ˆ
ˆ
ˆ
ˆ
2
1
(13)
ˆ
ˆ
ˆ
2
1
min
min
2
2
ˆ
2
2
2
1
W
W
R
W
W
R
W
W
RV
V
W
P
V
P
W
R
W
RV
W
W
R
V
RV
V
d
E
W
V
P
W
V
R
W
V
d
E
T
T
T
T
T
T
V
P
RV
R
P
RV
W
T
T
T
k
T
T
k
T
T
T
73. 73/120
Note that since the mean-square error is
non-negative, we must have . This
implies that R is positive semi-definite.
Usually however, R is positive definite.
Computing R(2/2)
0
RV
VT
74. 74/120
Assume that R has distinct eigenvalues . Then we
can construct a matrix Q whose columns are
corresponding eigenvectors of R.
R can be diagonalized using an orthogonal similarity
transformation as follows. Having constructed Q, and
knowing that:
i
i
(17)
n
1
0
Q
(18)
0
0
0
0
0
n
1
0
n
1
0
RQ
We have
(19)
,
,
,
, 2
1
0
1
n
diag
D
RQ
Q
Diagonalization of R
75. 75/120
It is usual to choose the eigenvectors of R to be
orthonormal and .
Then
From Eqn. (15) we know that the shape of is a
bowl-shaped paraboloid with a minimum at the
Weiner solution (V=0, the origin of V-space).
Slices of parallel to the W space yield elliptic
constant error contours which define the weights in
weight space that the specific value of the square-error
(say ) at which the slice is made:
I
QQT
T
Q
Q
1
1
QDQ
QDQ
R T
c
(20)
constant
2
1
min
ε
ε
RV
V c
T
Some Observations
76. 76/120
Also note from Eqn. (15) that we can compute
the MSE gradient in V-space as,
which defines a family of vectors in V-space.
Exactly n+1 of these pass through the origin of
V-space and these are the principal axes of the
ellipse.
Eigenvectors of R(1/2)
(21)
RV
,
'
V
77. 77/120
However, vectors passing through the origin
must take the . Therefore, for the principal
axes
Clearly, is an eigenvector of R.
Eigenvectors of R(2/2)
V
,
'
V
(22)
'
'
V
RV
'
V
78. 78/120
Steepest Descent Search with Exact
Gradient Information
Steepest descent search
uses exact gradient
information available
from the mean-square
error surface to direct
the search in weight
space.
The figure shows a
projection of the
square-error function on
k
i
w
79. 79/120
Steepest Descent Procedure
Summary(1/2)
Provide an appropriate weight increment to to
push the error towards the minimum which
occurs at .
Perturb the weight in a direction that depends on
which side of the optimal weight the current
weight value lies.
If the weight component lies to the left of ,
say at , where the error gradient is negative (as
indicated by the tangent) we need to increase .
k
i
w
i
ŵ
i
ŵ
k
i
w1
k
i
w
k
i
w
i
ŵ
k
i
w
80. 80/120
Steepest Descent Procedure
Summary(2/2)
If is on the right of , say at where the error
gradient is positive, we need to decrease .
This rule is summarized in the following
statement:
i
ŵ
k
i
w
k
i
w
k
i
w 2
k
i
i
k
i
k
i
k
i
i
k
i
k
i
w
increase
,
w
w
0,
w
ε
If
w
decrease
,
w
w
0,
w
ε
If
ˆ
ˆ
81. 81/120
It follows logically therefore, that the weight
component should be updated in proportion
with the negative of the gradient:
27
n
,
,
0,1
i
1
k
i
k
i
k
i
w
w
w
Weight Update Procedure(1/3)
82. 82/120
Vectorially we may write
where we now re-introduced the iteration time
dependence into the weight components:
(28)
1
k
k W
W
(29)
,
,
0
T
k
n
k
w
w
Weight Update Procedure(2/3)
83. 83/120
Equation (28) is the steepest descent update
procedure. Note that steepest descent uses
exact gradient information at each step to
decide weight changes.
Weight Update Procedure(3/3)
84. 84/120
Convergence of Steepest Descent –
1(1/2)
Question: What can one say about the
stability of the algorithm? Does it converge
for all values of ?
To answer this question consider the
following series of subtitutions and
transformations. From Eqns. (28) and (21)
(33)
ˆ
1
(32)
ˆ
(31)
(30)
1
W
R
W
R
W
W
R
W
RV
W
W
W
k
k
k
k
k
k
k
85. 85/120
Convergence of Steepest Descent –
1(2/2)
Transforming Eqn. (33) into V-space (the
principal coordinate system) by substitution
of for yields:
W
Vk
ˆ
k
W
(34)
1 k
k V
R
I
V
86. 86/120
Rotation to the principal axes of the elliptic
contours can be effected by using :
Steepest Descent Convergence –
2(1/2)
'
QV
V
(35)
'
'
1 k
k QV
R
I
QV
or
(37)
(36)
'
'
1
'
1
k
k
k
V
D
I
QV
R
I
Q
V
87. 87/120
where D is the diagonal eigenvalue matrix.
Recursive application of
Eqn. (37) yields:
It follows from this that for stability and
convergence of the algorithm:
Steepest Descent Convergence –
2(2/2)
or
(38)
'
0
'
V
D
I
V
k
k
(39)
0
lim
k
k
D
I
88. 88/120
This requires that
being the largest eigenvalue of R.
(40)
0
1
lim max
k
k
or
(41)
2
0
max
max
Steepest Descent Convergence –
3(1/2)
89. 89/120
If this condition is satisfied then we have
Steepest descent is guaranteed to converge to
the Weiner solution as long as the learning rate
is maintained within the limits defined by Eqn.
(41).
(43)
ˆ
lim
(42)
0
ˆ
lim
lim
k
1
'
W
W
W
W
Q
V
k
k
k
k
k
or
Steepest Descent Convergence –
3(2/2)
90. 90/120
This simulation example employs the fifth
order function data scatter with the data
shifted in the y direction by 0.5.
Consequently, the values of R,P and the
Weiner solution are respectively:
Computer Simulation
Example(1/2)
(45)
3834
.
1
5303
.
1
ˆ
(44)
4625
.
1
8386
.
0
,
1.61
0.500
0.500
1
W
P
R
91. 91/120
Exact gradient information is available
since the correlation matrix R and the cross-
correlation matrix P are known.
The weights are updated using the equation:
Computer Simulation
Example(2/2)
(46)
ˆ
2
1 k
k
k W
W
R
W
W
92. 92/120
eta = .01; % Set learning rate
R=zeros(2,2); % Initialize correlation matrix
X = [ones(1,max_points);x]; % Augment input vectors
P = (sum([d.*X(1,:); d.*X(2,:)],2))/max_points; % Cross-correlations
D = (sum(d.^2))/max_points; %squares; % target expectations
for k =1:max_points
R = R + X(:,k)*X(:,k)'; % Compute R
end
R = R/max_points;
weiner=inv(R)*P; % Compute the Weiner solution
errormin = D - P'*inv(R)*P; % Find the minimum error
MATLAB Simulation
Example(1/3)
93. 93/120
shift1 = linspace(-12,12, 21); % Generate a weight space matrix
shift2 = linspace(-9,9, 21);
for i = 1:21 % Compute a weight matrix about
shiftwts(1,i) = weiner(1)+shift1(i); % Weiner solution
shiftwts(2,i) = weiner(2)+shift2(i);
end
for i=1:21 % Compute the error matrix
for j = 1:21 % to plot the error contours
error(i,j) = sum((d - (shiftwts(1,i) + x.*shiftwts(2,j))).^2);
end
end
error = error/max_points;
figure; hold on; % Plot the error contours
plot(weiner(1),weiner(2),'*k') % Labelling statements no shown
MATLAB Simulation Example (2/3)
94. 94/120
w = 10*(2*rand(2,1)-1); % Randomize weights
w0 = w; % Remember the initial weights
for loop = 1:500 % Perform descent for 500 iters
w = w + eta*(-2*(R*w-P));
wts1(loop)=w(1); wts2(loop)=w(2);
End
% Set up weights for plotting
wts1=[w0(1) wts1]; wts2=[w0(2) wts2];
plot(wts1,wts2,'r') % Plot the weight trajectory
MATLAB Simulation Example (3/3)
95. 95/120
Smooth Trajectory towards the
Weiner Solution
Steepest descent
uses exact
gradient
information to
search the
Weiner solution
in weight space.
-10 -5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
8
55.3
90.1
125
160
160
160
160
194
194
229
229
264
264
299
333
368
w
0
w
1
Weiner solution
(1.53, -1.38)
W0
= (-3.9, 6.27)T
96. 96/120
-LMS: Approximate Gradient
Descent(1/2)
The problem with steepest descent is that
true gradient information is only available
in situations where the data set is
completely specified in advance.
It is then possible to compute R and P
exactly, and thus the true gradient at
iteration
P
RW
k k
:
97. 97/120
-LMS: Approximate Gradient
Descent(2/2)
However, when the data set comprises a
random stream of patterns (drawn from a
stationary distribution), R and P cannot be
computed accurately. To find a correct
approximation one might have to examine
the data stream for a reasonably large period
of time and keep averaging out.
How long should we examine the stream to
get reliable estimates of R and P ?
98. 98/120
Definition
The μ-LMS algorithm is convergent in the
mean if the average of the weight vector Wk
approaches the optimal solution Wopt as the
number of iterations k, approaches infinity:
E[Wk] → Wopt as k → ∞
99. 99/120
The gradient computation modifies to:
where , and since we
are dealing with linear neurons. Note
therefore that the recursive update equation
then becomes
-LMS employs k for k(1/2)
(47)
,
,
,
~
0
0
k
k
T
k
n
k
k
k
k
T
k
n
k
k
k
k X
e
w
e
w
e
e
w
w
k
k
k s
d
e
k
T
k
k W
X
s
(49)
(48)
)
~
(
1
k
k
k
k
k
k
k
k
k
k
X
e
W
X
s
d
W
W
W
100. 100/120
What value does the long term average of
converge to? Taking the Expectation of both
sides of Eqn. (47):
-LMS employs k for k(2/2)
k
~
(52)
(51)
(50)
~
P
RW
W
X
X
X
d
E
X
e
E
E T
k
k
k
k
k
k
k
101. 101/120
Since the long term average of
approaches , we can safely use as an
unbiased estimate. That’s what makes -
LMS work!
Since approaches in the long run,
one could keep collecting for a
sufficiently large number of iterations
(while keeping the weights fixed), and then
make a weight change collectively for all
those iterations together.
k
~
k
~
k
~
k
~
k
X
Observations(1/8)
102. 102/120
If the data set is finite (deterministic), then
one can compute accurately by first
collecting the different gradients over all
training patterns for the same set of
weights. This accurate measure of the
gradient could then be used to change the
weights. In this situation -LMS is identical
to the steepest descent algorithm.
k
~
k
X
Observations(2/8)
103. 103/120
Even if the data set is deterministic, we still use
to update the weights. After all if the data set
becomes large, collection of all the gradients
becomes expensive in terms of storage. Much
easier to just go ahead and use
Be clear about the approximation made: we are
estimating the true gradient (which should be
computed from ) by a gradient computed
from the instantaneous sample error .
Although this may seem to be a rather drastic
approximation, it works.
k
~
k
E
k
Observations(3/8)
104. 104/120
In the deterministic case we can justify this
as follows: if the learning rate , is kept
small, the weight change in each iteration
will be small and consequently the weight
vector W will remain “somewhat constant”
over Q iterations where Q is the number of
patterns in the training set.
Observations(4/8)
105. 105/120
Of course this is provided that Q is a small
number! To see this, observe the total weight
change , over Q iterations from the
iteration:
W
th
k
(56)
(55)
1
(54)
1
(53)
1
0
1
0
1
0
k
Q
i
i
k
k
Q
i k
i
k
Q
i i
k
i
k
W
Q
Q
W
Q
W
Q
Q
W
W
Observations(5/8)
106. 106/120
Where denotes the mean-square error. Thus
the weight updates follow the true gradient on
average.
Observations(6/8)
107. 107/120
Observe that steepest descent search is
guaranteed to search the Weiner solution
provided the learning rate condition (41) is
satisfied.
Now since is an unbiased estimate of
one can expect that -LMS too will search out
the Weiner solution. Indeed it does—but not
smoothly. This is to be expected since we are
only using an estimate of the true gradient for
immediate use.
k
~
Observations(7/8)
108. 108/120
Although -LMS and - LMS are similar
algorithms, -LMS works on the normalizing
training set. What this simply means is that -
LMS also uses gradient information, and will
eventually search out the Weiner solution-of the
normalized training set. However, in one case
the two algorithms are identical: the case when
input vectors are bipolar. (Why?)
Observations(8/8)
109. 109/120
Definition 0.1 The -LMS algorithm is convergent in
the mean if the average of the weight vector
approaches the optimal solution as the number of
iterations k, approaches infinity:
Definition 0.2 The -LMS algorithm is convergent in
the mean square if the average of the squared error
approaches a constant as the Number of iterations, k,
approaches infinity:
-LMS Algorithm:
Convergence in the Mean (1)(1/2)
k
W
Ŵ
(57)
k
as
ˆ
W
W
E k
k
(58)
as
constant
k
E k
110. 110/120
Convergence in the mean square is a
stronger criterion than convergence in the
mean. In this section we discuss
convergence in the mean and merely state
the result for convergence in the mean
square.
-LMS Algorithm:
Convergence in the Mean (1)(2/2)
111. 111/120
Consider the -LMS weight update equation:
-LMS Algorithm:
Convergence in the Mean (2)(1/2)
(63)
-
I
(62)
-
(61)
(60)
(59)
1
k
k
k
T
k
k
k
k
k
T
k
k
k
k
k
T
k
k
k
k
k
k
T
k
k
k
k
k
k
k
k
X
d
W
X
X
X
d
W
X
X
W
X
W
X
X
d
W
X
W
X
d
W
X
s
d
W
W
112. 112/120
Taking the expectation of both sides of Eqn. (63)
yields,
where P and R are as already defined
-LMS Algorithm:
Convergence in the Mean (2)(2/2)
(65)
(64)
1
P
W
E
R
I
X
d
E
W
E
X
X
E
I
W
E
k
k
k
k
T
k
k
k
113. 113/120
Appropriate substitution yields:
Pre-multiplication throughout by results in:
-LMS Algorithm:
Convergence in the Mean (3)(1/2)
W
QDQ
W
E
Q
D
I
Q
W
QDQ
W
E
QDQ
QQ
W
QDQ
W
E
QDQ
I
W
E
T
k
T
T
k
T
T
T
k
T
k
ˆ
(66)
ˆ
ˆ
1
T
Q
(67)
ˆ
1 W
DQ
W
E
Q
D
I
W
E
Q T
k
T
k
T
114. 114/120
And subtraction of from both sides
gives:
We will re-write Eqn. (69) in familiar terms:
-LMS Algorithm:
Convergence in the Mean (3)(2/2)
W
QT ˆ
(69)
ˆ
(68)
ˆ
ˆ
1
W
W
E
Q
D
I
W
Q
D
I
W
E
Q
D
I
W
W
E
Q
k
T
T
k
T
k
T
(70)
~
~
1 k
T
k
T
V
Q
D
I
V
Q
115. 115/120
And
where and . Eqn. (71)
represents a set of n+1Decoupled difference
equations :
-LMS Algorithm:
Convergence in the Mean (4)(1/2)
(71)
~
~ '
'
1 k
k V
D
I
V
W
W
E
V k
k
ˆ
~
k
T
k V
Q
V
~
~'
(72)
n.
,
0,1,
i
~
1
~ '
1
'
k
i
i
k
i v
v
116. 116/120
Recursive application of Eqn. (72) yields,
To ensure convergence in the mean, as
since this condition requires that the
deviation of from should tend to 0.
-LMS Algorithm:
Convergence in the Mean (4)(2/2)
(73)
n
,
0,1,
i
~
1
~ 0
'
'
i
k
i
k
i v
v
0
~'
k
i
v
k
k
W
E Ŵ
117. 117/120
Therefore from Eqn. (73):
If this condition is satisfied for the largest
eigenvalue then it will be satisfied for all
other eigenvalues. We therefore conclude that if
then the -LMS algorithm is convergent in the
mean.
-LMS Algorithm:
Convergence in the Mean (5)(1/2)
(74)
n.
,
0,1,
i
1
|
1
|
i
max
(75)
2
0
max
118. 118/120
Further, since tr = (where tr
is the trace of R) convergence is assured if
-LMS Algorithm:
Convergence in the Mean (5)(2/2)
R max
0
n
i
i
R
(76)
2
0
R
tr