The document provides an introduction to TreeNet, a machine learning algorithm developed by Jerome Friedman. TreeNet builds regression and classification models in a stagewise fashion, using small regression trees at each stage to model residuals from the previous stage. It employs techniques like learning small trees, subsampling data, and using a small learning rate to minimize overfitting. TreeNet models can be very accurate while remaining resistant to overfitting.
This document discusses analyzing the time efficiency of recursive algorithms. It provides a general 5-step plan: 1) choose a parameter for input size, 2) identify the basic operation, 3) check if operation count varies, 4) set up a recurrence relation, 5) solve the relation to determine growth order. It then gives two examples - computing factorial recursively and solving the Tower of Hanoi puzzle recursively - to demonstrate applying the plan. The document also briefly discusses algorithm visualization using static or dynamic images to convey information about an algorithm's operations and performance.
The document describes a novel mixed method for order reduction of discrete linear systems. The method uses particle swarm optimization (PSO) to determine the denominator polynomials of the reduced order model. It then uses a polynomial technique to derive the numerator coefficients by equating the original and reduced order transfer functions. This leads to a set of equations that can be solved for the numerator coefficients. The proposed method is illustrated on an 8th order example system from literature. It is found to provide a stable 2nd order reduced model. A lead compensator is then designed and connected to improve the steady state response of the original and reduced order systems.
This document discusses radial basis function networks. It begins by introducing the basic structure of RBF networks, which typically involve an input layer, a hidden layer that applies a nonlinear transformation using radial basis functions, and an output layer with a linear transformation. The document then discusses Cover's theorem, which states that pattern classification problems are more likely to be linearly separable when mapped to a higher-dimensional space through a nonlinear transformation. Several key concepts are introduced, including dichotomies, phi-separable functions, and using hidden functions to map patterns to a hidden feature space.
The document discusses machine learning techniques for multivariate data analysis using the TMVA toolkit. It describes several common classification problems in high energy physics (HEP) and summarizes several machine learning algorithms implemented in TMVA for supervised learning, including rectangular cut optimization, likelihood methods, neural networks, boosted decision trees, support vector machines and rule ensembles. It also discusses challenges like nonlinear correlations between input variables and techniques for data preprocessing and decorrelation.
The document discusses two principal approaches to solving intractable problems: exact algorithms that guarantee an optimal solution but may not run in polynomial time, and approximation algorithms that can find a suboptimal solution in polynomial time. It focuses on exact algorithms like exhaustive search, backtracking, and branch-and-bound. Backtracking constructs a state space tree and prunes non-promising nodes to reduce search space. Branch-and-bound uses bounding functions to determine if nodes are promising or not, pruning those that are not. The traveling salesman problem is used as an example to illustrate branch-and-bound, discussing different bounding functions that can be used.
This document provides an overview of key algorithm analysis concepts including:
- Common algorithmic techniques like divide-and-conquer, dynamic programming, and greedy algorithms.
- Data structures like heaps, graphs, and trees.
- Analyzing the time efficiency of recursive and non-recursive algorithms using orders of growth, recurrence relations, and the master's theorem.
- Examples of specific algorithms that use techniques like divide-and-conquer, decrease-and-conquer, dynamic programming, and greedy strategies.
- Complexity classes like P, NP, and NP-complete problems.
This document discusses analyzing the time efficiency of recursive algorithms. It provides a general 5-step plan: 1) choose a parameter for input size, 2) identify the basic operation, 3) check if operation count varies, 4) set up a recurrence relation, 5) solve the relation to determine growth order. It then gives two examples - computing factorial recursively and solving the Tower of Hanoi puzzle recursively - to demonstrate applying the plan. The document also briefly discusses algorithm visualization using static or dynamic images to convey information about an algorithm's operations and performance.
The document describes a novel mixed method for order reduction of discrete linear systems. The method uses particle swarm optimization (PSO) to determine the denominator polynomials of the reduced order model. It then uses a polynomial technique to derive the numerator coefficients by equating the original and reduced order transfer functions. This leads to a set of equations that can be solved for the numerator coefficients. The proposed method is illustrated on an 8th order example system from literature. It is found to provide a stable 2nd order reduced model. A lead compensator is then designed and connected to improve the steady state response of the original and reduced order systems.
This document discusses radial basis function networks. It begins by introducing the basic structure of RBF networks, which typically involve an input layer, a hidden layer that applies a nonlinear transformation using radial basis functions, and an output layer with a linear transformation. The document then discusses Cover's theorem, which states that pattern classification problems are more likely to be linearly separable when mapped to a higher-dimensional space through a nonlinear transformation. Several key concepts are introduced, including dichotomies, phi-separable functions, and using hidden functions to map patterns to a hidden feature space.
The document discusses machine learning techniques for multivariate data analysis using the TMVA toolkit. It describes several common classification problems in high energy physics (HEP) and summarizes several machine learning algorithms implemented in TMVA for supervised learning, including rectangular cut optimization, likelihood methods, neural networks, boosted decision trees, support vector machines and rule ensembles. It also discusses challenges like nonlinear correlations between input variables and techniques for data preprocessing and decorrelation.
The document discusses two principal approaches to solving intractable problems: exact algorithms that guarantee an optimal solution but may not run in polynomial time, and approximation algorithms that can find a suboptimal solution in polynomial time. It focuses on exact algorithms like exhaustive search, backtracking, and branch-and-bound. Backtracking constructs a state space tree and prunes non-promising nodes to reduce search space. Branch-and-bound uses bounding functions to determine if nodes are promising or not, pruning those that are not. The traveling salesman problem is used as an example to illustrate branch-and-bound, discussing different bounding functions that can be used.
This document provides an overview of key algorithm analysis concepts including:
- Common algorithmic techniques like divide-and-conquer, dynamic programming, and greedy algorithms.
- Data structures like heaps, graphs, and trees.
- Analyzing the time efficiency of recursive and non-recursive algorithms using orders of growth, recurrence relations, and the master's theorem.
- Examples of specific algorithms that use techniques like divide-and-conquer, decrease-and-conquer, dynamic programming, and greedy strategies.
- Complexity classes like P, NP, and NP-complete problems.
The document introduces the EM algorithm, which allows maximum likelihood estimates (MLEs) to be made when data is incomplete. The EM algorithm consists of an Expectation (E)-step, where expected values of sufficient statistics are computed based on current parameter estimates, and a Maximization (M)-step, where new parameter estimates are calculated as the MLE given the sufficient statistics from the E-step. The algorithm iterates between these steps until convergence. As an example, the document shows how the EM algorithm can be used to estimate the parameter of a multinomial distribution even when some category counts are unknown.
This document provides an overview of dimensionality reduction techniques including PCA and manifold learning. It discusses the objectives of dimensionality reduction such as eliminating noise and unnecessary features to enhance learning. PCA and manifold learning are described as the two main approaches, with PCA using projections to maximize variance and manifold learning assuming data lies on a lower dimensional manifold. Specific techniques covered include LLE, Isomap, MDS, and implementations in scikit-learn.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
This document provides an overview of various machine learning algorithms and concepts, including supervised learning techniques like linear regression, logistic regression, decision trees, random forests, and support vector machines. It also discusses unsupervised learning methods like principal component analysis and kernel-based PCA. Key aspects of linear regression, logistic regression, and random forests are summarized, such as cost functions, gradient descent, sigmoid functions, and bagging. Kernel methods are also introduced, explaining how the kernel trick can allow solving non-linear problems by mapping data to a higher-dimensional feature space.
Understanding variable importances in forests of randomized treesGilles Louppe
This document discusses variable importances in random forests. It begins by introducing random forests and their strengths and weaknesses, specifically their loss of interpretability. It then discusses how variable importances can help recover interpretability by providing two main importance measures: mean decrease in impurity (MDI) and mean decrease in accuracy (MDA). The document focuses on MDI and presents three key results: 1) variable importances provide a three-level decomposition of information about the output, 2) importances only depend on relevant variables, and 3) most properties are lost when K > 1 in non-totally randomized trees.
The document discusses the divide-and-conquer algorithm design paradigm. It defines divide-and-conquer as breaking a problem down into smaller subproblems, solving those subproblems recursively, and combining the solutions to solve the original problem. Examples of algorithms that use this approach include merge sort, quicksort, and matrix multiplication. Divide-and-conquer allows for problems to be solved in parallel and more efficiently uses memory caches. The closest pair problem is then presented as a detailed example of how a divide-and-conquer algorithm works to solve this problem in O(n log n) time compared to the brute force O(n2) approach.
X01 Supervised learning problem linear regression one feature theorieMarco Moldenhauer
1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
Lesson 27: Integration by Substitution (Section 041 handout)Matthew Leingang
This document is notes from a Calculus I class covering integration by substitution. It discusses using substitution to evaluate both indefinite and definite integrals. Examples are provided to demonstrate how to use substitution to evaluate integrals involving polynomials, trigonometric functions, and exponentials. The key concept is that substitution allows one to transform integrals into simpler equivalent forms through a change of variables.
Machine Learning Algorithms Review(Part 2)Zihui Li
This document provides an overview of machine learning algorithms and techniques. It discusses classification and regression metrics, naive Bayesian classifiers, clustering methods like k-means, ensemble learning techniques like bagging and boosting, the expectation maximization algorithm, restricted Boltzmann machines, neural networks including convolutional and recurrent neural networks, and word embedding techniques like Word2Vec, GloVe, and matrix factorization. Key algorithms and their applications are summarized at a high level.
Bias-variance decomposition in Random ForestsGilles Louppe
This document discusses bias-variance decomposition in random forests. It explains that combining predictions from multiple randomized models can achieve better results than a single model by reducing variance. Random forests work by constructing decision trees on randomly selected subsets of data and features, averaging their predictions. This randomization increases bias but reduces variance, providing an effective bias-variance tradeoff. The document provides theorems on how the expected generalization error of random forests and individual trees can be decomposed into noise, bias, and variance components.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
This document discusses the divide-and-conquer algorithm design strategy. It begins by defining divide-and-conquer as dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions. Examples covered include sorting algorithms like mergesort and quicksort, tree traversals, binary search, integer multiplication, matrix multiplication using Strassen's algorithm, and closest pair problems. Analysis techniques like recursion trees and the master theorem are introduced.
The document discusses numerical methods and provides examples of how to implement them in Smalltalk. It covers frameworks for iterative processes, Newton's method for finding zeros, eigenvalue and eigenvector computation using the Jacobi method, and cluster analysis. Code examples and class diagrams are provided.
The document discusses different machine learning algorithms for instance-based learning. It describes k-nearest neighbor classification which classifies new instances based on the labels of the k closest training examples. It also covers locally weighted regression which approximates the target function based on nearby training data. Radial basis function networks are discussed as another approach using localized kernel functions to provide a global approximation of the target function. Case-based reasoning is presented as using rich symbolic representations of instances and reasoning over retrieved similar past cases to solve new problems.
Branch and bound is a state space search method that generates all children of a node before expanding any children. It associates a cost or profit with each node and uses a min or max heap to select the next node to expand. For the travelling salesman problem, it constructs a permutation tree representing all possible routes and uses lower bounds and reduced cost matrices at each node to prune the search space and find an optimal solution.
It covers knowledge representation techniques using propositional and predicate logic. It also discusses about the knowledge inference using resolution refutation process, rule based system and bayesian network.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
This document discusses differentiation and its applications in business. It covers the basic rules of differentiation including the power rule, constant multiple property, and sum and difference rules. It also explains how to use differentiation to find maximum and minimum points by setting the first derivative equal to zero and checking the second derivative. Examples are provided to demonstrate how differentiation can be used to find marginal costs, profit maximization, and other business optimization problems.
The document introduces the EM algorithm, which allows maximum likelihood estimates (MLEs) to be made when data is incomplete. The EM algorithm consists of an Expectation (E)-step, where expected values of sufficient statistics are computed based on current parameter estimates, and a Maximization (M)-step, where new parameter estimates are calculated as the MLE given the sufficient statistics from the E-step. The algorithm iterates between these steps until convergence. As an example, the document shows how the EM algorithm can be used to estimate the parameter of a multinomial distribution even when some category counts are unknown.
This document provides an overview of dimensionality reduction techniques including PCA and manifold learning. It discusses the objectives of dimensionality reduction such as eliminating noise and unnecessary features to enhance learning. PCA and manifold learning are described as the two main approaches, with PCA using projections to maximize variance and manifold learning assuming data lies on a lower dimensional manifold. Specific techniques covered include LLE, Isomap, MDS, and implementations in scikit-learn.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
The document discusses the divide and conquer algorithm design paradigm. It begins by defining divide and conquer as recursively breaking down a problem into smaller sub-problems, solving the sub-problems, and then combining the solutions to solve the original problem. Some examples of problems that can be solved using divide and conquer include binary search, quicksort, merge sort, and the fast Fourier transform algorithm. The document then discusses control abstraction, efficiency analysis, and uses divide and conquer to provide algorithms for large integer multiplication and merge sort. It concludes by defining the convex hull problem and providing an example input and output.
This document provides an overview of various machine learning algorithms and concepts, including supervised learning techniques like linear regression, logistic regression, decision trees, random forests, and support vector machines. It also discusses unsupervised learning methods like principal component analysis and kernel-based PCA. Key aspects of linear regression, logistic regression, and random forests are summarized, such as cost functions, gradient descent, sigmoid functions, and bagging. Kernel methods are also introduced, explaining how the kernel trick can allow solving non-linear problems by mapping data to a higher-dimensional feature space.
Understanding variable importances in forests of randomized treesGilles Louppe
This document discusses variable importances in random forests. It begins by introducing random forests and their strengths and weaknesses, specifically their loss of interpretability. It then discusses how variable importances can help recover interpretability by providing two main importance measures: mean decrease in impurity (MDI) and mean decrease in accuracy (MDA). The document focuses on MDI and presents three key results: 1) variable importances provide a three-level decomposition of information about the output, 2) importances only depend on relevant variables, and 3) most properties are lost when K > 1 in non-totally randomized trees.
The document discusses the divide-and-conquer algorithm design paradigm. It defines divide-and-conquer as breaking a problem down into smaller subproblems, solving those subproblems recursively, and combining the solutions to solve the original problem. Examples of algorithms that use this approach include merge sort, quicksort, and matrix multiplication. Divide-and-conquer allows for problems to be solved in parallel and more efficiently uses memory caches. The closest pair problem is then presented as a detailed example of how a divide-and-conquer algorithm works to solve this problem in O(n log n) time compared to the brute force O(n2) approach.
X01 Supervised learning problem linear regression one feature theorieMarco Moldenhauer
1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
Lesson 27: Integration by Substitution (Section 041 handout)Matthew Leingang
This document is notes from a Calculus I class covering integration by substitution. It discusses using substitution to evaluate both indefinite and definite integrals. Examples are provided to demonstrate how to use substitution to evaluate integrals involving polynomials, trigonometric functions, and exponentials. The key concept is that substitution allows one to transform integrals into simpler equivalent forms through a change of variables.
Machine Learning Algorithms Review(Part 2)Zihui Li
This document provides an overview of machine learning algorithms and techniques. It discusses classification and regression metrics, naive Bayesian classifiers, clustering methods like k-means, ensemble learning techniques like bagging and boosting, the expectation maximization algorithm, restricted Boltzmann machines, neural networks including convolutional and recurrent neural networks, and word embedding techniques like Word2Vec, GloVe, and matrix factorization. Key algorithms and their applications are summarized at a high level.
Bias-variance decomposition in Random ForestsGilles Louppe
This document discusses bias-variance decomposition in random forests. It explains that combining predictions from multiple randomized models can achieve better results than a single model by reducing variance. Random forests work by constructing decision trees on randomly selected subsets of data and features, averaging their predictions. This randomization increases bias but reduces variance, providing an effective bias-variance tradeoff. The document provides theorems on how the expected generalization error of random forests and individual trees can be decomposed into noise, bias, and variance components.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
This document discusses the divide-and-conquer algorithm design strategy. It begins by defining divide-and-conquer as dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions. Examples covered include sorting algorithms like mergesort and quicksort, tree traversals, binary search, integer multiplication, matrix multiplication using Strassen's algorithm, and closest pair problems. Analysis techniques like recursion trees and the master theorem are introduced.
The document discusses numerical methods and provides examples of how to implement them in Smalltalk. It covers frameworks for iterative processes, Newton's method for finding zeros, eigenvalue and eigenvector computation using the Jacobi method, and cluster analysis. Code examples and class diagrams are provided.
The document discusses different machine learning algorithms for instance-based learning. It describes k-nearest neighbor classification which classifies new instances based on the labels of the k closest training examples. It also covers locally weighted regression which approximates the target function based on nearby training data. Radial basis function networks are discussed as another approach using localized kernel functions to provide a global approximation of the target function. Case-based reasoning is presented as using rich symbolic representations of instances and reasoning over retrieved similar past cases to solve new problems.
Branch and bound is a state space search method that generates all children of a node before expanding any children. It associates a cost or profit with each node and uses a min or max heap to select the next node to expand. For the travelling salesman problem, it constructs a permutation tree representing all possible routes and uses lower bounds and reduced cost matrices at each node to prune the search space and find an optimal solution.
It covers knowledge representation techniques using propositional and predicate logic. It also discusses about the knowledge inference using resolution refutation process, rule based system and bayesian network.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
This document discusses differentiation and its applications in business. It covers the basic rules of differentiation including the power rule, constant multiple property, and sum and difference rules. It also explains how to use differentiation to find maximum and minimum points by setting the first derivative equal to zero and checking the second derivative. Examples are provided to demonstrate how differentiation can be used to find marginal costs, profit maximization, and other business optimization problems.
This document provides information about the key concepts and formulas related to differential calculus. It defines the derivative as the rate of change of a function with respect to an independent variable. Some common derivative formulas are presented for various standard functions like polynomial, exponential, logarithmic and trigonometric functions. Examples are given to demonstrate calculating derivatives using the formulas. The concepts of maxima and minima of functions are also introduced along with an example to find the maximum and minimum values of a given function without using derivatives.
This document provides an overview of various techniques for text categorization, including decision trees, maximum entropy modeling, perceptrons, and K-nearest neighbor classification. It discusses the data representation, model class, and training procedure for each technique. Key aspects covered include feature selection, parameter estimation, convergence criteria, and the advantages/limitations of each approach.
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
- A function is a rule that maps an input number (independent variable) to a unique output number (dependent variable).
- To determine if a rule describes a valid function, you can plot points from the rule on a graph and check that each input only maps to one output using a vertical ruler.
- For a rule to describe a valid function, its domain must be restricted if multiple outputs are possible for any single input. The domain is the set of possible inputs, and the range is the set of corresponding outputs.
Introduction to Machine Learning Lecturesssuserfece35
This lecture discusses ensemble methods in machine learning. It introduces bagging, which trains multiple models on random subsets of the training data and averages their predictions, in order to reduce variance and prevent overfitting. Bagging is effective because it decreases the correlation between predictions. Random forests apply bagging to decision trees while also introducing more randomness by selecting a random subset of features to consider at each node. The next lecture will cover boosting, which aims to reduce bias by training models sequentially to focus on examples previously misclassified.
This document summarizes a machine learning homework assignment with 4 problems:
1) Probability questions about constructing random variables.
2) Questions about Poisson generalized linear models (GLM) including log-likelihood, prediction, and regularization.
3) Questions comparing square loss and logistic loss for an outlier point.
4) Questions about batch normalization including its effect on model expressivity and gradients.
https://telecombcn-dl.github.io/2017-dlai/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks or Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles of deep learning from both an algorithmic and computational perspectives.
This document contains instructions for a final exam with multiple choice and written response questions about object-oriented programming, state machines, linear systems, circuits, and state estimation. The exam covers topics taught in the 6.01 course including defining classes to represent mathematical functions, implementing state machines, analyzing system stability from poles, designing circuits to produce buffered voltage levels, and applying Bayes' rule to update beliefs based on experimental observations.
I am Britney. I am a Differential Equations Assignment Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from London, UK. I have been helping students with their assignments for the past 10 years. I solved assignments related to Differential Equations Assignment.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Differential Equations Assignment.
I am Steven M. I am a Maths Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from Ryerson University. I have been helping students with their assignments for the past 10 years. I solve assignments related to Maths.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call +1 678 648 4277 for any assistance with Maths Assignments.
This document provides an overview of supervised learning and linear regression. It introduces supervised learning problems using an example of predicting house prices based on living area. Linear regression is discussed as an initial approach to model this relationship. The cost function is defined as the mean squared error between predictions and targets. Gradient descent and stochastic gradient descent are presented as algorithms to minimize this cost function and learn the parameters of the linear regression model.
Artificial intelligence is the simulation of human intelligence processes by machines, especially computer systems. Specific applications of AI include expert systems, natural language processing, speech recognition and machine vision.
Artificial intelligence is the simulation of human intelligence processes by machines, especially computer systems. Specific applications of AI include expert systems, natural language processing, speech recognition and machine vision.
Artificial intelligence is the simulation of human intelligence processes by machines, especially computer systems. Specific applications of AI include expert systems, natural language processing, speech recognition and machine vision.
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
This document summarizes support vector machines (SVMs), a machine learning technique for classification and regression. SVMs find the optimal separating hyperplane that maximizes the margin between positive and negative examples in the training data. This is achieved by solving a convex optimization problem that minimizes a quadratic function under linear constraints. SVMs can perform non-linear classification by implicitly mapping inputs into a higher-dimensional feature space using kernel functions. They have applications in areas like text categorization due to their ability to handle high-dimensional sparse data.
This document provides an overview of optimization techniques for deep learning models. It begins with challenges in neural network optimization such as saddle points and vanishing gradients. It then discusses various optimization algorithms including gradient descent, stochastic gradient descent, momentum, Adagrad, RMSProp, and Adam. The goal of optimization algorithms is to train deep learning models by minimizing the loss function through iterative updates of the model parameters. Learning rate, batch size, and other hyperparameters of the algorithms affect how quickly and accurately they can find the minimum.
Improve Your Regression with CART and RandomForestsSalford Systems
Why You Should Watch: Learn the fundamentals of tree-based machine learning algorithms and how to easily fine tune and improve your Random Forest regression models.
Abstract: In this webinar we'll introduce you to two tree-based machine learning algorithms, CART® decision trees and RandomForests®. We will discuss the advantages of tree based techniques including their ability to automatically handle variable selection, variable interactions, nonlinear relationships, outliers, and missing values. We'll explore the CART algorithm, bootstrap sampling, and the Random Forest algorithm (all with animations) and compare their predictive performance using a real world dataset.
Improved Predictions in Structure Based Drug Design Using Cart and Bayesian M...Salford Systems
The document discusses using in silico methods like virtual screening and predictive modeling to improve drug discovery. It presents results from applying techniques like receptor docking, machine learning algorithms, and Bayesian modeling to develop improved scoring functions that better distinguish active from inactive compounds. These scoring functions helped identify key molecular properties that correlated with active hits. The methods showed improved ability to find active hits compared to previous scoring functions.
Churn Modeling-For-Mobile-Telecommunications Salford Systems
This document summarizes a study on predicting customer churn for a major mobile provider. TreeNet models were used to predict the probability of customers churning (switching providers) within a 30-60 day period. TreeNet models significantly outperformed other methods, increasing accuracy and the proportion of high-risk customers identified. Applying the most accurate TreeNet models could translate to millions in additional annual revenue by helping the provider preemptively retain more customers.
This document provides dos and don'ts for data mining based on experiences from various practitioners. It lists important steps like clearly defining objectives, simplifying solutions, preparing data, using multiple techniques, and checking models. It warns against underestimating preparation, overfitting models, and collecting excessive unhelpful data. Practitioners emphasize the importance of domain knowledge, transparency, and creating models that are understandable to stakeholders.
9 Data Mining Challenges From Data Scientists Like YouSalford Systems
The document outlines 9 challenges faced by data scientists: 1) poor quality data issues like dirty, missing, or inadequate data, 2) lack of understanding of data mining techniques, 3) lack of good literature on important topics and techniques, 4) difficulty for academic institutions accessing commercial-grade software at reasonable costs, 5) accommodating data from different sources and formats, 6) updating models constantly with new incoming data for online machine learning, 7) dealing with huge datasets requiring distributed approaches, 8) determining the right questions to ask of the data, and 9) remaining objective and letting the data lead rather than preconceptions.
This document contains a collection of quotes related to statistics and data. Some key quotes emphasize that while data and information are important, they must be used carefully and combined with human intelligence, judgement, and insight. Other quotes note that statistics can be flexible and misleading if not interpreted carefully, and that collecting quality data over long periods of time is important for analysis. The overall message is that statistics are a useful tool but have limitations, and human discernment is still needed.
Using CART For Beginners with A Teclo Example DatasetSalford Systems
Familiarize yourself with CART Decision Tree technology in this beginner's tutorial using a telecommunications example dataset from the 1990s. By the end of this tutorial you should feel comfortable using CART on your own with sample or real-world data.
The document provides an overview of a 4-part webinar covering the evolution of regression techniques from classical least squares to more advanced machine learning methods like random forests and gradient boosting. It outlines the topics to be covered in each part, including classical regression, regularized regression techniques like ridge regression, LASSO, and MARS, and ensemble methods like random forests and TreeNet gradient boosted trees. Examples using the Boston housing data set are provided to illustrate some of these techniques.
This document discusses how educational institutions can use data mining software to better understand and support their students. It outlines several areas where data analysis can provide insights, such as predicting student performance based on more than just grades, understanding factors that lead to success or failure and graduation, determining the effectiveness of support programs, identifying which recruitment strategies and financial packages attract students, and predicting those most at risk of dropping out or defaulting on loans. The overall goal is to enhance student outcomes and institutional management through analytics.
Comparison of statistical methods commonly used in predictive modelingSalford Systems
This document compares four statistical methods commonly used in predictive modelling: Logistic Multiple Regression (LMR), Principal Component Regression (PCR), Classification and Regression Tree analysis (CART), and Multivariate Adaptive Regression Splines (MARS). It applies these methods to two ecological data sets to test their accuracy, reliability, ease of use, and implementation in a geographic information system (GIS). The results show that independent data is needed to validate models, and that MARS and CART achieved the best prediction success, although CART models became too complex for cartographic purposes with a large number of data points.
This document discusses Dr. Wayne Danter's research using artificial intelligence tools to predict biological activity of molecular structures. His method involves using CART to analyze public HIV data and build predictive models. CART generates decision trees to identify important variables that predict if a molecule is biologically active against HIV. Dr. Danter then uses MARS and NeuroShell Classifier to further improve prediction accuracy. His proprietary CHEMSASTM algorithm teaches neural networks to relate molecular structure to function for screening potential HIV drugs. Using these methods, Dr. Danter has achieved over 96% accuracy in classifying 311 drugs' activity against HIV.
TreeNet Tree Ensembles & CART Decision Trees: A Winning CombinationSalford Systems
Understand CART decision tree pros/cons, how TreeNet stochastic gradient boosting ca n help overcome single-tree challenges, and what the advantages are when using CART and TreeNet in combination for predictive modeling success.
Salford Systems offers several products for data mining and predictive modeling. The table compares features of their Basic, Pro, ProEx, and Ultra components. The Basic component includes basic modeling, reporting, and automation features. Pro adds additional modeling engines and missing data handling capabilities. ProEx further expands the supported modeling techniques and automations. Ultra provides the most extensive set of features, including additional modeling pipelines, ensemble methods, and tree-based algorithms.
This document provides an introduction to MARS (Multivariate Adaptive Regression Splines), an automated regression modeling tool. MARS can build accurate predictive models for continuous and binary dependent variables by automatically selecting variables, determining transformations and interactions between variables, and handling missing data. It efficiently searches through all possible models to identify an optimal solution. The document explains how MARS works, provides settings to configure MARS, and uses the Boston housing dataset to demonstrate the basic steps of building a MARS model.
The document discusses combining CART (Classification and Regression Tree) and logistic regression models to take advantage of their respective strengths in classification and data mining tasks. It describes how running a logistic regression on the entire dataset using CART terminal node assignments as dummy variables allows the logistic model to find effects across nodes that CART cannot detect. This improves CART's predictions by imposing slopes on cases within nodes and providing a more granular, continuous response than CART alone. The approach also allows compensating for some of CART's weaknesses like coarse-grained responses.
When building a predictive model in SPM, you'll want to know exactly what you did to get your results. This short slide deck will show you how to review your work in the session logs.
The document discusses techniques for compressing and extracting rules from TreeNet models. It describes how TreeNet has achieved high predictive performance but its models can be refined further. Regularized regression can be applied to the trees or nodes in a TreeNet model to combine similar trees, reweight trees, and select a compressed subset of trees without much loss in accuracy. This "model compression" technique aims to simplify TreeNet models for improved deployment while maintaining good predictive performance.
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?Speck&Tech
ABSTRACT: A prima vista, un mattoncino Lego e la backdoor XZ potrebbero avere in comune il fatto di essere entrambi blocchi di costruzione, o dipendenze di progetti creativi e software. La realtà è che un mattoncino Lego e il caso della backdoor XZ hanno molto di più di tutto ciò in comune.
Partecipate alla presentazione per immergervi in una storia di interoperabilità, standard e formati aperti, per poi discutere del ruolo importante che i contributori hanno in una comunità open source sostenibile.
BIO: Sostenitrice del software libero e dei formati standard e aperti. È stata un membro attivo dei progetti Fedora e openSUSE e ha co-fondato l'Associazione LibreItalia dove è stata coinvolta in diversi eventi, migrazioni e formazione relativi a LibreOffice. In precedenza ha lavorato a migrazioni e corsi di formazione su LibreOffice per diverse amministrazioni pubbliche e privati. Da gennaio 2020 lavora in SUSE come Software Release Engineer per Uyuni e SUSE Manager e quando non segue la sua passione per i computer e per Geeko coltiva la sua curiosità per l'astronomia (da cui deriva il suo nickname deneb_alpha).
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
Introduction of Cybersecurity with OSS at Code Europe 2024Hiroshi SHIBATA
I develop the Ruby programming language, RubyGems, and Bundler, which are package managers for Ruby. Today, I will introduce how to enhance the security of your application using open-source software (OSS) examples from Ruby and RubyGems.
The first topic is CVE (Common Vulnerabilities and Exposures). I have published CVEs many times. But what exactly is a CVE? I'll provide a basic understanding of CVEs and explain how to detect and handle vulnerabilities in OSS.
Next, let's discuss package managers. Package managers play a critical role in the OSS ecosystem. I'll explain how to manage library dependencies in your application.
I'll share insights into how the Ruby and RubyGems core team works to keep our ecosystem safe. By the end of this talk, you'll have a better understanding of how to safeguard your code.
Project Management Semester Long Project - Acuityjpupo2018
Acuity is an innovative learning app designed to transform the way you engage with knowledge. Powered by AI technology, Acuity takes complex topics and distills them into concise, interactive summaries that are easy to read & understand. Whether you're exploring the depths of quantum mechanics or seeking insight into historical events, Acuity provides the key information you need without the burden of lengthy texts.
OpenID AuthZEN Interop Read Out - AuthorizationDavid Brossard
During Identiverse 2024 and EIC 2024, members of the OpenID AuthZEN WG got together and demoed their authorization endpoints conforming to the AuthZEN API
Taking AI to the Next Level in Manufacturing.pdfssuserfac0301
Read Taking AI to the Next Level in Manufacturing to gain insights on AI adoption in the manufacturing industry, such as:
1. How quickly AI is being implemented in manufacturing.
2. Which barriers stand in the way of AI adoption.
3. How data quality and governance form the backbone of AI.
4. Organizational processes and structures that may inhibit effective AI adoption.
6. Ideas and approaches to help build your organization's AI strategy.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Building Production Ready Search Pipelines with Spark and MilvusZilliz
Spark is the widely used ETL tool for processing, indexing and ingesting data to serving stack for search. Milvus is the production-ready open-source vector database. In this talk we will show how to use Spark to process unstructured data to extract vector representations, and push the vectors to Milvus vector database for search serving.
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
National Security Agency - NSA mobile device best practices
Introduction to TreeNet (2004)
1. An Introduction to TreeNetTM
Salford Systems
http://www.salford-systems.com
golomi@salford-systems.com
Mikhail Golovnya, Dan Steinberg, Scott Cardell
2. New approaches to machine learning/function
◦ Approximation developed by Jerome H. Friedman at
Stanford University
Co-author of CART® with Breiman, Olshen and Stone
Author of MARSTM, PRIM, Projection Pursuit
Good for classification and regression problems
Builds on the notions of committees of experts
and boosting but is substantially different in
implementation details
3. Stagewise function approximation in which each stage models
residuals from the last step model
◦ Conventional boosting models original target each stage
Each stage uses a very small tree, as small as two nodes and
typically in the range of 4-8 nodes
◦ Conventional bagging and boosting use full size trees and even
massively large trees
Each stage learns from a fraction of the available training data.
Typically less than 50% to start and falling into 20% or less by the
last stage
Each stage learns only a little: Severely down weight contribution of
each new tree (learning rate is typically 0.10 or less)
Focus in classification is on points near decision boundary, ignore
points far away from boundary even if the points are on the wrong
4. Built on CART trees and thus
◦ Immune to outliers
◦ Handles missing values automatically
◦ Selects variables
◦ Results invariant wrt monotone transformations of variables
Trains very rapidly: many small trees do not take
much longer run times than one large tree
Resistant to over training- generalizes very well
Can be remarkably accurate with little effort
BUT resulting model may be very complex
5. An intuitive introduction
TreeNet Mathematical Basics
◦ Specifications of the TreeNet model as a series expansion
◦ Non-parametric approach to steepest descent optimization
TreeNet at work
◦ Small trees, learning rates, sub-sample fractions, regression types
◦ Reading the output: reports and diagnostics
Comparing to AdaBoost and other methods
6. Consider the basic problem of estimating continuous
outcome y based on a vector of predictors X
Running a step-wise multiple linear regression will produce
an estimate f1 (X) and associated residuals
A simple intuitive idea: run a second-stage regression model
to produce an estimate of the residuals f2 (X) and the
associated updated residuals r^2=(y-f1f2)
Repeating this process multiple times results to the following
series expansion: y=f1+f2+f3+…
7. The above idea can be easily implemented
Unfortunately, the direct implementation suffers from the
overfitting issues
The residuals from the previous model essentially communicate
information about where this model fails the most- hence, the
next stage model effectively tries to improve the previous model
where it failed
This is generally known as boosting
We may want to replace individual regressions with something
simpler- regression trees, for example
It is not yet known whether this simple idea actually works nor it
is clear how to generalize it for various types of loss functions or
classifications
8. For any given set of inputs X we want to predict some
outcome y
Thus we want to construct a “nice” function f(X) which in turn
can be used to express an estimate of y
We need to define how “nice” can be measured
9. In regression, when y is continuous, the easiest is to assume
that f(X) itself is the estimate of y
We may then define the loss function as the loss incurred
when y is estimated by f(X)
For example, least square loss (LS) is defined as (LOA¯0’:f)^2
Formally, a “nicely” defined f(X) will have the smallest
expected loss (over the entire population) within the
boundaries of its construction (for example, in multiple linear
regressions, f(X) belongs to the class of linear functions)
10. In reality, we have a set of N observed pairs (x,y) from the
population, not the entire population
Hence, the expected loss WU/A can be replaced with an
estimate
Here Fi=f(x)
The problem thus reduces to finding a function f(X) that
minimizes R
Unfortunately, classification will demand additional treatment
11. Consider binary classification and assume that y is coded as +1
or -1
The most detailed solution would then give us the associated
probabilities p(y)
Since probabilities are naturally constrained to the [0,1] interval,
we assume that the function f(X) is transformed
p(y)=1/(1+exp(-2fy))
Note that p(+1)+p(-1)=1
The “trick” here is finding an unconstrained estimate f instead of
constrained estimate p
Also note that f is simply half log-odds ratio of y=+1
12. (insert graph)
This graph shows the one-to-one correspondence between f
and p for y=+1
Note that the most significant probability change occurs when
f is between -3 and +3
13. Again, the main question is what “nice” f means given that we observed
N pairs (x,y) from the population
Approaching this problem from the maximum likelihood point of view,
one may show that the negative log-likelihood in this case becomes
(insert equation)
The problem once again reduces to finding f that minimizes R above
We could obtain the same result formally by introducing a special loss
function for classification (insert equation)
The above likelihood considerations show a “natural” way to arrive to
such a peculiar loss function
14. Other approaches to defining the loss functions for binary
classification are possible
For example, by throwing away the log term in the previous
equation one would arrive to the following loss L=exp(2yf)
It is possible to show that this loss function is effectively used
in the “classical” AdaBoost algorithm
AdaBoost could be considered as a predecessor of gradient
boosting, we will defer the comparison until later
15. To summarize we are looking for a function
f(X) that minimizes the estimate of loss
The typical loss functions are
(insert equations)
16. The function f(X) is introduced as a known function of a
fixed set of unknown parameters
The problem then reduces to finding a set of optimal
parameter estimates using non-linear optimization
techniques
Multiple linear regression and logistic regression: 1(X) is a
linear combination of fixed predictors; parameters being
the intercept term and the slope coefficients
Major problem: the function and predictors need to be
specified beforehand- this usually results to a lengthy
trial-and-error process
17. Construct f(X) using stage-wise approach
Start with a constant, then at each stage adjust the values off
(X) in various regions of data
It is important to keep the adjustment rate low- the resulting
model will become smoother and usually less subject to
overfitting
Note that we are effectively treating the values f=f(X) at all
individual observed data points as separate parameters
18. More specifically, assume that we have gone through k-1
stages and obtained the current version fK-1 (X)
We want to construct an updated version fk(x) resulting to
a smaller value of R
Treating individual (insert equation) as parameters, we
proceed by computing the anti-gradient (insert equation)
The individual components mark the “directions” in which
individual fK-1 must be changed to obtain a smaller R
To induce smoothness lets limit our “freedom” by allowing
only M (a smaller number, say between 2 and 10) distinct
constant adjustments at any given stage
19. The optimal strategy is then to group individual
components gk, into M mutually exclusive groups, such
that the variance within each group is minimized
But this is equivalent to growing a fixed-size (M terminal
nodes) regression tree using gk, as the target
Suppose we found M subsets (insert equation) of cases
(insert equation)
The constant adjustments a kj are computed to minimize
(insert equation)
Finally the updated f(X) is (insert equation)
20. For the given loss function L[y,IV],M, and MaxTrees
◦ Make an initial guess f(X)=f
◦ For K=0 to MaxTrees-1
◦ Compute the anti-gradient Gk by taking the derivative of the loss with
respect to f(X) and substitute y and current fk (X)
◦ Fit an M-node regression tree to the components of the negative gradient
1this will partition observations into M mutually exclusive groups
◦ Find the within node updates a5 by performing M univariate optimizations
of the node contributions to the estimated loss
◦ Do the update (insert equation)
◦ End for
21. For L[y,IV]=(y-f)^2, M, and MaxTrees
Initial guess f(X)=f= mean(y)
For K=0 to MaxTrees-1
The anti-gradient component (insert equation) which is the traditional
definition of the current residual
Fit an M-node regression tree to the current residuals 1* this will partition
observations into M mutually exclusive groups
The within-node updates a k, simply become node averages of the current
residuals
Do the update: (insert equation)
End for
22. For L[Y,fiX]=1 y-fl,M, and MaxTrees
Initial guess f(X)=f=median(y)
For k=0 to MaxTrees-1
The anti-gradient component (insert equation) which is the sign of the
current residuals
Fit an M-node regression tree to the sign of the current residuals 1* this
will partition observations into M mutually exclusive groups
The within-node updates a ki now become node medians of the current
residuals
Do the update (insert equation)
End for
23. For L[y,f(X)]=log[1-exp(-2yf)], M, and MaxTrees
Initial guess f(X)=f= half log- odds of y=+1
For k=0 to MaxTrees-1
Recall that (insert equation) we call these generalize residuals
Fit an M-node regression tree to the generalized residuals 1* this will
partition observations into M mutually exclusive groups
The within-node updates ak, are somewhat complicated (insert
equation) where all measures are taken with respect to the node and
variance (insert equation)
Do the update (insert equation)
End for
24. Consider the following simple data set with
single predictor X and 1000 observations
Here and in the following slides negative
response observations are marked in blue
whereas positive response observations are
marked in red
The general tendency is to have positive
response in the middle of the range of X
(insert table)
25. The dataset was generated using the
following model described by f(X) and the
corresponding p(X) for y=+1
(insert graphs)
26. (insert graph)
TreeNet fits constant probability 0.55
The residuals are positive for y=+1 and
negative for y=-1
27. (insert graph)
The dataset was partitioned into 3 regions:
low X (negative adjustment), middle X
(positive), and large X (negative)
The residuals “reflect” the directions of the
adjustments
28. (insert graph)
This graph shows predictors f(X) after 1000
iterations and a very small learning rate of
0.002
Note how the true shape was nearly perfectly
recovered
29. The purpose of running a regression tree is to group observations into
homogenous subsets
Once we have the right partition the adjustments for each terminal node
are computed separately to optimize the given loss function- these are
generally different from the predictions generated by the regression tree
itself (they are the same only for the LS Loss)
Thus, the procedure is no longer as simple as the initial intuitive
recursive regression approach we started with
Nonetheless, the tree is used to define the actual form of (X) over the
range of X and not only for the individual data points observed
This becomes important in the final model deployment and scoring
30. Up to this point we guarded against overfitting only by allowing a small
number of adjustments at each stage
We may further enhance this subject by forcing the adjustments to be
smaller
This is done by introducing a new parameter called “shrinkage” (learning
rate) that is set to a constant value between 0 and 1
Small learning rates result to smoother models: a rate of 0.1 means that
TreeNet will take 10 times more iterations to extract the same signal-
more variables will be tried, finer partitions will result, smaller boundary
jumps will take place
Ideally, one might ultimately want to keep the learning rate close to zero
and the number of stages (trees) close to infinity
However, rates below 0.001 usually become impractical
31. (insert graph)
This graph shows predictor f(X) after 100
iterations and a learning rate of 1
Note the roughness of the shape and the
presence of abrupt strong jumps
32. (insert graph)
This graph shows predicted f(X) after 1000
iterations and a very small learning rate of
0.0002
Note how the true shape was nearly perfectly
recovered
It may be further approved
33. At each stage, instead of working with the entire learn dataset,
consider taking a random sample of a fixed size
Typical sampling rates are set to 50% of the learn data (the
default) and even smaller for very large datasets
In the long run, the entire learn dataset is exploited but the
running time is reduced by the factor of two with the 50%
sampling rate
Sampling forces TreeNet to “rethink” optimal partition points
from run to run due to random fluctuations of the residuals
This, combined with the shrinkage and a large number of
iterations, results to the overall improvement of the captured
signal shape
34. (insert graph)
This graph shows predicted f(X) after 1000
stages, learning rate of 0.002, and 50%
sampling
Note the minor fluctuations in the average
loss
The resulting model is nice and smooth but
there is still room for improvement
35. (insert graph)
All previous allowed as few as 10 cases for
individual region/node (the default)
Here we have increased this limit up to 50
This immediately resulted to an even smoother
shape
In practice, various node size limits should be
tried
36. In classification problems, it is possible to further
reduce the amount of data processed as each stage
We ignore data points “too far” from the decision
boundary to be usefully considered
◦ Well correctly classified points are ignored (just like
conventional boosting)
◦ Badly misclassified data points are also ignored (very
different from conventional boosting)
◦ The focus is on the cases most difficult to classify correctly:
those near the decision boundary
37. (insert graph)
2-dimensional predictor space
Red dots represent cases with +1 target
Green dots represent cases with -1 target
Black curve represents the decision boundary
38. The remaining slides present TreeNet runs on real data as
well as give examples of GUI controls
We start with the Boston Housing dataset to illustrate
regression
Then we proceed with the Cell Phone dataset to illustrate
classification