This document discusses radial basis function networks and forward selection heuristics for neural networks. It begins by outlining topics to be covered, including predicting the variance of weights and outputs, selecting the regularization parameter, and forward selection algorithms. It then derives an expression for the variance of the weight vector w when noise is assumed to be normally distributed. Next, it discusses how to calculate the variance matrix and selects the regularization parameter λ. Finally, it introduces how to determine the number of dimensions and provides an overview of forward selection algorithms.
Artificial Intelligence 06.2 More on Causality Bayesian NetworksAndres Mendez-Vazquez
Here, I talk more about the causality idea in order to define D-Separation and the initial algorithm for finding it. I put some extra example of the algorithm...
Hopefully, It is good enough for you...
Combining Models - In this slides, we look at a way to combine the answers form various weak classifiers to build a robust classifier. At the slides we look at the following subjects:
1.- Model Combination Vs Bayesian Model
2.- Bootstrap Data Sets
And the cherry on the top the AdaBoost
Here a Review of the Combination of Machine Learning models from Bayesian Averaging, Committees to Boosting... Specifically An statistical analysis of Boosting is done
Artificial Intelligence 06.2 More on Causality Bayesian NetworksAndres Mendez-Vazquez
Here, I talk more about the causality idea in order to define D-Separation and the initial algorithm for finding it. I put some extra example of the algorithm...
Hopefully, It is good enough for you...
Combining Models - In this slides, we look at a way to combine the answers form various weak classifiers to build a robust classifier. At the slides we look at the following subjects:
1.- Model Combination Vs Bayesian Model
2.- Bootstrap Data Sets
And the cherry on the top the AdaBoost
Here a Review of the Combination of Machine Learning models from Bayesian Averaging, Committees to Boosting... Specifically An statistical analysis of Boosting is done
Universal Approximation Theorem
Here, we prove that the perceptron multi-layer can approximate all continuous functions in the hypercube [0,1]. For this, we used the Cybenko proof... I tried to include the basic in topology and mathematical analysis to make the slides more understandable. However, they still need some work to be done. In addition, I am a little bit rusty in my mathematical analysis, so I am still not so convinced with my linear functional I defined for the proof...!!! Back to the Rudin and Apostol!!! So expect changes in the future.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Introduction to machine learning terminology.
Applications within High Energy Physics and outside HEP.
* Basic problems: classification and regression.
* Nearest neighbours approach and spacial indices
* Overfitting (intro)
* Curse of dimensionality
* ROC curve, ROC AUC
* Bayes optimal classifier
* Density estimation: KDE and histograms
* Parametric density estimation
* Mixtures for density estimation and EM algorithm
* Generative approach vs discriminative approach
* Linear decision rule, intro to logistic regression
* Linear regression
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A comparison has been made between approximation of triangular fuzzy systems and the corresponding fuzzy triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
Tree models with Scikit-Learn: Great models with little assumptionsGilles Louppe
This talk gives an introduction to tree-based methods, both from a theoretical and practical point of view. It covers decision trees, random forests and boosting estimators, along with concrete examples based on Scikit-Learn about how they work, when they work and why they work.
Universal Approximation Theorem
Here, we prove that the perceptron multi-layer can approximate all continuous functions in the hypercube [0,1]. For this, we used the Cybenko proof... I tried to include the basic in topology and mathematical analysis to make the slides more understandable. However, they still need some work to be done. In addition, I am a little bit rusty in my mathematical analysis, so I am still not so convinced with my linear functional I defined for the proof...!!! Back to the Rudin and Apostol!!! So expect changes in the future.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
Introduction to machine learning terminology.
Applications within High Energy Physics and outside HEP.
* Basic problems: classification and regression.
* Nearest neighbours approach and spacial indices
* Overfitting (intro)
* Curse of dimensionality
* ROC curve, ROC AUC
* Bayes optimal classifier
* Density estimation: KDE and histograms
* Parametric density estimation
* Mixtures for density estimation and EM algorithm
* Generative approach vs discriminative approach
* Linear decision rule, intro to logistic regression
* Linear regression
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A comparison has been made between approximation of triangular fuzzy systems and the corresponding fuzzy triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
AGGREGATION OF OPINIONS FOR SYSTEM SELECTION USING APPROXIMATIONS OF FUZZY NU...mathsjournal
In this article we assume that experts express their view points by way of approximation of Triangular
fuzzy numbers. We take the help of fuzzy set theory concept to model the situation and present a method to
aggregate these approximations of triangular fuzzy numbers to obtain an overall approximation of
triangular fuzzy number for each system and then linear ordering done before the best system is chosen. A
comparison has been made betweenapproximation of triangular fuzzy systems and the corresponding fuzzy
triangular numbers systems. The notions like fuzziness and ambiguity for the approximation of triangular
fuzzy numbers are also found.
Tree models with Scikit-Learn: Great models with little assumptionsGilles Louppe
This talk gives an introduction to tree-based methods, both from a theoretical and practical point of view. It covers decision trees, random forests and boosting estimators, along with concrete examples based on Scikit-Learn about how they work, when they work and why they work.
Radial basis function network ppt bySheetal,Samreen and Dhanashrisheetal katkar
Radial Basis Functions are nonlinear activation functions used by artificial neural networks.Explained commonly used RBFs ,cover's theorem,interpolation problem and learning strategies.
In machine learning, support vector machines (SVMs, also support vector networks[1]) are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis. The basic SVM takes a set of input data and predicts, for each given input, which of two possible classes forms the output, making it a non-probabilistic binary linear classifier.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
Very often one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (\pdf) and/or by the corresponding probability characteristic functions (\pcf), or by a function representation. Here the interest is mainly to compute characterisations like the entropy, or
relations between two distributions, like their Kullback-Leibler divergence, or more general measures such as $f$-divergences,
among others. These are all computed from the \pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. It is an even stronger numerical challenge to then actually compute said characterisations in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
Contemporary communication systems 1st edition mesiya solutions manualto2001
Contemporary Communication Systems 1st Edition Mesiya Solutions Manual
Download:https://goo.gl/DmVRQ4
contemporary communication systems mesiya pdf download
contemporary communication systems mesiya download
contemporary communication systems pdf
contemporary communication systems mesiya solutions
HJB Equation and Merton's Portfolio ProblemAshwin Rao
Deriving the solution to Merton's Portfolio Problem (Optimal Asset Allocation and Consumption) using the elegant formulation of Hamilton-Jacobi-Bellman equation.
Lecture 2 from https://irdta.eu/deeplearn/2022su/
Covers concepts from Part 2 of my new book, https://meyn.ece.ufl.edu/2021/08/01/control-systems-and-reinforcement-learning/
Focus on algorithm design in general
Slides FIS5.pdfOutline1 Fixed Income DerivativesThe .docxbudabrooks46239
Slides FIS5.pdf
Outline
1 Fixed Income Derivatives
The Forward-Risk Adjusted Measure
2 Example
Dr Lara Cathcart () 2015 2 / 28
The problem
Consider a fixed-income derivative with a single payo↵ at time T which depends
on the term-structure. In particular, we will look at options on zero-coupon
bonds. For a call option on a zero-coupon bond maturing at time T
1
, the time T
payo↵ and hence the value of the derivative is given by
V
T
= max(P(T, T
1
) � K, 0) (1)
Dr Lara Cathcart () 2015 3 / 28
The problem
By the no-arbitrage theorem, the price today (t=0)is
V
0
= EQ
0
[e�
R
T
0
rsds
V
T
] (2)
where the expectation is taken under the risk-neutral distribution (also called the
Q measure). Thus the price depends on the stochastic process for the short rate
and the contractual specification of the security (i.e how the payo↵ is linked to the
term structure).
Dr Lara Cathcart () 2015 4 / 28
The problem
The price V
0
in equation (2) is given by the expectation of the product of two
dependent random variables, and calculating this expectation is often quite
di�cult. The purpose of this note is presenting a change-of measure technique
which considerably simplifies the evaluation of V
0
.
Dr Lara Cathcart () 2015 5 / 28
The problem
Specifically we are going to calculate V
0
as
V
0
= P(0, T)EQ
T
0
(V
T
) (3)
where QT is a new probability measure (distribution), the so-called forward-risk
adjusted measure. This technique was introduced in the fixed-income literature by
Jamishidian (1991).
Dr Lara Cathcart () 2015 6 / 28
Model setup and notation
Our term-structure is a general one-factor HJM model see Heath, Jarrow and
Morton (1992). Under the Q-measure, forwards rates are governed by
df (t, T) = ��(t, T)�
P
(t, T)dt + �(t, T)dW Q
t
(4)
where
�
P
(t, T) = �
Z
T
t
�(t, u)du (5)
Dr Lara Cathcart () 2015 7 / 28
The problem
Bond prices evolve according to the SDE
dP(t, T) = r
t
P(t, T)dt + �
P
(t, T)P(t, T)dW Q
t
(6)
so �
P
(t, T) is the time t volatility of the zero maturing at time T.
Dr Lara Cathcart () 2015 8 / 28
The Forward-Risk Adjusted Measure
The price of derivative security follows the SDE
dV
t
= r
t
V
t
dt + �
V
(t)V
t
dW
Q
t
(7)
This means that, under the risk-neutral distribution, the expected rate of return
equals the short rate (just like any other security), and the return volatility is
�
V
(t). So far neither V
t
nor �
V
(t) are known, but this is not essential for the
following arguments. In fact, the only thing that matters is that the process has
the form (7) since this facilitates pricing by the forward-risk adjusted measure.
Dr Lara Cathcart () 2015 9 / 28
The Forward-Risk Adjusted Measure
We begin by defining the deflated price process
F
t
⌘ V
t
/P(t, T) (8)
for t 2 [0, T]. We can interpret F
t
as the price of V
t
in units of the T-maturity
bond price (i.e., as a relative price).
Dr Lara Cathcart () 2015 10 / 28
The Forward-Ri.
Here the interest is mainly to compute characterisations like the entropy,
the Kullback-Leibler divergence, more general $f$-divergences, or other such characteristics based on
the probability density. The density is often not available directly,
and it is a computational challenge to just represent it in a numerically
feasible fashion in case the dimension is even moderately large. It
is an even stronger numerical challenge to then actually compute said characteristics
in the high-dimensional case.
The task considered here was the numerical computation of characterising statistics of
high-dimensional pdfs, as well as their divergences and distances,
where the pdf in the numerical implementation was assumed discretised on some regular grid.
We have demonstrated that high-dimensional pdfs,
pcfs, and some functions of them
can be approximated and represented in a low-rank tensor data format.
Utilisation of low-rank tensor techniques helps to reduce the computational complexity
and the storage cost from exponential $\C{O}(n^d)$ to linear in the dimension $d$, e.g.\
$O(d n r^2 )$ for the TT format. Here $n$ is the number of discretisation
points in one direction, $r<<n$ is the maximal tensor rank, and $d$ the problem dimension.
This slide set is a work in progress and is embedded in my Principles of Finance course site (under construction) that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
The time scale Fibonacci sequences satisfy the Friedmann-Lema\^itre-Robertson-Walker (FLRW) dynamic equation on time scale, which are an exact solution of Einstein's field equations of general relativity for an expanding homogeneous and isotropic universe. We show that the equations of motion correspond to the one-dimensional motion of a particle of position $F(t)$ in an inverted harmonic potential. For the dynamic equations on time scale describing the Fibonacci numbers $F(t)$, we present the Lagrangian and Hamiltonian formalism. Identifying these with the equations that describe factor scales, we conclude that for a certain granulation, for both the continuous and the discrete universe, we have the same dynamics.
Senior data scientist and founder of the company Intelligentia Data I+D SA de CV. We are offering consultancy services, development of projects and products in Machine Learning, Big Data, Data Sciences and Artificial Intelligence.
My first set of slides (The NN and DL class I am preparing for the fall)... I included the problem of Vanishing Gradient and the need to have ReLu (Mentioning btw the saturation problem inherited from Hebbian Learning)
It has been almost 62 years since the invention of the term Artificial Intelligence by Samuel and Minsky et al. at the Dartmouth workshop College in 1956 (“Dartmouth Summer Research Project on Artificial Intelligence”) where this new area of Computer Science was invented. However, the history of Artificial Intelligence goes back to previous millennia, when the Greeks in their Myths spoke about golden robots at Hephaestus, and the Galatea of Pygmalion. They were the first automatons known at the dawn of history, and although these first attempts were only myths, automatons were invented and built through multiple civilizations in history. Nevertheless, these automatons resembled in quite limited way their final objectives, representing animals and humans. In spite of that, the greatest illusion of an automaton, the Turk by Wolfgang von Kempelen, inspired many people, trough its exhibitions, as Alexander Graham Bell and Charles Babbage to develop inventions that would change forever human history. Thus, the importance of the concept “Artificial Intelligence” as a driver of our technological dreams. And although Artificial Intelligence has never been defined in a precise practical way, the amount of research and methods that have been developed to tackle some of its basics tasks have been and are quite humongous. Thus, the importance of having an introduction to the concepts of Artificial Intelligence, thus the dream can continue.
A review of one of the most popular methods of clustering, a part of what is know as unsupervised learning, K-Means. Here, we go from the basic heuristic used to solve the NP-Hard problem to an approximation algorithm K-Centers. Additionally, we look at variations coming from the Fuzzy Set ideas. In the future, we will add more about On-Line algorithms in the line of Stochastic Gradient Ideas...
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Final project report on grocery store management system..pdf
18 Machine Learning Radial Basis Function Networks Forward Heuristics
1. Neural Networks
Radial Basis Function Networks - Forward Heuristics
Andres Mendez-Vazquez
December 10, 2015
1 / 58
2. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
2 / 58
3. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
3 / 58
4. What is the variance of the weight vector w?
The meaning
If the weight have been calculated upon the basis an estimation of a
stochastic variable d:
What is the corresponding uncertainty in the estimation of w?
Assume that the noise affecting d is normal and independently,
identically distributed
ED d − d
T
d − d = σ2
I (1)
Where σ is the Standard Deviation of the noise and d the mean value
of d
Thus, we have that the noise N d, σ2 .
4 / 58
5. What is the variance of the weight vector w?
The meaning
If the weight have been calculated upon the basis an estimation of a
stochastic variable d:
What is the corresponding uncertainty in the estimation of w?
Assume that the noise affecting d is normal and independently,
identically distributed
ED d − d
T
d − d = σ2
I (1)
Where σ is the Standard Deviation of the noise and d the mean value
of d
Thus, we have that the noise N d, σ2 .
4 / 58
6. What is the variance of the weight vector w?
The meaning
If the weight have been calculated upon the basis an estimation of a
stochastic variable d:
What is the corresponding uncertainty in the estimation of w?
Assume that the noise affecting d is normal and independently,
identically distributed
ED d − d
T
d − d = σ2
I (1)
Where σ is the Standard Deviation of the noise and d the mean value
of d
Thus, we have that the noise N d, σ2 .
4 / 58
7. What is the variance of the weight vector w?
The meaning
If the weight have been calculated upon the basis an estimation of a
stochastic variable d:
What is the corresponding uncertainty in the estimation of w?
Assume that the noise affecting d is normal and independently,
identically distributed
ED d − d
T
d − d = σ2
I (1)
Where σ is the Standard Deviation of the noise and d the mean value
of d
Thus, we have that the noise N d, σ2 .
4 / 58
8. Remember
We are using a linear model
f (xi) =
d1
j=1
wjφj (xi) (2)
Thus, solving the error under regularization
w = ΦT
Φ + Λ
−1
ΦT
d (3)
5 / 58
9. Remember
We are using a linear model
f (xi) =
d1
j=1
wjφj (xi) (2)
Thus, solving the error under regularization
w = ΦT
Φ + Λ
−1
ΦT
d (3)
5 / 58
10. Thus
Getting the Expected Value
w = ED (w) = ED ΦT
Φ + Λ
−1
ΦT
d
= ΦT
Φ + Λ
−1
ΦT
ED [d]
= ΦT
Φ + Λ
−1
ΦT
d
6 / 58
11. Thus
Getting the Expected Value
w = ED (w) = ED ΦT
Φ + Λ
−1
ΦT
d
= ΦT
Φ + Λ
−1
ΦT
ED [d]
= ΦT
Φ + Λ
−1
ΦT
d
6 / 58
12. Thus
Getting the Expected Value
w = ED (w) = ED ΦT
Φ + Λ
−1
ΦT
d
= ΦT
Φ + Λ
−1
ΦT
ED [d]
= ΦT
Φ + Λ
−1
ΦT
d
6 / 58
13. Thus, we have
The Variance of w is
W = ED (w − w) (w − w)T
= ED ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d × ...
ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d
T
= ED ΦT
Φ + Λ
−1
ΦT
d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
ED d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
σ2
IΦ ΦT
Φ + Λ
−1
= σ2
ΦT
Φ + Λ
−1
ΦT
Φ ΦT
Φ + Λ
−1
7 / 58
14. Thus, we have
The Variance of w is
W = ED (w − w) (w − w)T
= ED ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d × ...
ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d
T
= ED ΦT
Φ + Λ
−1
ΦT
d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
ED d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
σ2
IΦ ΦT
Φ + Λ
−1
= σ2
ΦT
Φ + Λ
−1
ΦT
Φ ΦT
Φ + Λ
−1
7 / 58
15. Thus, we have
The Variance of w is
W = ED (w − w) (w − w)T
= ED ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d × ...
ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d
T
= ED ΦT
Φ + Λ
−1
ΦT
d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
ED d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
σ2
IΦ ΦT
Φ + Λ
−1
= σ2
ΦT
Φ + Λ
−1
ΦT
Φ ΦT
Φ + Λ
−1
7 / 58
16. Thus, we have
The Variance of w is
W = ED (w − w) (w − w)T
= ED ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d × ...
ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d
T
= ED ΦT
Φ + Λ
−1
ΦT
d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
ED d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
σ2
IΦ ΦT
Φ + Λ
−1
= σ2
ΦT
Φ + Λ
−1
ΦT
Φ ΦT
Φ + Λ
−1
7 / 58
17. Thus, we have
The Variance of w is
W = ED (w − w) (w − w)T
= ED ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d × ...
ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d
T
= ED ΦT
Φ + Λ
−1
ΦT
d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
ED d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
σ2
IΦ ΦT
Φ + Λ
−1
= σ2
ΦT
Φ + Λ
−1
ΦT
Φ ΦT
Φ + Λ
−1
7 / 58
18. Thus, we have
The Variance of w is
W = ED (w − w) (w − w)T
= ED ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d × ...
ΦT
Φ + Λ
−1
ΦT
d − ΦT
Φ + Λ
−1
ΦT
d
T
= ED ΦT
Φ + Λ
−1
ΦT
d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
ED d − d d − d
T
Φ ΦT
Φ + Λ
−1
= ΦT
Φ + Λ
−1
ΦT
σ2
IΦ ΦT
Φ + Λ
−1
= σ2
ΦT
Φ + Λ
−1
ΦT
Φ ΦT
Φ + Λ
−1
7 / 58
19. The Least Squared Error Case
We have
Λ = 0 =⇒ W = σ2
ΦT
Φ + Λ
−1
(4)
The following matrix is known as the variance matrix
A−1
= ΦT
Φ + Λ
−1
(5)
For the standard Ridge Regression when ΦT
Φ = A − λId1
W = σ2
A−1
[A − λId1 ] A−1
= σ2
A−1
− λA−2
8 / 58
20. The Least Squared Error Case
We have
Λ = 0 =⇒ W = σ2
ΦT
Φ + Λ
−1
(4)
The following matrix is known as the variance matrix
A−1
= ΦT
Φ + Λ
−1
(5)
For the standard Ridge Regression when ΦT
Φ = A − λId1
W = σ2
A−1
[A − λId1 ] A−1
= σ2
A−1
− λA−2
8 / 58
21. The Least Squared Error Case
We have
Λ = 0 =⇒ W = σ2
ΦT
Φ + Λ
−1
(4)
The following matrix is known as the variance matrix
A−1
= ΦT
Φ + Λ
−1
(5)
For the standard Ridge Regression when ΦT
Φ = A − λId1
W = σ2
A−1
[A − λId1 ] A−1
= σ2
A−1
− λA−2
8 / 58
22. The Least Squared Error Case
We have
Λ = 0 =⇒ W = σ2
ΦT
Φ + Λ
−1
(4)
The following matrix is known as the variance matrix
A−1
= ΦT
Φ + Λ
−1
(5)
For the standard Ridge Regression when ΦT
Φ = A − λId1
W = σ2
A−1
[A − λId1 ] A−1
= σ2
A−1
− λA−2
8 / 58
23. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
9 / 58
24. How to select the regularization parameter λ?
We know that
f = Φ ΦT
Φ + λId1
−1
ΦT
d (6)
Thus, we have the following projection matrix assuming the difference
between d and f
d − f = d − Φ ΦT
Φ + λId1
−1
ΦT
d (7)
Thus, we have tha Projection Matrix P
d − f =
IN − Φ ΦT
Φ + λId1
−1
ΦT
P
d (8)
10 / 58
25. How to select the regularization parameter λ?
We know that
f = Φ ΦT
Φ + λId1
−1
ΦT
d (6)
Thus, we have the following projection matrix assuming the difference
between d and f
d − f = d − Φ ΦT
Φ + λId1
−1
ΦT
d (7)
Thus, we have tha Projection Matrix P
d − f =
IN − Φ ΦT
Φ + λId1
−1
ΦT
P
d (8)
10 / 58
26. How to select the regularization parameter λ?
We know that
f = Φ ΦT
Φ + λId1
−1
ΦT
d (6)
Thus, we have the following projection matrix assuming the difference
between d and f
d − f = d − Φ ΦT
Φ + λId1
−1
ΦT
d (7)
Thus, we have tha Projection Matrix P
d − f =
IN − Φ ΦT
Φ + λId1
−1
ΦT
P
d (8)
10 / 58
27. We can use this to rewrite the cost function
Cost function
C (w, λ) =
N
i=1
(di − f (xi))2
+
d1
j=1
λjw2
j (9)
We have then
C (w, λ) = Hw − d
T
Hw − d + wT
Λw
= d
T
Φ [A]−1
ΦT
− Id1 Φ [A]−1
ΦT
− Id1 d + ...
dT
Φ [A]−1
Λ [A]−1
ΦT
d
11 / 58
28. We can use this to rewrite the cost function
Cost function
C (w, λ) =
N
i=1
(di − f (xi))2
+
d1
j=1
λjw2
j (9)
We have then
C (w, λ) = Hw − d
T
Hw − d + wT
Λw
= d
T
Φ [A]−1
ΦT
− Id1 Φ [A]−1
ΦT
− Id1 d + ...
dT
Φ [A]−1
Λ [A]−1
ΦT
d
11 / 58
29. We can use this to rewrite the cost function
Cost function
C (w, λ) =
N
i=1
(di − f (xi))2
+
d1
j=1
λjw2
j (9)
We have then
C (w, λ) = Hw − d
T
Hw − d + wT
Λw
= d
T
Φ [A]−1
ΦT
− Id1 Φ [A]−1
ΦT
− Id1 d + ...
dT
Φ [A]−1
Λ [A]−1
ΦT
d
11 / 58
30. However
We have
Φ [A]−1
Λ [A]−1
Φ = Φ [A]−1
A − ΦT
Φ [A]−1
ΦT
= Φ [A]−1
ΦT
− Φ [A]−1
ΦT
2
= P − P2
Simplifying the minimum cost
C (w, λ) = d
T
P2
d+ d
T
P − P2
d = d
T
Pd (10)
12 / 58
31. However
We have
Φ [A]−1
Λ [A]−1
Φ = Φ [A]−1
A − ΦT
Φ [A]−1
ΦT
= Φ [A]−1
ΦT
− Φ [A]−1
ΦT
2
= P − P2
Simplifying the minimum cost
C (w, λ) = d
T
P2
d+ d
T
P − P2
d = d
T
Pd (10)
12 / 58
32. However
We have
Φ [A]−1
Λ [A]−1
Φ = Φ [A]−1
A − ΦT
Φ [A]−1
ΦT
= Φ [A]−1
ΦT
− Φ [A]−1
ΦT
2
= P − P2
Simplifying the minimum cost
C (w, λ) = d
T
P2
d+ d
T
P − P2
d = d
T
Pd (10)
12 / 58
33. However
We have
Φ [A]−1
Λ [A]−1
Φ = Φ [A]−1
A − ΦT
Φ [A]−1
ΦT
= Φ [A]−1
ΦT
− Φ [A]−1
ΦT
2
= P − P2
Simplifying the minimum cost
C (w, λ) = d
T
P2
d+ d
T
P − P2
d = d
T
Pd (10)
12 / 58
34. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
35. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
36. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
37. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
38. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
39. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
40. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
41. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
42. In summary, we have for the Ridge Regression
Something Notable
A =ΦT
Φ + λId1
w =A−1
ΦT
d
P =IN − ΦA−1
ΦT
Important Observation
Some sort of model selection must be used to choose a value for the
regularisation parameter .
The value chosen is the one associated with the lowest prediction
error.
Question
Which method should be used to predict the error and how is the optimal
value found?
13 / 58
43. Answer
Something Notable
The answer to the rst question is that nobody knows for sure.
There are many methods to train to obtain that value
Leave-one-out cross-validation.
Generalized cross-validation.
Final prediction error.
Bayesian information criterion.
Bootstrap methods.
14 / 58
44. Answer
Something Notable
The answer to the rst question is that nobody knows for sure.
There are many methods to train to obtain that value
Leave-one-out cross-validation.
Generalized cross-validation.
Final prediction error.
Bayesian information criterion.
Bootstrap methods.
14 / 58
45. Answer
Something Notable
The answer to the rst question is that nobody knows for sure.
There are many methods to train to obtain that value
Leave-one-out cross-validation.
Generalized cross-validation.
Final prediction error.
Bayesian information criterion.
Bootstrap methods.
14 / 58
46. Answer
Something Notable
The answer to the rst question is that nobody knows for sure.
There are many methods to train to obtain that value
Leave-one-out cross-validation.
Generalized cross-validation.
Final prediction error.
Bayesian information criterion.
Bootstrap methods.
14 / 58
47. Answer
Something Notable
The answer to the rst question is that nobody knows for sure.
There are many methods to train to obtain that value
Leave-one-out cross-validation.
Generalized cross-validation.
Final prediction error.
Bayesian information criterion.
Bootstrap methods.
14 / 58
48. Answer
Something Notable
The answer to the rst question is that nobody knows for sure.
There are many methods to train to obtain that value
Leave-one-out cross-validation.
Generalized cross-validation.
Final prediction error.
Bayesian information criterion.
Bootstrap methods.
14 / 58
49. We will use an iterative method
We have the following iterative process from Generalized
Cross-Validation
λ =
dT P2
dtrace A−1 − λA−2
wT
A−1wtrace (P)
(11)
To see the development, please take a look to Appendix A.10
“Introduction to Radial Basis Function Networks” by Mark J.L. Orr
An iterative process started with an initial λ
The value is updated until convergence.
15 / 58
50. We will use an iterative method
We have the following iterative process from Generalized
Cross-Validation
λ =
dT P2
dtrace A−1 − λA−2
wT
A−1wtrace (P)
(11)
To see the development, please take a look to Appendix A.10
“Introduction to Radial Basis Function Networks” by Mark J.L. Orr
An iterative process started with an initial λ
The value is updated until convergence.
15 / 58
51. We will use an iterative method
We have the following iterative process from Generalized
Cross-Validation
λ =
dT P2
dtrace A−1 − λA−2
wT
A−1wtrace (P)
(11)
To see the development, please take a look to Appendix A.10
“Introduction to Radial Basis Function Networks” by Mark J.L. Orr
An iterative process started with an initial λ
The value is updated until convergence.
15 / 58
52. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
16 / 58
53. How many dimensions for the mapping to high dimensions?
We have the following for ordinary least squares - no regularization
A−1
= ΦT
Φ
−1
(12)
Now, you suppose that
You are given a set of numbers {xi}N
i=1 randomly drawn from a Gaussian
distribution and you are asked to estimate the variance without told the
mean.
We can calculate the sample mean
x =
1
N
N
i=1
xi (13)
17 / 58
54. How many dimensions for the mapping to high dimensions?
We have the following for ordinary least squares - no regularization
A−1
= ΦT
Φ
−1
(12)
Now, you suppose that
You are given a set of numbers {xi}N
i=1 randomly drawn from a Gaussian
distribution and you are asked to estimate the variance without told the
mean.
We can calculate the sample mean
x =
1
N
N
i=1
xi (13)
17 / 58
55. How many dimensions for the mapping to high dimensions?
We have the following for ordinary least squares - no regularization
A−1
= ΦT
Φ
−1
(12)
Now, you suppose that
You are given a set of numbers {xi}N
i=1 randomly drawn from a Gaussian
distribution and you are asked to estimate the variance without told the
mean.
We can calculate the sample mean
x =
1
N
N
i=1
xi (13)
17 / 58
56. Thus
This allows to calculate the sample variance
ˆσ2
=
1
N − 1
N
i=1
(xi − x)2
(14)
Problem, from Where the parameter N − 1 comes from?
It comes from the fact that the parameter x is fitting the noise.
The system has N degrees of freedom
Thus the underestimation of the variance is restored by reducing the
remaining degrees of freedom by one.
18 / 58
57. Thus
This allows to calculate the sample variance
ˆσ2
=
1
N − 1
N
i=1
(xi − x)2
(14)
Problem, from Where the parameter N − 1 comes from?
It comes from the fact that the parameter x is fitting the noise.
The system has N degrees of freedom
Thus the underestimation of the variance is restored by reducing the
remaining degrees of freedom by one.
18 / 58
58. Thus
This allows to calculate the sample variance
ˆσ2
=
1
N − 1
N
i=1
(xi − x)2
(14)
Problem, from Where the parameter N − 1 comes from?
It comes from the fact that the parameter x is fitting the noise.
The system has N degrees of freedom
Thus the underestimation of the variance is restored by reducing the
remaining degrees of freedom by one.
18 / 58
59. In Supervised Learning
Similarly
It would be a mistake to divide the sum-squared-training-error by the
number of patterns in order to estimate the noise variance since some
degrees of freedom will have been used up in fitting the model.
In our linear model there are d1 weights and N patterns in the
training set
It leaves N − d1 degrees of freedom.
The estimation of the variance is then
ˆσ2
=
ˆS
N − d1
(15)
Remark: ˆS is the sum-squared-error over the training set at the
optimal weight vector and ˆσ2 is called the unbiased estimate
of variance.
19 / 58
60. In Supervised Learning
Similarly
It would be a mistake to divide the sum-squared-training-error by the
number of patterns in order to estimate the noise variance since some
degrees of freedom will have been used up in fitting the model.
In our linear model there are d1 weights and N patterns in the
training set
It leaves N − d1 degrees of freedom.
The estimation of the variance is then
ˆσ2
=
ˆS
N − d1
(15)
Remark: ˆS is the sum-squared-error over the training set at the
optimal weight vector and ˆσ2 is called the unbiased estimate
of variance.
19 / 58
61. In Supervised Learning
Similarly
It would be a mistake to divide the sum-squared-training-error by the
number of patterns in order to estimate the noise variance since some
degrees of freedom will have been used up in fitting the model.
In our linear model there are d1 weights and N patterns in the
training set
It leaves N − d1 degrees of freedom.
The estimation of the variance is then
ˆσ2
=
ˆS
N − d1
(15)
Remark: ˆS is the sum-squared-error over the training set at the
optimal weight vector and ˆσ2 is called the unbiased estimate
of variance.
19 / 58
62. First, standard least squared error
Although there is still d1 weights in the model
The effective number of parameters(John Moody), γ, is less than d1
and it depends on the size of the regularization parameters.
We have the following (Moody and MacKay)
γ = N − trace (P) (16)
20 / 58
63. First, standard least squared error
Although there is still d1 weights in the model
The effective number of parameters(John Moody), γ, is less than d1
and it depends on the size of the regularization parameters.
We have the following (Moody and MacKay)
γ = N − trace (P) (16)
20 / 58
64. First, standard least squared error
In the standard least squared error without regularization,
A−1
= ΦT
Φ
−1
γ = N − trace IN − ΦA−1
ΦT
= trace ΦA−1
ΦT
= trace A−1
ΦT
Φ
= trace A−1
ΦT
Φ
= trace (Id1 )
= d1
21 / 58
65. First, standard least squared error
In the standard least squared error without regularization,
A−1
= ΦT
Φ
−1
γ = N − trace IN − ΦA−1
ΦT
= trace ΦA−1
ΦT
= trace A−1
ΦT
Φ
= trace A−1
ΦT
Φ
= trace (Id1 )
= d1
21 / 58
66. First, standard least squared error
In the standard least squared error without regularization,
A−1
= ΦT
Φ
−1
γ = N − trace IN − ΦA−1
ΦT
= trace ΦA−1
ΦT
= trace A−1
ΦT
Φ
= trace A−1
ΦT
Φ
= trace (Id1 )
= d1
21 / 58
67. First, standard least squared error
In the standard least squared error without regularization,
A−1
= ΦT
Φ
−1
γ = N − trace IN − ΦA−1
ΦT
= trace ΦA−1
ΦT
= trace A−1
ΦT
Φ
= trace A−1
ΦT
Φ
= trace (Id1 )
= d1
21 / 58
68. First, standard least squared error
In the standard least squared error without regularization,
A−1
= ΦT
Φ
−1
γ = N − trace IN − ΦA−1
ΦT
= trace ΦA−1
ΦT
= trace A−1
ΦT
Φ
= trace A−1
ΦT
Φ
= trace (Id1 )
= d1
21 / 58
69. First, standard least squared error
In the standard least squared error without regularization,
A−1
= ΦT
Φ
−1
γ = N − trace IN − ΦA−1
ΦT
= trace ΦA−1
ΦT
= trace A−1
ΦT
Φ
= trace A−1
ΦT
Φ
= trace (Id1 )
= d1
21 / 58
70. Now, with the regularization term
We have A−1
= ΦT
Φ − λId1
−1
γ = trace A−1
ΦT
Φ
= trace A−1
(A − λId1 )
= trace Id1 − λA−1
= d1 − λ A−1
22 / 58
71. Now, with the regularization term
We have A−1
= ΦT
Φ − λId1
−1
γ = trace A−1
ΦT
Φ
= trace A−1
(A − λId1 )
= trace Id1 − λA−1
= d1 − λ A−1
22 / 58
72. Now, with the regularization term
We have A−1
= ΦT
Φ − λId1
−1
γ = trace A−1
ΦT
Φ
= trace A−1
(A − λId1 )
= trace Id1 − λA−1
= d1 − λ A−1
22 / 58
73. Now, with the regularization term
We have A−1
= ΦT
Φ − λId1
−1
γ = trace A−1
ΦT
Φ
= trace A−1
(A − λId1 )
= trace Id1 − λA−1
= d1 − λ A−1
22 / 58
74. Now
If the eigenvalues of the matrix ΦT
Φ are {µj}d1
j=1
γ = d1 − λtrace A−1
= d1 − λ
d1
i=1
1
λ + µj
=
d1
i=1
µj
λ + µj
23 / 58
75. Now
If the eigenvalues of the matrix ΦT
Φ are {µj}d1
j=1
γ = d1 − λtrace A−1
= d1 − λ
d1
i=1
1
λ + µj
=
d1
i=1
µj
λ + µj
23 / 58
76. Now
If the eigenvalues of the matrix ΦT
Φ are {µj}d1
j=1
γ = d1 − λtrace A−1
= d1 − λ
d1
i=1
1
λ + µj
=
d1
i=1
µj
λ + µj
23 / 58
77. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
24 / 58
78. About Ridge Regression
Remark
Ridge regression is used as a way to balance bias and variance by varying
the effective number of parameters in a linear model.
An alternative strategy
It is to to compare models made up of different subsets of basis functions
drawn from the same fixed set of candidates.
This is called
Subset selection in statistics and machine learning.
25 / 58
79. About Ridge Regression
Remark
Ridge regression is used as a way to balance bias and variance by varying
the effective number of parameters in a linear model.
An alternative strategy
It is to to compare models made up of different subsets of basis functions
drawn from the same fixed set of candidates.
This is called
Subset selection in statistics and machine learning.
25 / 58
80. About Ridge Regression
Remark
Ridge regression is used as a way to balance bias and variance by varying
the effective number of parameters in a linear model.
An alternative strategy
It is to to compare models made up of different subsets of basis functions
drawn from the same fixed set of candidates.
This is called
Subset selection in statistics and machine learning.
25 / 58
81. Problem
This is normally intractable when you have N
2N
− 1 subsets to test (17)
We could use different methods
1 K-means which is explained in the book.
2 Forward Selection heuristics that we will explain here.
Forward Selection
It starts with an empty subset to which a basis function is added at a time:
The one that reduces the sum-squared error the most.
Until a chosen criterion, such that GCV stops decreasing.
26 / 58
82. Problem
This is normally intractable when you have N
2N
− 1 subsets to test (17)
We could use different methods
1 K-means which is explained in the book.
2 Forward Selection heuristics that we will explain here.
Forward Selection
It starts with an empty subset to which a basis function is added at a time:
The one that reduces the sum-squared error the most.
Until a chosen criterion, such that GCV stops decreasing.
26 / 58
83. Problem
This is normally intractable when you have N
2N
− 1 subsets to test (17)
We could use different methods
1 K-means which is explained in the book.
2 Forward Selection heuristics that we will explain here.
Forward Selection
It starts with an empty subset to which a basis function is added at a time:
The one that reduces the sum-squared error the most.
Until a chosen criterion, such that GCV stops decreasing.
26 / 58
84. Problem
This is normally intractable when you have N
2N
− 1 subsets to test (17)
We could use different methods
1 K-means which is explained in the book.
2 Forward Selection heuristics that we will explain here.
Forward Selection
It starts with an empty subset to which a basis function is added at a time:
The one that reduces the sum-squared error the most.
Until a chosen criterion, such that GCV stops decreasing.
26 / 58
85. Problem
This is normally intractable when you have N
2N
− 1 subsets to test (17)
We could use different methods
1 K-means which is explained in the book.
2 Forward Selection heuristics that we will explain here.
Forward Selection
It starts with an empty subset to which a basis function is added at a time:
The one that reduces the sum-squared error the most.
Until a chosen criterion, such that GCV stops decreasing.
26 / 58
86. Problem
This is normally intractable when you have N
2N
− 1 subsets to test (17)
We could use different methods
1 K-means which is explained in the book.
2 Forward Selection heuristics that we will explain here.
Forward Selection
It starts with an empty subset to which a basis function is added at a time:
The one that reduces the sum-squared error the most.
Until a chosen criterion, such that GCV stops decreasing.
26 / 58
87. Subset Selection Vs. Optimization
Classic Neural Network Optimization
It involves the optimization, by gradient descent, of a nonlinear
sum-squared-error surface in a high-dimensional space defined by the
network parameters.
In specific in RBF
The network parameters are the centers, sizes and hidden-to-output
weights.
27 / 58
88. Subset Selection Vs. Optimization
Classic Neural Network Optimization
It involves the optimization, by gradient descent, of a nonlinear
sum-squared-error surface in a high-dimensional space defined by the
network parameters.
In specific in RBF
The network parameters are the centers, sizes and hidden-to-output
weights.
27 / 58
89. Subset Selection Vs. Optimization
In Subset Selection
The heuristic searches in a discrete space of subsets of a set of hidden
units with fixed centers and sizes while finding a subset with the
lowest prediction error.
It uses, a minimization criteria as the variance of the GVC:
ˆσ2
GCV =
N ˆd
T
P2 ˆd
trace (P)2 (18)
28 / 58
90. Subset Selection Vs. Optimization
In Subset Selection
The heuristic searches in a discrete space of subsets of a set of hidden
units with fixed centers and sizes while finding a subset with the
lowest prediction error.
It uses, a minimization criteria as the variance of the GVC:
ˆσ2
GCV =
N ˆd
T
P2 ˆd
trace (P)2 (18)
28 / 58
91. Subset Selection Vs. Optimization
In Subset Selection
The heuristic searches in a discrete space of subsets of a set of hidden
units with fixed centers and sizes while finding a subset with the
lowest prediction error.
It uses, a minimization criteria as the variance of the GVC:
ˆσ2
GCV =
N ˆd
T
P2 ˆd
trace (P)2 (18)
28 / 58
92. In addition
Hidden-to-Output Weights
They are not selected, they are slaved to the centers and sizes of the
chosen subset.
Forward selection is a non-linear type of heuristic with the following
advantages
There is no need to fix the number of hidden units in advance.
The model selection criteria are tractable.
The computational requirements are relatively low.
29 / 58
93. In addition
Hidden-to-Output Weights
They are not selected, they are slaved to the centers and sizes of the
chosen subset.
Forward selection is a non-linear type of heuristic with the following
advantages
There is no need to fix the number of hidden units in advance.
The model selection criteria are tractable.
The computational requirements are relatively low.
29 / 58
94. In addition
Hidden-to-Output Weights
They are not selected, they are slaved to the centers and sizes of the
chosen subset.
Forward selection is a non-linear type of heuristic with the following
advantages
There is no need to fix the number of hidden units in advance.
The model selection criteria are tractable.
The computational requirements are relatively low.
29 / 58
95. In addition
Hidden-to-Output Weights
They are not selected, they are slaved to the centers and sizes of the
chosen subset.
Forward selection is a non-linear type of heuristic with the following
advantages
There is no need to fix the number of hidden units in advance.
The model selection criteria are tractable.
The computational requirements are relatively low.
29 / 58
96. Thus, under the classic least squared error
Something Notable
In forward selection each step involves growing the network by one basis
function.
Therefore
Adding a new basis function is one of the incremental operations by using
the equation
Pd1+1 = Pd1 −
Pd1 φjφT
j Pd1
φT
j Pd1 φj
(19)
30 / 58
97. Thus, under the classic least squared error
Something Notable
In forward selection each step involves growing the network by one basis
function.
Therefore
Adding a new basis function is one of the incremental operations by using
the equation
Pd1+1 = Pd1 −
Pd1 φjφT
j Pd1
φT
j Pd1 φj
(19)
30 / 58
98. Thus
Where
Pm+1 is the succeeding projection matrix if the J-th member of the
set is added.
Pm the projection matrix for the m−hidden units.
The vectors φj
N
j=1
are the column vectors of the matrix Φ with
N d1.
31 / 58
99. Thus
Where
Pm+1 is the succeeding projection matrix if the J-th member of the
set is added.
Pm the projection matrix for the m−hidden units.
The vectors φj
N
j=1
are the column vectors of the matrix Φ with
N d1.
31 / 58
100. Thus
Where
Pm+1 is the succeeding projection matrix if the J-th member of the
set is added.
Pm the projection matrix for the m−hidden units.
The vectors φj
N
j=1
are the column vectors of the matrix Φ with
N d1.
31 / 58
101. Thus
We have that
ΦN = [φ1 φ2 ... φN ] (20)
If we take in account all the possible centers given by all the basis
32 / 58
102. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
33 / 58
103. What are we going to do?
This is what we want to do
1 Adding a new basis function
2 Removing an old basis function
34 / 58
104. Given a matrix
Given a squared matrix of size d1, we have the following
B−1
B = Id1
BB−1
= Id1
Inverse of matrix with small-rank adjustment
Suppose that an n × n matrix B1 is obtained by adding a small-rank
adjustment XRY T to matrix B0,
B1 = B0 + XRYT
(21)
Where
B0 ∈ Rd1×d1 is the known inverse, X, Y ∈ Rd1×r are known with d1 > r,
R ∈ Rr×r and the inverse of B1 is sought.
35 / 58
105. Given a matrix
Given a squared matrix of size d1, we have the following
B−1
B = Id1
BB−1
= Id1
Inverse of matrix with small-rank adjustment
Suppose that an n × n matrix B1 is obtained by adding a small-rank
adjustment XRY T to matrix B0,
B1 = B0 + XRYT
(21)
Where
B0 ∈ Rd1×d1 is the known inverse, X, Y ∈ Rd1×r are known with d1 > r,
R ∈ Rr×r and the inverse of B1 is sought.
35 / 58
106. Given a matrix
Given a squared matrix of size d1, we have the following
B−1
B = Id1
BB−1
= Id1
Inverse of matrix with small-rank adjustment
Suppose that an n × n matrix B1 is obtained by adding a small-rank
adjustment XRY T to matrix B0,
B1 = B0 + XRYT
(21)
Where
B0 ∈ Rd1×d1 is the known inverse, X, Y ∈ Rd1×r are known with d1 > r,
R ∈ Rr×r and the inverse of B1 is sought.
35 / 58
107. We can do the following
We have the following formula
B−1
1 = B−1
0 − B−1
0 X YT
B−1
0 X + R−1
−1
YB−1
0 (22)
Something Notable
This is quite more efficient because involves inverting a r−matrix
YT
B−1
0 X + R−1
.
36 / 58
108. We can do the following
We have the following formula
B−1
1 = B−1
0 − B−1
0 X YT
B−1
0 X + R−1
−1
YB−1
0 (22)
Something Notable
This is quite more efficient because involves inverting a r−matrix
YT
B−1
0 X + R−1
.
36 / 58
109. Thus, we can then partition the matrix A
We have the following partition
B =
B11 B12
B21 B22
(23)
We have that
B−1
=
B11 − B12B−1
22 B21
−1
B−1
11 B12 B21B−1
11 B12 − A22
−1
B21B−1
11 A12 − A22
−1
B21B−1
11 B22 − B21B−1
11 B12
−1 (24)
37 / 58
110. Thus, we can then partition the matrix A
We have the following partition
B =
B11 B12
B21 B22
(23)
We have that
B−1
=
B11 − B12B−1
22 B21
−1
B−1
11 B12 B21B−1
11 B12 − A22
−1
B21B−1
11 A12 − A22
−1
B21B−1
11 B22 − B21B−1
11 B12
−1 (24)
37 / 58
111. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
112. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
113. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
114. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
115. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
116. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
117. Finally, we get using ∆ = B22 − B21B−1
11 B12
We have
B−1
=
B−1
11 + B−1
11 B12∆−1B21B−1
11 −B−1
11 B12∆−1
−∆−1B21B−1
11 ∆−1 (25)
Using this equation we obtain the following improvements
Because if we retrain the network, we need to do the following:
Involving constructing the new design matrix.
Multiplying it with itself.
Adding the regularizer (if there is one).
Taking the inverse to obtain the variance matrix.
Recomputing the projection matrix.
38 / 58
118. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
39 / 58
119. Complexity of calculation of P
We have the following approximate number of multiplications
Operation Completely Retrain Using Operation
Add a new basis d3
1 + Nd2
1 + N2d1 N2
Remove an old basis d3
1 + Nd2
1 + N2d1 N2
Add a new pattern d3
1 + Nd2
1 + N2d1 2d2
1 + d1N + N2
Remove an old pattern d3
1 + Nd2
1 + N2d1 2d2
1 + d1N + N2
40 / 58
120. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
41 / 58
121. Adding a Basis Function
We do the following
If the J−th basis function is chosen then φj is appended to the last
column of Φd1 and renamed m + 1.
Thus, incrementing to the new matrix
Φd1+1 = Φd1 φd1+1 (26)
42 / 58
122. Adding a Basis Function
We do the following
If the J−th basis function is chosen then φj is appended to the last
column of Φd1 and renamed m + 1.
Thus, incrementing to the new matrix
Φd1+1 = Φd1 φd1+1 (26)
42 / 58
123. Where
We have that
φd1+1 =
φd1+1 (x1)
φd1+1 (x2)
...
φd1+1 (xN )
(27)
43 / 58
124. Using our variance matrix
We have the following variance for the general case
Ad1+1 = ΦT
d1+1Φd1+1 + Λd1+1 (28)
We have
Ad1+1 = ΦT
d1+1Φd1+1 + Λd1+1 =
ΦT
d1
φT
d1+1
Φd1 φd1+1 +
Λd1 0
0T
λd1+1
44 / 58
125. Using our variance matrix
We have the following variance for the general case
Ad1+1 = ΦT
d1+1Φd1+1 + Λd1+1 (28)
We have
Ad1+1 = ΦT
d1+1Φd1+1 + Λd1+1 =
ΦT
d1
φT
d1+1
Φd1 φd1+1 +
Λd1 0
0T
λd1+1
44 / 58
132. We have then that
We can use the previous result for A−1
m+1
Pd1+1 =IN − Φd1+1A−1
d1+1ΦT
d1+1
=Pd1 −
Pd1 φd1+1 φT
d1+1Pd1
λd1+1 + φT
d1+1Pd1 φd1+1
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133. We have then that
We can use the previous result for A−1
m+1
Pd1+1 =IN − Φd1+1A−1
d1+1ΦT
d1+1
=Pd1 −
Pd1 φd1+1 φT
d1+1Pd1
λd1+1 + φT
d1+1Pd1 φd1+1
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134. How do we select the new basis
We can use the greatest in sum-squared error difference
ˆSd1 = ˆyT
P2
d1
ˆy (29)
In addition, we have
ˆSd1+1 = ˆyT
P2
d1+1ˆy (30)
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135. How do we select the new basis
We can use the greatest in sum-squared error difference
ˆSd1 = ˆyT
P2
d1
ˆy (29)
In addition, we have
ˆSd1+1 = ˆyT
P2
d1+1ˆy (30)
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140. An alternative is is to seek to maximize the decrease in the
cost function
We have
ˆCd1 − ˆCd1+1 = ˆyT
Pd1
ˆy − ˆyT
Pd1+1ˆy
= ˆyT Pd1 φd1+1 φT
d1+1Pd1
λd1+1 + φT
d1+1Pd1 φd1+1
ˆy
=
ˆyT
Pd1 φd1+1
2
λd1+1 + φT
d1+1Pd1 φd1+1
51 / 58
141. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
52 / 58
142. Removing an Old Basis Function under Regularization
Here, we can remove any column
Process:
1 Move the selected j-th column at the end (Permutation).
2 Apply our well known equation with Pd1 in place of Pd1+1 and
Pd1−1 in place of Pd1 .
3 In addition φj in place of φd1+1 .
4 And λj in place of λd1+1.
Thus, we have
Pd1 = Pd1−1 −
Pd1−1 φj φT
j Pd1−1
λj + φT
j Pd1−1 φj
(31)
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143. Removing an Old Basis Function under Regularization
Here, we can remove any column
Process:
1 Move the selected j-th column at the end (Permutation).
2 Apply our well known equation with Pd1 in place of Pd1+1 and
Pd1−1 in place of Pd1 .
3 In addition φj in place of φd1+1 .
4 And λj in place of λd1+1.
Thus, we have
Pd1 = Pd1−1 −
Pd1−1 φj φT
j Pd1−1
λj + φT
j Pd1−1 φj
(31)
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144. Removing an Old Basis Function under Regularization
Here, we can remove any column
Process:
1 Move the selected j-th column at the end (Permutation).
2 Apply our well known equation with Pd1 in place of Pd1+1 and
Pd1−1 in place of Pd1 .
3 In addition φj in place of φd1+1 .
4 And λj in place of λd1+1.
Thus, we have
Pd1 = Pd1−1 −
Pd1−1 φj φT
j Pd1−1
λj + φT
j Pd1−1 φj
(31)
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145. Removing an Old Basis Function under Regularization
Here, we can remove any column
Process:
1 Move the selected j-th column at the end (Permutation).
2 Apply our well known equation with Pd1 in place of Pd1+1 and
Pd1−1 in place of Pd1 .
3 In addition φj in place of φd1+1 .
4 And λj in place of λd1+1.
Thus, we have
Pd1 = Pd1−1 −
Pd1−1 φj φT
j Pd1−1
λj + φT
j Pd1−1 φj
(31)
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146. Removing an Old Basis Function under Regularization
Here, we can remove any column
Process:
1 Move the selected j-th column at the end (Permutation).
2 Apply our well known equation with Pd1 in place of Pd1+1 and
Pd1−1 in place of Pd1 .
3 In addition φj in place of φd1+1 .
4 And λj in place of λd1+1.
Thus, we have
Pd1 = Pd1−1 −
Pd1−1 φj φT
j Pd1−1
λj + φT
j Pd1−1 φj
(31)
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147. Thus
If λj = 0
We can first post- and then pre-multiplying by φj to obtain expressions of
Pd1−1 φj and φT
j Pd1−1 φj in terms of Pd1
Thus, we have
Pd1−1 = Pd1 +
Pd1 φj φT
j Pd1
λj − φT
j Pd1 φj
(32)
However
For small λj, the round-off error can be problematic!!!
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148. Thus
If λj = 0
We can first post- and then pre-multiplying by φj to obtain expressions of
Pd1−1 φj and φT
j Pd1−1 φj in terms of Pd1
Thus, we have
Pd1−1 = Pd1 +
Pd1 φj φT
j Pd1
λj − φT
j Pd1 φj
(32)
However
For small λj, the round-off error can be problematic!!!
54 / 58
149. Thus
If λj = 0
We can first post- and then pre-multiplying by φj to obtain expressions of
Pd1−1 φj and φT
j Pd1−1 φj in terms of Pd1
Thus, we have
Pd1−1 = Pd1 +
Pd1 φj φT
j Pd1
λj − φT
j Pd1 φj
(32)
However
For small λj, the round-off error can be problematic!!!
54 / 58
150. Outline
1 Predicting Variance of w and the output d
The Variance Matrix
Selecting Regularization Parameter
2 How many dimensions?
How many dimensions?
3 Forward Selection Algorithms
Introduction
Incremental Operations
Complexity Comparison
Adding Basis Function Under Regularization
Removing an Old Basis Function under Regularization
A Possible Forward Algorithm
55 / 58
151. Based in the previous ideas
We are ready for a basic algorithm
However, this can be improved.
56 / 58
152. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
153. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
154. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
155. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
156. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
157. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
158. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
159. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
160. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
161. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
162. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
163. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
164. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
165. We have the following pseudocode
Forward-Regularization(D)
1 Select the functions d1 to used as basis based on the data D
This can be done randomly or using the clustering method described in
Haykin
2 Select an > 0 stopping criteria
3 ˆCd1 = ˆyT
Pd1 ˆy
4 Do
5 ˆCd1 = ˆCd1+1
6 d1 = d1 + 1
7 Do
8 Select a new base element and generate φd1+1 . Several strategies exist
9 Generate A−1
d1+1 and Pd1+1
10 Calculate ˆCd1+1
11 Until
ˆyT
Pd1
φd1+1
2
λd1+1+ φT
d1+1Pd1
φd1+1
> 0
12 Until ˆCd1 − ˆCd1+1
2
<
57 / 58
166. For more on this
Please Read the following
Introduction to Radial Basis Function Networks by Mark J. L. Orr
And there is much more
Look At the book Bootstrap Methods and their Application by A. C.
Davison and D. V. Hinkley
58 / 58
167. For more on this
Please Read the following
Introduction to Radial Basis Function Networks by Mark J. L. Orr
And there is much more
Look At the book Bootstrap Methods and their Application by A. C.
Davison and D. V. Hinkley
58 / 58