The document discusses using eigenvector decomposition to model northern spotted owl populations. It provides background on threats to the owl's habitat and its endangered status. Mathematical terms like eigenvalues and eigenvectors are defined. Survival and fecundity rates from recent data are used to construct a matrix, from which eigenvalues are calculated using decomposition. The resulting principal eigenvalue of 1.06528 indicates notable growth in the owl population.
This document provides an introduction to the Expectation Maximization (EM) algorithm. EM is used to estimate parameters in statistical models when data is incomplete or has missing values. It is a two-step process: 1) Expectation step (E-step), where the expected value of the log likelihood is computed using the current estimate of parameters; 2) Maximization step (M-step), where the parameters are re-estimated to maximize the expected log likelihood found in the E-step. EM is commonly used for problems like clustering with mixture models and hidden Markov models. Applications of EM discussed include clustering data using mixture of Gaussian distributions, and training hidden Markov models for natural language processing tasks. The derivation of the EM algorithm and
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.
Sampling based approximation of confidence intervals for functions of genetic...prettygully
Approximate lower bound sampling errors of maximum likelihood estimates of covariance components and their linear functions can be obtained from the inverse of the
information matrix. For non-linear functions, sampling variances are commonly determined as the variance of their first order Taylor series expansions. This is used to obtain sampling errors for estimates of heritabilities and correlations, and these quantities can be computed
with most software performing such analyses. In other instances, however, more complicated functions are of interest or the linear approximation is difficult or inadequate. A pragmatic alternative then is to evaluate sampling characteristics by repeated sampling of parameters from their asymptotic, multivariate normal distribution, calculating the function(s) of interest for each sample and inspecting the distribution across replicates. This paper demonstrates the use of this approach and examines the quality of
approximation obtained.
This document summarizes a journal article that proposes an alternative approach to variable selection called the KL adaptive lasso. The KL adaptive lasso replaces the squared error loss used in traditional adaptive lasso with Kullback-Leibler divergence loss. The paper shows that the KL adaptive lasso enjoys oracle properties, meaning it performs as well as if the true underlying model was given. Specifically, it consistently selects the true variables and estimates their coefficients at optimal rates. The KL adaptive lasso can also be solved using efficient algorithms like LARS. The approach is extended to generalized linear models, and theoretical properties are discussed.
Numerical Methods - Power Method for Eigen valuesDr. Nirav Vyas
The document discusses the power method, an iterative method for estimating the largest or smallest eigenvalue and corresponding eigenvector of a matrix. It begins by introducing the power method and notes it is useful when a matrix's eigenvalues can be ordered by magnitude. It then provides the working rules for determining a matrix's largest eigenvalue using the power method, which involves iteratively computing the matrix-vector product and rescaling the vector. Finally, it includes an example applying the power method to estimate the largest eigenvalue and eigenvector of a 2x2 matrix.
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
This document presents a comparison of dimension reduction techniques for survival analysis, including principal component analysis (PCA), partial least squares (PLS), and random matrix approaches. Simulation data with 100 observations and 1000 covariates was generated to test the ability of each method to minimize bias and mean squared error in estimating survival functions. PCA and PLS were able to capture 50% of the variance by reducing the dimensions to 37. The estimated survival functions were compared to the true function over 5000 iterations. PLS had the lowest bias and mean squared error, followed by PCA, with the random matrix approaches performing worse.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This document provides an overview of eigenvalues and their applications. It discusses:
1) Eigenvalues arise in applications across science and engineering, including mechanics, control theory, and quantum mechanics. Numerical methods are used to solve increasingly large eigenvalue problems.
2) Common methods for small problems include the QR and power methods. For large, sparse problems, techniques like the Krylov subspace and Arnoldi methods are used to compute a few desired eigenvalues/eigenvectors.
3) The document outlines the structure of the thesis, which will investigate methods for finding eigenvalues like Krylov subspace, power, and QR. It will also explore applications in areas like biology, statistics, and engineering.
This document provides an introduction to the Expectation Maximization (EM) algorithm. EM is used to estimate parameters in statistical models when data is incomplete or has missing values. It is a two-step process: 1) Expectation step (E-step), where the expected value of the log likelihood is computed using the current estimate of parameters; 2) Maximization step (M-step), where the parameters are re-estimated to maximize the expected log likelihood found in the E-step. EM is commonly used for problems like clustering with mixture models and hidden Markov models. Applications of EM discussed include clustering data using mixture of Gaussian distributions, and training hidden Markov models for natural language processing tasks. The derivation of the EM algorithm and
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.
Sampling based approximation of confidence intervals for functions of genetic...prettygully
Approximate lower bound sampling errors of maximum likelihood estimates of covariance components and their linear functions can be obtained from the inverse of the
information matrix. For non-linear functions, sampling variances are commonly determined as the variance of their first order Taylor series expansions. This is used to obtain sampling errors for estimates of heritabilities and correlations, and these quantities can be computed
with most software performing such analyses. In other instances, however, more complicated functions are of interest or the linear approximation is difficult or inadequate. A pragmatic alternative then is to evaluate sampling characteristics by repeated sampling of parameters from their asymptotic, multivariate normal distribution, calculating the function(s) of interest for each sample and inspecting the distribution across replicates. This paper demonstrates the use of this approach and examines the quality of
approximation obtained.
This document summarizes a journal article that proposes an alternative approach to variable selection called the KL adaptive lasso. The KL adaptive lasso replaces the squared error loss used in traditional adaptive lasso with Kullback-Leibler divergence loss. The paper shows that the KL adaptive lasso enjoys oracle properties, meaning it performs as well as if the true underlying model was given. Specifically, it consistently selects the true variables and estimates their coefficients at optimal rates. The KL adaptive lasso can also be solved using efficient algorithms like LARS. The approach is extended to generalized linear models, and theoretical properties are discussed.
Numerical Methods - Power Method for Eigen valuesDr. Nirav Vyas
The document discusses the power method, an iterative method for estimating the largest or smallest eigenvalue and corresponding eigenvector of a matrix. It begins by introducing the power method and notes it is useful when a matrix's eigenvalues can be ordered by magnitude. It then provides the working rules for determining a matrix's largest eigenvalue using the power method, which involves iteratively computing the matrix-vector product and rescaling the vector. Finally, it includes an example applying the power method to estimate the largest eigenvalue and eigenvector of a 2x2 matrix.
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
This document presents a comparison of dimension reduction techniques for survival analysis, including principal component analysis (PCA), partial least squares (PLS), and random matrix approaches. Simulation data with 100 observations and 1000 covariates was generated to test the ability of each method to minimize bias and mean squared error in estimating survival functions. PCA and PLS were able to capture 50% of the variance by reducing the dimensions to 37. The estimated survival functions were compared to the true function over 5000 iterations. PLS had the lowest bias and mean squared error, followed by PCA, with the random matrix approaches performing worse.
Numerical solution of eigenvalues and applications 2SamsonAjibola
This document provides an overview of eigenvalues and their applications. It discusses:
1) Eigenvalues arise in applications across science and engineering, including mechanics, control theory, and quantum mechanics. Numerical methods are used to solve increasingly large eigenvalue problems.
2) Common methods for small problems include the QR and power methods. For large, sparse problems, techniques like the Krylov subspace and Arnoldi methods are used to compute a few desired eigenvalues/eigenvectors.
3) The document outlines the structure of the thesis, which will investigate methods for finding eigenvalues like Krylov subspace, power, and QR. It will also explore applications in areas like biology, statistics, and engineering.
The document discusses eigenvalue problems and algorithms for solving them. Eigenvalue problems involve finding the eigenvalues and eigenvectors of a matrix and occur across science and engineering. The properties of the eigenvalue problem, like whether the matrix is real or complex, affect the choice of algorithm. The Power Method is described as an iterative technique for determining the dominant eigenvalue and eigenvector of a matrix. It works by successively applying the matrix to a starting vector to isolate the component in the direction of the dominant eigenvector. Variants can find other eigenvalues like the smallest. General projection methods approximate eigenvectors within a subspace, while subspace iteration generalizes Power Method to compute multiple eigenvalues.
The document provides an overview of goodness-of-fit tests for multinomial experiments and contingency tables, which are used to test if observed frequency distributions fit expected distributions. It defines multinomial experiments, goodness-of-fit tests, and contingency tables, and explains how to perform tests of independence and homogeneity using chi-square tests on contingency tables. Sample problems are provided to test claims about categories of outcomes and the independence of variables in contingency tables.
This document discusses eigenvalues and eigenvectors. It introduces eigenvalues and eigenvectors and some of their applications in areas like engineering, science, control theory and physics. It defines diagonal matrices and explains how eigenvalues and eigenvectors are used to transform a given matrix into a diagonal matrix. It also discusses how this process can be used to solve coupled differential equations. It provides background on linear independence and explains that the eigenvectors of a matrix must be linearly independent for diagonalization.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The document discusses eigenvectors and eigenvalues. It begins by defining diagonal matrices and provides examples. It then states that the goal is to understand diagonalization using eigenvectors and eigenvalues. Diagonalization involves finding a matrix such that when it transforms another matrix, the result is a diagonal matrix. This requires the eigenvectors of the original matrix to be linearly independent. The document provides examples of calculating eigenvectors and eigenvalues from matrices and shows how this relates to diagonalization. It also gives a brief introduction to linear independence and its implications for diagonalization.
Linear Regression and Logistic Regression in MLKumud Arora
Linear regression and logistic regression are statistical modeling techniques. Linear regression predicts continuous dependent variables using independent variables, while logistic regression predicts binary dependent variables. Both aim to model relationships between variables by estimating coefficients. Logistic regression models the log odds of the dependent variable rather than the variable directly. Key evaluation metrics for regression include accuracy, precision, recall, and F1 score, which are calculated using a confusion matrix.
1) Students will assemble different types of salads and salad dressings in groups by grade level. Third grade will make honey-tahini and oil-vinegar dressings, fourth grade a fruit salad, and fifth grade a main green salad.
2) After assembling, the salads will be displayed buffet style for all students to enjoy samples of each other's work.
3) The lesson aims to teach students about healthy eating, teamwork, and bringing their new salad-making skills home to share with families and communities.
This document contains a resume for Ashim Saha Roy. It summarizes his professional experience working for DHL for over 8 years in various finance roles such as financial reporting and accounting. It also lists his education including an MBA in Finance and Banking. The resume highlights his technical skills in Microsoft Office, SAP and other finance systems. It provides details of his career history and achievements working in finance roles in Bangladesh and Malaysia.
This document provides a summary of Graham Winsor's professional qualifications and experience. It outlines his contact information and highlights his strong communication, problem-solving, and leadership skills. His experience includes over 13 years as a Maintenance Supervisor managing facilities and staff. He also has experience as a Building Science Technologist and over 20 years of experience in the Canadian Forces working on communication and information systems.
This document summarizes research on the potential adverse effects of the ADHD medication Adderall XR on neurocognitive brain health. It discusses that while Adderall is effective in treating ADHD symptoms, research is limited on long-term effects. The medication increases dopamine levels in the brain and can alter the autonomic nervous system. Some studies have found psychiatric issues and brain abnormalities in long-term Adderall users. However, more research is still needed to determine if Adderall causes permanent neurological damage or if symptoms are due to other factors like sleep deprivation. The document calls for more research using neurological imaging to properly diagnose ADHD and avoid inappropriate exposure to medications.
El documento presenta una lista de las 10 herramientas TIC más utilizadas por Kevin Cano Quintero, un estudiante de grado 9-3. Estas herramientas incluyen redes sociales como Twitter y Facebook, servicios de almacenamiento como Google Drive, sitios para compartir videos como YouTube, motores de búsqueda como Google Search, software de presentación como Power Point, y herramientas de comunicación como Skype y Google Hangouts.
The RNLI needed to upgrade its communications system to reduce crew call-out times for its 24/7 search and rescue operations. Arqiva implemented a new digital paging system that provided immediate call-out of over 6,000 volunteer crew members. The system included automatic integrity checks, secure radio links between crews and coastguard, and remote control of station equipment from the pagers. Arqiva also provides 24-hour monitoring of the system from their Customer Service Centre with a commitment to respond to issues within two hours and fix problems within 24 hours. The new system has reduced crew call-out times and improved information available to lifeboat crews.
Everyone knows that exercise improves your health, but did you know that it can boost your productivity as well? Research shows that a regular workout routine can make you happier, more energetic, and smarter. Sitting for long periods of time at work is physically and mentally draining and causes tension to build up in your neck, shoulders, back, and legs. Allow yourself mental breaks throughout the day to get your second and third winds – by providing your body and mind with regular breaks, you will counteract the negative effects of prolonged periods in an efficient manner.
It is often classified as a compact exchanger of heat to emphasize its relatively high heat conversion surface area to quantity ratio. The plate-fin is widely used in several industries, like the aerospace business for its compact size and light-weight properties, as well as in cryogenics exactly where its ability to facilitate warmth transfer with little temperature differences is required.A plate-fin exchanger consists of layers of corrugated sheets divided via flat metal dishes, usually aluminum, to establish a number of finned chambers.Separate cold and hot fluid streams circulation through alternating levels of the exchanger and are generally enclosed in the sides through bars at the sides. Heat is moved from one stream with the fin interface towards the separator plate and with the next set of fins to the adjacent fluid. The actual fins also serve to boost the structural integrity from the heat exchanger and allow this to withstand high demands while providing a long surface area for the exchange of heat.
With the traditional method of cleaning heat exchangers retube contaminants accumulates once again in a very short span of time. Constant cleaning is also not a solution because in order to clean heat exchanger tubes they must be stopped to perform any other function. And if they are stopped for cleaning purposes, machinery itself cannot function, for obvious reasons. And if our equipment is not function able for longer period of time it can lead to other serious problems.
The document discusses eigenvalue problems and algorithms for solving them. Eigenvalue problems involve finding the eigenvalues and eigenvectors of a matrix and occur across science and engineering. The properties of the eigenvalue problem, like whether the matrix is real or complex, affect the choice of algorithm. The Power Method is described as an iterative technique for determining the dominant eigenvalue and eigenvector of a matrix. It works by successively applying the matrix to a starting vector to isolate the component in the direction of the dominant eigenvector. Variants can find other eigenvalues like the smallest. General projection methods approximate eigenvectors within a subspace, while subspace iteration generalizes Power Method to compute multiple eigenvalues.
The document provides an overview of goodness-of-fit tests for multinomial experiments and contingency tables, which are used to test if observed frequency distributions fit expected distributions. It defines multinomial experiments, goodness-of-fit tests, and contingency tables, and explains how to perform tests of independence and homogeneity using chi-square tests on contingency tables. Sample problems are provided to test claims about categories of outcomes and the independence of variables in contingency tables.
This document discusses eigenvalues and eigenvectors. It introduces eigenvalues and eigenvectors and some of their applications in areas like engineering, science, control theory and physics. It defines diagonal matrices and explains how eigenvalues and eigenvectors are used to transform a given matrix into a diagonal matrix. It also discusses how this process can be used to solve coupled differential equations. It provides background on linear independence and explains that the eigenvectors of a matrix must be linearly independent for diagonalization.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The document discusses eigenvectors and eigenvalues. It begins by defining diagonal matrices and provides examples. It then states that the goal is to understand diagonalization using eigenvectors and eigenvalues. Diagonalization involves finding a matrix such that when it transforms another matrix, the result is a diagonal matrix. This requires the eigenvectors of the original matrix to be linearly independent. The document provides examples of calculating eigenvectors and eigenvalues from matrices and shows how this relates to diagonalization. It also gives a brief introduction to linear independence and its implications for diagonalization.
Linear Regression and Logistic Regression in MLKumud Arora
Linear regression and logistic regression are statistical modeling techniques. Linear regression predicts continuous dependent variables using independent variables, while logistic regression predicts binary dependent variables. Both aim to model relationships between variables by estimating coefficients. Logistic regression models the log odds of the dependent variable rather than the variable directly. Key evaluation metrics for regression include accuracy, precision, recall, and F1 score, which are calculated using a confusion matrix.
1) Students will assemble different types of salads and salad dressings in groups by grade level. Third grade will make honey-tahini and oil-vinegar dressings, fourth grade a fruit salad, and fifth grade a main green salad.
2) After assembling, the salads will be displayed buffet style for all students to enjoy samples of each other's work.
3) The lesson aims to teach students about healthy eating, teamwork, and bringing their new salad-making skills home to share with families and communities.
This document contains a resume for Ashim Saha Roy. It summarizes his professional experience working for DHL for over 8 years in various finance roles such as financial reporting and accounting. It also lists his education including an MBA in Finance and Banking. The resume highlights his technical skills in Microsoft Office, SAP and other finance systems. It provides details of his career history and achievements working in finance roles in Bangladesh and Malaysia.
This document provides a summary of Graham Winsor's professional qualifications and experience. It outlines his contact information and highlights his strong communication, problem-solving, and leadership skills. His experience includes over 13 years as a Maintenance Supervisor managing facilities and staff. He also has experience as a Building Science Technologist and over 20 years of experience in the Canadian Forces working on communication and information systems.
This document summarizes research on the potential adverse effects of the ADHD medication Adderall XR on neurocognitive brain health. It discusses that while Adderall is effective in treating ADHD symptoms, research is limited on long-term effects. The medication increases dopamine levels in the brain and can alter the autonomic nervous system. Some studies have found psychiatric issues and brain abnormalities in long-term Adderall users. However, more research is still needed to determine if Adderall causes permanent neurological damage or if symptoms are due to other factors like sleep deprivation. The document calls for more research using neurological imaging to properly diagnose ADHD and avoid inappropriate exposure to medications.
El documento presenta una lista de las 10 herramientas TIC más utilizadas por Kevin Cano Quintero, un estudiante de grado 9-3. Estas herramientas incluyen redes sociales como Twitter y Facebook, servicios de almacenamiento como Google Drive, sitios para compartir videos como YouTube, motores de búsqueda como Google Search, software de presentación como Power Point, y herramientas de comunicación como Skype y Google Hangouts.
The RNLI needed to upgrade its communications system to reduce crew call-out times for its 24/7 search and rescue operations. Arqiva implemented a new digital paging system that provided immediate call-out of over 6,000 volunteer crew members. The system included automatic integrity checks, secure radio links between crews and coastguard, and remote control of station equipment from the pagers. Arqiva also provides 24-hour monitoring of the system from their Customer Service Centre with a commitment to respond to issues within two hours and fix problems within 24 hours. The new system has reduced crew call-out times and improved information available to lifeboat crews.
Everyone knows that exercise improves your health, but did you know that it can boost your productivity as well? Research shows that a regular workout routine can make you happier, more energetic, and smarter. Sitting for long periods of time at work is physically and mentally draining and causes tension to build up in your neck, shoulders, back, and legs. Allow yourself mental breaks throughout the day to get your second and third winds – by providing your body and mind with regular breaks, you will counteract the negative effects of prolonged periods in an efficient manner.
It is often classified as a compact exchanger of heat to emphasize its relatively high heat conversion surface area to quantity ratio. The plate-fin is widely used in several industries, like the aerospace business for its compact size and light-weight properties, as well as in cryogenics exactly where its ability to facilitate warmth transfer with little temperature differences is required.A plate-fin exchanger consists of layers of corrugated sheets divided via flat metal dishes, usually aluminum, to establish a number of finned chambers.Separate cold and hot fluid streams circulation through alternating levels of the exchanger and are generally enclosed in the sides through bars at the sides. Heat is moved from one stream with the fin interface towards the separator plate and with the next set of fins to the adjacent fluid. The actual fins also serve to boost the structural integrity from the heat exchanger and allow this to withstand high demands while providing a long surface area for the exchange of heat.
With the traditional method of cleaning heat exchangers retube contaminants accumulates once again in a very short span of time. Constant cleaning is also not a solution because in order to clean heat exchanger tubes they must be stopped to perform any other function. And if they are stopped for cleaning purposes, machinery itself cannot function, for obvious reasons. And if our equipment is not function able for longer period of time it can lead to other serious problems.
Swarm Intelligence Based Algorithms: A Critical AnalysisXin-She Yang
This document summarizes and analyzes swarm intelligence based algorithms. It discusses how these algorithms can be viewed as iterative processes, self-organizing systems, or Markov chains. The key components of exploration and exploitation are also analyzed. Evolutionary algorithms like genetic algorithms are discussed in terms of their crossover, mutation, and selection operators. Overall, the document provides a critical analysis of swarm intelligence algorithms from different perspectives to understand how they work and can be improved.
Eigen-Decomposition: Eigenvalues and Eigenvectors.pdfNehaVerma933923
Eigenvalues and eigenvectors are numbers and vectors associated with square matrices. Together they provide the eigen-decomposition, which analyzes a matrix's structure. The eigen-decomposition expresses a matrix as a linear combination of orthogonal eigenvectors multiplied by the corresponding eigenvalues. For positive semi-definite matrices like correlation and covariance matrices, the eigen-decomposition is particularly useful because the eigenvalues are positive and the eigenvectors are orthogonal.
Mathematical Methods for Engineers 2 (MATH1064)Leslie matr.docxandreecapon
Mathematical Methods for Engineers 2 (MATH1064)
Leslie matrix Matlab group project
Due no later than 2 pm on Friday 10th October, 2014
Graduate Qualities: This project is designed to help the student achieve course objective 4: solve
simple applied problems using software such as Matlab , and to develop Graduate Qualities 1 & 3,
namely operating effectively with and upon a body of knowledge, and effective problem solving.
Assessment:
The assessment will take into account all of your documentation of the mathematical analysis of the
problem, your Matlab m-file(s), your Matlab output, the correctness of the final solutions and the
presentation of your whole report.
Groups should contain two or three people. It will be assumed that each member of the team
contributed equally and will be awarded individually the mark allocated to the report. If this is
not the case, then a lesser percentage for one or more members must be agreed by the team and
clearly indicated. This especially will apply to absences from the practical class or non-attendance
at agreed team meetings. The University policy on plagiarism will apply between different groups.
Students who wish to can submit a peer assessment form which can be found on the
course webpage.
How to divide the work: Each team member must participate in all aspects of the project: math-
ematical calculations, Matlab work and report writing.
Only one copy of your project report is required for each group.
Summary: In this project you will:
• Investigate the Leslie matrix model for a population
• Explain how a Leslie matrix can be used to calculate the population in each age class from time
to time
• Use Matlab to draw plots of age class populations evolving over time
• Use Matlab to study the long term behaviour of population numbers
Your report must be typed, and submitted through LearnOnline by one member of your
group. It should include:
• Written worked answers to all questions where this is required.
• Appropriately labeled figures where required.
• A listing of your Matlab script file should be included at the end of your report in an appendix.
• A coversheet is not needed but your report must have a title page that lists the names and
student identification numbers of all members of the group.
• The group’s .m file must be submitted as a saparate file via LearnOnline. Be sure to list all
group members at the top of the file; only one copy per group is required. There will be marks
awarded for submitting this file, so don’t forget. Your .m file may be run and checked during
the marking process.
1
Leslie Matrix Model
Invented by Patrick H. Leslie in the 1940s, the Leslie Matrix is a mathematical model of population
growth for a species. Time is divided into discrete periods, with individual memebers of the population
progressing through discrete age classes at given survival rates. Here is a simplified example:
The Central Australian Budgericoot (CAB) cannot live beyond five y ...
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
This document summarizes key concepts regarding eigenvalues and eigenvectors of matrices:
- Eigenvalues are scalars such that there exist non-zero eigenvectors satisfying Ax = λx.
- The characteristic equation states that λ is an eigenvalue if and only if it satisfies det(A - λI) = 0.
- A matrix is diagonalizable if it can be written as A = PDP-1, where D is a diagonal matrix of eigenvalues and P is a matrix of corresponding eigenvectors. Diagonalizable matrices can easily compute powers by raising the eigenvalues to powers.
The document summarizes the Whale Optimization Algorithm (WOA), which is a meta-heuristic optimization algorithm inspired by the hunting behavior of humpback whales. It describes how WOA simulates the bubble-net feeding mechanism of humpback whales to optimize problem solutions. The algorithm includes steps of encircling prey to find the best solution, then exploiting and exploring further to update positions and potentially find an even better solution. WOA iterates through these steps until a termination criterion is met, at which point it outputs the best found solution.
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
This document provides an introduction to matrix algebra concepts needed for a systems biology course, including matrices, determinants, inverses, eigenvalues and eigenvectors. It discusses how matrices first arose from solving systems of linear equations and how the modern approach is to transform linear systems into matrix equations. Key concepts introduced include matrix operations like addition and multiplication, properties of the matrix multiplication like it being non-commutative, and how determinants are important for solving linear systems. The document also notes how complex numbers allow solving equations that have no real number solutions.
Forecasting With An Adaptive Control Algorithmshwetakarsh
This working paper describes using a recursive least squares estimation method with exponential forgetting to estimate coefficients of a state space model of the US macroeconomy. The model uses 12 state variables including consumption, investment, industrial production, and interest rates. Out-of-sample forecasts show lower root mean square errors than OLS forecasts, indicating the adaptive method provides better tracking. Sensitivity analysis found a forgetting factor of 0.96411 produced the most accurate in-sample and out-of-sample forecasts compared to other values tested.
Stuck with your Regression Assignment? Get 24/7 help from tutors with Phd in the subject. Email us at support@helpwithassignment.com
Reach us at http://www.HelpWithAssignment.com
This document summarizes a directed research report on using singular value decomposition (SVD) to reconstruct images with missing pixel values. It describes how images can be represented as matrices and SVD is commonly used for matrix completion problems. The report explores using an alternating least squares (ALS) algorithm based on SVD to fill in missing pixel values by finding feature matrices that approximate the rank k reconstruction of an image matrix. The ALS algorithm works by alternating between optimizing one feature matrix while holding the other fixed, minimizing the reconstruction error between the known pixel values and predicted values from multiplying the feature matrices.
This document presents Alacart, a SAS macro system for generating classification trees similar to Breiman's CART methodology. It summarizes the core CART tree classification methodology, which involves recursively splitting data into purer subsets based on minimizing impurity at each node. Alacart generates the maximal tree on a training set and then prunes it back using either cross-validation or a test set to select the optimal size tree. An example application to customer classification is provided, showing the maximal 21-node tree and optimal 8-node pruned tree.
This document presents a discrete valuation methodology for swing options using a forest model approach. It develops numerical implementations of swing options on one-factor and two-factor mean-reverting underlying processes using binomial trees. It establishes convergence via finite difference methods and considers qualitative properties and sensitivity analysis. The methodology values swing options as a system of coupled European options and allows for various discrete models of the underlying process.
This document discusses the eigenvalue-eigenvector problem, which is important for solving differential equations, modeling population growth, and calculating matrix powers. It defines eigenvalues and eigenvectors and provides examples of solving eigenvalue problems. The power method, an iterative approach, is described for finding the dominant or lowest eigenvalue of a matrix. Solving eigenvalue problems has applications in many fields including physics, biology, economics and statistics.
This document discusses eigenvectors and eigenvalues. It defines eigenvectors as non-zero vectors that satisfy the equation AX = λX, where λ is the eigenvalue. Properties of eigenpairs are described, such as how eigenvectors can be scaled and how eigenvalues relate to the determinant and trace of the matrix. Methods for finding eigenpairs are presented, including solving the characteristic equation. Applications in areas like Google's PageRank algorithm are also mentioned.
This document proposes a method for estimating k sample survival functions under stochastic ordering constraints. It begins by reviewing existing work on estimating survival functions for two samples and extends this to k samples. The proposed method uses benchmark functions to estimate survival curves in a way that maintains stochastic ordering. It was tested on both uncensored and censored data and was shown to have low mean squared error and bias. The method was also applied to a real-world dataset with results comparable to previous work.
COVARIANCE ESTIMATION AND RELATED PROBLEMS IN PORTFOLIO OPTIMICruzIbarra161
COVARIANCE ESTIMATION AND RELATED PROBLEMS IN PORTFOLIO OPTIMIZATION
Ilya Pollak
Purdue University
School of Electrical and Computer Engineering
West Lafayette, IN 47907
USA
ABSTRACT
This overview paper reviews covariance estimation problems and re-
lated issues arising in the context of portfolio optimization. Given
several assets, a portfolio optimizer seeks to allocate a fixed amount
of capital among these assets so as to optimize some cost function.
For example, the classical Markowitz portfolio optimization frame-
work defines portfolio risk as the variance of the portfolio return,
and seeks an allocation which minimizes the risk subject to a target
expected return. If the mean return vector and the return covariance
matrix for the underlying assets are known, the Markowitz problem
has a closed-form solution.
In practice, however, the expected returns and the covariance
matrix of the returns are unknown and are therefore estimated from
historical data. This introduces several problems which render the
Markowitz theory impracticable in real portfolio management appli-
cations. This paper discusses these problems and reviews some of
the existing literature on methods for addressing them.
Index Terms— Covariance, estimation, portfolio, market, fi-
nance, Markowitz
1. INTRODUCTION
The return of a security between trading day t1 and trading day t2
is defined as the change in the closing price over this time period,
divided by the closing price on day t1. For example, the daily (i.e.,
one-day) return on trading day t is defined as (p(t)−p(t−1))/p(t−
1) where p(t) is the closing price on day t and p(t−1) is the closing
price on the previous trading day. Note that if t is a Monday or the
day after a holiday, the previous trading day will not be the same as
the previous calendar day.
Suppose an investment is made into N assets whose return vec-
tor is R, modeled as a random vector with expected return µ =
E[R] and covariance matrix Λ = E[(R − µ)(R − µ)T ]. In other
words, R = (R(1), . . . , R(N))T where R(n) is the return of the n-th
asset. It is assumed throughout the paper that the covariance matrix
Λ is invertible. This assumption is realistic, since it is quite unusual
in practice to have a set of assets whose linear combination has re-
turns exactly equal to zero. Even if an investment universe contained
such a set, the number of assets in the universe could be reduced to
eliminate the linear dependence and make the covariance matrix in-
vertible.
Out of these N assets, a portfolio is formed with allocation
weights w = (w(1), . . . , w(N))T . The n-th weight is defined as the
amount invested into the n-th asset, as a fraction of the overall invest-
ment into the portfolio: if the overall investment into the portfolio is
$D, and $D(n) is invested into the n-th asset, then w(n) = D(n)/D.
Therefore, by definition, the weights sum to one:
w
T
1 = 1, (1)
where 1 is an N -vector of ones. Note that some of the weights may
be negative, ...
Transportation and logistics modeling 2karim sal3awi
This document discusses statistical concepts like variability, random experiments, descriptive statistics, probability distributions, and statistical data analysis. It provides examples of different probability distributions like binomial, Poisson, normal, exponential, and Weibull distributions. It also discusses the four basic steps of statistical data analysis: defining the problem, collecting the data, analyzing the data, and reporting results. Methods like hypothesis testing are discussed as part of data analysis.
2. The northern spotted owl is a bird that resides around the Pacific Northwest, with its habitat
spanning approximately from northern California to southern Canada, and occupying a substantial
portion of Washington and Oregon. A good part of the economy in this region had been rooted in
logging and timber for over a hundred years, so the natural habitat of the northern spotted owl
subsequently diminished. As a result of the ever-shrinking natural habitat, the population of the
northern spotted owl had dropped to a point where the bird was listed as a threatened species under
the Endangered Species Act in 1990. (Conservation Northwest)
After being listed as a threatened species, several organizations such as the Northwest
Ecosystem Alliance, and the United States Fish and Wildlife Service studied the populations of
the northern spotted owl, and sought a way to further conservatory efforts. In 1991, Charles Biles,
a mathematician at Humboldt State University, and Barry R. Noon, a conservationist with the US
Forest Service published an article in the UMAP Journal outlining the mathematics used for the
purpose of spotted owl conservation. In The Spotted Owl, Biles demonstrates how systems of
equations involving the spotted owl populations at different stages in life, and their tendency to
survive and reproduce can yield the percentage of either growth or decline in northern spotted owl
population using an eigenvector decomposition (which will be explained further in this paper).
The method of eigenvector decomposition has since become standard practice in the study and
modeling of northern spotted owl conservation efforts.
Throughout this paper, there will be a variety of terminology used for the purpose of
explaining the processes involved in northern spotted owl conservation. It seems appropriate to
provide definitions for some of the less common terminology associated with the mathematical
processes outlined herein, as well as some terminology used in explaining conservatory efforts.
Two mathematical terms that will be seen regularly in this paper are eigenvalue and eigenvector.
3. An eigenvalue for an arbitrary matrix A is a scalar λ such that the matrix equation Ax = λx has a
solution for some non-zero vector x. An eigenvector for an arbitrary matrix A is a non-zero vector
x such that Ax = λx for some scalar λ. (Lay, A9) How these operations work specifically will be
shown later in a basic walkthrough. It should be noted that this paper is written with the assumption
that the reader minimally has a fundamental understanding of basic linear algebraic concepts and
a several other mathematical operations which are essential in understanding the applications of
this particular study.
A non-mathematical term that seems like a good idea to define is fecundity, as this is a
term that is frequently used in the development of systems of equations for northern spotted owl
conservation study, and will be used several times throughout the course of this paper. Fecundity,
as it pertains to the northern spotted owl, refers to the ability of owls in a particular age bracket to
produce offspring. This is important because fecundity is tantamount in determining the rate of
growth or decline of this particular population.
The next part of this paper will demonstrate the basic process of an eigenvector
decomposition. The first part will consist of a detailed outline using an arbitrary 3 x 3 matrix, the
scalar λ, and the identity matrix, denoted I. The process of eigenvector decomposition is described
by the following characteristic equation:
det (A- λI) = 0
For those unfamiliar with the terminology, this means that the eigenvalues can be found by
finding the product of λ and the identity matrix, I, subtracting that product from the matrix A,
finding the determinant of the resultant matrix thereafter, and factoring to find the values of λ. In
most cases, the determinant will yield a polynomial that must be factored to find the values for λ,
4. which are the eigenvalues. Consider the first problem, with an arbitrary 3 x 3 matrix. Underneath
each of the units in this matrix, there is a symbol to describe what each matrix is corresponding to
the in explanations listed earlier.
det [
1 2 3
4 5 6
7 8 9
] – λ [
1 0 0
0 1 0
0 0 1
] = 0
determinant (A - λ I)
The first order of business is to multiply the scalar λ by the identity matrix, I. This yields:
det [
1 2 3
4 5 6
7 8 9
] - [
λ 0 0
0 λ 0
0 0 λ
] = 0
Next is a basic matrix subtraction problem, which will consolidate these two matrices into one
matrix yielding:
det [
1 − λ 2 3
4 5 − λ 6
7 8 9 − λ
] = 0
After the matrix is consolidated, we must then find the determinant as follows:
1-λ |
5 − λ 6
8 9 − λ
| – 2 |
4 6
7 9 − λ
| + 3 |
4 5 − λ
7 8
|
1-λ (λ2
-14 λ-3) – 2 (-λ-6) + 3 (7 λ-3)
After completing the process of finding the determinant, combining like terms, and setting the
result equal to zero, the resulting polynomial equation for λ is as follows:
−λ3
+ 15 𝜆2
+ 18λ = 0
λ2
- 15 λ -18 = 0
5. To find the eigenvalues of this equation, we must solve the equation for λ. The values of λ are the
eigenvalues. This can be accomplished by factoring, or utilizing the quadratic formula. For the
sake of making this paper look more mathematically adept, the next step will outline the use of the
quadratic formula. It must be noted, however, that later problems demonstrated will show the
eigenvalues after having been calculated using the assistance of Maple software.
λ =
15 ± √225+72
2
λ =
15
2
±
3
2
√33
In this case, the result is two real eigenvalues. In further examples, the results may not be as nice
looking, and approximations to several significant figures may be used. It must also be reiterated
that the previous case was completely arbitrary and meant only as a primer to the problems which
will be demonstrated later in this writing, and as a basis for how to obtain eigenvalues via
eigenvector decomposition.
Now, consider the second problem, which is listed in a supplement to the Lay text available
online. The problem in review is question number one from Case Study: Dynamical Systems and
Spotted Owls. Let’s first review the information provided in the case study, an explanation will
then be provided as to how to construct the dynamical system needed to perform the subsequent
eigenvector decomposition. “The most recent spotted owl data available gives the following for
the matrix A: Juvenile Survival .33, Sub-adult Survival .85, Adult Survival .85, Sub-adult
Fecundity .125, Adult Fecundity .26.” (Lay, pp. 3) After inputting the data in a stage-matrix, an
explanation for the transcription will be afforded. The transcribed matrix yields:
[
0 . 125 . 26
. 33 0 0
0 . 85 . 85
]
6. The first column of the matrix describes the behavior and tendencies of juvenile spotted owls.
Since juveniles do not tend to reproduce, their fecundity is described in the matrix as zero. The
middle column describes the survival rate and fecundity of the sub-adult owls. The top row
describes the fecundity, and the bottom two rows describe the survival rate. The same holds for
the third column as it applies to adult owl population. According to Lay in the spotted owl
supplement, the model for this matrix is called the stage-matrix model for a population. (Lay, 1)
Next, the previous matrix will be applied to the eigenvector decomposition formula, recall the
formula is det (A-λI), yielding:
[
0 . 125 . 26
. 33 0 0
0 . 85 . 85
] – λ [
1 0 0
0 1 0
0 0 1
]
The process outlined here is the same as the process outlined in the previous example with the
arbitrary matrix. Therefore, the next step is to multiply the λ vector by the identity matrix I.
[
0 . 125 . 26
. 33 0 0
0 . 85 . 85
] - [
λ 0 0
0 λ 0
0 0 λ
]
After subtracting the λ identity matrix from the fecundity and survival matrix, the remaining matrix
yields:
[
−𝜆 . 125 . 26
. 33 −𝜆 0
0 . 85 . 85 − 𝜆
]
Now, the determinant of the matrix must be found.
- 𝜆 |
−𝜆 0
. 85 . 85 − 𝜆
| - .125 |
. 33 0
0 . 85 − 𝜆
| + .26 |
. 33 −𝜆
0 . 85
|
.85 𝜆2
- 𝜆3
+ .0378675 + .04125 𝜆
7. With the assistance of Maple software, the eigenvalues from the polynomial shown above are
listed as:
0.9371354439
0.04356772197 - 0.1962385450i
0.04356772197 - 0.1962385450i
Note, the bottom two eigenvalues are complex, which are beyond the scope of this paper, though
they are listed here to show the complete result of the decomposition. The eigenvalue of import is
the one listed at the top. According to the Spotted Owl Case Study, “If λ is a real number greater
than 1 and all the other eigenvalues are less than 1 in magnitude, then the population is increasing
exponentially. In this case the eigenvector gives the stable distribution of the population between
classes, and yields the percentages found in each class if scaled so that its entries sum to 1.” (Lay,
pp. 2)
According to Lay, the important number in this decomposition is 0.9371354439. This
number corresponds to the increase or decrease in spotted owl population. If we were to decompose
a data set resulting with a real eigenvalue greater than one, the conclusion would be that the
population is increasing, or has increased. In the case of this particular problem, the resulting
eigenvalue is less than 1, which means that the data set for this particular case indicates a decrease
in spotted owl population.
The last example will review the most recent available set of data, whereupon the current
state of affairs of northern spotted owl population will be determined. In an article entitled Status
and Trends of Northern Spotted Owl Populations, there is a plethora of data which will be utilized
to find the current survival rate of northern spotted owls. The following images are tables that are
8. included in the aforementioned articles and will be used as the primary reference for constructing
the matrix which will be the basis for impending eigenvector decomposition. The first image is a
table of average survival rates for 1, 2, and 3+ year old northern spotted owls:
(Davis, Dugger, Mohoric, Evers, Aney, pp. 11)
It should be noted that the survival rates listed are for several different regions, and vary
from region to region. Because of this variance, the survival rates for the three groups will be
averaged to obtain an overall population growth or decline model in a basic stage-matrix. The next
table is taken from the same journal, and outlines fecundity:
9. (Davis, Dugger, Mohoric, Evers, Aney, pp. 12).
By utilizing the method of finding an arithmetic mean with the data provided by the
aforementioned authors, we can conclude that the average survival rate of 1 year old spotted owls
is .70422, for 2 years old: .80144, and 3+ years old: .84567. The corresponding fecundity averages
are as follows: 1 year old: .06467, 2 years old: .25911, and for 3+ years old: .34378. The
eigenvector decomposition model from the previous examples with the survival and fecundity
averages taken from recent data yields the following stage-matrix model:
[
. 06467 . 25911 . 34378
. 70422 0 0
0 . 80144 . 84567
]
With this matrix, the formula for eigenvector decomposition will be applied, and the necessary
mathematical operations will be applied to find the eigenvalues, and thereafter determine the
overall population growth and decline.
10. [
. 06467 . 25911 . 34378
. 70422 0 0
0 . 80144 . 84567
] – λ [
1 0 0
0 1 0
0 0 1
]
[
. 06467 . 25911 . 34378
. 70422 0 0
0 . 80144 . 84567
]- [
𝜆 0 0
0 𝜆 0
0 0 𝜆
]
[
. 06467 − 𝜆 . 25911 . 34378
. 70422 −𝜆 0
0 . 80144 . 84567 − 𝜆
]
.06467- 𝜆 |
−𝜆 0
. 80144 . 84567 − 𝜆
| - .25911|
. 70422 0
0 . 84567 − 𝜆
| + .34378|
. 70422 −𝜆
0 . 80144
|
-𝜆3
+ .91034𝜆2
+ .12778𝜆 + .03972
-𝜆3
+ .91034𝜆2
+ .12778𝜆 + .03972 = 0
After inputting the above matrix in Maple, the software results yielded the following eigenvalues,
shortened to five significant figures:
1.06528
-.07747 + .01768i
-.07747 + .01768i
After reviewing the data, and the resulting eigenvalues after decomposition, it can be seen
that for the most recent data set that could be acquired the 𝜆 value for the real eigenvalue is
1.06528, and the subsequent eigenvalues are less than 1, indicating a notable growth in northern
spotted owl population. This is good. Thanks to the wit of mathematicians, and the diligent efforts
of conservationists in the Pacific Northwest, the northern spotted owl population can hopefully
11. live to thrive and grow another year. More information about northern spotted owl conservation
efforts can be found on the references page of this paper.
12. References
Biles, C., & Noon, B. (1990). The Spotted Owl. The UMAP Journal, 11.2, 99-109.
Conservation Northwest (2015). The Northern Spotted Owl. Retrieved from:
http://www.conservationnw.org/what-we-do/wildlife-habitat/northern-spotted-owl
Davis, R., Dugger, K., Mohoric, S., Evers, L., & Aney, W. (2011) Status and Trends of Northern
Spotted Owl Populations and Habitats. United States Department of Agriculture, United
States Forest Service, Pacific Northwest Research Station.
Lay, D. (2012). Eigenvalues and Eigenvectors. Linear Algebra and its applications, (pp. 265-324).
United States: Addison-Wesley.
Lay, D. (2003). Case Study: Dynamical Systems and Spotted Owls. United States: Addison-
Wesley. Retrieved from:
http://media.pearsoncmg.com/aw/aw_lay_linearalg_updated_cw_3/cs_apps/lay03_05_c
s.pdf