MACHINE LEARNING IS GOING TO DEFINE THE NEXT ERA; PEOPLE SAY IT WILL HAVE SIMILAR EFFECTS AS ELECTRICITY OR INTERNET HAD IN their TIMES .ARITIFICIAL INTELLIGENCE IS THE NEXT BEST THING IN THE WORLD OF TECHNOLOGY , BLOCKCHAIN AFTER THAT
The document discusses dynamic programming and its application to the matrix chain multiplication problem. It begins by explaining dynamic programming as a bottom-up approach to solving problems by storing solutions to subproblems. It then details the matrix chain multiplication problem of finding the optimal way to parenthesize the multiplication of a chain of matrices to minimize operations. Finally, it provides an example applying dynamic programming to the matrix chain multiplication problem, showing the construction of cost and split tables to recursively build the optimal solution.
MATLAB/SIMULINK for Engineering Applications day 2:Introduction to simulinkreddyprasad reddyvari
The document provides an introduction to MATLAB and Simulink through a presentation. It discusses what MATLAB and Simulink are, their basic functions and capabilities, and how to get started using them. The presentation covers topics such as vectors, matrices, plotting, control structures, M-files, and writing user-defined functions. The goal is to help attendees gain basic knowledge of MATLAB/Simulink and be able to explore them on their own.
This document provides an overview of support vector machines (SVMs) for machine learning. It explains that SVMs find the optimal separating hyperplane that maximizes the margin between examples of separate classes. This is achieved by formulating SVM training as a convex optimization problem that can be solved efficiently. The document discusses how SVMs can handle non-linear decision boundaries using the "kernel trick" to implicitly map examples to higher-dimensional feature spaces without explicitly performing the mapping.
Exact Matrix Completion via Convex Optimization Slide (PPT)Joonyoung Yi
Slide of the paper "Exact Matrix Completion via Convex Optimization" of Emmanuel J. Candès and Benjamin Recht. We presented this slide in KAIST CS592 Class, April 2018.
- Code: https://github.com/JoonyoungYi/MCCO-numpy
- Abstract of the paper: We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys
𝑚≥𝐶𝑛1.2𝑟log𝑛
for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
Matlab is a high-level programming language and environment used for numerical computation, visualization, and programming. The document outlines key Matlab concepts including the Matlab screen, variables, arrays, matrices, operators, plotting, flow control, m-files, and user-defined functions. Matlab allows users to analyze data, develop algorithms, and create models and applications.
Clustering:k-means, expect-maximization and gaussian mixture modeljins0618
This document discusses K-means clustering, Expectation Maximization (EM), and Gaussian mixture models (GMM). It begins with an overview of unsupervised learning and introduces K-means as a simple clustering algorithm. It then describes EM as a general algorithm for maximum likelihood estimation that can be applied to problems like GMM. GMM is presented as a density estimation technique that models data using a weighted sum of Gaussian distributions. EM is described as a method for estimating the parameters of a GMM from data.
This document provides an outline for a course on Optimization and Economics of Integrated Power Systems. The course will cover topics such as optimization basics, power systems basics, MATLAB review, and examples of optimization techniques applied to power systems, including linear, nonlinear, integer, and mixed integer programming. It also provides details on using MATLAB, including basics of variables, matrices, plotting, and loops. Equations and concepts from calculus relevant to optimization are defined, such as gradients, Hessians, and Taylor series.
The document discusses dynamic programming and its application to the matrix chain multiplication problem. It begins by explaining dynamic programming as a bottom-up approach to solving problems by storing solutions to subproblems. It then details the matrix chain multiplication problem of finding the optimal way to parenthesize the multiplication of a chain of matrices to minimize operations. Finally, it provides an example applying dynamic programming to the matrix chain multiplication problem, showing the construction of cost and split tables to recursively build the optimal solution.
MATLAB/SIMULINK for Engineering Applications day 2:Introduction to simulinkreddyprasad reddyvari
The document provides an introduction to MATLAB and Simulink through a presentation. It discusses what MATLAB and Simulink are, their basic functions and capabilities, and how to get started using them. The presentation covers topics such as vectors, matrices, plotting, control structures, M-files, and writing user-defined functions. The goal is to help attendees gain basic knowledge of MATLAB/Simulink and be able to explore them on their own.
This document provides an overview of support vector machines (SVMs) for machine learning. It explains that SVMs find the optimal separating hyperplane that maximizes the margin between examples of separate classes. This is achieved by formulating SVM training as a convex optimization problem that can be solved efficiently. The document discusses how SVMs can handle non-linear decision boundaries using the "kernel trick" to implicitly map examples to higher-dimensional feature spaces without explicitly performing the mapping.
Exact Matrix Completion via Convex Optimization Slide (PPT)Joonyoung Yi
Slide of the paper "Exact Matrix Completion via Convex Optimization" of Emmanuel J. Candès and Benjamin Recht. We presented this slide in KAIST CS592 Class, April 2018.
- Code: https://github.com/JoonyoungYi/MCCO-numpy
- Abstract of the paper: We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys
𝑚≥𝐶𝑛1.2𝑟log𝑛
for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
Matlab is a high-level programming language and environment used for numerical computation, visualization, and programming. The document outlines key Matlab concepts including the Matlab screen, variables, arrays, matrices, operators, plotting, flow control, m-files, and user-defined functions. Matlab allows users to analyze data, develop algorithms, and create models and applications.
Clustering:k-means, expect-maximization and gaussian mixture modeljins0618
This document discusses K-means clustering, Expectation Maximization (EM), and Gaussian mixture models (GMM). It begins with an overview of unsupervised learning and introduces K-means as a simple clustering algorithm. It then describes EM as a general algorithm for maximum likelihood estimation that can be applied to problems like GMM. GMM is presented as a density estimation technique that models data using a weighted sum of Gaussian distributions. EM is described as a method for estimating the parameters of a GMM from data.
This document provides an outline for a course on Optimization and Economics of Integrated Power Systems. The course will cover topics such as optimization basics, power systems basics, MATLAB review, and examples of optimization techniques applied to power systems, including linear, nonlinear, integer, and mixed integer programming. It also provides details on using MATLAB, including basics of variables, matrices, plotting, and loops. Equations and concepts from calculus relevant to optimization are defined, such as gradients, Hessians, and Taylor series.
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It is applicable to problems that exhibit optimal substructure and overlapping subproblems. The matrix chain multiplication problem can be solved using dynamic programming in O(n^3) time by defining the problem recursively, computing the costs of subproblems in a bottom-up manner using dynamic programming, and tracing the optimal solution back from the computed information. Similarly, the longest common subsequence problem exhibits optimal substructure and can be solved using dynamic programming.
This document describes solving a 1D Poisson equation using the finite element method. It provides the analytical solution and defines the problem. MATLAB functions are implemented to generate the mesh, system matrices and vectors, apply boundary conditions, and solve the linear system. Plots compare the analytical and numerical solutions, showing better agreement for finer meshes. Optional sections explore changing the analytical solution, calculating errors for different mesh sizes, and implementing a Neumann boundary condition.
MIT OpenCourseWare provides course materials for the free online course 6.094 Introduction to MATLAB taught in January 2009. The course covers topics like linear algebra, polynomials, optimization, differentiation and integration, and solving differential equations using MATLAB. Lecture 3 focuses on solving systems of linear equations, matrix operations, polynomial fitting to data, nonlinear root finding, function minimization, and numerical methods for differentiation, integration, and solving ordinary differential equations.
Principal Components Analysis (PCA) is an exploratory technique used to reduce the dimensionality of data sets while retaining as much information as possible. It transforms a number of correlated variables into a smaller number of uncorrelated variables called principal components. PCA is commonly used for applications like face recognition, image compression, and gene expression analysis by reducing the dimensions of large data sets and finding patterns in the data.
The following ppt is about principal component analysisSushmit8
Principal Components Analysis (PCA) is an exploratory technique used to reduce the dimensionality of data sets while retaining as much information as possible. It transforms a number of correlated variables into a smaller number of uncorrelated variables called principal components. PCA is commonly used for applications like face recognition, image compression, and gene expression analysis by reducing the dimensions of large data sets and finding patterns in the data.
This document contains the course calendar for a machine learning course covering topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, independent component analysis, and clustering algorithms. The calendar lists 15 classes over the semester, the topics to be covered in each class, and any dates where there will be no class. It also includes lecture plans and slides on principal component analysis, linear discriminant analysis, and comparing PCA and LDA.
Here are the steps to solve this problem in MATLAB:
1. Create a matrix A with the marks data:
A = [24 44 36 36;
52 57 68 76;
66 53 69 73;
85 40 86 72;
15 47 25 28;
79 72 82 91];
2. Define the column labels:
subjects = {'Math','Programming','Thermodynamics','Mechanics'};
3. Find the total marks of each subject:
totals = sum(A)
4. Find the average marks of each subject:
averages = mean(A)
5. Find the highest marks scored in each subject:
maxMarks = max
This document discusses various machine learning concepts including notations, statistical learning, support vector machines (SVM), principal component analysis (PCA), and kernel principal component analysis (K-PCA). It provides an overview of supervised and unsupervised learning. It also explains how SVM can be used for linear and non-linear classification problems using kernels to project data into higher dimensions. PCA is described as finding the directions of maximum variance in high-dimensional data. K-PCA is presented as a way to apply PCA in kernel-induced feature spaces to handle non-linear problems.
The name MATLAB stands for MATrix LABoratory.MATLAB is a high-performance language for technical computing.
It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming.
These factor make MATLAB an excellent tool for teaching and research.
2012 mdsp pr12 k means mixture of gaussiannozomuhamada
The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.
This document provides an introduction to MATLAB. It covers MATLAB basics like arithmetic, variables, vectors, matrices and built-in functions. It also discusses plotting graphs, programming in MATLAB by creating functions and scripts, and solving systems of linear equations. The document is compiled by Endalkachew Teshome from the Department of Mathematics at AMU for teaching MATLAB.
optimization methods by using matlab.pptxabbas miry
This document discusses optimization techniques in MATLAB. It describes how to perform both unconstrained and constrained optimization. For unconstrained problems, the fminunc function is used to find the minimum of an objective function. For constrained problems, fmincon is used to minimize an objective function subject to inequality, equality, and bound constraints. The document provides an example of using these functions to solve a sample constrained optimization problem.
A machine learning method for efficient design optimization in nano-opticsJCMwave
The document describes a machine learning method for efficient design optimization in nano-optics using Gaussian process regression and Bayesian optimization. It discusses how Gaussian process regression can be used to build regression models from expensive black-box functions to enable model-based optimization and integration. Bayesian optimization is then used to iteratively query the black-box function at points of maximum expected improvement to find its minimum. The method can incorporate derivative observations to speed up optimization by providing additional training data for the Gaussian process. Differential evolution is also utilized to efficiently maximize the expected improvement at each iteration. The approach is demonstrated on benchmark optimization problems, showing it outperforms other algorithms like L-BFGS and particle swarm optimization.
Here are the steps to solve this ODE problem:
1. Define the ODE function:
function dydt = odefun(t,y)
dydt = -t.*y/10;
end
2. Solve the ODE:
[t,y] = ode45(@odefun,[0 10],10);
3. Plot the result:
plot(t,y)
xlabel('t')
ylabel('y(t)')
This uses ode45 to solve the ODE dy/dt = -t*y/10 on the interval [0,10] with initial condition y(0)=10.
This document discusses using MATLAB to solve differential equations related to electric circuits. It begins by explaining some advantages of MATLAB, such as its use of matrices, vectorized operations, and graphical output capabilities. It then provides an example of using MATLAB to solve the first order differential equation iR+Ldi/dt=E(t), which models an LCR circuit. The document also discusses solving second order differential equations manually and with MATLAB code. It provides an example of solving the second order equation d2q/dt2+10dq/dt+250q=0 both manually and using MATLAB code.
The document discusses the Remainder Theorem, which provides a way to factorize polynomials by dividing them by factors and obtaining a remainder. There are two methods for finding the remainder: long division/evaluation and synthetic division. Evaluation involves substituting the factor value into the polynomial, while synthetic division arranges the coefficients and repeatedly multiplies and adds down the line. The document provides examples of using both methods and notes that synthetic division allows determining the full quotient polynomial.
1) The document describes the divide-and-conquer algorithm design paradigm. It can be applied to problems where the input can be divided into smaller subproblems, the subproblems can be solved independently, and the solutions combined to solve the original problem.
2) Binary search is provided as an example divide-and-conquer algorithm. It works by recursively dividing the search space in half and only searching the subspace containing the target value.
3) Finding the maximum and minimum elements in an array is also solved using divide-and-conquer. The array is divided into two halves, the max/min found for each subarray, and the overall max/min determined by comparing the subsolutions.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
This document provides an overview of MATLAB, including:
- MATLAB is a software package for numerical computation, originally designed for linear algebra problems using matrices. It has since expanded to include other scientific computations.
- MATLAB treats all variables as matrices and supports various matrix operations like addition, multiplication, element-wise operations, and matrix manipulation functions.
- MATLAB allows plotting of 2D and 3D graphics, importing/exporting of data from files and Excel, and includes flow control statements like if/else, for loops, and while loops to structure code execution.
- Efficient MATLAB programming involves using built-in functions instead of custom functions, preallocating arrays, and avoiding nested loops where possible through matrix operations.
car rentals in nassau bahamas | atv rental nassau bahamasjustinwilson0857
At Dash Auto Sales & Car Rentals, we take pride in providing top-notch automotive services to residents and visitors alike in Nassau, Bahamas. Whether you're looking to purchase a vehicle, rent a car for your vacation, or embark on an exciting ATV adventure, we have you covered with our wide range of options and exceptional customer service.
Website: www.dashrentacarbah.com
Charging and Fueling Infrastructure Grant: Round 2 by Brandt HertensteinForth
Brandt Hertenstein, Program Manager of the Electrification Coalition gave this presentation at the Forth and Electrification Coalition CFI Grant Program - Overview and Technical Assistance webinar on June 12, 2024.
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It is applicable to problems that exhibit optimal substructure and overlapping subproblems. The matrix chain multiplication problem can be solved using dynamic programming in O(n^3) time by defining the problem recursively, computing the costs of subproblems in a bottom-up manner using dynamic programming, and tracing the optimal solution back from the computed information. Similarly, the longest common subsequence problem exhibits optimal substructure and can be solved using dynamic programming.
This document describes solving a 1D Poisson equation using the finite element method. It provides the analytical solution and defines the problem. MATLAB functions are implemented to generate the mesh, system matrices and vectors, apply boundary conditions, and solve the linear system. Plots compare the analytical and numerical solutions, showing better agreement for finer meshes. Optional sections explore changing the analytical solution, calculating errors for different mesh sizes, and implementing a Neumann boundary condition.
MIT OpenCourseWare provides course materials for the free online course 6.094 Introduction to MATLAB taught in January 2009. The course covers topics like linear algebra, polynomials, optimization, differentiation and integration, and solving differential equations using MATLAB. Lecture 3 focuses on solving systems of linear equations, matrix operations, polynomial fitting to data, nonlinear root finding, function minimization, and numerical methods for differentiation, integration, and solving ordinary differential equations.
Principal Components Analysis (PCA) is an exploratory technique used to reduce the dimensionality of data sets while retaining as much information as possible. It transforms a number of correlated variables into a smaller number of uncorrelated variables called principal components. PCA is commonly used for applications like face recognition, image compression, and gene expression analysis by reducing the dimensions of large data sets and finding patterns in the data.
The following ppt is about principal component analysisSushmit8
Principal Components Analysis (PCA) is an exploratory technique used to reduce the dimensionality of data sets while retaining as much information as possible. It transforms a number of correlated variables into a smaller number of uncorrelated variables called principal components. PCA is commonly used for applications like face recognition, image compression, and gene expression analysis by reducing the dimensions of large data sets and finding patterns in the data.
This document contains the course calendar for a machine learning course covering topics like Bayesian estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, independent component analysis, and clustering algorithms. The calendar lists 15 classes over the semester, the topics to be covered in each class, and any dates where there will be no class. It also includes lecture plans and slides on principal component analysis, linear discriminant analysis, and comparing PCA and LDA.
Here are the steps to solve this problem in MATLAB:
1. Create a matrix A with the marks data:
A = [24 44 36 36;
52 57 68 76;
66 53 69 73;
85 40 86 72;
15 47 25 28;
79 72 82 91];
2. Define the column labels:
subjects = {'Math','Programming','Thermodynamics','Mechanics'};
3. Find the total marks of each subject:
totals = sum(A)
4. Find the average marks of each subject:
averages = mean(A)
5. Find the highest marks scored in each subject:
maxMarks = max
This document discusses various machine learning concepts including notations, statistical learning, support vector machines (SVM), principal component analysis (PCA), and kernel principal component analysis (K-PCA). It provides an overview of supervised and unsupervised learning. It also explains how SVM can be used for linear and non-linear classification problems using kernels to project data into higher dimensions. PCA is described as finding the directions of maximum variance in high-dimensional data. K-PCA is presented as a way to apply PCA in kernel-induced feature spaces to handle non-linear problems.
The name MATLAB stands for MATrix LABoratory.MATLAB is a high-performance language for technical computing.
It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming.
These factor make MATLAB an excellent tool for teaching and research.
2012 mdsp pr12 k means mixture of gaussiannozomuhamada
The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.
This document provides an introduction to MATLAB. It covers MATLAB basics like arithmetic, variables, vectors, matrices and built-in functions. It also discusses plotting graphs, programming in MATLAB by creating functions and scripts, and solving systems of linear equations. The document is compiled by Endalkachew Teshome from the Department of Mathematics at AMU for teaching MATLAB.
optimization methods by using matlab.pptxabbas miry
This document discusses optimization techniques in MATLAB. It describes how to perform both unconstrained and constrained optimization. For unconstrained problems, the fminunc function is used to find the minimum of an objective function. For constrained problems, fmincon is used to minimize an objective function subject to inequality, equality, and bound constraints. The document provides an example of using these functions to solve a sample constrained optimization problem.
A machine learning method for efficient design optimization in nano-opticsJCMwave
The document describes a machine learning method for efficient design optimization in nano-optics using Gaussian process regression and Bayesian optimization. It discusses how Gaussian process regression can be used to build regression models from expensive black-box functions to enable model-based optimization and integration. Bayesian optimization is then used to iteratively query the black-box function at points of maximum expected improvement to find its minimum. The method can incorporate derivative observations to speed up optimization by providing additional training data for the Gaussian process. Differential evolution is also utilized to efficiently maximize the expected improvement at each iteration. The approach is demonstrated on benchmark optimization problems, showing it outperforms other algorithms like L-BFGS and particle swarm optimization.
Here are the steps to solve this ODE problem:
1. Define the ODE function:
function dydt = odefun(t,y)
dydt = -t.*y/10;
end
2. Solve the ODE:
[t,y] = ode45(@odefun,[0 10],10);
3. Plot the result:
plot(t,y)
xlabel('t')
ylabel('y(t)')
This uses ode45 to solve the ODE dy/dt = -t*y/10 on the interval [0,10] with initial condition y(0)=10.
This document discusses using MATLAB to solve differential equations related to electric circuits. It begins by explaining some advantages of MATLAB, such as its use of matrices, vectorized operations, and graphical output capabilities. It then provides an example of using MATLAB to solve the first order differential equation iR+Ldi/dt=E(t), which models an LCR circuit. The document also discusses solving second order differential equations manually and with MATLAB code. It provides an example of solving the second order equation d2q/dt2+10dq/dt+250q=0 both manually and using MATLAB code.
The document discusses the Remainder Theorem, which provides a way to factorize polynomials by dividing them by factors and obtaining a remainder. There are two methods for finding the remainder: long division/evaluation and synthetic division. Evaluation involves substituting the factor value into the polynomial, while synthetic division arranges the coefficients and repeatedly multiplies and adds down the line. The document provides examples of using both methods and notes that synthetic division allows determining the full quotient polynomial.
1) The document describes the divide-and-conquer algorithm design paradigm. It can be applied to problems where the input can be divided into smaller subproblems, the subproblems can be solved independently, and the solutions combined to solve the original problem.
2) Binary search is provided as an example divide-and-conquer algorithm. It works by recursively dividing the search space in half and only searching the subspace containing the target value.
3) Finding the maximum and minimum elements in an array is also solved using divide-and-conquer. The array is divided into two halves, the max/min found for each subarray, and the overall max/min determined by comparing the subsolutions.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
This document provides an overview of MATLAB, including:
- MATLAB is a software package for numerical computation, originally designed for linear algebra problems using matrices. It has since expanded to include other scientific computations.
- MATLAB treats all variables as matrices and supports various matrix operations like addition, multiplication, element-wise operations, and matrix manipulation functions.
- MATLAB allows plotting of 2D and 3D graphics, importing/exporting of data from files and Excel, and includes flow control statements like if/else, for loops, and while loops to structure code execution.
- Efficient MATLAB programming involves using built-in functions instead of custom functions, preallocating arrays, and avoiding nested loops where possible through matrix operations.
car rentals in nassau bahamas | atv rental nassau bahamasjustinwilson0857
At Dash Auto Sales & Car Rentals, we take pride in providing top-notch automotive services to residents and visitors alike in Nassau, Bahamas. Whether you're looking to purchase a vehicle, rent a car for your vacation, or embark on an exciting ATV adventure, we have you covered with our wide range of options and exceptional customer service.
Website: www.dashrentacarbah.com
Charging and Fueling Infrastructure Grant: Round 2 by Brandt HertensteinForth
Brandt Hertenstein, Program Manager of the Electrification Coalition gave this presentation at the Forth and Electrification Coalition CFI Grant Program - Overview and Technical Assistance webinar on June 12, 2024.
Charging Fueling & Infrastructure (CFI) Program by Kevin MillerForth
Kevin Miller, Senior Advisor, Business Models of the Joint Office of Energy and Transportation gave this presentation at the Forth and Electrification Coalition CFI Grant Program - Overview and Technical Assistance webinar on June 12, 2024.
Dahua provides a comprehensive guide on how to install their security camera systems. Learn about the different types of cameras and system components, as well as the installation process.
Top-Quality AC Service for Mini Cooper Optimal Cooling PerformanceMotor Haus
Ensure your Mini Cooper stays cool and comfortable with our top-quality AC service. Our expert technicians provide comprehensive maintenance, repairs, and performance optimization, guaranteeing reliable cooling and peak efficiency. Trust us for quick, professional service that keeps your Mini Cooper's air conditioning system in top condition, ensuring a pleasant driving experience year-round.
Charging Fueling & Infrastructure (CFI) Program Resources by Cat PleinForth
Cat Plein, Development & Communications Director of Forth, gave this presentation at the Forth and Electrification Coalition CFI Grant Program - Overview and Technical Assistance webinar on June 12, 2024.
9. Machine Learning 9
💡 Principal Component Analysis
It Reduces overfitting problem
It reduces dimension (Features) to solve problem of overfitting
Number of Principle component can be less than or equal to Number of features
PC1 will have highest Priority
All Principle components should be independent
Example➖
Steps
Find mean
Find covarainace matrix
10. Machine Learning 10
Steps➖
Find Eigen Values Lambda1 and Lambda2
For each eigen value , find eigen vector
This 2 eigen vectors are our final answer
11. Machine Learning 11
PRACTICE PROBLEMS BASED ON PRINCIPAL
COMPONENT ANALYSIS-
Consider the two dimensional patterns (2,
1), (3, 5), (4, 3), (5, 6), (6, 7), (7, 8). Compute
the principal component using PCA
Algorithm.
Solution-
We use the above discussed PCA Algorithm-
Step-01:
Get data.
The given feature vectors are-
x = (2, 1)
1
x = (3, 5)
2
x = (4, 3)
3
12. Machine Learning 12
x = (5, 6)
4
x = (6, 7)
5
x = (7, 8)
6
Step-02:
Calculate the mean vector (µ).
Mean vector (µ)
= ((2 + 3 + 4 + 5 + 6 + 7) / 6, (1 + 5 + 3 + 6 + 7 + 8) / 6)
= (4.5, 5)
Thus,
Step-03:
Subtract mean vector (µ) from the given feature vectors.
x – µ = (2 – 4.5, 1 – 5) = (-2.5, -4)
1
x – µ = (3 – 4.5, 5 – 5) = (-1.5, 0)
2
x – µ = (4 – 4.5, 3 – 5) = (-0.5, -2)
3
x – µ = (5 – 4.5, 6 – 5) = (0.5, 1)
4
x – µ = (6 – 4.5, 7 – 5) = (1.5, 2)
5
x – µ = (7 – 4.5, 8 – 5) = (2.5, 3)
6
Feature vectors (xi) after subtracting mean vector (µ) are-
Step-04:
Calculate the covariance matrix.
Covariance matrix is given by-
13. Machine Learning 13
Now,
Now,
Covariance matrix
= (m1 + m2 + m3 + m4 + m5 + m6) / 6
On adding the above matrices and dividing by 6, we get-
Step-05:
Calculate the eigen values and eigen vectors of the covariance matrix.
λ is an eigen value for a matrix M if it is a solution of the characteristic equation |M – λI|
= 0.
So, we have-
From here,
(2.92 – λ)(5.67 – λ) – (3.67 x 3.67) = 0
16.56 – 2.92λ – 5.67λ + λ2 – 13.47 = 0
λ2 – 8.59λ + 3.09 = 0
Solving this quadratic equation, we get λ = 8.22, 0.38
Thus, two eigen values are λ1 = 8.22 and λ2 = 0.38.
Clearly, the second eigen value is very small compared to the first eigen value.
14. Machine Learning 14
So, the second eigen vector can be left out.
Eigen vector corresponding to the greatest eigen value is the principal component for
the given data set.
So. we find the eigen vector corresponding to eigen value λ1.
We use the following equation to find the eigen vector-
MX = λX
where-
M = Covariance Matrix
X = Eigen vector
λ = Eigen value
Substituting the values in the above equation, we get-
Solving these, we get-
2.92X1 + 3.67X2 = 8.22X1
3.67X1 + 5.67X2 = 8.22X2
On simplification, we get-
5.3X1 = 3.67X2 ………(1)
3.67X1 = 2.55X2 ………(2)
From (1) and (2), X1 = 0.69X2
From (2), the eigen vector is-
Thus, principal component for the given data set is-
Lastly, we project the data points onto the new subspace as-
46. ENSEMBLE TECHNIQUE
In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain
better predictive performance than could be obtained from any of the constituent learning
algorithms alone.