ExcelR offers 160 hours classroom training on Business Analytics / Data Scientist / Data Analytics. We are considered as one of the best training institutes on Business Analytics in Pune. “Faculty and vast course agenda is our differentiator”. The training is conducted by alumni of premier institutions such as IIT & ISB
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
This document provides an overview of Project Management Professional (PMP) certification. It discusses why the PMP is valuable for career advancement and increased success in project management. It outlines the eligibility requirements, including education, training hours, and experience leading projects. The examination process is also summarized, including the distribution of questions across different process groups, grading procedures, and scheduling an exam.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
This document provides an overview of Project Management Professional (PMP) certification. It discusses why the PMP is valuable for career advancement and increased success in project management. It outlines the eligibility requirements, including education, training hours, and experience leading projects. The examination process is also summarized, including the distribution of questions across different process groups, grading procedures, and scheduling an exam.
This document contains an agenda for a training program on data science using R, Minitab, and XLMiner. The agenda covers topics like data visualization using Tableau, supervised and unsupervised data mining techniques, text mining and natural language processing, statistical analysis, data mining, forecasting, and an introduction to big data. It emphasizes that data scientists need skills in domains, statistics, data mining, visualization, and forecasting. Brief sections provide overviews of data types, random variables, probability, probability distributions, sampling, and how to contact the training providers.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
This document provides an overview of multinomial regression. It begins with an agenda that includes multinomial regression, zero-inflated Poisson regression, and negative binomial regression. It then discusses multinomial regression in more detail, explaining that it is an extension of logistic regression used when the output has more than two categories. It also discusses model outputs such as iteration history, log odds ratios, and goodness of fit measures like residual deviance and AIC criterion.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
https://www.excelr.com/pmi-agile-certified-practitioner/
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
https://www.excelr.com/pmi-agile-certified-practitioner/
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
This document provides information about the Project Management Professional (PMP) certification. It discusses the eligibility requirements to take the PMP exam, including education and experience requirements. It also describes the exam pattern, including the 4 hour duration, 200 questions with 25 being pre-test questions, grading across five process groups, and no negative marking. Finally, it provides a high-level introduction to project management concepts like the project life cycle, development life cycles, and phase gates.
https://www.excelr.com/pmi-agile-certified-practitioner/
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
This document provides information about the Project Management Professional (PMP) certification. It discusses the eligibility requirements to take the PMP exam, including education and experience requirements. It also describes the exam pattern, including the 4 hour duration, 200 questions with 25 being pre-test questions, grading across five process groups, and no negative marking. Finally, it provides a high-level introduction to project management concepts like the project life cycle, development life cycle, and phase gates.
This document provides an overview of Project Management Professional (PMP) certification. It discusses why the PMP is valuable for career advancement and increased success in project management. The eligibility requirements are presented, including education, training hours, and experience leading projects. The exam pattern is covered, including the 4 hour duration, question distribution across process groups, and grading system. An introduction to fundamental project management concepts is also provided, such as the definition of a project, relationship to programs and portfolios, and typical project constraints and life cycles.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
This document contains an agenda for a training program on data science using R, Minitab, and XLMiner. The agenda covers topics like data visualization using Tableau, supervised and unsupervised data mining techniques, text mining and natural language processing, statistical analysis, data mining, forecasting, and an introduction to big data. It emphasizes that data scientists need skills in domains, statistics, data mining, visualization, and forecasting. Brief sections provide overviews of data types, random variables, probability, probability distributions, sampling, and how to contact the training providers.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
This document provides an overview of multinomial regression. It begins with an agenda that includes multinomial regression, zero-inflated Poisson regression, and negative binomial regression. It then discusses multinomial regression in more detail, explaining that it is an extension of logistic regression used when the output has more than two categories. It also discusses model outputs such as iteration history, log odds ratios, and goodness of fit measures like residual deviance and AIC criterion.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
https://www.excelr.com/pmi-agile-certified-practitioner/
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
https://www.excelr.com/pmi-agile-certified-practitioner/
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
This document provides information about the Project Management Professional (PMP) certification. It discusses the eligibility requirements to take the PMP exam, including education and experience requirements. It also describes the exam pattern, including the 4 hour duration, 200 questions with 25 being pre-test questions, grading across five process groups, and no negative marking. Finally, it provides a high-level introduction to project management concepts like the project life cycle, development life cycles, and phase gates.
https://www.excelr.com/pmi-agile-certified-practitioner/
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
This document provides information about the Project Management Professional (PMP) certification. It discusses the eligibility requirements to take the PMP exam, including education and experience requirements. It also describes the exam pattern, including the 4 hour duration, 200 questions with 25 being pre-test questions, grading across five process groups, and no negative marking. Finally, it provides a high-level introduction to project management concepts like the project life cycle, development life cycle, and phase gates.
This document provides an overview of Project Management Professional (PMP) certification. It discusses why the PMP is valuable for career advancement and increased success in project management. The eligibility requirements are presented, including education, training hours, and experience leading projects. The exam pattern is covered, including the 4 hour duration, question distribution across process groups, and grading system. An introduction to fundamental project management concepts is also provided, such as the definition of a project, relationship to programs and portfolios, and typical project constraints and life cycles.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
PMI® Agile Certified Practitioner (PMI-ACP)® certification is the most sort after in the agile world. This includes the agile practices from various methodologies such as Scrum, Extreme Programming (XP), Lean, Kanban, DSDM Atern, etc. The World has realized that “one size does not fit all”, hence (PMI-ACP)® helps you perform a fit-gap analysis and use practices from various agile methodologies which best addresses the pain area. The training offered by ExcelR Solutions includes the content which is aligned with PMI®.
ExcelR is a proud partner of University Malaysia Sarawak (UNIMAS), Malaysia's 1st public University and ranked 8th top university in Malaysia and ranked among top 200th in Asian University Rankings 2017 by QS World University Rankings. Participants will be awarded Data Science International Certification from UNIMAS after successfully clearing the online examination.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
2. Application of dimension
reduction
• Computational advantage for other
algorithms
• Face recognition— image data (pixels)
along new axes works better for
recognizing faces
• Image compression
4. Input Output
Univ SAT Top10 Accept SFRatio Expenses GradRate PC1 PC2 PC3 PC4 PC5 PC6
Brown 1310 89 22 13 22,704 94
CalTech 1415 100 25 6 63,575 81
CMU 1260 62 59 9 25,026 72
Columbia 1310 76 24 12 31,510 88
Cornell 1280 83 33 13 21,864 90
Dartmouth 1340 89 23 10 32,162 95
Duke 1315 90 30 12 31,585 95
Georgetown 1255 74 24 12 20,126 92
Harvard 1400 91 14 11 39,525 97
JohnsHopkins 1305 75 44 7 58,691 87
Hope is that a fewer columns may
capture most of the information
from the original dataset
PCA
5. The Primitive Idea – Intuition
First
How to compress the data
loosing the least amount of information?
6. Input
• p measurements/
original columns
• Correlated
Output
• p principal components
(= p weighted averages
of original
measurements)
• Uncorrelated
• Ordered by variance
• Keep top principal
components; drop rest
PCA
7. Mechanism
The ith principal component is a weighted average of
original measurements/columns:
Weights (aij) are chosen such that:
1. PCs are ordered by their variance (PC1 has largest
variance, followed by PC2, PC3, and so on)
2. Pairs of PCs have correlation = 0
3. For each PC, sum of squared weights =1
pip2i21i1i XaXaXaPC +…++=
Univ SAT Top10 Accept SFRatio Expenses GradRate PC1 PC2 PC3 PC4 PC5 PC6
Brown 1310 89 22 13 22,704 94
CalTech 1415 100 25 6 63,575 81
CMU 1260 62 59 9 25,026 72
Columbia 1310 76 24 12 31,510 88
Cornell 1280 83 33 13 21,864 90
Dartmouth 1340 89 23 10 32,162 95
Duke 1315 90 30 12 31,585 95
Georgetown 1255 74 24 12 20,126 92
Harvard 1400 91 14 11 39,525 97
JohnsHopkins 1305 75 44 7 58,691 87
8. Demystifying weight computation
• Main idea: high variance = lots of information
• Goal: Find weights aij that maximize variance
of PCi, while keeping PCi uncorrelated to
other PCs.
• The covariance matrix of the X’s is needed.
jiPCPC
),XCov(Xaa),XCov(Xaa
)Var(X)Var(X)Var(X)Var(PC
ji
pp-ipi p-ii
pipiii aaa
≠=
+…++
++…++=
when,0),(Covarwant,Also
22 112121
2
2
2
21
2
1
pip2i21i1i XaXaXaPC +…++=
10. Standardization shortcut for PCA
• Rather than standardize the data
manually, you can use correlation
matrix instead of covariance matrix
as input
• PCA with and without standardization
gives different results!
11. PCA
Transform > Principal Components
(correlation matrix has been used here)
• PCs are uncorrelated
• Var(PC1) > Var (PC2)
> ...
pip2i21i1i XaXaXaPC +…++=
Scaled Data PC Scores
Principal
Components
12. Computing principal scores
• For each record, we can compute their score
on each PC.
• Multiply each weight (aij) by the appropriate
Xij
• Example for Brown University (using
standardized numbers):
• PC Score1 for Brown University = (–
0.458)(0.40) +(–0.427)(.64) +(0.424)(–0.87)
+(0.391)(.07) + (–0.363)(–0.32) + (–0.379)(.80) =
–0.989
13. R Code for PCA (Assignment)
OPTIONAL R Code
install.packages("gdata") ## for reading xls files
install.packages("xlsx") ## ” for reading xlsx files
mydata<-read.xlsx("University Ranking.xlsx",1) ## use read.csv for csv files
mydata ## make sure the data is loaded correctly
help(princomp) ## to understand the api for princomp
pcaObj<-princomp(mydata[1:25,2:7], cor = TRUE, scores = TRUE, covmat = NULL)
## the first column in mydata has university names
## princomp(mydata, cor = TRUE) not_same_as prcomp(mydata, scale=TRUE); similar, but different
summary(pcaObj)
loadings(pcaObj)
plot(pcaObj)
biplot(pcaObj)
pcaObj$loadings
pcaObj$scores
14. Goal #1: Reduce data dimension
• PCs are ordered by their variance (=information)
• Choose top few PCs and drop the rest!
Example:
• PC1 captures most ??% of the information.
• The first 2 PCs capture ??%
• Data reduction: use only two variables instead of
6.
15. Matrix Transpose
OPTIONAL: R code
help(matrix)
A<-matrix(c(1,2),nrow=1,ncol=2,byrow=TRUE)
A
t(A)
B<-matrix(c(1,2,3,4),nrow=2,ncol=2,byrow=TRUE)
B
t(B)
C<matrix(c(1,2,3,4,5,6),nrow=3,ncol=2,byrow=TRUE)
C
t(C)
16. Matrix Multiplication
OPTIONAL R Code
A<-
matrix(c(1,2,3,4,5,6),nrow=3,ncol=2,byrow=
TRUE)
A
B<-
matrix(c(1,2,3,4,5,6,7,8),nrow=2,ncol=4,byro
w=TRUE)
B
C<-A%*%B
D<-t(B)%*%t(A) ## note, B%*%A is not
possible; how does D look like?
18. Data Compression
[ ]
[ ]
[ ] [ ]
[ ]
[ ] [ ] [ ]omponentsPrincipalCPCScorestaedScaledDaApproximat
:ionApproximat
omponentsPrincipalCPCScores
omponentsPrincipalCPCScores
ScaledData
omponentsPrincipalCScaledDataPCScores
1
T
pc
cNpN
T
pppN
pp
pN
pN
pppNpN
×
××
××
−
×
×
×
×
××
×=
×=
×=
×=
c = Number of
components
kept; c ≤ p
19. Goal #2: Learn relationships with
PCA by interpreting the weights
• ai1,…, aip are the coefficients for PCi.
• They describe the role of original X
variables in computing PCi.
• Useful in providing context-specific
interpretation of each PC.
20. PC1 Scores
(choose one or more)
1. are approximately a simple average of
the 6 variables
2. measure the degree of high Accept &
SFRatio, but low Expenses, GradRate,
SAT, and Top10
21. Goal #3: Use PCA for visualization
• The first 2 (or 3) PCs provide a way to
project the data from a p-dimensional
space onto a 2D (or 3D) space
23. Monitoring batch processes
using PCA
• Multivariate data at different time points
• Historical database of successful batches
are used
• Multivariate trajectory data is projected
to low-dimensional space
>>> Simple monitoring charts to spot outlier
24. Your Turn!
1. If we use a subset of the principal
components, is this useful for
prediction? for explanation?
2. What are advantages and
weaknesses of PCA compared to
choosing a subset of the variables?
3. PCA vs. Clustering