Principal component analysis (PCA) is a technique used to reduce the dimensionality of large data sets by transforming correlated variables into a smaller number of uncorrelated variables called principal components. PCA identifies patterns in data and expresses the data in such a way as to highlight their similarities and differences. The main goals of PCA are data reduction and interpretation. It works by identifying the directions (principal components) along which the variation in the data is maximized.
Using several mathematical examples from three different authors in texts from different courses this paper illustrates the easier way to avoid confusions and always get the correct results with the least effort was to use the proposed Excel Gamma function explained in detail for the proper use of the Q(z) and ercf(x) functions in most communication courses. The paper serves as a tutorial and introduction for such functions
Using several mathematical examples from three different authors in texts from different courses this paper illustrates the easier way to avoid confusions and always get the correct results with the least effort was to use the proposed Excel Gamma function explained in detail for the proper use of the Q(z) and ercf(x) functions in most communication courses. The paper serves as a tutorial and introduction for such functions
Forecasting day ahead power prices in germany using fixed size least squares ...Niklas Ignell
By using less than 0.5% of the original training dataset the aggregated out-of sample result, over the
period the model was tested, 7 days in May 2017, shows that the average difference between the actual
and the forecasted average daily hourly German EPEX power prices differed 0.4%. The presence of
outliers, heteroskedastic residuals and sparseness of prices at lower price levels in the training data set
can explain that two of the days in the test period differed by more than +/- 10%.
The aim of this report is to use eigenvectors, eigenvalues, and orthogonality to understand the concept of Principal Component Analysis (PCA) and to show why PCA is useful.
Algorithmic entropy can be seen as a special case of entropy as studied in
statistical mechanics. This viewpoint allows us to apply many techniques
developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing
KNN and ARL Based Imputation to Estimate Missing Valuesijeei-iaes
Missing data are the absence of data items for a subject; they hide some information that may be important. In practice, missing data have been one major factor affecting data quality. Thus, Missing value imputation is needed. Methods such as hierarchical clustering and K-means clustering are not robust to missing data and may lose effectiveness even with a few missing values. Therefore, to improve the quality of data method for missing value imputation is needed. In this paper KNN and ARL based Imputation are introduced to impute missing values and accuracy of both the algorithms are measured by using normalized root mean sqare error. The result shows that ARL is more accurate and robust method for missing value estimation.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
Arthur B. Weglein, Hong Liang, and Chao Ma M-OSRP/Physics Dept./University o...Arthur Weglein
Arthur B. Weglein, a professor in the Department of Physics and the Department of Earth and Atmospheric Sciences in Houston TX. Read more on Research & Awards.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Forecasting day ahead power prices in germany using fixed size least squares ...Niklas Ignell
By using less than 0.5% of the original training dataset the aggregated out-of sample result, over the
period the model was tested, 7 days in May 2017, shows that the average difference between the actual
and the forecasted average daily hourly German EPEX power prices differed 0.4%. The presence of
outliers, heteroskedastic residuals and sparseness of prices at lower price levels in the training data set
can explain that two of the days in the test period differed by more than +/- 10%.
The aim of this report is to use eigenvectors, eigenvalues, and orthogonality to understand the concept of Principal Component Analysis (PCA) and to show why PCA is useful.
Algorithmic entropy can be seen as a special case of entropy as studied in
statistical mechanics. This viewpoint allows us to apply many techniques
developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing
KNN and ARL Based Imputation to Estimate Missing Valuesijeei-iaes
Missing data are the absence of data items for a subject; they hide some information that may be important. In practice, missing data have been one major factor affecting data quality. Thus, Missing value imputation is needed. Methods such as hierarchical clustering and K-means clustering are not robust to missing data and may lose effectiveness even with a few missing values. Therefore, to improve the quality of data method for missing value imputation is needed. In this paper KNN and ARL based Imputation are introduced to impute missing values and accuracy of both the algorithms are measured by using normalized root mean sqare error. The result shows that ARL is more accurate and robust method for missing value estimation.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
Numerical Study of Some Iterative Methods for Solving Nonlinear Equationsinventionjournals
In this paper we introduce, numerical study of some iterative methods for solving non linear equations. Many iterative methods for solving algebraic and transcendental equations is presented by the different formulae. Using bisection method , secant method and the Newton’s iterative method and their results are compared. The software, matlab 2009a was used to find the root of the function for the interval [0,1]. Numerical rate of convergence of root has been found in each calculation. It was observed that the Bisection method converges at the 47 iteration while Newton and Secant methods converge to the exact root of 0.36042170296032 with error level at the 4th and 5th iteration respectively. It was also observed that the Newton method required less number of iteration in comparison to that of secant method. However, when we compare performance, we must compare both cost and speed of convergence [6]. It was then concluded that of the three methods considered, Secant method is the most effective scheme. By the use of numerical experiments to show that secant method are more efficient than others.
Arthur B. Weglein, Hong Liang, and Chao Ma M-OSRP/Physics Dept./University o...Arthur Weglein
Arthur B. Weglein, a professor in the Department of Physics and the Department of Earth and Atmospheric Sciences in Houston TX. Read more on Research & Awards.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Strong differences in personalities and views can lead to robust discussions and decisions - or a dysfunctional Board of limited value. This article explores how directors can become a meaningful part of the dialogue. Phillip Ralph from The Leadership Sphere contributes a thought leadership piece in the article.
Our intention in writing a Leadership Declaration is to clearly state the case for leadership and its importance to all of us at this time. At an organisational level, there is little doubt that real leadership is the “engine room” of performance. In the absence of systemic, results-focused leadership, breakthrough performance and high commitment will not be achieved.
In an ever-changing world where we are all being asked to do more with less, the cracks are starting to show. Global studies of organisations highlight the unrelenting need for uplift in performance - yet many of those responsible for the uplift (managers and teams) indicate they don't have any more to give. Within this environment, the productivity and role of teams becomes critical to success. Experience tells us however that not only do many teams fail to reach their full potential; they can in fact unknowingly impede their own performance.
In this presentation, you will learn:
1. How focusing solely on improving performance can actually undermine its effectiveness.
2. Successfully coaching a team is actually like a marriage - it needs continued investment, honest dialogue and support to make it a success.
3. Practical tools and tips to coach your team or the teams you are responsible for to new levels by focusing on the drivers of exceptional performance.
TAKEON! IS A PROGRAM FOR IMPROVING BUSINESS PERFORMANCE THAT GETS PEOPLE WORKING TOGETHER ON WHAT MATTERS MOST.
The results are immediate and measurable.
TakeON! resources and concepts are easily woven into existing practices.
You own it, you lead it, it’s your take on what matters to your business now.
Imagine people across your organisation coming together regularly, discussing what’s already working, what could be improved, and how they can contribute. Dozens of suggestions are generated and acted upon. The power comes not from a single silver-bullet idea, but in creating a culture of constant incremental change.
TakeON! enables these conversations at leader level or across your whole business. What’s more, it focuses them on the specific challenges that you face today. This creates quick wins that build confidence and momentum across the business.
Identifying Partisan Slant in News Articles and Twitter during Political CrisesDima Karamshuk
In this paper, we are interested in understanding the interrelationships between mainstream and social media in forming public opinion during mass crises, specifically in regards to how events are framed in the mainstream news and on social networks and to how the language used in those frames may allow to infer political slant and partisanship. We study the lingual choices for political agenda setting in mainstream and social media by analyzing a dataset of more than 40M tweets and more than 4M news articles from the mass protests in Ukraine during 2013-2014 — known as "Euromaidan" — and the post-Euromaidan conflict between Russian, pro-Russian and Ukrainian forces in eastern Ukraine and Crimea. We design a natural language processing algorithm to analyze at scale the linguistic markers which point to a particular political leaning in online media and show that political slant in news articles and Twitter posts can be inferred with a high level of accuracy. These findings allow us to better understand the dynamics of partisan opinion formation during mass crises and the interplay between mainstream and social media in such circumstances.
Take-away TV: Recharging Work Commutes with Greedy and Predictive Preloading ...Dima Karamshuk
Mobile data offloading can greatly decrease the load on and usage of cellular data networks by exploiting opportunistic and frequent access to Wi- Fi connectivity. Unfortunately, Wi-Fi access from mobile devices can be difficult during typical work commutes, e.g., via trains or cars on highways. In this paper, we propose a new approach: to preload the mobile device with content that a user might be interested in, and thereby avoid the need for cellular data access. We demonstrate the feasibility of this approach by developing a supervised machine learning model that learns from user preferences for different types of content, and propensity to be guided by the UI of the player, and predictively preload entire TV shows. Testing on a dataset of nearly 3.9 million sessions from all over the UK to BBC TV shows, we find that predictive preloading can save significant share of the mobile data for an average user.
Robust Fuzzy Data Clustering In An Ordinal Scale Based On A Similarity MeasureIJRES Journal
This paper is devoted to processing data given in an ordinal scale. A new objective function of a
special type is introduced. A group of robust fuzzy clustering algorithms based on the similarity measure is
introduced.
Covariance matrices are central to many adaptive filtering and optimisation problems. In practice, they have to be estimated from a finite number of samples; on this, I will review some known results from spectrum estimation and multiple-input multiple-output communications systems, and how properties that are assumed to be inherent in covariance and power spectral densities can easily be lost in the estimation process. I will discuss new results on space-time covariance estimation, and how the estimation from finite sample sets will impact on factorisations such as the eigenvalue decomposition, which is often key to solving the introductory optimisation problems. The purpose of the presentation is to give you some insight into estimating statistics as well as to provide a glimpse on classical signal processing challenges such as the separation of sources from a mixture of signals.
This research paper demonstrates the invention of the kinetic bands, based on Romanian mathematician and statistician Octav Onicescu’s kinetic energy, also known as “informational energy”, where we use historical data of foreign exchange currencies or indexes to predict the trend displayed by a stock or an index and whether it will go up or down in the future. Here, we explore the imperfections of the Bollinger Bands to determine a more sophisticated triplet of indicators that predict the future movement of prices in the Stock Market. An Extreme Gradient Boosting Modelling was conducted in Python using historical data set from Kaggle, the historical data set spanning all current 500 companies listed. An invariable importance feature was plotted. The results displayed that Kinetic Bands, derived from (KE) are very influential as features or technical indicators of stock market trends. Furthermore, experiments done through this invention provide tangible evidence of the empirical aspects of it. The machine learning code has low chances of error if all the proper procedures and coding are in play. The experiment samples are attached to this study for future references or scrutiny.
Global Futures & Strategic Foresight (GFSF) program enhances and uses a coordinated suite of biophysical and socioeconomic models to assess potential returns to investments in new agricultural technologies and policies. These models include IFPRI’s International Model for Policy Analysis of Agricultural Commodities and Trade (IMPACT), hydrology and water supply-demand models, and the DSSAT suite of process-based crop models.
The program also provides tools and trainings to scientists and policy makers to undertake similar assessments.
GFSF program is a Consultative Group on International Agricultural Research (CGIAR) program led by the International Food Policy Research Institute (IFPRI)
Similar to Mva 06 principal_component_analysis_2010_11 (20)
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Neuro-symbolic is not enough, we need neuro-*semantic*
Mva 06 principal_component_analysis_2010_11
1. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 1
9 Principal Component Analysis
A principal component analysis (PCA) is concerned with explaining the variance-covariance
structure of a set of variables through a few linear combinations of these variables, called
principal components. Its general objectives are:
• data reduction and
• interpretation.
In general, p principal components are required to reproduce the total system of variability of
the original data set (n measurements on p variables). Fortunatelly, much of this variability can
often be accounted for by a small number of k of principal components. If so, there is (almost)
as much information in the first k components as there is in the original p variables. The k first
principal components can then replace the initial p variables, and the original n p× data set is
reduced to n k× data set consisting of n measurements on k principal components.
An analysis of principal components often reveals relationships that were not previously
suspected and thereby allows interpretations that would not ordinarily result.
Principal components also frequently serve as intermediate steps in much larger investigations,
e.g. as inputs to a multiple regression, cluster analysis, etc.
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 2
Example
Suppose one would like to investigate the level of the socio-economic development of some
European countries in the year 1981. An investigation will take into account the following set of
economic, demographic, health, social security and level of living indicators:
• Per capita gross domestic product in $
• Share of agriculture in gross domestic product (%)
• Share of service activities in gross domestic product (%)
• Export/import ratio
• Per capita fuel consumption in kilograms of coal
• Natural change of population (rates per 1000 inhabitants)
• Share of urban population (%)
• Infant mortality per 1000 live birth
• Number of students per 1000 inhabitants
• Number of TV sets per 1000 inhabitants
2. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 3
9.1 Geometry of Principal Component Analysis
Example
Suppose we have a data set of 12 measurements on 2 variables 1X and 2X for 12 randomly
selected units (Sharma, 1966, p. 59). Let us calculate their mean-corrected values.
Table 1
1ix 2ix 1,cix 2,cix
16 8 8 5
12 10 4 7
13 6 5 3
11 2 3 -1
10 8 2 5
9 -1 1 -4
8 4 0 1
7 6 -1 3
5 -3 -3 -6
3 -1 -5 -4
2 -3 -6 -6
0 0 -8 -3
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 4
The position of units can be presented with points in the two-dimensional space. The
coordinates of the points are the values of mean-corrected variables 1,cX and 2,cX :
X1C
1086420-2-4-6-8-10
X2C
10
8
6
4
2
0
-2
-4
-6
-8
-10
12
11
10
9
8
7
6
5
4
3
2
1
3. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 5
9.1.1 Identification of Alternative Axes and Forming New Variables
Let *
1,cX be any axis in the two dimensional space that goes through the origin of the two
rectangular axes 1,cX and 2,cX 1. Axis *
1,cX is making an angle of θ degrees with 1,cX . The
perpendicular projections of the units (observations) onto *
1,cX will give the coordinates of the
observations with respect to *
1,cX . These new coordinates are linear combinations of the
coordinates of the points with respect to the original set of axes 1,cX and 2,cX :
*
1,c 1,c 2,ccos sinX X Xθ θ= ⋅ + ⋅
There is one and only one new axis 1, cξ that results in a new variable accounting for the
maximum variance in the data. In our case this axis makes an angle of o
43,261 with 1,cX . The
corresponding equation for computing the values of 1,cξ is
o o
1,c 1,c 2,c 1,c 2,ccos43,261 sin 43,261 0,728 0,685X X X Xξ = ⋅ + ⋅ = + ,
while its values are
1,c 1,c 2,c0,728 0,685i i ix xξ = + , 1,2, ,i n= … .
1 The origin 1, 2,( , ) (0,0)c cx x ′ ′= , i.e. the centroid, is always part of the optimal subspace in the sence of
least squares.
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 6
Of course, a one-dimensional space represented by the new axis 1, cξ (in general) does not
account for all the variance of the investigated phenomena, that has been originally presented by
the values of the two variables 1,cX and 2,cX in a two-dimensional space. Therefore, it is
possible to identify a second axis 2, cξ such that the corresponding new variable accounts for the
maximum of the variance that is not accounted for by 1, cξ . Let 2, cξ be the second new axis that
is orthogonal to 1, cξ . Thus, if the angle between 1, cξ and 1,cX is θ then the angle between 2, cξ
and 2,cX will also be θ.
The equation for computing the values of 2,cξ is
o o
2,c 1,c 2,c 1,c 2,csin 43,261 cos43,261 0,685 0,728X X X Xξ = − ⋅ + ⋅ = − + ,
while its values are
2,c 1,c 2,c0,685 0,728i i ix xξ = − + , 1,2, ,i n= … .
4. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 7
The following conclusions can be made from the above figure and the statistical measures:
• the perpendicular projections of the points onto the original axes give the values of the
original variables 1,cX and 2,cX , and the perpendicular projections of the points onto the
new axes give the values for the new variables 1, cξ and 2, cξ . The new axes and the
corresponding variables are called principal components and the values of the new variables
are called principal component scores. Each of the new variables are linear combinations of
the original variables and remain mean-corrected.
• The total variance of the principal components is the same as the total variance of the
original variables.The variance accounted for by the first principal component is greater than
the variance accounted for by any one of the original variables.
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 8
The geometrical illustration of principal component analysis can be easily extended to more
than two variables. An n p× data set now consists of p variables and each unit (observation)
can be represented as a point in a p-dimensional space with respect to the p new axes – principal
components. The projections of points on principal components are called principal component
scores.
If a substantial amount of the total variance in the data set is accounted for by a few first
principal components, than we can use these principal components for further analysis or for
interpretations instead of the original variables. This would result in a substantial data reduction
– an n k× data set ( k p ) of principal component scores is sufficient for further analysis.
Hence, principal component analysis is commonly referred to as a data-reduction technique.
9.2 Analytical Approach
Let us form the following p linear combinations:
1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
p p
p p
p p p pp p
w X w X w X
w X w X w X
w X w X w X
ξ
ξ
ξ
= + + +
= + + +
= + + +
…
…
…
where 1 2, , , pξ ξ ξ… are the p principal components and jkw ( , 1,2, , )j k p= … is the weight of the
k-th variable for the j-th principal component.
5. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 9
The principal component weights are estimated in such a way that:
1. The first principal component, 1ξ , accounts for the maximum variance in the data, the
second principal component, 2ξ , accounts for the maximum variance that has not been
accounted for by the first principal component, and so on
2. For each principal component, the sum of squares of its weights should be equal to 1
2
1
1
p
jk
k
w
=
=∑ , 1,2, ,j p= …
3. Sum of the products of the corresponding weights of two principal components should be
equal to 0
1
0
p
jk j k
k
w w ′′
=
=∑ , j j′′≠
The last condition ensures that principal components are ortogonal to each other.
How do we obtain the weights such that the above listed conditions are satisfied? We are
dealing with an optimization problem, usually based on covariance or correlation matrix. We
need to calculate eigen vectors, that define principal component weights, and eigenvalues that
represent variances of principal components.
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 10
9.3 Issues Relating to the Use of Principal Component Analysis
9.3.1 Effect of Type of Data on Principal Component Analysis
Principal component analysis can either be done on raw or mean-corrected data on one hand or
on standardised data on the other. Each data set could give a different solution depending upon
the extent to which the variances of the variables differ.
In case of raw or mean-corrected data, the basis for principal component analysis is covariance
matrix. The influence of an individual variable on principal components is determined by the
magnitude of its variance. The higher the variance of the variable, the stronger the effect of a
variable on principal components.
In case of standardized data, the basis for principal component analysis is correlation matrix. All
the variances are equal to 1 and therefore they all have the same influence on principal
components.
In cases for which there is a reason to believe that the variances of the variables do indicate the
importance of given variable and the units of measure are commensurable, the raw or the mean-
corrected data should be used. In all other cases standardised data are preferable alternative.
6. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 11
9.3.2 Is Principal Component Analysis the Appropriate Technique
The use of principal component analysis is appropriate at least in two cases:
• if principal components have meaningful interpretation, what is particularly important for
their further use in other statistical techniques and/or
• if the objective is to reduce the number of variables in the data set to a few principal
components without a substantial loss of information.
Principal component analysis is most appropriate if the variables are interrelated, for only then
will it be possible to reduce a number of variables to a manageble few without much loss of
information.
Many statistical tests are available for determining if the variables are significantly correlated
among themselves. For standardised data we can use Bartlett's test, but we should keep in mind
that it is very sensitive on the sample size:
0H : =P I , 1H : ≠P I
2 1
6
( 1) (2 5) lnn pχ = − − − +⎡ ⎤⎣ ⎦ R
2
( )/ 2m p p= −
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 12
9.3.3 Number of Principal Components to Extract
We suggest the use of the following two empirical rules :
1. Kaiser's rule
In the case of standardised data, retain only those components whose eigenvalues (variances)
are greater than 1.
, s
2
1jj ξλ σ= ≥
The rationale for this rule is that for standardised data the amount of variance extracted by
each component should, at minimum, be equal to the variance of at least one variable.
2. Scree plot (Cattell, 1966)
Plot the percentage of variance (or the eigenvalue) accounted for by each of principal
components (on vertical axis) against the ordinal number of the components (on horizontal
axis) and look for an elbow.
However, no one rule is best under all circumstances. One should take into consideration the
purpose of the study, the type of data, and the trade-off between parsimony and the amount of
variation in the data that the researcher is willing to sacrifice in order to achieve parsimony.
Lastly, and more importantly, one should determine the interpretability of the principal
components in deciding upon how many principal components should be retained (Sharma,
1996, p. 79)
7. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 13
9.3.4 Interpreting Principal Components
Since principal components are linear combinations of the original variables, one can use
loadings (simple correlations between the original variables and principal components) for
interpreting the principal components. The higher the loading of a variable, the more influence it
has in the formation of the principal component score and vice versa. Traditionally, a loading of
0.5 or above is used as the cutoff point.
9.3.5 Use of Principal Component Scores
The principal component scores can be plotted for further interpreting the results. Based on
visual examination of the plot, clusters can be defined.
Principal component scores can also be used as input variables for further analysing the data
using other multivariate techniques such as cluster analysis, multiple regression, and
discriminant analysis. The advantage of using principal component scores is that they are not
correlated and the problem of multicollinearity is avoided. Unfortunatelly, a new problem can
arise due to the inability to meaningfully interpret the principal components.
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 14
Example (the level of the socio-economic development of some European countries - continued)
GET
FILE='F:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.sav'.
EXECUTE .
LIST .
List
country gdp agric service expimp energy growth urban infmort student tv
Austria 8725 4,4 55,7 ,753 4160,00 ,0 54 14,7 15,7 290
Belgium 9702 2,1 61,2 ,896 6037,00 ,1 72 11,1 20,3 296
Bulgaria 4150 16,9 25,4 1,136 5678,00 ,7 64 19,8 13,2 200
Czechslovakia 5820 8,4 16,9 ,983 6482,00 ,7 63 15,8 12,5 252
Denmark 10874 4,8 66,7 ,898 5225,00 ,3 84 8,8 20,5 361
Finland 10028 8,2 56,2 ,987 5135,00 ,3 62 7,6 17,4 318
France 12214 4,2 60,1 ,840 4351,00 ,4 78 10,1 19,0 299
Greece 3887 15,5 56,7 ,477 2137,00 1,1 62 18,7 12,4 151
Italy 6085 6,4 50,7 ,826 3318,00 ,5 69 15,3 19,1 232
Yugoslavia 2620 13,3 34,8 ,694 2049,00 ,9 42 34,0 20,0 195
Hungary 4180 14,3 26,8 ,954 3850,00 ,4 54 23,7 9,9 251
GDR (East Germany) 7180 9,1 22,1 ,893 7408,00 -,2 77 13,0 23,0 344
Netherlands 9760 4,0 63,0 1,040 6183,00 ,7 76 8,7 23,2 298
Norway 13522 4,5 55,1 1,150 6434,00 ,4 53 8,8 18,5 294
Poland 3900 15,3 20,6 ,856 5590,00 ,9 57 21,3 16,9 218
Portugal 2370 13,0 41,0 ,423 1097,00 1,1 31 39,0 8,6 126
Romania 1904 11,0 25,0 ,904 4593,00 1,0 50 31,6 8,6 166
Spain 5678 8,0 55,0 ,632 2530,00 1,1 74 15,0 17,7 267
Sweden 13326 3,1 65,5 ,991 5296,00 ,3 87 7,5 23,9 375
Switzerland 15069 6,1 55,0 ,881 3708,00 -,3 58 10,0 12,6 320
United Kingdom 9358 1,9 63,5 1,003 4835,00 ,0 91 12,8 13,6 336
Sowiet Union 4550 15,1 23,5 1,115 5598,00 ,9 62 25,6 19,1 307
FRG (West Germany) 11135 2,2 49,9 1,074 5727,00 -,2 85 13,5 18,0 343
Number of cases read: 23 Number of cases listed: 23
8. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 15
FACTOR
/VARIABLES gdp agric service expimp energy growth urban infmort student tv
/MISSING LISTWISE /ANALYSIS gdp agric service expimp energy growth urban infmort st
udent tv
/PRINT UNIVARIATE INITIAL CORRELATION SIG DET KMO EXTRACTION FSCORE
/PLOT EIGEN
/CRITERIA FACTORS(10) ITERATE(25)
/EXTRACTION PC
/ROTATION NOROTATE
/SAVE REG(ALL)
/METHOD=CORRELATION .
_
- - - - - - - - - - - - F A C T O R A N A L Y S I S - - - - - - - - - - - -
Factor Analysis
F:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.sav
Descriptive Statistics
7653,78 3941,192 23
8,339 4,9609 23
45,670 17,0293 23
,88722 ,190763 23
4670,4783 1613,28267 23
,483 ,4448 23
65,43 14,981 23
16,800 8,8066 23
16,683 4,5156 23
271,26 69,072 23
Per capita gross domestic product in $
Share of agriculture in gross domestic product (%)
Share of services activities in gross domestic product (%)
Export/import ratio
Per capita fuel consumption in kilograms of coal
Natural change of population (rates per 1000 inhabitants)
Share of urban population (%)
Infant mortality per 1000 live birth
Number of students per 1000 inhabitants
Number of TV sets per 1000 inhabitants
Mean Std. Deviation Analysis N
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 16
Correlation Matrixa
1,000 -,801 ,686 ,410 ,389 -,728 ,542 -,842 ,460 ,799
-,801 1,000 -,723 -,261 -,314 ,655 -,611 ,704 -,447 -,704
,686 -,723 1,000 -,103 -,151 -,332 ,465 -,606 ,359 ,438
,410 -,261 -,103 1,000 ,817 -,409 ,406 -,445 ,290 ,573
,389 -,314 -,151 ,817 1,000 -,449 ,479 -,544 ,437 ,595
-,728 ,655 -,332 -,409 -,449 1,000 -,476 ,622 -,278 -,751
,542 -,611 ,465 ,406 ,479 -,476 1,000 -,735 ,554 ,744
-,842 ,704 -,606 -,445 -,544 ,622 -,735 1,000 -,549 -,784
,460 -,447 ,359 ,290 ,437 -,278 ,554 -,549 1,000 ,635
,799 -,704 ,438 ,573 ,595 -,751 ,744 -,784 ,635 1,000
,000 ,000 ,026 ,033 ,000 ,004 ,000 ,014 ,000
,000 ,000 ,115 ,072 ,000 ,001 ,000 ,016 ,000
,000 ,000 ,319 ,245 ,061 ,013 ,001 ,046 ,018
,026 ,115 ,319 ,000 ,026 ,027 ,017 ,090 ,002
,033 ,072 ,245 ,000 ,016 ,010 ,004 ,019 ,001
,000 ,000 ,061 ,026 ,016 ,011 ,001 ,100 ,000
,004 ,001 ,013 ,027 ,010 ,011 ,000 ,003 ,000
,000 ,000 ,001 ,017 ,004 ,001 ,000 ,003 ,000
,014 ,016 ,046 ,090 ,019 ,100 ,003 ,003 ,001
,000 ,000 ,018 ,002 ,001 ,000 ,000 ,000 ,001
Per capita gross domestic product in $
Share of agriculture in gross domestic produ
Share of services activities in gross domestic
Export/import ratio
Per capita fuel consumption in kilograms of c
Natural change of population (rates per 1000
Share of urban population (%)
Infant mortality per 1000 live birth
Number of students per 1000 inhabitants
Number of TV sets per 1000 inhabitants
Per capita gross domestic product in $
Share of agriculture in gross domestic produ
Share of services activities in gross domestic
Export/import ratio
Per capita fuel consumption in kilograms of c
Natural change of population (rates per 1000
Share of urban population (%)
Infant mortality per 1000 live birth
Number of students per 1000 inhabitants
Number of TV sets per 1000 inhabitants
Correlation
Sig. (1-taile
Per capita
gross
domestic
product in $
Share of
agriculture
in gross
domestic
product (%)
Share of
services
activities in
gross
domestic
product (%)
Export/import
ratio
Per capita fue
consumption
in kilograms
of coal
Natural
change of
population
(rates per
1000
inhabitants)
Share of
urban
population
(%)
nfant mortality
per 1000 live
birth
Number of
students per
1000
inhabitants
Number of TV
sets per 1000
inhabitants
Determinant = 2,926E-05a.
KMO and Bartlett's Test
,769
186,166
45
,000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity
9. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 17
Communalities
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
Per capita gross domestic product in $
Share of agriculture in gross domestic product (%)
Share of services activities in gross domestic product (%)
Export/import ratio
Per capita fuel consumption in kilograms of coal
Natural change of population (rates per 1000 inhabitants)
Share of urban population (%)
Infant mortality per 1000 live birth
Number of students per 1000 inhabitants
Number of TV sets per 1000 inhabitants
Initial Extraction
Extraction Method: Principal Component Analysis.
Total Variance Explained
5,879 58,787 58,787 5,879 58,787 58,787
1,751 17,514 76,302 1,751 17,514 76,302
,830 8,305 84,607 ,830 8,305 84,607
,437 4,367 88,973 ,437 4,367 88,973
,399 3,995 92,968 ,399 3,995 92,968
,260 2,603 95,570 ,260 2,603 95,570
,224 2,237 97,808 ,224 2,237 97,808
,106 1,062 98,870 ,106 1,062 98,870
6,090E-02 ,609 99,479 6,090E-02 ,609 99,479
5,207E-02 ,521 100,000 5,207E-02 ,521 100,000
Component
1
2
3
4
5
6
7
8
9
10
Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings
Extraction Method: Principal Component Analysis.
J. Rovan: Multivariate Analysis 9 Principal Component Analysis 18
Component Matrixa
,890 -,220 -,226 -,074 ,226 -,118 ,047 ,094 -,091 ,136
-,831 ,341 ,131 -,007 -,024 -,374 ,161 -,061 ,053 ,056
,589 -,745 ,060 ,135 ,186 ,027 ,076 -,124 ,137 ,024
,572 ,705 -,117 ,153 ,242 ,113 ,236 -,094 -,040 -,044
,623 ,719 ,031 ,029 ,075 ,031 -,259 ,003 ,112 ,077
-,764 -,024 ,486 ,261 ,290 ,049 ,006 ,160 -,002 ,000
,795 ,005 ,296 ,367 -,368 ,018 ,051 -,019 -,050 ,065
-,909 ,077 -,035 -,159 -,098 ,297 ,166 ,007 ,039 ,122
,651 ,035 ,647 -,382 ,052 ,022 ,012 -,073 -,042 -,001
,931 ,098 ,005 -,122 -,133 -,029 ,193 ,195 ,106 -,054
Per capita gross domestic product in $
Share of agriculture in gross domestic product (%)
Share of services activities in gross domestic
product (%)
Export/import ratio
Per capita fuel consumption in kilograms of coal
Natural change of population (rates per 1000
inhabitants)
Share of urban population (%)
Infant mortality per 1000 live birth
Number of students per 1000 inhabitants
Number of TV sets per 1000 inhabitants
1 2 3 4 5 6 7 8 9 10
Component
Extraction Method: Principal Component Analysis.
10 components extracted.a.