Normal equations for linear regression?
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This document discusses the normal equations method and QR decomposition method for solving linear regression problems. It notes that while the normal equations method is simpler, the QR decomposition method is numerically more stable. Specifically, it notes that the normal equations method is only backward stable if the data matrix X is well-conditioned, whereas the QR method is always backward stable. It concludes that the normal equations method is useful for teaching but to be aware of its disadvantages compared to the QR method.
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Principal component analysis (PCA) is a dimensionality reduction technique that identifies important contributing components in big data. It works by transforming the data to a new coordinate system such that the greatest variance by some projection of the data lies on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Specifically, PCA analyzes the covariance matrix of the data to obtain its eigenvectors and eigenvalues. It then chooses principal components corresponding to the largest eigenvalues, which identify the directions with the most variance in the data.
Similar to Normal equations for linear regression? (20)
Random Matrix Theory and Machine Learning - Part 3
Random Matrix Theory and Machine Learning - Part 3
Solving High-order Non-linear Partial Differential Equations by Modified q-Ho...
Solving High-order Non-linear Partial Differential Equations by Modified q-Ho...
QMC: Operator Splitting Workshop, Using Sequences of Iterates in Inertial Met...
QMC: Operator Splitting Workshop, Using Sequences of Iterates in Inertial Met...
2018 MUMS Fall Course - Statistical Representation of Model Input (EDITED) - ...
2018 MUMS Fall Course - Statistical Representation of Model Input (EDITED) - ...
More from Hamed Zakerzadeh
Can you count on your computer?
Floating-point arithmetic is more complicated than it seems, and it may bring very surprising results!
Asymptotic Preserving Schemes
This document discusses the origins and concepts of asymptotic preserving (AP) schemes for numerically solving multiscale differential equations. AP schemes allow for a parameter-independent grid size while maintaining stability and convergence to the correct solution even as the parameter approaches zero. They originated from work studying boundary layers in advection-diffusion equations and the interaction between grid size and multiscale parameters. Key properties of AP schemes are that they use a parameter-independent grid, remain stable for any parameter value, and converge to the proper asymptotic limit as the parameter goes to zero.
Is “Principal Component Analysis” different from SVD?
“Principal Component Analysis” or "Singular Value Decomposition"? I compare them from a mathematical and historical point of view!
Can you count on your computer for the Gaussian elimination?
Gaussian elimination is a reliable method for solving linear systems of equations when done by hand, but its reliability comes into question when performed on a computer with finite precision arithmetic. This project aims to get familiar with how finite precision affects Gaussian elimination, observe the effects, and investigate remedies by coding examples in MATLAB. Basic MATLAB coding skills are required.
Congrès SMAI 2019
We provide a comprehensive convergence analysis of the asymptotic preserving implicit-explicit particle-in-cell (IMEX-PIC) methods for the Vlasov–Poisson system with a strong magnetic field. This study is of utmost importance for understanding the behavior of plasmas in magnetic fusion devices such as tokamaks, where such a large magnetic field needs to be applied in order to keep the plasma particles on desired tracks.
Higham
1. This document summarizes key concepts in numerical analysis related to floating point arithmetic, including: the floating point number system; relative error; rounding; IEEE standard; precision versus accuracy; exceptional results; cancellation; and fused multiply-add instructions.
2. It discusses common misconceptions around floating point arithmetic, including the ideas that innocuous calculations are always accurate; increasing precision improves accuracy; and cancellation eliminates rounding errors.
3. The document uses examples to illustrate concepts like loss of accuracy from repeated operations, sensitivity to precision levels, and how cancellation can expose existing errors.
ENTROPY STABLE ENO SCHEMES
The document discusses the development of entropy stable essentially non-oscillatory (ENO) schemes of arbitrary order for systems of conservation laws. Key points include:
1) Existing ENO schemes are combined with entropy conservative fluxes to create new entropy stable ENO schemes.
2) It is shown that the jump of an ENO reconstruction has the same sign and is upper-bounded by the underlying cell averages.
3) Numerical experiments demonstrate the entropy stability and non-oscillatory properties of the new schemes.
More from Hamed Zakerzadeh (7)
Is “Principal Component Analysis” different from SVD?
Is “Principal Component Analysis” different from SVD?
Can you count on your computer for the Gaussian elimination?
Can you count on your computer for the Gaussian elimination?
Recently uploaded
8.Isolation of pure cultures and preservation of cultures.pdf
Isolation of pure culture, its various method.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
11.1 Role of physical biological in deterioration of grains.pdf
Storagedeteriorationisanyformoflossinquantityandqualityofbio-materials.
Themajorcausesofdeteriorationinstorage
•Physical
•Biological
•Mechanical
•Chemical
Storageonlypreservesquality.Itneverimprovesquality.
Itisadvisabletostartstoragewithqualityfoodproduct.Productwithinitialpoorqualityquicklydepreciates
Direct Seeded Rice - Climate Smart Agriculture
Direct Seeded Rice - Climate Smart AgricultureInternational Food Policy Research Institute- South Asia Office
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Farming systems analysis: what have we learnt?.pptx
Presentation given at the official farewell of Prof Ken Gillet at Wageningen on 13 June 2024
在线办理(salfor毕业证书)索尔福德大学毕业证毕业完成信一模一样
学校原件一模一样【微信：741003700 】《(salfor毕业证书)索尔福德大学毕业证》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
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二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
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◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
Gadgets for management of stored product pests_Dr.UPR.pdf
Insectsplayamajorroleinthedeteriorationoffoodgrainscausingbothquantitativeandqualitativelosses
Wellprovedthatnogranariescanbefilledwithgrainswithoutinsectsastheharvestedproducecontainegg(or)larvae(or)pupae(or)adultinsectinthembecauseoffieldcarryoverinfestationwhichcannotbeavoidedindevelopingcountrieslikeIndia
Simpletechnologiesfortimelydetectionofinsectsinthestoredproduceandtherebyplantimelycontrolmeasures
waterlessdyeingtechnolgyusing carbon dioxide chemicalspdf
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
Micronuclei test.M.sc.zoology.fisheries.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdf
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Applied Science: Thermodynamics, Laws & Methodology.pdf
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
ESR spectroscopy in liquid food and beverages.pptx
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
The debris of the ‘last major merger’ is dynamically young
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Randomised Optimisation Algorithms in DAPHNE
Slides from talk:
Aleš Zamuda: Randomised Optimisation Algorithms in DAPHNE .
Austrian-Slovenian HPC Meeting 2024 – ASHPC24, Seeblickhotel Grundlsee in Austria, 10–13 June 2024
https://ashpc.eu/
快速办理(UAM毕业证书)马德里自治大学毕业证学位证一模一样
学校原件一模一样【微信：741003700 】《(UAM毕业证书)马德里自治大学毕业证学位证》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
Recently uploaded (20)
8.Isolation of pure cultures and preservation of cultures.pdf
8.Isolation of pure cultures and preservation of cultures.pdf
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...
GBSN - Biochemistry (Unit 6) Chemistry of Proteins
GBSN - Biochemistry (Unit 6) Chemistry of Proteins
11.1 Role of physical biological in deterioration of grains.pdf
11.1 Role of physical biological in deterioration of grains.pdf
Farming systems analysis: what have we learnt?.pptx
Farming systems analysis: what have we learnt?.pptx
Gadgets for management of stored product pests_Dr.UPR.pdf
Gadgets for management of stored product pests_Dr.UPR.pdf
waterlessdyeingtechnolgyusing carbon dioxide chemicalspdf
waterlessdyeingtechnolgyusing carbon dioxide chemicalspdf
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdf
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdf
Applied Science: Thermodynamics, Laws & Methodology.pdf
Applied Science: Thermodynamics, Laws & Methodology.pdf
ESR spectroscopy in liquid food and beverages.pptx
ESR spectroscopy in liquid food and beverages.pptx
The debris of the ‘last major merger’ is dynamically young
The debris of the ‘last major merger’ is dynamically young
Normal equations for linear regression?
- 1. Normal equations for linear regression? No! Please no! Hamed Zakerzadeh Hamed Zakerzadeh Normal equations 1 / 3
- 2. Linear regression Find the best parameters β describing the linear relation between n variables x1, ⋯, xn (columns of X ∈ Rm×n ) and y, having m data points, that is, to minimize ε: y = Xβ + ε How to solve minβ ∥Xβ − y∥? using normal equation Apply first-order optimality condition: (Xt X) β = Xt y solve n × n full symmetric system using Cholesky factorization using QR decomposition X = Q R the product of orthogonal Q ∈ Rm×m and upper-triangular R ∈ Rm×n min β ∥Xβ − y∥ = min β ∥R β − Qt y∥ solve simple n × n upper-triangular system using backward substitution If m ≫ n Ô⇒ NE method is 2x faster (flops: O(n3 + mn2 ) vs O(2mn2 ) for QR) Hamed Zakerzadeh Normal equations 2 / 3
- 3. Achilles heel: numerical stability Information may be lost forming Xt X: [ 1 1 0 ] Ô⇒ [ 1 + 2 1 1 1 ] Condition number being denoted by κ, forward error bound is proportional to κ(X)2 for NE method while it was only κ(X) for the original least squares problem. QR method is always backward stable while NE method is guaranteed to be backward stable only if X is well-conditioned. The last word NE method is simple for teaching machine learning and, sometimes, useful in practice. But be aware of its disadvantages! “Although numerical analysts almost invariably solve the full rank LS problem by QR factor- ization, statisticians frequently use the normal equations (though perhaps less frequently than they used to, thanks to the influence of numer- ical analysts).” Hamed Zakerzadeh Normal equations 3 / 3