Jan 30

Misc. Math. Data Science. Machine Learning. Optimization. Vector, PCA, Basis, Covariance

Misc. Math. Data Science. Machine Learning. Optimization. Vector, PCA, Basis, Covariance

Orthonormality: Orthonormal Vectors

“In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.”
https://en.wikipedia.org/wiki/Orthonormality

Basis for a Vector Space
“A vector space’s basis is a subset of vectors within the space that are linearly independent and span the space. A basis is linearly independent because the vectors in it cannot be defined as a linear combination of any of the other vectors in the basis.”

https://study.com/academy/lesson/finding-the-basis-of-a-vector-space.html

Vector Space
“In linear algebra, you might find yourself working with a set of vectors. When the operations of scalar multiplication and vector addition hold for a set of vectors, we call it a vector space.”
https://study.com/academy/lesson/finding-the-basis-of-a-vector-space.html

Explain the concept of covariance matrices based on the shape of data.

Variance:

covariance captures: “The diagonal spread of the data is captured by the covariance.”

“The covariance matrix defines the shape of the data. Diagonal spread is captured by the covariance, while axis-aligned spread is captured by the variance.”

https://www.visiondummy.com/2014/04/geometric-interpretation-covariance-matrix/

https://www.cs.rutgers.edu/~elgammal/classes/cs536/lectures/i2ml-chap6.pdf

https://pathmind.com/wiki/eigenvector

How to derive variance-covariance matrix of coefficients in linear regression

https://stats.stackexchange.com/questions/68151/how-to-derive-variance-covariance-matrix-of-coefficients-in-linear-regression

“The matrix {\displaystyle \operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}} is known as the matrix of regression coefficients, while in linear algebra {\displaystyle \operatorname {K} _{\mathbf {Y|X} }} is the Schur complement of {\displaystyle \operatorname {K} _{\mathbf {XX} }} in {\displaystyle \mathbf {\Sigma } }.
The matrix of regression coefficients may often be given in transpose form, {\displaystyle \operatorname {K} _{\mathbf {XX} }^{-1}\operatorname {K} _{\mathbf {XY} }}, suitable for post-multiplying a row vector of explanatory variables {\displaystyle \mathbf {X} ^{\rm {T}}} rather than pre-multiplying a column vector {\mathbf {X}}. In this form they correspond to the coefficients obtained by inverting the matrix of the normal equations of ordinary least squares (OLS).”
https://en.wikipedia.org/wiki/Covariance_matrix

Statistics 512: Applied Linear Models Topic 3

https://www.stat.purdue.edu/~boli/stat512/lectures/topic3.pdf

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*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

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If you want to contribute to occasional free and/or low cost online/offline training or charitable/non-profit work in the education/health/social service sector, you can financially contribute to: safoundation at salearningschool.com using Paypal or Credit Card (on http://sitestree.com/training/enrol/index.php?id=114 ).

Jan 29

Misc Math, Data Science, Machine Learning, PCA, FA

“In mathematics, a set B of elements (vectors) in a vector space V is called a basis, if every element of V may be written in a unique way as a (finite) linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates on B of the vector. The elements of a basis are called basis vectors.”

”

Equivalently B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.[1] In more general terms, a basis is a linearly independent spanning set.

A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.

”

https://en.wikipedia.org/wiki/Basis_(linear_algebra)

Positive Semidefinite Matrix
“A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative. SEE ALSO: Negative Definite Matrix, Negative Semidefinite Matrix, Positive Definite Matrix, Positive Eigenvalued Matrix, Positive Matrix.”

http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html

”

Hermitian Matrix

A square matrix is called Hermitian if it is self-adjoint. Therefore, a Hermitian matrix A=(a_(ij)) is defined as one for which

A=A^(H), (1)

where A^(H) denotes the conjugate transpose. This is equivalent to the condition

a_(ij)=a^__(ji),

”

http://mathworld.wolfram.com/HermitianMatrix.html

Definiteness of a matrix

“In linear algebra, a symmetric n\times n real matrix M is said to be positive definite if the scalar {\displaystyle z^{\textsf {T}}Mz} is strictly positive for every non-zero column vector z of n real numbers. Here {\displaystyle z^{\textsf {T}}} denotes the transpose of z.[1] When interpreting {\displaystyle Mz} as the output of an operator, M, that is acting on an input, z, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes.”
https://en.wikipedia.org/wiki/Definiteness_of_a_matrix

”

Singular value decomposition

From Wikipedia, the free encyclopedia

Jump to navigationJump to search
Illustration of the singular value decomposition UΣV* of a real 2×2 matrix M.

  • Top: The action of M, indicated by its effect on the unit disc D and the two canonical unit vectors e1 and e2.
  • Left: The action of V*, a rotation, on D, e1, and e2.
  • Bottom: The action of Σ, a scaling by the singular values σ1 horizontally and σ2 vertically.
  • Right: The action of U, another rotation.

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any {\displaystyle m\times n}m\times n matrix via an extension of the polar decomposition.

Specifically, the singular value decomposition of an m\times n real or complex matrix \mathbf {M} is a factorization of the form {\displaystyle \mathbf {U\Sigma V^{*}} }, where \mathbf {U} is an m\times m real or complex unitary matrix, \mathbf{\Sigma} is an m\times n rectangular diagonal matrix with non-negative real numbers on the diagonal, and \mathbf {V} is an n\times n real or complex unitary matrix. If \mathbf {M} is real, \mathbf {U} and {\displaystyle \mathbf {V} =\mathbf {V^{*}} } are real orthonormal matrices.”

https://en.wikipedia.org/wiki/Singular_value_decomposition

PCA using Python (scikit-learn)

https://towardsdatascience.com/pca-using-python-scikit-learn-e653f8989e60

Random R code in relation to PCA

#calculate covariance matrix
cov_mat = cov(normalized_mat)

#Calculation of eigen values using built in eigen function
#no need here to do our own eigen
eig <- eigen(cov_mat)

#verify with prcomp from R (principal components function)
prcomp(pca_data)

eig$vectors

t(eig$vectors)

Some more information on PCA and FA (Factor Analysis)
https://www.cs.rutgers.edu/~elgammal/classes/cs536/lectures/i2ml-chap6.pdf

*** *** ***
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*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

Facebook Group/Form to discuss (Q & A): https://www.facebook.com/banglasalearningschool

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Get access to courses on Big Data, Data Science, AI, Cloud, Linux, System Admin, Web Development and Misc. related. Also, create your own course to sell to others. http://sitestree.com/training/

Jan 29

‘Scam’ fundraisers reap millions in the name of heart-tugging causes

"The call centers in Alabama, along with others in Nevada, New Jersey, and Florida, raise money on behalf of “scam PACs,” slang among critics for political action committees that purport to support worthy causes but in reality hand over little of the money for political – or charitable – purposes. Instead, the bulk of the money is kept by fundraising firms or the people running the PACs."
https://www.reuters.com/investigates/special-report/usa-fundraisers-scampacs/


*** *** *** .***
Note: Older short-notes from this site are posted on Medium: https://medium.com/@SayedAhmedCanada

*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

Facebook Group/Form to discuss (Q & A): https://www.facebook.com/banglasalearningschool

Our free or paid training events: https://www.facebook.com/justetcsocial

Get access to courses on Big Data, Data Science, AI, Cloud, Linux, System Admin, Web Development and Misc. related. Also, create your own course to sell to others. http://sitestree.com/training/

If you want to contribute to occasional free and/or low cost online/offline training or charitable/non-profit work in the education/health/social service sector, you can financially contribute to: safoundation at salearningschool.com using Paypal or Credit Card (on http://sitestree.com/training/enrol/index.php?id=114 ).

Jan 28

Optimization, Data Science, Math

Optimization Problem:

Advances in Missile Guidance, Control, and Estimation

Preview:
https://play.google.com/books/reader?id=A2PMBQAAQBAJ&hl=en_GB&pg=GBS.PR14

https://books.google.ca/books?id=A2PMBQAAQBAJ&pg=PA595&lpg=PA595&dq=force+moment+interaction+with+thrusters&source=bl&ots=BruxnXwLzp&sig=ACfU3U39G-l3xDzbotOBJHcMV5uR7DkciQ&hl=en&sa=X&ved=2ahUKEwjZpsT44afnAhXRJt8KHfPYCroQ6AEwCnoECAoQAQ#v=onepage&q=force%20moment%20interaction%20with%20thrusters&f=false

"What is the difference between affine and linear?
4 Answers. A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.Sep 15, 2014"

"If you choose bases for vector spaces ? and ? of dimensions ? and ? respectively, and consider functions ?:?→?, then ? is linear if ?(?)=?? for some ?×? matrix ? and ? is affine if ?(?)=??+? for some matrix ? and vector ?, where coordinate representations are used with respect to the bases chosen."

https://math.stackexchange.com/questions/275310/what-is-the-difference-between-linear-and-affine-function/275327



*** *** ***
Note: Older short-notes from this site are posted on Medium: https://medium.com/@SayedAhmedCanada

*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

Facebook Group/Form to discuss (Q & A): https://www.facebook.com/banglasalearningschool

Our free or paid training events: https://www.facebook.com/justetcsocial

Get access to courses on Big Data, Data Science, AI, Cloud, Linux, System Admin, Web Development and Misc. related. Also, create your own course to sell to others. http://sitestree.com/training/

If you want to contribute to occasional free and/or low cost online/offline training or charitable/non-profit work in the education/health/social service sector, you can financially contribute to: safoundation at salearningschool.com using Paypal or Credit Card (on http://sitestree.com/training/enrol/index.php?id=114 ).

Jan 28

Misc. Math for Data Science, Engineering, and/or Optimization

What is the Inverse of a Matrix?

https://www.mathsisfun.com/algebra/matrix-inverse.html

What is Norm?
“In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that satisfies certain properties pertaining to scalability and additivity, and assigns a strictly positive real number to each vector in a vector space over the field of real or complex numbers—except for the zero vector, which is assigned zero.[1]

A pseudonorm (seminorm), on the other hand, is allowed to assign zero to some non-zero vectors (in addition to the zero vector).[2]

The term “norm” is commonly used to refer to the vector norm in Euclidean space. It is known as the “Euclidean norm” (see below) which is technically called the L2-norm. The Euclidean norm maps a vector to its length in Euclidean space. Because of this, the Euclidean norm is often known as the magnitude.”

“A vector space on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a semi normed vector space. It is often possible to supply a norm for a given vector space in more than one way.”

https://en.wikipedia.org/wiki/Norm_(mathematics)

What is Linear programming?

“Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. ”

“More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists.

Linear programs are problems that can be expressed in canonical form as

{\displaystyle {\begin{aligned}&{\text{Maximize}}&&\mathbf {c} ^{\mathrm {T} }\mathbf {x} \\&{\text{subject to}}&&A\mathbf {x} \leq \mathbf {b} \\&{\text{and}}&&\mathbf {x} \geq \mathbf {0} \end{aligned}}}”

https://en.wikipedia.org/wiki/Linear_programming

*** . *** . ***
Note: Older short-notes from this site are posted on Medium: https://medium.com/@SayedAhmedCanada

*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

Facebook Group/Form to discuss (Q & A): https://www.facebook.com/banglasalearningschool

Our free or paid training events: https://www.facebook.com/justetcsocial

Get access to courses on Big Data, Data Science, AI, Cloud, Linux, System Admin, Web Development and Misc. related. Also, create your own course to sell to others. http://sitestree.com/training/

If you want to contribute to occasional free and/or low cost online/offline training or charitable/non-profit work in the education/health/social service sector, you can financially contribute to: safoundation at salearningschool.com using Paypal or Credit Card (on http://sitestree.com/training/enrol/index.php?id=114 ).

Jan 26

Investing in an ever-changing world

"Anything can happen in the short term but over a few decades the equity market has always produced very significant gains including periods of war, countless recessions and economic shocks. The best financial advice you can give to your children, grandchildren, nieces and nephews is to start as early as possible, stay invested and continue buying over time. Maximize tax advantaged accounts and stuff TFSAs with growth ETFs. Most importantly don’t lose money.

By Sinan Terzioglu, CFA, CIM, is a financial advisor with Turner Investments, Private Client Group, Raymond James Ltd."

https://www.greaterfool.ca/2020/01/26/investing-in-a-world-like-this/

Note: Older short-notes from this site are posted on Medium: https://medium.com/@SayedAhmedCanada

*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

Facebook Group/Form to discuss (Q & A): https://www.facebook.com/banglasalearningschool

Our free or paid training events: https://www.facebook.com/justetcsocial

Get access to courses on Big Data, Data Science, AI, Cloud, Linux, System Admin, Web Development and Misc. related. Also, create your own course to sell to others. http://sitestree.com/training/

If you want to contribute to occasional free and/or low cost online/offline training or charitable/non-profit work in the education/health/social service sector, you can financially contribute to: safoundation at salearningschool.com using Paypal or Credit Card (on http://sitestree.com/training/enrol/index.php?id=114 ).

Jan 26

Misc. Math. Might Relate to Optimization

find the equation for a line

http://www.webmath.com/_answer.php

Parametric forms for lines and vectors

https://www.futurelearn.com/courses/maths-linear-quadratic-relations/0/steps/12128

Solving Systems of Linear Equations Using Matrices

https://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html

Affine Space
”

”

Subspace
https://www.wolframalpha.com/input/?i=subspace

“What is an affine set?
A set is called “affine” iff for any two points in the set, the line through them is contained in the set. In other words, for any two points in the set, their affine combinations are in the set itself. Theorem 1. A set is affine iff any affine combination of points in the set is in the set itself.”
https://www.cse.iitk.ac.in/users/rmittal/prev_course/s14/notes/lec3.pdf [good one to check]

linear/conic/affine/convex combination

https://observablehq.com/@eliaskal/point-combinations-linear-conic-affine-convex

Related Course:
https://www.cse.iitk.ac.in/users/rmittal/prev_course/s14/course_s14.html

“In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
en.wikipedia.org › wiki › Row_and_column_spaces

Row and column spaces – Wikipedia

“

Row and column spaces

https://en.wikipedia.org/wiki/Row_and_column_spaces

“Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:”

”

Infimum and supremum

From Wikipedia, the free encyclopedia

Jump to navigationJump to search
A set T of real numbers (hollow and filled circles), a subset S of T (filled circles), and the infimum of S. Note that for finite, totally ordered sets the infimum and the minimum are equal.


A set A of real numbers (blue circles), a set of upper bounds of A (red diamond and circles), and the smallest such upper bound, that is, the supremum of A (red diamond).

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.[1] Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.[1]

The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists.[1] Consequently, the supremum is also referred to as the least upper bound (or LUB).[1]

”
https://en.wikipedia.org/wiki/Infimum_and_supremum

Jan 24

Industry Job Prospect for Graph Mining

Industry Job Prospect for Graph Mining

Sample Jobs

https://www.careerbuilder.com/jobs-graph-mining

https://www.indeed.com/q-Graph-Mining-jobs.html

For example, Google works in the following areas of Graph Mining. Google has jobs for such. Also, Facebook and any other social networking site will have jobs in relation to Graph Mining. Computational Biology, Medicine Research, Drug Discovery, Disease Diagnosis, Transportation, Scheduling, Shipping Scheduling will have applications and jobs for Graph Mining.

Job Areas:

The general Mining (data based) jobs and Machine/Deep/Reinforcement Learning jobs will require Graph Mining expertise sometimes such as positions (real) : Research Intern – Deep Learning for Graphs, ML Engineer – Siri Knowledge Graph

Computer Networks, Network/Cyber Security application development (also R & D) positions might ask for Graph Mining expertise.

Graph Mining will have applications and jobs in Biological, Chemistry, Drug Design areas also in Transportation

Social Network Mining will always involve Graph Mining. Applications: Friend Recommendation

Trajectory Data Mining Jobs at Microsoft

https://www.microsoft.com/en-us/research/publication/trajectory-data-mining-an-overview/

Graph Mining Jobs (areas) at Google:
https://ai.google/research/teams/algorithms-optimization/graph-mining/

"Large-Scale Balanced Partitioning: Example Google Maps Driving Directions, Large-Scale Clustering:clustering graphs at Google scale, Large-Scale Connected Components, Large-Scale Link Modeling: similarity ranking and centrality metrics: link prediction and anomalous link discovery., Large-Scale Similarity Ranking: Personalized PageRank, Egonet similarity, Adamic Adar, and others, Public-private Graph Computation, Streaming and Dynamic Graph Algorithms, ASYMP: Async Message Passing Graph Mining, Large-Scale Centrality Ranking, Large-Scale Graph Building"

More Related Jobs:

Tools in Jobs/Jobs–https://www.researchgate.net/post/Can_you_suggest_a_graph_mining_tool

—https://www.linkedin.com/jobs/gephi-jobs/

—https://bit.ly/2Nwlnrp Data Scientist 2

Jan 23

Part X: Engineering Optimization: Mathematical Optimization

Good intro to: Quadratic Forms and Convexity
https://www.dr-eriksen.no/teaching/GRA6035/2010/lecture4.pdf

Concave Upward and Downward

https://www.mathsisfun.com/calculus/concave-up-down-convex.html

Convex functions and K-Convexityhttps://ljk.imag.fr/membres/Anatoli.Iouditski/cours/convex/chapitre_3.pdf



*** . *** . *** . ***
Note: Older short-notes from this site are posted on Medium: https://medium.com/@SayedAhmedCanada

*** . *** *** . *** . *** . ***

Sayed Ahmed

BSc. Eng. in Comp. Sc. & Eng. (BUET)
MSc. in Comp. Sc. (U of Manitoba, Canada)
MSc. in Data Science and Analytics (Ryerson University, Canada)
Linkedin: https://ca.linkedin.com/in/sayedjustetc

Blog: http://Bangla.SaLearningSchool.com, http://SitesTree.com
Online and Offline Training: http://Training.SitesTree.com (Also, can be free and low cost sometimes)

Facebook Group/Form to discuss (Q & A): https://www.facebook.com/banglasalearningschool

Our free or paid training events: https://www.facebook.com/justetcsocial

Get access to courses on Big Data, Data Science, AI, Cloud, Linux, System Admin, Web Development and Misc. related. Also, create your own course to sell to others. http://sitestree.com/training/

If you want to contribute to occasional free and/or low cost online/offline training or charitable/non-profit work in the education/health/social service sector, you can financially contribute to: safoundation at salearningschool.com using Paypal or Credit Card (on http://sitestree.com/training/enrol/index.php?id=114 ).

Jan 23

How Styline is Building the #FashionTech Company Where You Would Love to Work

"

How Styline is Building the #FashionTech Company Where You Would Love to Work

0
Since Marc Andreessen penned his famous “Why Software Is Eating the World” essay in The Wall Street Journal 8 years ago, the world of business has changed fundamentally. Today, the idea that every company needs to become a technology company is considered almost a cliché. No matter your industry, you’re expected to be reimagining your business. Fashion is no different. A new breed of companies that calls themselves fashion-tech is now slowly shaping the present and the future of fashion across the world. Some of these companies are mere marketplaces. Others are changing how and what people shop, wear and when. Others are launching new products using a combination of tech and common sense. We have seen the meteoric rise of companies like Rent the Runway to All Birds to a long list of other fashion-tech companies. "https://futurestartup.com/2019/12/25/how-styline-is-building-the-fashiontech-company-where-you-would-love-to-work/