{"id":16536,"date":"2019-12-29T14:56:24","date_gmt":"2019-12-29T19:56:24","guid":{"rendered":"https:\/\/bangla.salearningschool.com\/recent-posts\/?p=16536"},"modified":"2020-01-30T21:04:24","modified_gmt":"2020-01-31T02:04:24","slug":"part-2-some-basic-math-statistics-concepts-that-data-scientists-the-true-ones-will-usually-know-use","status":"publish","type":"post","link":"http:\/\/bangla.sitestree.com\/?p=16536","title":{"rendered":"Part 2: Some basic Math\/Statistics concepts that Data Scientists (the true ones) will usually know\/use"},"content":{"rendered":"

Part 2: Some basic Math\/Statistics concepts that Data Scientists (the true ones) will usually know\/use (came across, studied, learned, used)<\/p>\n

Covariance and Correlation<\/strong><\/p>\n

“Covariance<\/strong> is a measure of how two variables change together, but its magnitude is unbounded, so it is difficult to interpret. By dividing covariance<\/strong> by the product of the two standard deviations, one can calculate the normalized version of the statistic. This is the correlation coefficient<\/strong>.” https:\/\/www.investopedia.com\/terms\/c\/correlationcoefficient.asp<\/a> on Investing and Covariance\/Correlation<\/p>\n

Covariance and expected value<\/strong><\/p>\n

“<\/strong>Covariance is calculated as expected value or average of the product of the differences of each random variable from their expected values, where E[X] is the expected value for X and E[Y] is the expected value of y.“<\/strong>
cov(X, Y) = E[(X – E[X]) . (Y – E[Y])]
cov(X, Y) = sum (x – E[X]) * (y – E[Y]) * 1\/n<\/p>\n

Sample: covariance: cov(X, Y) = sum (x – E[X]) * (y – E[Y]) * 1\/(n – 1)
Ref: https:\/\/machinelearningmastery.com\/introduction-to-expected-value-variance-and-covariance\/<\/a><\/p>\n

Formula for continuous variables<\/p>\n

\"[eq4]\"<\/a><\/p>\n

where \"[eq5]\" is the joint probability density function<\/a> of \"X\" and \"Y\".<\/p>\n

Formula for Discrete Variables<\/p>\n

\"[eq1]\"<\/p>\n

Reference: https:\/\/www.statlect.com\/glossary\/covariance-formula<\/a><\/p>\n

Correlation:<\/strong><\/p>\n

What is Correlation?<\/h3>\n

“Correlation, in the finance and investment industries, is a statistic that measures the degree to which two securities move in relation to each other. Correlations are used in advanced portfolio management<\/a>, computed as the correlation coefficient<\/a>, which has a value that must fall between -1.0 and +1.0.” Ref: https:\/\/www.investopedia.com\/terms\/c\/correlation.asp<\/a><\/p>\n

Correlation formula:<\/strong><\/p>\n

\"\"<\/a><\/p>\n

Ref:<\/strong> http:\/\/www.stat.yale.edu\/Courses\/1997-98\/101\/correl.htm<\/a><\/p>\n

Correlation in Linear Regression:<\/h2>\n

“The square of the correlation coefficient, r\u00b2<\/em>, is a useful value in linear regression<\/a>. This value represents the fraction of the variation in one variable that may be explained by the other variable. Thus, if a correlation of 0.8 is observed between two variables (say, height and weight, for example), then a linear regression model attempting to explain either<\/em> variable in terms of the other variable will account for 64% of the variability in the data.”<\/p>\n

http:\/\/www.stat.yale.edu\/Courses\/1997-98\/101\/correl.htm<\/a><\/h2>\n

http:\/\/sphweb.bumc.bu.edu\/otlt\/MPH-Modules\/BS\/BS704_Multivariable\/BS704_Multivariable5.html<\/a><\/p>\n

Ref: http:\/\/ci.columbia.edu\/ci\/premba_test\/c0331\/s7\/s7_5.html<\/a><\/p>\n

Independent Events<\/strong><\/p>\n

\"\"<\/a><\/p>\n

www.wyzant.com
<\/a><\/p>\n

“In probability<\/strong>, two events<\/strong> are independent<\/strong> if the incidence of one event<\/strong> does not affect the probability<\/strong> of the other event<\/strong>. If the incidence of one event<\/strong> does affect the probability<\/strong> of the other event<\/strong>, then the events<\/strong> are dependent.”<\/p>\n

Ref: https:\/\/brilliant.org\/wiki\/probability-independent-events\/<\/a><\/p>\n

How do you know if an event is independent?<\/strong><\/p>\n

“To test whether<\/strong> two events<\/strong> A and B are independent<\/strong>, calculate P(A), P(B), and P(A \u2229 B), and then check whether<\/strong> P(A \u2229 B) equals P(A)P(B). If<\/strong> they are equal, A and B are independent<\/strong>; if<\/strong> not, they are dependent. 1. You throw two fair dice, one green and one red, and observe the numbers uppermost.”
Ref: https:\/\/www.zweigmedia.com\/RealWorld\/tutorialsf15e\/frames7_5C.html<\/a>
With Examples: https:\/\/www.mathsisfun.com\/data\/probability-events-independent.html<\/a><\/p>\n

Joint Distributions and Independence<\/h2>\n

The joint PMF of X1X1, X2X2, \u22ef\u22ef, XnXn is defined asPX1,X2,…,Xn(x1,x2,…,xn)=P(X1=x1,X2=x2,…,Xn=xn).<\/p>\n

For continuous case:
P((X1,X2,\u22ef,Xn)\u2208A)=\u222b\u22ef\u222bA\u22ef\u222bfX1X2\u22efXn(x1,x2,\u22ef,xn)dx1dx2\u22efdxn.

marginal PDF of XiXi
fX1(x1)=\u222b\u221e\u2212\u221e\u22ef\u222b\u221e\u2212\u221efX1X2…Xn(x1,x2,…,xn)dx2\u22efdxn.<\/p>\n

Ref: <\/strong><\/em>https:\/\/www.probabilitycourse.com\/chapter6\/6_1_1_joint_distributions_independence.php<\/a><\/p>\n

Random Vectors, Random Matrices, and Their Expected Values<\/p>\n

http:\/\/www.statpower.net\/Content\/313\/Lecture%20Notes\/MatrixExpectedValue.pdf<\/a><\/p>\n

Random Variables and Probability Distributions: https:\/\/www.stat.pitt.edu\/stoffer\/tsa4\/intro_prob.pdf<\/a><\/p>\n

What are moments of a random variable?<\/strong><\/p>\n

“The \u201cmoments<\/strong>\u201d of a random variable<\/strong> (or of its distribution) are expected values of powers or related functions of the random variable<\/strong>. The rth moment<\/strong> of X is E(Xr). In particular, the first moment<\/strong> is the mean, \u00b5X = E(X). The mean is a measure of the \u201ccenter\u201d or \u201clocation\u201d of a distribution. Ref: http:\/\/homepages.gac.edu\/~holte\/courses\/mcs341\/fall10\/documents\/sect3-3a.pdf<\/a>“<\/p>\n

Characteristic function (probability theory)<\/h1>\n

Jump to navigation<\/a>Jump to search<\/a>\"\"<\/a>The characteristic function of a uniform U<\/em>(\u20131,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued.<\/p>\n

“In probability theory<\/a> and statistics<\/a>, the characteristic function<\/strong> of any real-valued<\/a> random variable<\/a> completely defines its probability distribution<\/a>. If a random variable admits a probability density function<\/a>, then the characteristic function is the Fourier transform<\/a> of the probability density function.”<\/p>\n

Ref: <\/strong><\/em>https:\/\/en.wikipedia.org\/wiki\/Characteristic_function_(probability_theory)<\/a><\/p>\n

Functions of random vectors and their distribution<\/h1>\n

https:\/\/www.statlect.com\/fundamentals-of-probability\/functions-of-random-vectors<\/a><\/p>\n

\"\"<\/a><\/p>\n\n\n

\u00a0<\/p>\n

Ref: <\/strong><\/em>https:\/\/books.google.com\/books?id=xz9nQ4wdXG4C&pg=PA42&lpg=PA42&dq=the+characteristic+functions+of+a+vector+random+variable+is+shalom+kiruba&source=bl&ots=VqY-t6i-u2&sig=ACfU3U09k0CHqK_9Lowd8MkLoyjo1ela1Q&hl=en&sa=X&ved=2ahUKEwiaw9mQ0dvmAhXBmeAKHaSgDUgQ6AEwCXoECAkQAQ#v=onepage&q=the%20characteristic%20functions%20of%20a%20vector%20random%20variable%20is%20shalom%20kiruba&f=false<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Part 2: Some basic Math\/Statistics concepts that Data Scientists (the true ones) will usually know\/use (came across, studied, learned, used) Covariance and Correlation “Covariance is a measure of how two variables change together, but its magnitude is unbounded, so it is difficult to interpret. By dividing covariance by the product of the two standard deviations, … <\/p>\n

Continue reading<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1908,182],"tags":[],"class_list":["post-16536","post","type-post","status-publish","format-standard","hentry","category-math-and-statistics-for-data-science-and-engineering","category---blog","item-wrap"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack-related-posts":[{"id":16701,"url":"http:\/\/bangla.sitestree.com\/?p=16701","url_meta":{"origin":16536,"position":0},"title":"Misc. Math. Data Science. Machine Learning. Optimization. Vector, PCA, Basis, Covariance","author":"Sayed","date":"January 30, 2020","format":false,"excerpt":"Misc. Math. Data Science. Machine Learning. Optimization. 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Tracking and Fusion https:\/\/www.mathworks.com\/support\/search.html?q=covariance%20intersection&page=1","rel":"","context":"In "\u09ac\u09cd\u09b2\u0997 \u0964 Blog"","block_context":{"text":"\u09ac\u09cd\u09b2\u0997 \u0964 Blog","link":"http:\/\/bangla.sitestree.com\/?cat=182"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":16575,"url":"http:\/\/bangla.sitestree.com\/?p=16575","url_meta":{"origin":16536,"position":3},"title":"Part 4: Some Basic Math\/Stat Concepts for the wanna be Data Scientists","author":"Sayed","date":"December 31, 2019","format":false,"excerpt":"Part 4: Some Basic Math\/Stat Concepts for the wanna be Data Scientists Also for the Engineers in General Quadratic form \"In multivariate statistics, if is a vector of random variables, and is an -dimensional symmetric matrix, then the scalar quantity is known as a quadratic form in . \" Ref:\u2026","rel":"","context":"In "AI ML DS RL DL NN NLP Data Mining Optimization"","block_context":{"text":"AI ML DS RL DL NN NLP Data Mining Optimization","link":"http:\/\/bangla.sitestree.com\/?cat=1910"},"img":{"alt_text":"","src":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3c\/Polynomialdeg4.svg\/233px-Polynomialdeg4.svg.png","width":350,"height":200},"classes":[]},{"id":24967,"url":"http:\/\/bangla.sitestree.com\/?p=24967","url_meta":{"origin":16536,"position":4},"title":"Math and Probability for Data Science #Root","author":"Author-Check- Article-or-Video","date":"April 14, 2021","format":false,"excerpt":"Math and Probability for Data Science Esp. for Reinforcement Learning Probability: cov(x,y): https:\/\/math.stackexchange.com\/questions\/1581773\/computing-the-covariance-of-flipping-a-coin-in-the-first-and-last-flips Fundamental Concept in Machine Learning https:\/\/faculty.washington.edu\/tamre\/BayesTheorem.pdf ---- Lecture Slides - old book https:\/\/www.inf.ed.ac.uk\/teaching\/courses\/rl\/lecturelist.html -- Color : https:\/\/digitalsynopsis.com\/design\/color-thesaurus-correct-names-of-shades\/ Random Sum: http:\/\/www.math.unl.edu\/~sdunbar1\/ProbabilityTheory\/Lessons\/Conditionals\/RandomSums\/randsum.shtml Simple probability examples: http:\/\/jwilson.coe.uga.edu\/emt668\/emat6680.folders\/dickerson\/probability\/titled.html https:\/\/faculty.math.illinois.edu\/~kkirkpat\/SampleSpace.pdf Series and Sum: http:\/\/www.mathcentre.ac.uk\/resources\/uploaded\/mc-ty-convergence-2009-1.pdf Reference: http:\/\/www.sjsu.edu\/faculty\/watkins\/runs.htm https:\/\/mathinsight.org\/image\/two_dice_distribution http:\/\/www.mathguide.com\/lessons2\/ExpectedValues.html http:\/\/www.mathguide.com\/lessons2\/ExpectedValues.html https:\/\/www.texasgateway.org\/resource\/44-geometric-distribution-optional https:\/\/www.me.utexas.edu\/~jensen\/ORMM\/computation\/unit\/rvadd\/discrete_dist\/geometric.html http:\/\/www.mathguide.com\/lessons2\/ExpectedValues.html\u2026","rel":"","context":"In 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https:\/\/www.cse.iitk.ac.in\/users\/rmittal\/prev_course\/s14\/course_s14.html http:\/\/bangla.salearningschool.com\/recent-posts\/misc-math-might-relate-to-optimization\/ http:\/\/bangla.salearningschool.com\/recent-posts\/part-x-engineering-optimization-mathematical-optimization\/ https:\/\/www.dr-eriksen.no\/teaching\/GRA6035\/2010\/lecture4.pdf https:\/\/www.mathsisfun.com\/calculus\/concave-up-down-convex.html https:\/\/www-ljk.imag.fr\/membres\/Anatoli.Iouditski\/cours\/convex\/chapitre_3.pdf http:\/\/bangla.salearningschool.com\/recent-posts\/optimization-and-linear-algebra-math-from-the-internet\/ https:\/\/www.thestudentroom.co.uk\/showthread.php?t=1247928 https:\/\/en.wikipedia.org\/wiki\/Hessian_matrix https:\/\/en.wikipedia.org\/wiki\/Newton%27s_method_in_optimization https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/9781118733639.app6 http:\/\/bangla.salearningschool.com\/recent-posts\/optimization-and-linear-algebra-math-from-the-internet\/ https:\/\/en.wikipedia.org\/wiki\/Interior-point_method https:\/\/web.stanford.edu\/~boyd\/papers\/rt_cvx_sig_proc.html Must Read https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/ivantash-optimization_methods_and_their_applications_in_dsp.pdf https:\/\/www.researchgate.net\/publication\/269211254_The_analysis_and_optimization_algorithms_of_the_electronic_circuits_design good one http:\/\/www.bcamath.org\/documentos_public\/courses\/Nogales_2012-13_02_18-22.pdf https:\/\/en.wikipedia.org\/wiki\/Convex_analysis https:\/\/www.khanacademy.org\/search?page_search_query=Optimization%20problems%20(calculus) http:\/\/bangla.salearningschool.com\/recent-posts\/overview-on-optimization-concepts-from-the-internet\/ http:\/\/bangla.salearningschool.com\/recent-posts\/misc-optimization-machine-learning\/ http:\/\/bangla.salearningschool.com\/recent-posts\/misc-math-data-science-machine-learning-optimization-vector-pca-basis-covariance\/ https:\/\/www.mathsisfun.com\/algebra\/matrix-inverse-row-operations-gauss-jordan.html http:\/\/bangla.salearningschool.com\/recent-posts\/misc-math-for-data-science-engineering-and-or-optimization\/","rel":"","context":"In "\u09ac\u09cd\u09b2\u0997 \u0964 Blog"","block_context":{"text":"\u09ac\u09cd\u09b2\u0997 \u0964 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