Math for Data Science PDF

Math for data science Lesson 1 Data Science [Matrematic & Statistics for Data Science] Data Science > Our goal...

Math for data science Lesson 1 Data Science [Matrematic & Statistics for Data Science] Data Science > Our goal - ↳ 4 Collect (audience) I what's Competitiveness & use data about customers , trending , discover insights (pattern) that can help us improve the market. ourselves way what is Data Analytics ? A of data for I systematic analysis discovery , interpetation communication of meaningful patterns. · , focuses happen & what future explaining why something · will in happen. · It involves computer skills math I statistics. , Y Stages of Data Analytics Maturity > - [DDPP) Descriptive Diagnostic Prescriptive · ·. Predictive making Decision Data science always begin with Business Understanding Data Science Lifestyle - [DCPEMU] Define Collect Extract Model · · · Process · ·. Use ↳Collect you've set your goal , next step collect data After Traditionally Obtain is data explicity through · : , · data can be : structured (customer records transaction) sales questionnaires and , survey. : Unstructured (images , videos , text messages ↳ [for machine Filtering idea to make data more understandable : human user [FT] · Typical tasks : · Filtering Cleaning up noise : I stuff irrelvent to your question Tranformation making your meaningful : data more · ↳ Extract A task often very important negleted = involves exploring I extracting important element from data to reduce complexity = Typical tasks : Selection Choose which feature = · : to use [SAD] form Aggregation Combining two feature to · a new are : or more Performing math statistical technique to convert data into something meaningful : Decomposition more · ↳ Model statical/machine to learning technique get data to answer Apply your question = ↳ 3 common Clustering : ways · [CCR] Classification · Regression ·  1) Use = use the result of hard work to meet our goal in visualization Usage = ~ use result to automate task like : Recommend content Monitor 1 improve customer engagement · to users · · Alert important activities /event · Take action on behalf of users Lesson 2 [Basic Tool for Data science) check Lesson 2 Powerpoint Python Libraries [MONTPP] Matrics & number MatplotLib MATLAB Numpy · : · : Pandas Data analysis OpenCV : images · Pillow · : , Flow Machine learning Tens or Pytorch · : , Os Files , pathlib : system · - Excel Formulas [CLAST1] formulas Sum addition subtant multiply division Logical Function And , or, not cases · Common · = = , , , , function · Text = Left concat , Function if right Ian Average median mode · mean average = , , , , , · function Vertical/horizontal lookup Statistical Function Min Lookup quartile , standard deviation , correlation · = data to get the = , max , [Basic Lesson 3 Probability ineory statistical methods about make inference by studying relatively small from it. · Basic idea : a population a sample chosen objects / outcomes about which information Apulation is entire collection is sought · A sample subset of containing objects/outcomes that actually observed. · is - a population ,  Preliminary concept understand probability , start sample space & event for observation by understanding · terms an The · set of all possible outcomes of experiment is called the space le for the experiment - · A subset of is called event. a sample space an - V = Union Intersect union everything 1 : = S Intersection : common probabilities · Given & event A any experiment : any PCA) denotes event A · Expression the probability that occurs · PCA) is the proportion times that event A would occurs the long run, the experiment were be repeated over I over again. Axioms of Probability · : Let s be a sample space. Then P(s) = 1 For event A 01P(A) E any · , · If A lb exclusive events the PCAUB) p(A) PCB) mutually are = + , (10 + 3 + 5) only = i (10 + 3+ 5+ y + 13) = 35 Addition Rule If A 1 PCAUB) B are mutually exclusive events then : P(A) + p(B) · = , let A & B be then : P(AUB) p(B) P(AnB) generally events P(A) · more + any = = , , [counting] ? Permutation of collection of A permutation is ordering objects · an The number of permutation of object is ! · n n of of from object : permutation objects of is given by · The number R chosen group a n /ka (n n - ! 1) ! P(n, r) : nPr = 0! = 1 - -  - Combination ! (n) n A combination is selection of distinct of without objects, regards order · a to group. = The number of combinations of objects of objects K! (n 1) ! given by - from is · : chosen grap a n -permutation conditional Probability P(AnB) Probability · is based on part of a sample vace P(A(B) = PCB) events with P(B) Conditional of A denoted PCAIB) is : probability Let A & B = 0 given B · be , ,. Independence that one event has occurred does not knowledge change probability that another event occurs · In this events said to be independent · case are , If PCA) 0 and PCB) independent if · + + O the A l B are : , P(BIA) = P(B) and equally , PLAIB) = PCA)  Theorem Bayes · Provides a formula that allows us to calculate one of the conditional probabilities if we know other one of conditional PCAB) By definition probability PCA(B) · : = P(B) P(BIA) PCA) · We can subsitiute PCBJA) PCA) for PCANB) : PCAIB) = P(B) · Lesson 4 [Descriptive Statistics] statistical measures [MMMSRV] Median : Middle value tendency (Average) of the values in dataset · Mean : Central. · in a list of dataset. Mode : value that the most in dataset standard Deviation : distance with value Average · appear a mean value lowest number variance : Squared average distance from the mean Range : Difference highest&. · · between - Type of Data Series [DIF] Frequency Individual Distribution series. 7 Stage. 2 Discrete Series. 3 Statistics for Individual Descriptive Series  Statistics for Individual Descriptive Series raw data-mean to get Remember - M Population mean = - = sample mean Population Standard Deviation = - s Sample Standard Deviation = Descriptive Statistics for Frequency Distribution Series Graphical Representation for Numeric Data values Box far from most of data data plots : Diagram show how the extreme values how close values are to them. · or : of numerical data histogram Representation the distribution ·  Box Plots (Whisker Plots · based five-number of dataset [Minimum, first quartile median third quartile , I maximum) summary on , , Suitable Easy · multiple datasets interpret comparing to · · Interquartile Range (IQR) : Difference between the Third quartile & first quartile. Also represent middle 50 % of total data ↳ 1QR = Q3 - Q) - Outilers significantly · : Data that different from most of other values· ↳ Outilers = Greater Than 93 + (1 5. x 1QR) or Less than Q7-(1 5X1QR). Box Plot Distribution Symmetric /Normal Distribution distance from both maximum equal = = median is values. Distance from median skewed Positively : to maximum is greater than distance from the median to minimum = Negatively = skewed : Distance from median to minimum is greater than distance from the median to maximum Histogram Similar to charts but not the same Represent the distribution of numerical data · bar · ,. ↳Class of intervals , frequency each class Easier to box plots for datasets Suitable shape & spread of distribution · displaying · interpret than large. a Distribution [MBSTENDC) Histogram Normal Distribution : A symmetric, bell shaped symmetrical data arand · curve average · Skewed Distribution : data concentrated side with tail More on one , extending to the right different data dataset. Bimodal Distribution Two distinct groups/ processes : peaks representing within the · : (Plateau) Distribution contain several peaks, dataset with multiple data processes/source of data Multimodal indicating · a additional often Edge Peak Distribution Similar normal but with large peak tail de data groping histogram : an at one in construction , Comb Distribution Exhibits alternating high & low bars typically caused randing/interval histogram construction · : data issue in , by Truncated (Heart-Cut) Distribution Similar to normal with its tail sampling/quality : often selective data control missing due to · processes. Food Distribution lack of data around the with concentrations near the Dog shows mean · upper....... - , When to Use Box plots or Histogram useful for dataset 1 Box plots comparing multiple displaying the spread I skewness of the data · are distribution of data & of distribution Histogram determining shape I spread are better for the · display the a Limitations of Box plots & Histograms Box detailed than I not show the full distribution of the data be less histogram · plots can may affected bin size & not accurately represent the choice of distribution if the bins underlying · be Histogram are can by may tren or small. too large Lesson 5 [probability Distribution Random variables Random variables numerical assigns value to each outcome in sample space · a a discrete & continues · There are types 2 : A discrete random variable is whose possible values can be ordered I there are between adjacent value one gaps - , Possible values of continous random variable always contain interval all the points between some two numbers - a an , (Discrete) Probability Distribution list of of discrete random variable X for complete description of · possible value a , along with probabilities each , provides a the from which X population is drawn This is known distribution probability · as distribution of x) Probability discrete random variable X the function p(x) P(X · a is = = A cumulative distribution function that * is less than / equal to value i e F(x) P(X(x) specifies probability given. · = the a. ,  Distribution (Continuous Probability Random variable continuous if probabilities areas under given by · is are a. curve The distribution) for random variable called probability density function (or probability · curve a ⑧ Cumulative Distribution Function Let be a continuous random variable with function f(x) cumulative distribution function of X is: probability density · X , Bernalli Trial for "failure" · experiment that can result in one of 2 outcomes, example "success" & of "success" probability is denoted by - p of "failure" probability is therefore 1-p -. trial called Bernoulli trial with success Such a probability p. · a  Bernalli Distribution Otherwise X. 0 For if experiment result 1 = Bernalli trial define random variable X is success then X= any ; · we a were , , , It follow that X is discrete random variable with probability distribution p(X) defined by : · a , p(0) = P(X = 0) = 7 - P P(X p() 1) = = = p · The random variable X is said to have the Bernalli distribution N Bernalli Trials "success" practice, might several samples / can't the number of them · In we very large take from lot a among This amants several independent Bernalli number of successes conducting trails & canting the · to · The number of success is then a random variable which is said to have a binomial distribution Binominal Distribution · If a total ofn Bernalli trials are conducted and: , trials independent· Each trial The are has the same probability - success p - X is the number of successes in the trials X binominal distribution with parameter no · has the p  Normal Distribution · Normal distribution also called Gaussian distribution is by far the most commonly used distribution in statistic. This distribution model not all, populations provides many , although · for continuous a good l a given by : The function of variance is · normal random variable with probability density a mean it The 2-score normal items Dealing with population, we often convert from unit which the population were originally · in measured to standard mits · If x is an item sampled from a normal population with mean & variance -2 the C , standard mit equivalent & of X is number 2 where : number-mean & - sd number is called the z-score - z X = 10 8. 10 8-16. > 1 3 M = 10. 0 = 1. 3  Central Limit Theorem draw large enough sample from a population , then distribution of sample mean is approximately normal, no matter · a drawn from what population sample was though population from for compute probabilities table sample using · Allows us to sample means 2 which the was , drawn is not normal. Jointly Discrete · If x& Y are jointly discrete random variables, joint probability distribution of X1T is : p(u y) P(X xmdY y) = = = , distribution of XSofY marginal probability · : PX(x) = P(X = x) = Ep(x y) , Pi(y) P(i y) = p(u, y) = = = Jointly Continuous If X & are jointly continous random variable the joint probability distribution of X & Y is : · y , Marginal probability · distribution of X & of Y :  Point Estimators calculated from data generall , quantity is called statistic · a a statistic that used to estimate unknown constant, parameter , is called point estimator/point estimate · an or For if X Xn random sample from population example · ,..... is a a. , y,9 is often used to estimate the The sample mean X population mean - The sample variance s is often used to estimate population variance a - Confidence Interval Xn (n > 3) random sample from with Let X 1 be large population · a a...... normal. Then pl standard deviation that approximately mean o so X is , a level 700 (1-x) % confidence interval for is · where UX = /2n When the value of a is unknown it can be replaced with the sample standard ,. deviation. S  Hypothesis Testing test to determine certain we can be about · A now a hypothesis. be about : hypothesis · can test population mean -Paired data-Chi-square - population proportion variance difference between two / proportions - - - means. Alternative Null VS Hypothesis possibilities population mean is /equal I sample is lover because : Two actually greater than to 100 mean only · · , of random variation from population mean than 100 & sample reflects this fact Cie emission really reduced population mean is actually less · mean. , Null The effect indicated & hypothesis by sample is due only to random variation between sample The population · - indicated represent whole Alternative hypothesis the effect by sample isreal, that it accurately the population - · steps of Hypothesis Testing Define null I alternative HOSHI hypotheses · , · Assume Ho to be true Compute test statistic Test statistic is statistic used to access the strength of evidence against HO. · a. of test statistic P-value is the that test statistic would Compute the P-value value disagreement · probability have a whose. with MO is great as / greater than that actually observed concluded about of evidence strength against · no.  Lesson 6 [Exploratory Data Analysis] what is Data & Collection of data objects& their attributes · of An Atribute is property characteristic of object e g eye color temperatue · · or an person,.. · A collection of attributes describes an object also known objects · are as records , sample /instances Attribute Properties [MOAD] · Distinctness = # · Addition + · Order · multiplication Properties of Attributes [ROIN] · Nominal : - distinctiness · Interval : - distinctness orders addition , · Ordinal : - distinctness & order · Ratio : - all 4 properties Discrete VS Continuous · Discrete Attributes Continuas Attributes attribute value finite/cantably infinite set of values Has only Has real number · as · a set of words Example : temperature , height weight · Examples zip : codes cants or or · , , often variable represented integer Typically floating points · represented · as as Types of Dataset [ROG] t · Record Data Set : - DataMatrix- Document Data - Transaction Data · Graph Data Set : - Would wibe web - molecular structure Ordered Data Set : Spatial Data Temporal Data Sequential Data Sequence Genetic - ·