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El. knyga: Probability and Statistics for Data Science: Math + R + Data

4.33/5 (28 ratings by Goodreads)
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Probability and Statistics for Data Science: Math + R + Data covers math stat—distributions, expected value, estimation etc.—but takes the phrase Data Science in the title quite seriously:* Real datasets are used extensively. * All data analysis is supported by R coding. * Includes many Data Science applications, such as PCA, mixture distributions, random graph models, Hidden Markov models, linear and logistic regression, and neural networks.* Leads the student to think critically about the how and why of statistics, and to see the big picture.* Not theorem/proof-oriented, but concepts and models are stated in a mathematically precise manner.Prerequisites are calculus, some matrix algebra, and some experience in programming.Norman Matloff is a professor of computer science at the University of California, Davis, and was formerly a statistics professor there. He is on the editorial boards of the Journal of Statistical Software and The R Journal. His book Statistical Regression and Classification: From Linear Models to Machine Learning was the recipient of the Ziegel Award for the best book reviewed in Technometrics in 2017. He is a recipient of his universitys Distinguished Teaching Award.

Recenzijos

"I quite like this book. I believe that the book describes itself quite well when it says: Mathematically correct yet highly intuitiveThis book would be great for a class that one takes before one takes my statistical learning class. I often run into beginning graduate Data Science students whose background is not math (e.g., CS or Business) and they are not readyThe book fills an important niche, in that it provides a self-contained introduction to material that is useful for a higher-level statistical learning course. I think that it compares well with competing books, particularly in that it takes a more "Data Science" and "example driven" approach than more classical books." ~Randy Paffenroth, Worchester Polytechnic Institute

"This text by Matloff (Univ. of California, Davis) affords an excellent introduction to statistics for the data science studentIts examples are often drawn from data science applications such as hidden Markov models and remote sensing, to name a few All the models and concepts are explained well in precise mathematical terms (not presented as formal proofs), to help students gain an intuitive understanding." ~CHOICE

About the Author xxiii
To the Instructor xxv
To the Reader xxxi
I Fundamentals of Probability 1(146)
1 Basic Probability Models
3(32)
1.1 Example: Bus Ridership
3(1)
1.2 A "Notebook" View: the Notion of a Repeatable Experiment
4(3)
1.2.1 Theoretical Approaches
5(1)
1.2.2 A More Intuitive Approach
5(2)
1.3 Our Definitions
7(4)
1.4 "Mailing Tubes"
11(1)
1.5 Example: Bus Ridership Model (cont'd.)
11(3)
1.6 Example: ALOHA Network
14(5)
1.6.1 ALOHA Network Model Summary
16(1)
1.6.2 ALOHA Network Computations
16(3)
1.7 ALOHA in the Notebook Context
19(1)
1.8 Example: A Simple Board Game
20(3)
1.9 Bayes' Rule
23(1)
1.9.1 General Principle
23(1)
1.9.2 Example: Document Classification
23(1)
1.10 Random Graph Models
24(2)
1.10.1 Example: Preferential Attachment Model
25(1)
1.11 Combinatorics-Based Computation
26(5)
1.11.1 Which Is More Likely in Five Cards, One King or Two Hearts?
26(1)
1.11.2 Example: Random Groups of Students
27(1)
1.11.3 Example: Lottery Tickets
27(1)
1.11.4 Example: Gaps between Numbers
28(1)
1.11.5 Multinomial Coefficients
29(1)
1.11.6 Example: Probability of Getting Four Aces in a Bridge Hand
30(1)
1.12 Exercises
31(4)
2 Monte Carlo Simulation
35(10)
2.1 Example: Rolling Dice
35(4)
2.1.1 First Improvement
36(1)
2.1.2 Second Improvement
37(1)
2.1.3 Third Improvement
38(1)
2.2 Example: Dice Problem
39(1)
2.3 Use of runif() for Simulating Events
39(1)
2.4 Example: Bus Ridership (cont'd.)
40(1)
2.5 Example: Board Game (cont'd.)
40(1)
2.6 Example: Broken Rod
41(1)
2.7 How Long Should We Run the Simulation?
42(1)
2.8 Computational Complements
42(1)
2.8.1 More on the replicate() Function
42(1)
2.9 Exercises
43(2)
3 Discrete Random Variables: Expected Value
45(20)
3.1 Random Variables
45(1)
3.2 Discrete Random Variables
46(1)
3.3 Independent Random Variables
46(1)
3.4 Example: The Monty Hall Problem
47(3)
3.5 Expected Value
50(1)
3.5.1 Generality - Not Just for Discrete Random Variables
50(1)
3.5.2 Misnomer
50(1)
3.5.3 Definition and Notebook View
50(1)
3.6 Properties of Expected Value
51(7)
3.6.1 Computational Formula
51(3)
3.6.2 Further Properties of Expected Value
54(4)
3.7 Example: Bus Ridership
58(1)
3.8 Example: Predicting Product Demand
58(1)
3.9 Expected Values via Simulation
59(1)
3.10 Casinos, Insurance Companies and "Sum Users," Compared to Others
60(1)
3.11 Mathematical Complements
61(1)
3.11.1 Proof of Property E
61(1)
3.12 Exercises
62(3)
4 Discrete Random Variables: Variance
65(18)
4.1 Variance
65(6)
4.1.1 Definition
65(4)
4.1.2 Central Importance of the Concept of Variance
69(1)
4.1.3 Intuition Regarding the Size of Var(X)
69(5)
4.1.3.1 Chebychev's Inequality
69(1)
4.1.3.2 The Coefficient of Variation
70(1)
4.2 A Useful Fact
71(1)
4.3 Covariance
72(2)
4.4 Indicator Random Variables, and Their Means and Variances
74(5)
4.4.1 Example: Return Time for Library Books, Version I
75(1)
4.4.2 Example: Return Time for Library Books, Version II
76(1)
4.4.3 Example: Indicator Variables in a Committee Problem
77(2)
4.5 Skewness
79(1)
4.6 Mathematical Complements
79(2)
4.6.1 Proof of Chebychev's Inequality
79(2)
4.7 Exercises
81(2)
5 Discrete Parametric Distribution Families
83(30)
5.1 Distributions
83(3)
5.1.1 Example: Toss Coin Until First Head
84(1)
5.1.2 Example: Sum of Two Dice
85(1)
5.1.3 Example: Watts-Strogatz Random Graph Model
85(3)
5.1.3.1 The Model
85(1)
5.2 Parametric Families of Distributions
86(1)
5.3 The Case of Importance to Us: Parameteric Families of pmfs
86(2)
5.4 Distributions Based on Bernoulli Trials
88(10)
5.4.1 The Geometric Family of Distributions
88(6)
5.4.1.1 R Functions
91(1)
5.4.1.2 Example: A Parking Space Problem
92(2)
5.4.2 The Binomial Family of Distributions
94(2)
5.4.2.1 R Functions
95(1)
5.4.2.2 Example: Parking Space Model
96(1)
5.4.3 The Negative Binomial Family of Distributions
96(2)
5.4.3.1 R Functions
97(1)
5.4.3.2 Example: Backup Batteries
98(1)
5.5 Two Major Non-Bernoulli Models
98(8)
5.5.1 The Poisson Family of Distributions
99(1)
5.5.1.1 R Functions
99(1)
5.5.1.2 Example: Broken Rod
100(1)
5.5.2 The Power Law Family of Distributions
100(2)
5.5.2.1 The Model
100(2)
5.5.3 Fitting the Poisson and Power Law Models to Data
102(4)
5.5.3.1 Poisson Model
102(1)
5.5.3.2 Straight-Line Graphical Test for the Power Law
103(1)
5.5.3.3 Example: DNC E-mail Data
103(3)
5.6 Further Examples
106(2)
5.6.1 Example: The Bus Ridership Problem
106(1)
5.6.2 Example: Analysis of Social Networks
107(1)
5.7 Computational Complements
108(1)
5.7.1 Graphics and Visualization in R
108(1)
5.8 Exercises
109(4)
6 Continuous Probability Models
113(34)
6.1 A Random Dart
113(1)
6.2 Individual Values Now Have Probability Zero
114(1)
6.3 But Now We Have a Problem
115(1)
6.4 Our Way Out of the Problem: Cumulative Distribution Functions
115(4)
6.4.1 CDFs
115(4)
6.4.2 Non-Discrete, Non-Continuous Distributions
119(1)
6.5 Density Functions
119(4)
6.5.1 Properties of Densities
120(2)
6.5.2 Intuitive Meaning of Densities
122(1)
6.5.3 Expected Values
122(1)
6.6 A First Example
123(1)
6.7 Famous Parametric Families of Continuous Distributions
124(14)
6.7.1 The Uniform Distributions
125(2)
6.7.1.1 Density and Properties
125(1)
6.7.1.2 R Functions
125(1)
6.7.1.3 Example: Modeling of Disk Performance
126(1)
6.7.1.4 Example: Modeling of Denial-of-Service Attack
126(1)
6.7.2 The Normal (Gaussian) Family of Continuous Distributions
127(1)
6.7.2.1 Density and Properties
127(1)
6.7.2.2 R Functions
127(1)
6.7.2.3 Importance in Modeling
128(1)
6.7.3 The Exponential Family of Distributions
128(3)
6.7.3.1 Density and Properties
128(1)
6.7.3.2 R Functions
128(1)
6.7.3.3 Example: Garage Parking Fees
129(1)
6.7.3.4 Memoryless Property of Exponential Distributions
130(1)
6.7.3.5 Importance in Modeling
131(1)
6.7.4 The Gamma Family of Distributions
131(3)
6.7.4.1 Density and Properties
132(1)
6.7.4.2 Example: Network Buffer
133(1)
6.7.4.3 Importance in Modeling
133(1)
6.7.5 The Beta Family of Distributions
134(4)
6.7.5.1 Density Etc.
134(4)
6.7.5.2 Importance in Modeling
138(1)
6.8 Mathematical Complements
138(3)
6.8.1 Hazard Functions
138(1)
6.8.2 Duality of the Exponential Family with the Poisson Family
139(2)
6.9 Computational Complements
141(2)
6.9.1 R's integrate() Function
141(1)
6.9.2 Inverse Method for Sampling from a Density
141(1)
6.9.3 Sampling from a Poisson Distribution
142(1)
6.10 Exercises
143(4)
II Fundamentals of Statistics 147(96)
7 Statistics: Prologue
149(22)
7.1 Importance of This
Chapter
150(1)
7.2 Sampling Distributions
150(2)
7.2.1 Random Samples
150(2)
7.3 The Sample Mean - a Random Variable
152(4)
7.3.1 Toy Population Example
152(1)
7.3.2 Expected Value and Variance of X
153(1)
7.3.3 Toy Population Example Again
154(1)
7.3.4 Interpretation
155(1)
7.3.5 Notebook View
155(1)
7.4 Simple Random Sample Case
156(1)
7.5 The Sample Variance
157(2)
7.5.1 Intuitive Estimation of σ2
157(1)
7.5.2 Easier Computation
158(1)
7.5.3 Special Case: X Is an Indicator Variable
158(1)
7.6 To Divide by n or n-1?
159(2)
7.6.1 Statistical Bias
159(2)
7.7 The Concept of a "Standard Error"
161(1)
7.8 Example: Pima Diabetes Study
162(2)
7.9 Don't Forget: Sample not equal to Population!
164(1)
7.10 Simulation Issues
164(1)
7.10.1 Sample Estimates
164(1)
7.10.2 Infinite Populations?
164(1)
7.11 Observational Studies
165(1)
7.12 Computational Complements
165(5)
7.12.1 The *apply() Functions
165(3)
7.12.1.1 R's apply() Function
166(1)
7.12.1.2 The lapply() and sapply() Function
166(1)
7.12.1.3 The split() and tapply() Functions
167(1)
7.12.2 Outliers/Errors in the Data
168(2)
7.13 Exercises
170(1)
8 Fitting Continuous Models
171(26)
8.1 Why Fit a Parametric Model?
171(1)
8.2 Model-Free Estimation of a Density from Sample Data
172(8)
8.2.1 A Closer Look
172(1)
8.2.2 Example: BMI Data
173(1)
8.2.3 The Number of Bins
174(7)
8.2.3.1 The Bias-Variance Tradeoff
175(1)
8.2.3.2 The Bias-Variance Tradeoff in the Histogram Case
176(2)
8.2.3.3 A General Issue: Choosing the Degree of Smoothing
178(2)
8.3 Advanced Methods for Model-Free Density Estimation
180(1)
8.4 Parameter Estimation
181(6)
8.4.1 Method of Moments
181(1)
8.4.2 Example: BMI Data
182(1)
8.4.3 The Method of Maximum Likelihood
183(2)
8.4.4 Example: Humidity Data
185(2)
8.5 MM vs. MLE
187(1)
8.6 Assessment of Goodness of Fit
187(2)
8.7 The Bayesian Philosophy
189(2)
8.7.1 How Does It Work?
190(1)
8.7.2 Arguments For and Against
190(1)
8.8 Mathematical Complements
191(1)
8.8.1 Details of Kernel Density Estimators
191(1)
8.9 Computational Complements
192(2)
8.9.1 Generic Functions
192(1)
8.9.2 The gmm Package
193(4)
8.9.2.1 The gmm() Function
193(1)
8.9.2.2 Example: Bodyfat Data
193(1)
8.10 Exercises
194(3)
9 The Family of Normal Distributions
197(20)
9.1 Density and Properties
197(3)
9.1.1 Closure under Affine Transformation
198(1)
9.1.2 Closure under Independent Summation
199(1)
9.1.3 A Mystery
200(1)
9.2 R Functions
200(1)
9.3 The Standard Normal Distribution
200(1)
9.4 Evaluating Normal cdfs
201(1)
9.5 Example: Network Intrusion
202(1)
9.6 Example: Class Enrollment Size
203(1)
9.7 The Central Limit Theorem
204(3)
9.7.1 Example: Cumulative Roundoff Error
205(1)
9.7.2 Example: Coin Tosses
205(1)
9.7.3 Example: Museum Demonstration
206(1)
9.7.4 A Bit of Insight into the Mystery
207(1)
9.8 X Is Approximately Normal
207(2)
9.8.1 Approximate Distribution of X
207(1)
9.8.2 Improved Assessment of Accuracy of X
208(1)
9.9 Importance in Modeling
209(1)
9.10 The Chi-Squared Family of Distributions
210(2)
9.10.1 Density and Properties
210(1)
9.10.2 Example: Error in Pin Placement
211(1)
9.10.3 Importance in Modeling
211(1)
9.10.4 Relation to Gamma Family
212(1)
9.11 Mathematical Complements
212(1)
9.11.1 Convergence in Distribution, and the Precisely-Stated CLT
212(1)
9.12 Computational Complements
213(1)
9.12.1 Example: Generating Normal Random Numbers
213(1)
9.13 Exercises
214(3)
10 Introduction to Statistical Inference
217(26)
10.1 The Role of Normal Distributions
217(1)
10.2 Confidence Intervals for Means
218(2)
10.2.1 Basic Formulation
218(2)
10.3 Example: Pima Diabetes Study
220(1)
10.4 Example: Humidity Data
221(1)
10.5 Meaning of Confidence Intervals
221(2)
10.5.1 A Weight Survey in Davis
221(2)
10.6 Confidence Intervals for Proportions
223(3)
10.6.1 Example: Machine Classification of Forest Covers
224(2)
10.7 The Student-t Distribution
226(1)
10.8 Introduction to Significance Tests
227(1)
10.9 The Proverbial Fair Coin
228(1)
10.10 The Basics
229(2)
10.11 General Normal Testing
231(1)
10.12 The Notion of "p-Values"
231(1)
10.13 What's Random and What Is Not
232(1)
10.14 Example: The Forest Cover Data
232(2)
10.15 Problems with Significance Testing
234(3)
10.15.1 History of Significance Testing
234(1)
10.15.2 The Basic Issues
235(1)
10.15.3 Alternative Approach
236(1)
10.16 The Problem of "P-hacking"
237(2)
10.16.1 A Thought Experiment
238(1)
10.16.2 Multiple Inference Methods
238(1)
10.17 Philosophy of Statistics
239(2)
10.17.1 More about Interpretation of Cis
239(6)
10.17.1.1 The Bayesian View of Confidence Intervals
241(1)
10.18 Exercises
241(2)
III Multivariate Analysis 243(122)
11 Multivariate Distributions
245(20)
11.1 Multivariate Distributions: Discrete
245(1)
11.1.1 Example: Marbles in a Bag
245(1)
11.2 Multivariate Distributions: Continuous
246(2)
11.2.1 Motivation and Definition
246(1)
11.2.2 Use of Multivariate Densities in Finding Probabilities and Expected Values
247(1)
11.2.3 Example: Train Rendezvous
247(1)
11.3 Measuring Co-variation
248(3)
11.3.1 Covariance
248(2)
11.3.2 Example: The Committee Example Again
250(1)
11.4 Correlation
251(1)
11.4.1 Sample Estimates
252(1)
11.5 Sets of Independent Random Variables
252(2)
11.5.1 Mailing Tubes
252(2)
11.5.1.1 Expected Values Factor
253(1)
11.5.1.2 Covariance Is 0
253(1)
11.5.1.3 Variances Add
253(1)
11.6 Matrix Formulations
254(2)
11.6.1 Mailing Tubes: Mean Vectors
254(1)
11.6.2 Covariance Matrices
254(1)
11.6.3 Mailing Tubes: Covariance Matrices
255(1)
11.7 Sample Estimate of Covariance Matrix
256(1)
11.7.1 Example: Pima Data
257(1)
11.8 Mathematical Complements
257(5)
11.8.1 Convolution
257(2)
11.8.1.1 Example: Backup Battery
258(1)
11.8.2 Transform Methods
259(17)
11.8.2.1 Generating Functions
259(2)
11.8.2.2 Sums of Independent Poisson Random Variables Are Poisson Distributed
261(1)
11.9 Exercises
262(3)
12 The Multivariate Normal Family of Distributions
265(10)
12.1 Densities
265(1)
12.2 Geometric Interpretation
266(3)
12.3 R Functions
269(1)
12.4 Special Case: New Variable Is a Single Linear Combination of a Random Vector
270(1)
12.5 Properties of Multivariate Normal Distributions
270(2)
12.6 The Multivariate Central Limit Theorem
272(1)
12.7 Exercises
273(2)
13 Mixture Distributions
275(12)
13.1 Iterated Expectations
276(5)
13.1.1 Conditional Distributions
277(1)
13.1.2 The Theorem
277(2)
13.1.3 Example: Flipping Coins with Bonuses
279(1)
13.1.4 Conditional Expectation as a Random Variable
280(1)
13.1.5 What about Variance?
280(1)
13.2 A Closer Look at Mixture Distributions
281(3)
13.2.1 Derivation of Mean and Variance
281(2)
13.2.2 Estimation of Parameters
283(5)
13.2.2.1 Example: Old Faithful Estimation
283(1)
13.3 Clustering
284(1)
13.4 Exercises
285(2)
14 Multivariate Description and Dimension Reduction
287(22)
14.1 What Is Overfitting Anyway?
288(5)
14.1.1 "Desperate for Data"
288(1)
14.1.2 Known Distribution
289(1)
14.1.3 Estimated Mean
289(1)
14.1.4 The Bias/Variance Tradeoff: Concrete Illustration
290(2)
14.1.5 Implications
292(1)
14.2 Principal Components Analysis
293(4)
14.2.1 Intuition
293(2)
14.2.2 Properties of PCA
295(1)
14.2.3 Example: Turkish Teaching Evaluations
296(1)
14.3 The Log-Linear Model
297(3)
14.3.1 Example: Hair Color, Eye Color and Gender
297(2)
14.3.2 Dimension of Our Data
299(1)
14.3.3 Estimating the Parameters
299(1)
14.4 Mathematical Complements
300(2)
14.4.1 Statistical Derivation of PCA
300(2)
14.5 Computational Complements
302(4)
14.5.1 R Tables
302(1)
14.5.2 Some Details on Log-Linear Models
302(8)
14.5.2.1 Parameter Estimation
303(1)
14.5.2.2 The loglin() Function
304(1)
14.5.2.3 Informal Assessment of Fit
305(1)
14.6 Exercises
306(3)
15 Predictive Modeling
309(34)
15.1 Example: Heritage Health Prize
309(1)
15.2 The Goals: Prediction and Description
310(1)
15.2.1 Terminology
310(1)
15.3 What Does "Relationship" Mean?
311(3)
15.3.1 Precise Definition
311(2)
15.3.2 Parametric Models for the Regression Function m()
313(1)
15.4 Estimation in Linear Parametric Regression Models
314(1)
15.5 Example: Baseball Data
315(4)
15.5.1 R. Code
316(3)
15.6 Multiple Regression
319(1)
15.7 Example: Baseball Data (cont'd.)
320(1)
15.8 Interaction Terms
321(1)
15.9 Parametric Estimation
322(6)
15.9.1 Meaning of "Linear"
322(1)
15.9.2 Random-X and Fixed-X Regression
322(1)
15.9.3 Point Estimates and Matrix Formulation
323(3)
15.9.4 Approximate Confidence Intervals
326(2)
15.10 Example: Baseball Data (cont'd )
328(1)
15.11 Dummy Variables
329(1)
15.12 Classification
330(6)
15.12.1 Classification = Regression
331(1)
15.12.2 Logistic Regression
332(2)
15.12.2.1 The Logistic Model: Motivations
332(2)
15.12.2.2 Estimation and Inference for Logit
334(1)
15.12.3 Example: Forest Cover Data
334(1)
15.12.4 R Code
334(1)
15.12.5 Analysis of the Results
335(1)
15.12.5.1 Multiclass Case
336(1)
15.13 Machine Learning: Neural Networks
336(4)
15.13.1 Example: Predicting Vertebral Abnormalities
336(3)
15.13.2 But What Is Really Going On?
339(1)
15.13.3 R Packages
339(1)
15.14 Computational Complements
340(2)
15.14.1 Computational Details in Section 15.5.1
340(1)
15.14.2 More Regarding glm()
341(1)
15.15 Exercises
342(1)
16 Model Parsimony and Overfitting
343(6)
16.1 What Is Overfitting?
343(2)
16.1.1 Example: Histograms
343(1)
16.1.2 Example: Polynomial Regression
344(1)
16.2 Can Anything Be Done about It?
345(1)
16.2.1 Cross-Validation
345(1)
16.3 Predictor Subset Selection
346(1)
16.4 Exercises
347(2)
17 Introduction to Discrete Time Markov Chains
349(16)
17.1 Matrix Formulation
350(1)
17.2 Example: Die Game
351(1)
17.3 Long-Run State Probabilities
352(4)
17.3.1 Stationary Distribution
353(1)
17.3.2 Calculation of π
354(1)
17.3.3 Simulation Calculation of π
355(1)
17.4 Example: 3-Heads-in-a-Row Game
356(2)
17.5 Example: Bus Ridership Problem
358(1)
17.6 Hidden Markov Models
359(2)
17.6.1 Example: Bus Ridership
360(1)
17.6.2 Computation
361(1)
17.7 Google PageRank
361(1)
17.8 Computational Complements
361(1)
17.8.1 Initializing a Matrix to All 0s
361(1)
17.9 Exercises
362(3)
IV Appendices 365(26)
A R Quick Start
367(16)
A.1 Starting R
367(1)
A.2 Correspondences
368(1)
A.3 First Sample Programming Session
369(3)
A.4 Vectorization
372(1)
A.5 Second Sample Programming Session
372(2)
A.6 Recycling
374(1)
A.7 More on Vectorization
374(1)
A.8 Default Argument Values
375(1)
A.9 The R List Type
376(2)
A.9.1 The Basics
376(1)
A.9.2 S3 Classes
377(1)
A.10 Data Frames
378(2)
A.11 Online Help
380(1)
A.12 Debugging in R
380(3)
B Matrix Algebra
383(8)
B.1 Terminology and Notation
383(2)
B.1.1 Matrix Addition and Multiplication
383(2)
B.2 Matrix Transpose
385(1)
B.3 Matrix Inverse
385(1)
B.4 Eigenvalues and Eigenvectors
385(1)
B.5 Mathematical Complements
386(5)
B.5.1 Matrix Derivatives
386(5)
Bibliography 391(4)
Index 395
Norman Matloff is a professor of computer science at the University of California, Davis, and was formerly a statistics professor there. He is on the editorial boards of the Journal of Statistical Software and The R Journal. His book Statistical Regression and Classification: From Linear Models to Machine Learning was the recipient of the Ziegel Award for the best book reviewed in Technometrics in 2017. He is a recipient of his university's Distinguished Teaching Award.
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