Not many data scientists are formally trained in statistics. There are also very few good books and courses that teach these statistical methods from a data science perspective. Through this post, I intend to shed some light on the following: What is Statistics? Statistics in relation with machine learning.
How do I learn statistics & probability in data science?
If you do have a formal math background, this approach will help you translate theory into practice and give you some fun programming challenges. Here are the 3 steps to learning the statistics and probability required for data science: Core Statistics Concepts – Descriptive statistics, distributions, hypothesis testing, and regression.
What are the different types of statistical analysis?
11. Types of Statistical Analysis ● Descriptive Statistics - Describes data. ○ Common Tools - Central tendency, Data distribution, skewness ● Inferential Statistics - Draw conclusions from the sample & generalize for entire population ○ Common Tools - Hypothesis Testing, Confidence Intervals, Regression Analysis
Introduction
The first two decades of this century has witnessed the exposition of the data collection at a blossoming age of information and technology. The recent technological revolution has made information acquisition easy and inexpen-sive through automated data collection processes. The frontiers of scientific research and technological developments have
1.1.3 Computer and Information Sciences
The development of information and technology itself collects massive amounts of data. For example, there are billions of web pages on the internet, and an internet search engine needs to statistically learn the most likely out-comes of a query and fast algorithms need to evolve with empirical data. The input dimensionality of queries can be huge.
1.1.5 Business and Program Evaluation
Big data arises frequently in marketing and program evaluation. Multi-channel strategies are frequently used to market products, such as drugs and medical devices. Data from hundreds of thousands of doctors are collected with different marketing strategies over a period of time, resulting in big data. The design of marketing strategies and the eval
1.1.6 Earth Sciences and Astronomy
Spatial-temporal data have been widely available in the earth sciences. In meteorology and climatology studies, measurements such as temperatures and precipitations are widely available across many regions over a long period of time. They are critical for understanding climate changes, local and global warming, and weather forecasts, and provide an
1.3 Impact of Dimensionality
What makes high-dimensional statistical inference different from tradi-tional statistics? High-dimensionality has a significant impact on computa-tion, spurious correlation, noise accumulation, and theoretical studies. We now briefly touch these topics.
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1.3.1 Computation
Statistical inferences frequently involve numerical optimization. Optimiza-tions in millions and billions dimensional spaces are not unheard of and arise easily when interactions are considered. High-dimensional optimization is not only expensive in computation, but also slow in convergence. It also creates numerical instability. Algorithms can eas
1.3.4 Statistical theory
High dimensionality has a strong impact on statistical theory. The tradi-tional asymptotic theory assumes that sample size n tends to infinity while keeping p fixed. This does not reflect the reality of the high dimensionality and cannot explain the observed phenomena such as noise accumulation and spurious correlation. A more reasonable framework
1.5 What big data can do
Big Data hold great promise for the discovery of heterogeneity and search for personalized treatments and precision marketing. An important aim for big data analysis is to understand heterogeneity for personalized medicine or services from large pools of variables, factors, genes, environments and their interactions as well as latent factors. Such
1.6 Scope of the book
This book will provide a comprehensive and systematic account of theo-ries and methods in high-dimensional data analysis. The statistical problems range from high-dimensional sparse regression, compressed sensing, sparse likelihood-based models, supervised and unsupervised learning, large covari-ance matrix estimation and graphical models, high-dim
2.1 Introduction
In this chapter we discuss some popular linear methods for regression anal-ysis with continuous response variable. We call them linear regression models in general, but our discussion is not limited to the classical multiple linear regression. They are extended to multivariate nonparametric regression via the kernel trick. We first give a brief int
Y = X + .
β ε The matrix X is known as the design matrix and is of crucial importance to the whole theory of linear regression analysis. The RSS( ) can be written as β RSS( ) = Y X 2 = (Y X )T (Y X ). β − β − β − β Differentiating RSS( ) with respect to and setting the gradient vector to β β zero, we obtain the normal equations
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XT Y = XT X .
β Here we assume that p < n and X has rank p. Hence XT X is invertible and the normal equations yield the least-squares estimator of β
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P2 = P or P(In P) = 0,
− namely P is a projection matrix onto the space spanned by the columns of X. Proof. It follows from the direct calculation that
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⎛ B1(X1) . B .. = ⎜ ⎝ B1(Xn)
· · · · · · and the least-squares estimate is given by
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2 Penalized Least Squares
Define a penalized residual sum-of-squares (PRSS) as follows: n p p
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2.6.3 Bayesian Interpretation
Ridge regression has a neat Bayesian interpretation in the sense that it can be a formal Bayes estimator. We begin with the homoscedastic Gaussian error model: p Yi =
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2.7 Regression in Reproducing Kernel Hilbert Space
A Hilbert space is an abstract vector space endowed by the structure of an inner product. Let be an arbitrary set and be a Hilbert space of real-
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X H
valued functions on , endowed by the inner product , . The evaluation X · · H functional over the Hilbert space of functions is a linear functional that H evaluates each function at a point x: Lx : f f(x), f . ∀ ∈ H Hilbert space is called a reproducing kernel Hilbert space (RKHS) if, for H all x , the map Lx is continuous at any f , namely, there
H ∀ ∈ H
By the Riesz representation theorem, for all x , there exists a unique ∈ X element Kx with the reproducing property ∈ H f(x) = Lx(f) = f, Kx , f . H ∀ ∈ H Since Kx is itself a function in , it holds that for every x , there exists
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2.8 Leave-one-out and Generalized Cross-validation
We have seen that both ridge regression and the kernel ridge regression use a tuning parameter λ. In practice, we would like to use the data to pick a data-driven λ in order to achieve the “best” estimation/prediction performance. This problem is often called tuning parameter selection and is ubiquitous in modern statistics and machine learning. A
Theorem 2.7 For a linear smoother Y = SY with the self-stable property, we have
Yi f( − i)(Xi) = − and its leave-one-out CV error is equal to
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Introduction to Penalized Least-Squares
Variable selection is vital to high-dimensional statistical learning and in-ference, and is essential for scientific discoveries and engineering innovation. Multiple regression is one of the most classical and useful techniques in statis-tics. This chapter introduces penalized least-squares approaches to variable selection problems in multiple regr
Theorem 3.1 (Characterization of PLSE) Assume that pλ( θ ) is folded
concave. Then a necessary condition for Rp being a local minimizer of β ∈
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3.3 Lasso and L1 Regularization
Lasso gains its popularity due to its convexity and computational expe-dience. The predecessor of Lasso is the negative garrote. The study of Lasso also leads to the Dantzig selector, the adaptive Lasso and the elastic net. This section touches on the basis of these estimators in which the L1-norm regularization plays a central role.
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Practical Statistics for Data Scientists: 50+ Essential Concepts
