Maria Cristina Mariani - Data Science in Theory and Practice

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DATA SCIENCE IN THEORY AND PRACTICE delivers a comprehensive treatment of the mathematical and statistical models useful for analyzing data sets arising in various disciplines, like banking, finance, health care, bioinformatics, security, education, and social services. Written in five parts, the book examines some of the most commonly used and fundamental mathematical and statistical concepts that form the basis of data science. The authors go on to analyze various data transformation techniques useful for extracting information from raw data, long memory behavior, and predictive modeling. The book offers readers a multitude of topics all relevant to the analysis of complex data sets. Along with a robust exploration of the theory underpinning data science, it contains numerous applications to specific and practical problems. The book also provides examples of code algorithms in R and Python and provides pseudo-algorithms to port the code to any other language. Ideal for students and practitioners without a strong background in data science, readers will also learn from topics like: Analyses of foundational theoretical subjects, including the history of data science, matrix algebra and random vectors, and multivariate analysis A comprehensive examination of time series forecasting, including the different components of time series and transformations to achieve stationarity Introductions to both the R and Python programming languages, including basic data types and sample manipulations for both languages An exploration of algorithms, including how to write one and how to perform an asymptotic analysis A comprehensive discussion of several techniques for analyzing and predicting complex data sets Perfect for advanced undergraduate and graduate students in Data Science, Business Analytics, and Statistics programs,
will also earn a place in the libraries of practicing data scientists, data and business analysts, and statisticians in the private sector, government, and academia.

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361 Linear Combinations of Sample Means The sample mean of can be found - фото 426

3.6.1 Linear Combinations of Sample Means

The sample mean of can be found either by averaging the values or as a linear combination of - фото 427can be found either by averaging the values or as a linear combination of the sample mean vector of - фото 428values Data Science in Theory and Practice - изображение 429or as a linear combination of Data Science in Theory and Practice - изображение 430, the sample mean vector of Data Science in Theory and Practice - изображение 431.

(3.12) 362 Linear Combinations of Sample Variance and Covariance The sample - фото 432

3.6.2 Linear Combinations of Sample Variance and Covariance

The sample variance of Data Science in Theory and Practice - изображение 433can be found as the sample variance of Data Science in Theory and Practice - изображение 434or directly from картинка 435and Data Science in Theory and Practice - изображение 436, where Data Science in Theory and Practice - изображение 437is the sample covariance matrix of Data Science in Theory and Practice - изображение 438:

(3.13) We recall from Section 23that variance is always nonnegative Thus we have - фото 439

We recall from Section 2.3that variance is always nonnegative. Thus, we have картинка 440, and therefore, for every If we define another linear combination where - фото 441, for every Data Science in Theory and Practice - изображение 442.

If we define another linear combination Data Science in Theory and Practice - изображение 443, where Data Science in Theory and Practice - изображение 444is a vector of constants different from картинка 445, then the sample covariance of and is given by 314 where - фото 446and is given by 314 where is the number of measurements - фото 447is given by

(3.14) where is the number of measurements Please refer to Johnson and Wichern 2014 - фото 448

where картинка 449is the number of measurements.

Please refer to Johnson and Wichern (2014) for the proof of ( 3.14).

3.6.3 Linear Combinations of Sample Correlation

The sample correlation between and is obtained as follows 315 We note that the sample resu - фото 450and is obtained as follows 315 We note that the sample results in Section - фото 451is obtained as follows:

(3.15) We note that the sample results in Section 36have population counterparts We - фото 452

We note that the sample results in Section 3.6have population counterparts. We briefly state them below:

The population mean of is defined as follows where denotes the population mean vector The p - фото 453is defined as follows:

where denotes the population mean vector The population variance of is de - фото 454

where картинка 455denotes the population mean vector. The population variance of is defined as follows where denotes the population covariance matrix - фото 456is defined as follows:

where denotes the population covariance matrix which is defined in 35 as - фото 457

where denotes the population covariance matrix which is defined in 35 as Let - фото 458denotes the population covariance matrix which is defined in ( 3.5) as

Let where is a vector of constants different from - фото 459

Let картинка 460, where картинка 461is a vector of constants different from картинка 462. The population covariance of and is defined as where - фото 463and is defined as where denotes the population covariance matrix which is - фото 464is defined as

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