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**Tidy data** is an alternative name for the common statistical form called a *model matrix* or *data matrix*. A data matrix is defined in ^{ [1] } as follows:

A standard method of displaying a multivariate set of data is in the form of a data matrix in which rows correspond to sample individuals and columns to variables, so that the entry in the

ith row andjth column gives the value of thejth variate as measured or observed on theith individual.

Hadley Wickham later defined "Tidy Data" as data sets that are arranged such that each variable is a column and each observation (or *case*) is a row.^{ [2] } (Originally with additional per-table conditions that made the definition equivalent to the Boyce–Codd 3rd normal form.)

Data arrangement is an important consideration in data processing, but should not be confused with the also important task of data cleansing.

Other relevant formulations include denormalization prior to machine learning modeling (informally denoting moving data to a "wide form" where all possible measurements are in a given row), and use of semantic triples as intermediate representation (informally a "tall" or "long" form, where measurements about a single instance are spread across many rows).

The **principal components** of a collection of points in a real *p*-space are a sequence of direction vectors, where the vector is the direction of a line that best fits the data while being orthogonal to the first vectors. Here, a best-fitting line is defined as one that minimizes the average squared distance from the points to the line. These directions constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. **Principal component analysis** (**PCA**) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest.

**Factor analysis** is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called **factors**. For example, it is possible that variations in six observed variables mainly reflect the variations in two unobserved (underlying) variables. Factor analysis searches for such joint variations in response to unobserved latent variables. The observed variables are modelled as linear combinations of the potential factors, plus "error" terms.

**Analysis of covariance** (**ANCOVA**) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

The **general linear model** or **general multivariate regression model** is simply a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as

In statistics, a **contingency table** is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business intelligence, engineering, and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The term *contingency table* was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation", part of the *Drapers' Company Research Memoirs Biometric Series I* published in 1904.

In statistics, a **design matrix**, also known as **model matrix** or **regressor matrix** and often denoted by **X**, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables that indicate group membership in an ANOVA, or it can contain values of continuous variables.

In the analysis of multivariate observations designed to assess subjects with respect to an attribute, a **Guttman Scale** is a single (unidimensional) ordinal scale for the assessment of the attribute, from which the original observations may be reproduced. The discovery of a Guttman Scale in data depends on their multivariate distribution's conforming to a particular structure. Hence, a Guttman Scale is a *hypothesis* about the structure of the data, formulated with respect to a specified attribute and a specified population and cannot be constructed for any given set of observations. Contrary to a widespread belief, a Guttman Scale is not limited to dichotomous variables and does not necessarily determine an order among the variables. But if variables are all dichotomous, the variables are indeed ordered by their sensitivity in recording the assessed attribute, as illustrated by Example 1.

**Multivariate analysis** (**MVA**) is based on the principles of multivariate statistics, which involves observation and analysis of more than one statistical outcome variable at a time. Typically, MVA is used to address the situations where multiple measurements are made on each experimental unit and the relations among these measurements and their structures are important. A modern, overlapping categorization of MVA includes:

The **sample mean** or **empirical mean** and the **sample covariance** are statistics computed from a collection of data on one or more random variables. The sample mean and sample covariance are estimators of the population mean and population covariance, where the term *population* refers to the set from which the sample was taken.

In statistics, the **projection matrix**, sometimes also called the **influence matrix** or **hat matrix**, maps the vector of response values to the vector of fitted values. It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation.

The **multitrait-multimethod** (**MTMM**) **matrix** is an approach to examining construct validity developed by Campbell and Fiske (1959). It organizes convergent and discriminant validity evidence for comparison of how a measure relates to other measures.

In regression analysis, **partial leverage** (**PL**) is a measure of the contribution of the individual independent variables to the total leverage of each observation. That is, if *h*_{i} is the *i*^{th} element of the diagonal of the hat matrix, PL is a measure of how *h*_{i} changes as a variable is added to the regression model. It is computed as:

In mathematics, a **matrix** is a rectangular *array* or *table* of numbers, symbols, or expressions, arranged in *rows* and *columns*. For example, the dimension of the matrix below is 2 × 3, because there are two rows and three columns:

In statistics, **multiple correspondence analysis** (**MCA**) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables.

In probability and statistics, an **elliptical distribution** is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots.

**Linear least squares** (**LLS**) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

**Hadley Wickham** is a statistician from New Zealand who is currently Chief Scientist at RStudio and an adjunct Professor of statistics at the University of Auckland, Stanford University, and Rice University. He is best known for his development of open-source software packages for the R statistical programming language for data visualisation, notably ggplot2, and data transformation, notably his "tidyverse" packages, which support a tidy data approach to data import, analysis and modelling.

In the field of statistical learning theory, **matrix regularization** generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predictive functions. For example, in the more common vector framework, Tikhonov regularization optimizes over

In statistics, **linear regression** is a linear approach to modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called **multiple linear regression**. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

In statistics, the class of **vector generalized linear models** (**VGLMs**) was proposed to enlarge the scope of models catered for by generalized linear models (**GLMs**). In particular, VGLMs allow for response variables outside the classical exponential family and for more than one parameter. Each parameter can be transformed by a *link function*. The VGLM framework is also large enough to naturally accommodate multiple responses; these are several independent responses each coming from a particular statistical distribution with possibly different parameter values.

- ↑ Krzanowski, W. J., F. H. C. Marriott, Multivariate Analysis Part 1, Edward Arnold, 1994
- ↑ Wickham, Hadley (20 February 2013). "Tidy Data" (PDF).
*Journal of Statistical Software*.

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