GlossaryΒΆ
Term (s) | Description | Notation |
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Data object (point, observation, sample, example) | A unit of analysis. Typically, a data object is represented as a vector of features. | Typically denoted as lower case letter, often bold, e.g., \(\bf x\) or \({\bf x}_i\) or \({\bf x}^{(i)}\), where the subscript or superscript \(i\) denotes membership in a data set. |
Data set | A collection of data objects. | Typically denoted as upper case letter, often bold, e.g., \(\bf X\). |
Vector | A list (or array) of real values (\(\in \{-\infty,\infty\}\)). | Typically denoted as bold lower case letter, e.g., \(\bf x\). \(\bf x \in \mathbb{R}^d\), means that the vector represents a point in a \(d\)-dimensional vector space. An individual element of the vector is denoted as \(x_i\). |
Matrix | A 2-way array of real values (\(\in \{-\infty,\infty\}\)). | Typically denoted as bold upper case letter, e.g., \(\bf X\).
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Transpose | \({\bf X}^\top\), \({\bf x}^\top\) | A transpose of a matrix is an operator which flips a matrix over its diagonal, i.e., \(X_{ij} = X^\top_{ji}\). Transpose of a vector is a \(1 \times d\) matrix. |
Matrix multiplication | \({\bf X} = {\bf YZ}\) | Only valid if the number of columns in \({\bf Y}\) is equal to the number of rows in \({\bf Z}\).
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Vector dot (inner) product | \({\bf x}.{\bf x} = \sum_{i=1}^d x_i^2\) | In matrix notation, the dot product is expressed as \({\bf x}^\top{\bf x}\) |
Data Matrix | A \(n \times d\) matrix, \({\bf X}\) | If each data object in a data set can be represented as a vector in \(\mathbb{R}^d\), the data set of \(n\) such objects is typically arranged as a \(n \times d\) matrix, \({\bf X}\), where the transpose of each row of the matrix corresponds to a data object, i.e., \({\bf x}_i = X_{i*}^\top\). |
Random Variable | A variable whose possible values are outcomes of a random phenomena (distribution) | Typically denoted as an upper case letter, \(X\) (bold - \({\bf X}\), if multivariate) |
Probability | A measure of the likelihood of an event to occur | \(P(A)\) denotes the probability of an event \(A\) to occur.
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