Meta-pages: Notation¶
This page sets out the notational conventions used in the MUCM toolkit.
General Conventions¶
Vectors and matrices¶
Vectors in the toolkit can be either row \((1\times n)\) or column \((n\times 1)\) vectors. When not specified explicitly, a vector is a column vector by default.
We do not identify vectors or matrices by bold face.
Functions¶
Functions are denoted with a dot argument, e.g. \(f(\cdot)\), while their value at a point \(x\) is \(f(x)\).
The following compact notation is used for arrays of function values. Suppose that \(g(\cdot)\) is a function taking values \(x\) in some space, and let \(D\) be a vector with \(n\) elements \(D = \{x_1,x_2, \ldots, x_n\}\) in that space; then \(g(D)\) is made up of the function values \(g(x_1),g(x_2),\ldots,g(x_n)\). More specifically, we have the following cases:
- If \(g(x)\) is a scalar, then \(g(D)\) is a \((n\times 1)\) (column) vector by default, unless explicitly defined as a \((1\times n)\) (row) vector.
- If \(g(x)\) is a \((t\times 1)\) column vector, then \(g(D)\) is a matrix of dimension \(t\times n\).
- If \(g(x)\) is a \((1\times t)\) row vector, then \(g(D)\) is a matrix of dimension \((n\times t)\).
- If \(g(x,x')\) is a scalar, then \(g(D,x')\) is a \((n\times 1)\) column vector, \(g(x,D')\) is a \((1\times n)\) row vector and \(g(D,D')\) is a \((n\times n)\) matrix.
Other notation¶
The \(\,^*\) superscript denotes a posterior value or function.
The matrix/vector transpose is denoted with a roman superscript \(\,^{\textrm{T}}\).
Expectations, Variances and Covariances are denoted as \(\textrm{E}[\cdot], \textrm{Var}[\cdot], \textrm{Cov}[\cdot,\cdot]\).
The trace of a matrix is denoted as \(\textrm{tr}\).
Reserved symbols¶
The following is a list of symbols that represent fundamental quantities across the toolkit. These reserved symbols will always represent quantities of the same generic type in the toolkit. For instance, the symbol \(n\) will always denote a number of design points. Where two or more different designs are considered in a toolkit page, their sizes will be distinguished by subscripts or superscripts, e.g. \(n_{t}\) might be the size of a training sample design, while \(n_v\) is the size of a validation design. Notation should always be defined properly in toolkit pages, but the use of reserved symbols has mnemonic value to assist the reader in remembering the meanings of different symbols.
The reserved symbols comprise a relatively small set of symbols (and note that if only a lower-case symbol is reserved the corresponding upper-case symbol is not). Non-reserved symbols have no special meanings in the toolkit.
| Symbol | Meaning |
|---|---|
| Dimensions | |
| \(\strut n\) | Number of design points |
| \(\strut p\) | Number of active inputs |
| \(\strut q\) | Number of basis functions |
| \(\strut r\) | Number of outputs |
| \(\strut s\) | Number of hyperparameter sets in an emulator |
| Input - Output | |
| \(\strut x\) | Point in the simulator’s input space |
| \(\strut y\) | Reality - the actual system value |
| \(\strut z\) | Observation of reality \(y\) |
| \(\strut D\) | Design, comprising an ordered set of points in an input space |
| \(d(\cdot)\) | Model discrepancy function |
| \(f(\cdot)\) | The output(s) of a simulator |
| \(h(\cdot)\) | Vector of basis functions |
| Hyperparameters | |
| \(\beta\) | Hyperparameters of a mean function |
| \(\delta\) | Hyperparameters of a correlation function |
| \(\sigma^2\) | Scale hyperparameter for a covariance function |
| \(\theta\) | Collection of hyperparameters on which the emulator is conditioned |
| \(\nu\) | Nugget |
| \(\pi\) | Distribution of hyperparameters |
| Statistics | |
| \(m(\cdot)\) | Mean function |
| \(v(\cdot,\cdot)\) | Covariance function |
| \(m^*(\cdot)\) | Emulator’s posterior mean, conditioned on the hyperparameters and design points |
| \(v^*(\cdot,\cdot)\) | Emulator’s posterior covariance, conditioned on the hyperparameters and design points |
| \(c(\cdot,\cdot)\) | Correlation function |