0.7 Notation

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$x$ - input sequence
$n$ - length of $x$
$x = x_{1} x_{2} ... x_{n}$ , $x_{i}^{j} = x_{i} x_{i + 1} ... x_{j}$
$α$ - discrete alphabet
$r$ - cardinality of $α$
$n_{x} (a)$ - the number of times that $a \in α$ appears in $x$
$P_{x} (a)$ - empirical probability
$Q$ - the true i.i.d. distribution of $x$
$H$ - entropy
$D (\cdot ∥ \cdot)$ - divergence
$⌈ \cdot ⌉$ - rounding up
$R$ - rate of source code
$T_{Q} (δ)$ - set of inputs that are $δ$ -typical with respect to (w.r.t.) $Q$
$C$ - a code
${(\cdot)}^{C}$ - complement of set
$\cdot^{T}$ - transpose of vector or matrix
$l (x)$ - length of code in bits
$x^{'} = S_{x}$ - step $\forall n \in Z$ , $x_{n}^{'} = x_{n + 1}$
$θ$ - parameters of parametric source (can contain multiple scalar parameters)
$M$ - number of states of unifilar source
$S = {1, ..., M}$ - states of unifilar source
$s_{1}, ..., s_{n}$ , - state sequence
$q (s | s^{'})$ - transition probability
$S_{t} = g (S_{t - 1}, X_{t - 1})$ - next state function
$Λ$ - class of parametric models
$c_{n}$ - collection of lossless codes for length- $n$ inputs
$I (\cdot; \cdot)$ - mutual information
$r_{n} (l, θ)$ - how far a coding length function $l$ is from the entropy
$h_{2} (\cdot)$ - binary entropy
$I (\cdot)$ - Fisher information
$J (\cdot)$ - Jeffreys' prior
$θ_{M L}$ - maximum likelihood parameter
$R_{n}^{-}$ , $R_{n}^{+}$ - min-max and max-min redundancy
$T$ - set of leaves of a context tree source
$s$ - state of context tree
$n_{x} (s, a)$ - number of times that $a \in α$ appears in context $s$ in $x$
$T^{*}$ - optimal context tree source
$D$ - maximal depth of tree source
$T_{s}^{*}$ - optimal tree structure to encode symbols whose context ends with $s$
$MDL (s)$ - minimum description length required for encoding these symbols
$KT (\cdot, \cdot)$ - Krichevsky-Trofimov coding length
$(y, i)$ - BWT output consisting the permuted sequence and index
$\hat{x}$ - approximate version of $x$
$\hat{α}$ - reproduction alphabet
$d (\cdot, \cdot)$ - distortion metric
$\bar{d} (x, \hat{x})$ - distortion between sequences
$D$ - distortion level
$R (D)$ - minimal rate such that $\hat{x}$ can be described
$t$ - temperature in simulated annealing
$Z_{t}$ - normalization factor in Boltzmann pmf
$N_{r}$ - recurrence time

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Source: OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1

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