# Complex random vector

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In [probability theory](/source/probability_theory) and [statistics](/source/statistics), a '''complex random vector''' is typically a [tuple](/source/tuple) of [complex](/source/complex_number)-valued [random variable](/source/random_variable)s, and generally is a random variable taking values in a [vector space](/source/vector_space) over the [field](/source/field_(mathematics)) of complex numbers. If <math>Z_1,\ldots,Z_n</math> are complex-valued random variables, then the ''n''-tuple <math>\left( Z_1,\ldots,Z_n \right)</math> is a complex random vector. Complex random variables can always be considered as pairs of real random vectors: their real and imaginary parts.

Some concepts of real random vectors have a straightforward generalization to complex random vectors. For example, the definition of the mean of a complex random vector. Other concepts are unique to complex random vectors.

Applications of complex random vectors are found in [digital signal processing](/source/digital_signal_processing).

{{Probability fundamentals}}

==Definition==
A complex random vector <math> \mathbf{Z} = (Z_1,\ldots,Z_n)^T </math> on the [probability space](/source/probability_space) <math>(\Omega,\mathcal{F},P)</math> is a [function](/source/Function_(mathematics)) <math> \mathbf{Z} \colon \Omega \rightarrow \mathbb{C}^n </math> such that the vector <math>(\Re{(Z_1)},\Im{(Z_1)},\ldots,\Re{(Z_n)},\Im{(Z_n)})^T </math> is a [real random vector](/source/Multivariate_random_variable) on <math>(\Omega,\mathcal{F},P)</math> where <math>\Re{(z)}</math> denotes the real part of <math>z</math> and <math>\Im{(z)}</math> denotes the imaginary part of <math>z</math>.<ref name=Lapidoth>{{cite book |first=Amos |last=Lapidoth |year=2009 |title=A Foundation in Digital Communication |publisher=Cambridge University Press |isbn=978-0-521-19395-5}}</ref>{{rp|p. 292}}

==Cumulative distribution function==
The generalization of the [cumulative distribution function](/source/cumulative_distribution_function) from real to complex random variables is not obvious because expressions of the form <math> P(Z \leq 1+3i) </math> make no sense. However expressions of the form <math> P(\Re{(Z)} \leq 1, \Im{(Z)} \leq 3) </math> make sense. Therefore, the cumulative distribution function <math>F_{\mathbf{Z}} : \mathbb{C}^n \mapsto [0,1]</math> of a random vector <math>\mathbf{Z}=(Z_1,...,Z_n)^T </math> is defined as

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>F_{\mathbf{Z}}(\mathbf{z}) = \operatorname{P}(\Re{(Z_1)} \leq \Re{(z_1)} , \Im{(Z_1)} \leq \Im{(z_1)},\ldots,\Re{(Z_n)} \leq \Re{(z_n)} , \Im{(Z_n)} \leq \Im{(z_n)})</math>|{{EquationRef|Eq.1}}}}
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where <math>\mathbf{z} = (z_1,...,z_n)^T</math>.

==Expectation==
As in the real case the '''expectation''' (also called [expected value](/source/expected_value)) of a complex random vector is taken component-wise.<ref name=Lapidoth />{{rp|p. 293}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math> \operatorname{E}[\mathbf{Z}] = (\operatorname{E}[Z_1],\ldots,\operatorname{E}[Z_n])^T </math>|{{EquationRef|Eq.2}}}}
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==Covariance matrix and pseudo-covariance matrix==
{{See also|Covariance matrix#Complex random vector}}

The ''[covariance matrix](/source/covariance_matrix)'' (also called ''second central moment'') <math> \operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math> contains the covariances between all pairs of components. The covariance matrix of an <math>n \times 1</math> random vector is an <math>n \times n</math> [matrix](/source/Matrix_(mathematics)) whose <math>(i,j)</math><sup>th</sup> element is the [covariance](/source/covariance) between the ''i''<sup> th</sup> and the ''j''<sup> th</sup> random variables.<ref name=Gubner>{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}</ref>{{rp|p.372}} Unlike in the case of real random variables, the covariance between two random variables involves the [complex conjugate](/source/complex_conjugate) of one of the two. Thus the covariance matrix is a [Hermitian matrix](/source/Hermitian_matrix).<ref name=Lapidoth />{{rp|p. 293}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||
<math>
\begin{align}
& \operatorname{K}_{\mathbf{Z}\mathbf{Z}}
=
\operatorname{cov}[\mathbf{Z},\mathbf{Z}]
=
\operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{Z}-\operatorname{E}[\mathbf{Z}])}^H]
=
\operatorname{E}[\mathbf{Z}\mathbf{Z}^H]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{Z}^H] \\[12pt]
\end{align}
</math>
|{{EquationRef|Eq.3}}}}
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|border colour = #0073CF
|background colour=#F5FFFA}}

:<math>
\operatorname{K}_{\mathbf{Z}\mathbf{Z}}=
\begin{bmatrix}
 \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(Z_1 - \operatorname{E}[Z_1])}] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(Z_2 - \operatorname{E}[Z_2])}] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(Z_n - \operatorname{E}[Z_n])}] \\ \\
 \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(Z_1 - \operatorname{E}[Z_1])}] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(Z_2 - \operatorname{E}[Z_2])}] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(Z_n - \operatorname{E}[Z_n])}] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
 \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(Z_1 - \operatorname{E}[Z_1])}] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(Z_2 - \operatorname{E}[Z_2])}] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(Z_n - \operatorname{E}[Z_n])}]
\end{bmatrix}
</math>

The ''[pseudo-covariance matrix](/source/pseudo-covariance_matrix)'' (also called ''relation matrix'') is defined replacing Hermitian transposition by transposition in the definition above.

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||
<math>
\operatorname{J}_{\mathbf{Z}\mathbf{Z}}
=
\operatorname{cov}[\mathbf{Z},\overline{\mathbf{Z}}]
=
\operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{Z}-\operatorname{E}[\mathbf{Z}])}^T]
=
\operatorname{E}[\mathbf{Z}\mathbf{Z}^T]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{Z}^T]
</math>
|{{EquationRef|Eq.4}}}}
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:<math>
\operatorname{J}_{\mathbf{Z}\mathbf{Z}}=
\begin{bmatrix}
 \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(Z_1 - \operatorname{E}[Z_1])] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(Z_2 - \operatorname{E}[Z_2])] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(Z_n - \operatorname{E}[Z_n])] \\ \\
 \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(Z_1 - \operatorname{E}[Z_1])] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(Z_2 - \operatorname{E}[Z_2])] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(Z_n - \operatorname{E}[Z_n])] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
 \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(Z_1 - \operatorname{E}[Z_1])] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(Z_2 - \operatorname{E}[Z_2])] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(Z_n - \operatorname{E}[Z_n])]
\end{bmatrix}
</math>

;Properties
The covariance matrix is a [hermitian matrix](/source/hermitian_matrix), i.e.<ref name=Lapidoth/>{{rp|p. 293}}
:<math>\operatorname{K}_{\mathbf{Z}\mathbf{Z}}^H = \operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math>.

The pseudo-covariance matrix is a [symmetric matrix](/source/symmetric_matrix), i.e.
:<math>\operatorname{J}_{\mathbf{Z}\mathbf{Z}}^T = \operatorname{J}_{\mathbf{Z}\mathbf{Z}}</math>.

The covariance matrix is a [positive semidefinite matrix](/source/positive_semidefinite_matrix), i.e.
:<math>\mathbf{a}^H \operatorname{K}_{\mathbf{Z}\mathbf{Z}} \mathbf{a} \ge 0 \quad \text{for all } \mathbf{a} \in \mathbb{C}^n</math>.

===Covariance matrices of real and imaginary parts===
{{See also|Complex random variable#Covariance matrix of real and imaginary parts}}

By decomposing the random vector <math>\mathbf{Z}</math> into its real part <math>\mathbf{X} = \Re{(\mathbf{Z})}</math> and imaginary part <math>\mathbf{Y} = \Im{(\mathbf{Z})}</math> (i.e. <math>\mathbf{Z}=\mathbf{X}+i\mathbf{Y}</math>), the pair <math> (\mathbf{X},\mathbf{Y})</math> has a [covariance matrix](/source/covariance_matrix) of the form:

:<math>\begin{bmatrix} 
    \operatorname{K}_{\mathbf{X}\mathbf{X}} & \operatorname{K}_{\mathbf{X}\mathbf{Y}} \\ 
    \operatorname{K}_{\mathbf{Y}\mathbf{X}} & \operatorname{K}_{\mathbf{Y}\mathbf{Y}} 
  \end{bmatrix}</math>

The matrices <math>\operatorname{K}_{\mathbf{Z}\mathbf{Z}}</math> and <math>\operatorname{J}_{\mathbf{Z}\mathbf{Z}}</math> can be related to the covariance matrices of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> via the following expressions:
: <math>\begin{align}
  & \operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{E}[(\mathbf{X}-\operatorname{E}[\mathbf{X}])(\mathbf{X}-\operatorname{E}[\mathbf{X}])^\mathrm T] = \tfrac{1}{2}\operatorname{Re}(\operatorname{K}_{\mathbf{Z}\mathbf{Z}} + \operatorname{J}_{\mathbf{Z}\mathbf{Z}}) \\
  & \operatorname{K}_{\mathbf{Y}\mathbf{Y}} = \operatorname{E}[(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])^\mathrm T] = \tfrac{1}{2}\operatorname{Re}(\operatorname{K}_{\mathbf{Z}\mathbf{Z}} - \operatorname{J}_{\mathbf{Z}\mathbf{Z}}) \\
  & \operatorname{K}_{\mathbf{Y}\mathbf{X}} = \operatorname{E}[(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])(\mathbf{X}-\operatorname{E}[\mathbf{X}])^\mathrm T] = \tfrac{1}{2}\operatorname{Im}(\operatorname{J}_{\mathbf{Z}\mathbf{Z}} + \operatorname{K}_{\mathbf{Z}\mathbf{Z}}) \\
  & \operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X}-\operatorname{E}[\mathbf{X}])(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])^\mathrm T] = \tfrac{1}{2}\operatorname{Im}(\operatorname{J}_{\mathbf{Z}\mathbf{Z}} -\operatorname{K}_{\mathbf{Z}\mathbf{Z}}) \\
  \end{align}</math>

Conversely:
: <math>\begin{align}
  & \operatorname{K}_{\mathbf{Z}\mathbf{Z}} = \operatorname{K}_{\mathbf{X}\mathbf{X}} + \operatorname{K}_{\mathbf{Y}\mathbf{Y}} + i(\operatorname{K}_{\mathbf{Y}\mathbf{X}} - \operatorname{K}_{\mathbf{X}\mathbf{Y}}) \\
  & \operatorname{J}_{\mathbf{Z}\mathbf{Z}} = \operatorname{K}_{\mathbf{X}\mathbf{X}} - \operatorname{K}_{\mathbf{Y}\mathbf{Y}} + i(\operatorname{K}_{\mathbf{Y}\mathbf{X}} + \operatorname{K}_{\mathbf{X}\mathbf{Y}})
  \end{align}</math>

==Cross-covariance matrix and pseudo-cross-covariance matrix==
The '''cross-covariance matrix''' between two complex random vectors <math>\mathbf{Z},\mathbf{W}</math> is defined as:

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math> \operatorname{K}_{\mathbf{Z}\mathbf{W}} =
\operatorname{cov}[\mathbf{Z},\mathbf{W}] =
\operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{W}-\operatorname{E}[\mathbf{W}])}^H] = \operatorname{E}[\mathbf{Z}\mathbf{W}^H]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}^H] </math>|{{EquationRef|Eq.5}}}}
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:<math>\operatorname{K}_{\mathbf{Z}\mathbf{W}} =
\begin{bmatrix}
 \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(W_1 - \operatorname{E}[W_1])}] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(W_2 - \operatorname{E}[W_2])}] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])\overline{(W_n - \operatorname{E}[W_n])}] \\ \\
 \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(W_1 - \operatorname{E}[W_1])}] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(W_2 - \operatorname{E}[W_2])}] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])\overline{(W_n - \operatorname{E}[W_n])}] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
 \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(W_1 - \operatorname{E}[W_1])}] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(W_2 - \operatorname{E}[W_2])}] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])\overline{(W_n - \operatorname{E}[W_n])}]
\end{bmatrix}
</math>

And the '''pseudo-cross-covariance matrix''' is defined as:

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math> \operatorname{J}_{\mathbf{Z}\mathbf{W}} = \operatorname{cov}[\mathbf{Z},\overline{\mathbf{W}}] = \operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{W}-\operatorname{E}[\mathbf{W}])}^T] = \operatorname{E}[\mathbf{Z}\mathbf{W}^T]-\operatorname{E}[\mathbf{Z}]\operatorname{E}[\mathbf{W}^T] </math>|{{EquationRef|Eq.6}}}}
|cellpadding= 6
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|border colour = #0073CF
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:<math>\operatorname{J}_{\mathbf{Z}\mathbf{W}} = 
\begin{bmatrix}
 \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(W_1 - \operatorname{E}[W_1])] & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(W_2 - \operatorname{E}[W_2])] & \cdots & \mathrm{E}[(Z_1 - \operatorname{E}[Z_1])(W_n - \operatorname{E}[W_n])] \\ \\
 \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(W_1 - \operatorname{E}[W_1])] & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(W_2 - \operatorname{E}[W_2])] & \cdots & \mathrm{E}[(Z_2 - \operatorname{E}[Z_2])(W_n - \operatorname{E}[W_n])] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
 \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(W_1 - \operatorname{E}[W_1])] & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(W_2 - \operatorname{E}[W_2])] & \cdots & \mathrm{E}[(Z_n - \operatorname{E}[Z_n])(W_n - \operatorname{E}[W_n])]
\end{bmatrix}
</math>

Two complex random vectors <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> are called '''uncorrelated''' if
:<math>\operatorname{K}_{\mathbf{Z}\mathbf{W}}=\operatorname{J}_{\mathbf{Z}\mathbf{W}}=0</math>.

==Independence==
{{main|Independence (probability theory)}}
Two complex random vectors <math>\mathbf{Z}=(Z_1,...,Z_m)^T</math> and <math>\mathbf{W}=(W_1,...,W_n)^T</math> are called '''independent''' if

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>F_{\mathbf{Z,W}}(\mathbf{z,w}) = F_{\mathbf{Z}}(\mathbf{z}) \cdot F_{\mathbf{W}}(\mathbf{w}) \quad \text{for all } \mathbf{z},\mathbf{w}</math>|{{EquationRef|Eq.7}}}}
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where <math>F_{\mathbf{Z}}(\mathbf{z})</math> and <math>F_{\mathbf{W}}(\mathbf{w})</math> denote the cumulative distribution functions of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> as defined in {{EquationNote|Eq.1}} and <math>F_{\mathbf{Z,W}}(\mathbf{z,w})</math> denotes their joint cumulative distribution function. Independence of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> is often denoted by <math>\mathbf{Z} \perp\!\!\!\perp \mathbf{W}</math>.
Written component-wise, <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> are called independent if
:<math>F_{Z_1,\ldots,Z_m,W_1,\ldots,W_n}(z_1,\ldots,z_m,w_1,\ldots,w_n) = F_{Z_1,\ldots,Z_m}(z_1,\ldots,z_m) \cdot F_{W_1,\ldots,W_n}(w_1,\ldots,w_n) \quad \text{for all } z_1,\ldots,z_m,w_1,\ldots,w_n</math>.

==Circular symmetry==
A complex random vector <math> \mathbf{Z} </math> is called circularly symmetric if for every deterministic <math> \varphi \in [-\pi,\pi) </math> the distribution of <math> e^{\mathrm i \varphi}\mathbf{Z} </math> equals the distribution of <math> \mathbf{Z} </math>.<ref name=TseViswanath>{{cite book |first=David |last=Tse |year=2005 |title=Fundamentals of Wireless Communication |publisher=Cambridge University Press}}</ref>{{rp|pp. 500–501}}

;Properties
* The expectation of a circularly symmetric complex random vector is either zero or it is not defined.<ref name=TseViswanath />{{rp|p. 500}}
* The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.<ref name=TseViswanath />{{rp|p. 584}}

==Proper complex random vectors==
A complex random vector <math>\mathbf{Z}</math> is called '''proper''' if the following three conditions are all satisfied:<ref name=Lapidoth />{{rp|p. 293}}
* <math> \operatorname{E}[\mathbf{Z}] = 0 </math> (zero mean)
* <math> \operatorname{var}[Z_1] < \infty , \ldots , \operatorname{var}[Z_n] < \infty </math> (all components have finite variance)
* <math> \operatorname{E}[\mathbf{Z}\mathbf{Z}^T] = 0 </math>

Two complex random vectors <math>\mathbf{Z},\mathbf{W}</math> are called ''' jointly proper''' if the composite random vector <math>(Z_1,Z_2,\ldots,Z_m,W_1,W_2,\ldots,W_n)^T</math> is proper.

;Properties
* A complex random vector <math>\mathbf{Z}</math> is proper if, and only if, for all (deterministic) vectors <math> \mathbf{c} \in \mathbb{C}^n</math> the complex random variable <math>\mathbf{c}^T \mathbf{Z}</math> is proper.<ref name=Lapidoth />{{rp|p. 293}}
* Linear transformations of proper complex random vectors are proper, i.e. if <math>\mathbf{Z}</math> is a proper random vectors with <math>n</math> components and <math>A</math> is a deterministic <math>m \times n</math> matrix, then the complex random vector <math>A \mathbf{Z}</math> is also proper.<ref name=Lapidoth />{{rp|p. 295}}
* Every circularly symmetric complex random vector with finite variance of all its components is proper.<ref name=Lapidoth />{{rp|p. 295}}
* There are proper complex random vectors that are not circularly symmetric.<ref name=Lapidoth />{{rp|p. 504}}
* A real random vector is proper [if and only if](/source/if_and_only_if) it is constant.
* Two jointly proper complex random vectors are uncorrelated if and only if their covariance matrix is zero, i.e. if <math>\operatorname{K}_{\mathbf{Z}\mathbf{W}} = 0</math>.

==Cauchy–Schwarz inequality==
The [Cauchy–Schwarz inequality](/source/Cauchy%E2%80%93Schwarz_inequality) for complex random vectors is
:<math>\left| \operatorname{E}[\mathbf{Z}^H \mathbf{W}] \right|^2 \leq \operatorname{E}[\mathbf{Z}^H \mathbf{Z}] \operatorname{E}[|\mathbf{W}^H \mathbf{W}|]</math>.

==Characteristic function==
The [characteristic function](/source/Characteristic_function_(probability_theory)) of a complex random vector <math> \mathbf{Z} </math> with <math> n </math> components is a function <math> \mathbb{C}^n \to \mathbb{C} </math> defined by:<ref name=Lapidoth />{{rp|p. 295}}

: <math> \varphi_{\mathbf{Z}}(\mathbf{\omega}) = \operatorname{E} \left [ e^{i\Re{(\mathbf{\omega}^H \mathbf{Z})}} \right ] = \operatorname{E} \left [ e^{i( \Re{(\omega_1)}\Re{(Z_1)} + \Im{(\omega_1)}\Im{(Z_1)} + \cdots + \Re{(\omega_n)}\Re{(Z_n)} + \Im{(\omega_n)}\Im{(Z_n)} )} \right ]</math>

==See also==
* [Complex normal distribution](/source/Complex_normal_distribution)
* [Complex random variable](/source/Complex_random_variable) (scalar case)

==References==
{{reflist}}

Category:Probability theory
Category:Randomness
Category:Algebra of random variables

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Adapted from the Wikipedia article [Complex random vector](https://en.wikipedia.org/wiki/Complex_random_vector) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Complex_random_vector?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.