Kovariančna matrika

Označimo z ${\displaystyle {\vec {X}}\,}$ stolpični vektor

${\displaystyle {\vec {X}}={\begin{bmatrix}X_{1}\\\vdots \\X_{n}\end{bmatrix}}}$

kjer so ${\displaystyle X_{n}\,}$ posamezne komponente slučajne spremenljivke, ki imajo končno varianco.

Kovariančna matrika ${\displaystyle \Sigma \,}$ , ki ima za elemente kovariance tako, da je

${\displaystyle \Sigma _{ij}=\mathrm {cov} (X_{i},X_{j})=\mathrm {E} {\begin{bmatrix}(X_{i}-\mu _{i})(X_{j}-\mu _{j})\end{bmatrix}}}$

kjer je

• ${\displaystyle \mu _{i}=\mathrm {E} (X_{i})\,}$ pričakovana vrednost za i-to komponento vektorja ${\displaystyle X\,}$ .
• ${\displaystyle \mathrm {cov} (X_{i},X_{j})\,}$ kovarianca elementov ${\displaystyle X_{i}\,}$ in ${\displaystyle X_{j}\,}$ .

Iz tega sledi, da kovariančno matriko lahko zapišemo kot

${\displaystyle \Sigma ={\begin{bmatrix}\mathrm {E} [(X_{1}-\mu _{1})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{1}-\mu _{1})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{1}-\mu _{1})(X_{n}-\mu _{n})]\\\\\mathrm {E} [(X_{2}-\mu _{2})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{2}-\mu _{2})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{2}-\mu _{2})(X_{n}-\mu _{n})]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(X_{n}-\mu _{n})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{n}-\mu _{n})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{n}-\mu _{n})(X_{n}-\mu _{n})]\end{bmatrix}}.}$ .

Obratno matriko kovariančne matrike ${\displaystyle \Sigma ^{-1}\,}$ imenujejo tudi matrika natančnosti.

Kovariančno matriko imenujemo tudi variančno-kovariančna matrika, ker velja

${\displaystyle \Sigma _{X}=\operatorname {var} ({\vec {X}})=\operatorname {var} {\begin{pmatrix}X_{1}\\\vdots \\X_{p}\end{pmatrix}}={\begin{pmatrix}\operatorname {var} (X_{1})&\operatorname {cov} (X_{1}X_{2})&\cdots &\operatorname {cov} (X_{1}X_{p})\\\operatorname {cov} (X_{2}X_{1})&\ddots &\cdots &\vdots \\\vdots &\vdots &\ddots &\vdots \\\operatorname {cov} (X_{P}X_{1})&\cdots &\cdots &\operatorname {var} (X_{p})\end{pmatrix}}={\begin{pmatrix}\sigma _{x_{1}}^{2}&\sigma _{x_{1}x_{2}}&\cdots &\sigma _{x_{1}x_{p}}\\\sigma _{x_{2}x_{1}}&\ddots &\cdots &\vdots \\\vdots &\vdots &\ddots &\vdots \\\sigma _{x_{p}x_{1}}&\cdots &\cdots &\sigma _{x_{p}}^{2}\end{pmatrix}}}$

kjer je

• ${\displaystyle \operatorname {var} ({\vec {X}})\,}$ varianca vektorja ${\displaystyle {\vec {X}}\,}$
• ${\displaystyle \operatorname {cov} \,}$ kovarianca komponent ${\displaystyle X_{i}\,}$ in ${\displaystyle X_{j}\,}$
• ${\displaystyle \sigma _{n}\,}$ varianca n-te komponente vektorja (na glavni diagonali so same variance, izven diagonale pa so kovariance). Zaradi tega ima matrika tudi ime variančno-kovariančna matrika.

Zgornja definicija je enakovredna zapisu

${\displaystyle \Sigma =\mathrm {E} \left[\left({\textbf {X}}-\mathrm {E} [{\textbf {X}}]\right)\left({\textbf {X}}-\mathrm {E} [{\textbf {X}}]\right)^{\top }\right]}$ .

Ta zapis lahko smatramo za posplošitev skalarne oblike variance na višje razsežnosti. Pri tem velja za slučajno spremenljivko s skalarnimi vrednostmi

${\displaystyle \sigma ^{2}=\mathrm {var} (X)=\mathrm {E} [(X-\mu )^{2}],\,}$

kjer je

• ${\displaystyle \mu =\mathrm {E} (X).\,}$

Za kovariančno matriko ${\displaystyle \Sigma \,}$

• ${\displaystyle \Sigma =\mathrm {E} (\mathbf {XX^{\top }} )-\mathbf {\mu } \mathbf {\mu ^{\top }} }$
• ${\displaystyle \Sigma \,}$ je pozitivno semidefinitna matrika (to pomeni, da je simetrična).
• ${\displaystyle \operatorname {var} (\mathbf {AX} +\mathbf {a} )=\mathbf {A} \,\operatorname {var} (\mathbf {X} )\,\mathbf {A^{\top }} }$
• ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )^{\top }}$
• ${\displaystyle \operatorname {cov} (\mathbf {X} _{1}+\mathbf {X} _{2},\mathbf {Y} )=\operatorname {cov} (\mathbf {X} _{1},\mathbf {Y} )+\operatorname {cov} (\mathbf {X} _{2},\mathbf {Y} )}$
• kadar velja p = q, potem je ${\displaystyle \operatorname {var} (\mathbf {X} +\mathbf {Y} )=\operatorname {var} (\mathbf {X} )+\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )+\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )+\operatorname {var} (\mathbf {Y} )}$
• ${\displaystyle \operatorname {cov} (\mathbf {AX} ,\mathbf {B} ^{\top }\mathbf {Y} )=\mathbf {A} \,\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )\,\mathbf {B} }$
• kadar sta ${\displaystyle \mathbf {X} }$ in ${\displaystyle \mathbf {Y} }$ neodvisna, velja tudi ${\displaystyle \operatorname {cov} (\mathbf {X} \mathbf {Y} )=0\,}$ .

• kovarianca
• varianca
• seznam vrst matrik

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