4711:
3725:
4706:{\displaystyle \operatorname {cov} (\mathbf {X} )^{-1}={\begin{bmatrix}{\frac {1}{\sigma _{x_{1}|x_{2}...}}}&&&0\\&{\frac {1}{\sigma _{x_{2}|x_{1},x_{3}...}}}\\&&\ddots \\0&&&{\frac {1}{\sigma _{x_{n}|x_{1}...x_{n-1}}}}\end{bmatrix}}{\begin{bmatrix}1&-\rho _{x_{1},x_{2}\mid x_{3}...}&\cdots &-\rho _{x_{1},x_{n}\mid x_{2}...x_{n-1}}\\-\rho _{x_{2},x_{1}\mid x_{3}...}&1&\cdots &-\rho _{x_{2},x_{n}\mid x_{1},x_{3}...x_{n-1}}\\\vdots &\vdots &\ddots &\vdots \\-\rho _{x_{n},x_{1}\mid x_{2}...x_{n-1}}&-\rho _{x_{n},x_{2}\mid x_{1},x_{3}...x_{n-1}}&\cdots &1\\\end{bmatrix}}{\begin{bmatrix}{\frac {1}{\sigma _{x_{1}|x_{2}...}}}&&&0\\&{\frac {1}{\sigma _{x_{2}|x_{1},x_{3}...}}}\\&&\ddots \\0&&&{\frac {1}{\sigma _{x_{n}|x_{1}...x_{n-1}}}}\end{bmatrix}}}
3714:
3148:
3222:
2326:
3709:{\displaystyle \operatorname {cov} (\mathbf {X} )={\begin{bmatrix}\sigma _{x_{1}}&&&0\\&\sigma _{x_{2}}\\&&\ddots \\0&&&\sigma _{x_{n}}\end{bmatrix}}{\begin{bmatrix}1&\rho _{x_{1},x_{2}}&\cdots &\rho _{x_{1},x_{n}}\\\rho _{x_{2},x_{1}}&1&\cdots &\rho _{x_{2},x_{n}}\\\vdots &\vdots &\ddots &\vdots \\\rho _{x_{n},x_{1}}&\rho _{x_{n},x_{2}}&\cdots &1\\\end{bmatrix}}{\begin{bmatrix}\sigma _{x_{1}}&&&0\\&\sigma _{x_{2}}\\&&\ddots \\0&&&\sigma _{x_{n}}\end{bmatrix}}}
15846:
14875:
3143:{\displaystyle \operatorname {corr} (\mathbf {X} )={\begin{bmatrix}1&{\frac {\operatorname {E} }{\sigma (X_{1})\sigma (X_{2})}}&\cdots &{\frac {\operatorname {E} }{\sigma (X_{1})\sigma (X_{n})}}\\\\{\frac {\operatorname {E} }{\sigma (X_{2})\sigma (X_{1})}}&1&\cdots &{\frac {\operatorname {E} }{\sigma (X_{2})\sigma (X_{n})}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\operatorname {E} }{\sigma (X_{n})\sigma (X_{1})}}&{\frac {\operatorname {E} }{\sigma (X_{n})\sigma (X_{2})}}&\cdots &1\end{bmatrix}}.}
11535:
6070:
14861:
7454:
266:
2077:
11521:
52:
8763:
11074:
14899:
5910:
14887:
7266:
10209:
1903:
10363:
11354:
6774:
202:
5715:
8432:
1790:
10682:
6897:
5120:
10047:
4871:
10214:
1521:
11341:
6065:{\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{X}\\{\boldsymbol {\mu }}_{Y}\end{bmatrix}},\qquad {\boldsymbol {\Sigma }}={\begin{bmatrix}\operatorname {K} _{\mathbf {XX} }&\operatorname {K} _{\mathbf {XY} }\\\operatorname {K} _{\mathbf {YX} }&\operatorname {K} _{\mathbf {YY} }\end{bmatrix}}}
7851:
6642:
1316:
8150:
7449:{\displaystyle \operatorname {K} _{\mathbf {XY\mid I} }=\operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )-\operatorname {cov} (\mathbf {X} ,\mathbf {I} )\operatorname {cov} (\mathbf {I} ,\mathbf {I} )^{-1}\operatorname {cov} (\mathbf {I} ,\mathbf {Y} ).}
1609:
2072:{\displaystyle \operatorname {corr} (\mathbf {X} )={\big (}\operatorname {diag} (\operatorname {K} _{\mathbf {X} \mathbf {X} }){\big )}^{-{\frac {1}{2}}}\,\operatorname {K} _{\mathbf {X} \mathbf {X} }\,{\big (}\operatorname {diag} (\operatorname {K} _{\mathbf {X} \mathbf {X} }){\big )}^{-{\frac {1}{2}}},}
12482:
O Kornilov, M Eckstein, M Rosenblatt, C P Schulz, K Motomura, A RouzĂ©e, J Klei, L Foucar, M Siano, A LĂŒbcke, F. Schapper, P Johnsson, D M P Holland, T Schlatholter, T Marchenko, S DĂŒsterer, K Ueda, M J J Vrakking and L J Frasinski "Coulomb explosion of diatomic molecules in intense XUV fields mapped
9105:
10451:
a scalar factor and small random fluctuations (proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation). Intuitively, this result is supported by the rationale that the optimal covariance distribution can offer mutation steps
5582:
12367:
W J Krzanowski "Principles of
Multivariate Analysis" (Oxford University Press, New York, 1988), Chap. 14.4; K V Mardia, J T Kent and J M Bibby "Multivariate Analysis (Academic Press, London, 1997), Chap. 6.5.3; T W Anderson "An Introduction to Multivariate Statistical Analysis" (Wiley, New York,
11516:{\displaystyle \operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )-\operatorname {cov} (\mathbf {X} ,\mathbf {I} )\left(\operatorname {cov} (\mathbf {I} ,\mathbf {I} )\backslash \operatorname {cov} (\mathbf {I} ,\mathbf {Y} )\right),}
8416:
5246:
6259:
10439:, a particular family of Randomized Search Heuristics, fundamentally relies on a covariance matrix in its mechanism. The characteristic mutation operator draws the update step from a multivariate normal distribution using an evolving covariance matrix. There is a formal proof that the
8758:{\displaystyle {\begin{aligned}&w^{\mathsf {T}}\operatorname {E} \left)(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}\right]w=\operatorname {E} \left)(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}w\right]\\&=\operatorname {E} {\big ){\big )}^{2}{\big ]}\geq 0,\end{aligned}}}
6634:
5531:
11069:{\displaystyle \left={\begin{bmatrix}X_{1}(t_{1})&X_{2}(t_{1})&\cdots &X_{n}(t_{1})\\\\X_{1}(t_{2})&X_{2}(t_{2})&\cdots &X_{n}(t_{2})\\\\\vdots &\vdots &\ddots &\vdots \\\\X_{1}(t_{m})&X_{2}(t_{m})&\cdots &X_{n}(t_{m})\end{bmatrix}},}
6782:
11569:
shows that 10% overcorrection improves the map and makes ion-ion correlations clearly visible. Owing to momentum conservation these correlations appear as lines approximately perpendicular to the autocorrelation line (and to the periodic modulations which are caused by detector
1371:
11241:
7668:
10604:, the map shows statistical relations between different regions of the random functions. Statistically independent regions of the functions show up on the map as zero-level flatland, while positive or negative correlations show up, respectively, as hills or valleys.
6489:
8049:
9651:
5347:
5021:
1166:
4724:
402:
Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the
1039:
7188:
A covariance matrix with all non-zero elements tells us that all the individual random variables are interrelated. This means that the variables are not only directly correlated, but also correlated via other variables indirectly. Often such indirect,
10204:{\displaystyle \mathbf {Q} _{\mathbf {XX} }={\frac {1}{n-1}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {X} }^{\mathsf {T}},\qquad \mathbf {Q} _{\mathbf {XY} }={\frac {1}{n-1}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {Y} }^{\mathsf {T}}}
9570:
9824:
10358:{\displaystyle \mathbf {Q} _{\mathbf {XX} }={\frac {1}{n}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {X} }^{\mathsf {T}},\qquad \mathbf {Q} _{\mathbf {XY} }={\frac {1}{n}}\mathbf {M} _{\mathbf {X} }\mathbf {M} _{\mathbf {Y} }^{\mathsf {T}}.}
8058:
10367:
These empirical sample covariance matrices are the most straightforward and most often used estimators for the covariance matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.
8983:
8330:
6769:{\displaystyle {\boldsymbol {\mu }}_{\mathbf {Y} |\mathbf {X} }={\boldsymbol {\mu }}_{\mathbf {Y} }+\operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}\left(\mathbf {X} -{\boldsymbol {\mu }}_{\mathbf {X} }\right)}
11232:
7117:
6957:
12124:
show. The suppression of the uninteresting correlations is, however, imperfect because there are other sources of common-mode fluctuations than the laser intensity and in principle all these sources should be monitored in vector
11969:
5710:{\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} )}
6542:
6174:
6122:
12017:(note a change in the colour scale). Unfortunately, this map is overwhelmed by uninteresting, common-mode correlations induced by laser intensity fluctuating from shot to shot. To suppress such correlations the laser intensity
2123:
1785:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} )(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}]=\operatorname {R} _{\mathbf {X} \mathbf {X} }-\operatorname {E} \operatorname {E} ^{\mathsf {T}}}
12114:
10554:
1858:
8191:
5165:
6179:
4924:
779:
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260:
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of nitrogen molecules multiply ionised by a laser pulse. Since only a few hundreds of molecules are ionised at each laser pulse, the single-shot spectra are highly fluctuating. However, collecting typically
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4993:
12015:
10504:
5757:
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5156:
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correlations are trivial and uninteresting. They can be suppressed by calculating the partial covariance matrix, that is the part of covariance matrix that shows only the interesting part of correlations.
2157:
1604:
1567:
1352:
1109:
825:
545:
11917:
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reveals several nitrogen ions in a form of peaks broadened by their kinetic energy, but to find the correlations between the ionisation stages and the ion momenta requires calculating a covariance map.
865:
8255:
7492:
7893:
6892:{\displaystyle \operatorname {K} _{\mathbf {Y|X} }=\operatorname {K} _{\mathbf {YY} }-\operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}\operatorname {K} _{\mathbf {XY} }.}
8437:
6998:
5115:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} (\mathbf {XX^{\mathsf {T}}} )-{\boldsymbol {\mu }}_{\mathbf {X} }{\boldsymbol {\mu }}_{\mathbf {X} }^{\mathsf {T}}}
9426:
7978:
7524:
7034:
6325:
6293:
9676:
4866:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {var} (\mathbf {X} )=\operatorname {E} \left\right)\left(\mathbf {X} -\operatorname {E} \right)^{\mathsf {T}}\right]}
9595:
7641:
5291:
11745:
1516:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} \left)(\mathbf {Y} -\operatorname {E} )^{\mathsf {T}}\right].}
11773:
10042:
10008:
11336:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )\approx \langle \mathbf {XY^{\mathsf {T}}} \rangle -\langle \mathbf {X} \rangle \langle \mathbf {Y} ^{\mathsf {T}}\rangle ,}
9906:
9875:
9413:
7148:
10635:
7925:
7846:{\displaystyle \operatorname {f} (\mathbf {X} )=(2\pi )^{-n/2}|{\boldsymbol {\Sigma }}|^{-1/2}\exp \left(-{\tfrac {1}{2}}\mathbf {(X-\mu )^{\mathsf {T}}\Sigma ^{-1}(X-\mu )} \right),}
7951:
7663:
7056:
5861:
9828:
In contrast to the covariance matrix defined above, Hermitian transposition gets replaced by transposition in the definition. Its diagonal elements may be complex valued; it is a
5836:
2283:
11853:
11815:
11688:
1311:{\displaystyle \operatorname {var} (\mathbf {X} )=\operatorname {cov} (\mathbf {X} ,\mathbf {X} )=\operatorname {E} \left)(\mathbf {X} -\operatorname {E} )^{\mathsf {T}}\right].}
8145:{\displaystyle \mathbf {d} ^{\mathsf {T}}{\boldsymbol {\Sigma }}\mathbf {c} =\operatorname {cov} (\mathbf {d} ^{\mathsf {T}}\mathbf {X} ,\mathbf {c} ^{\mathsf {T}}\mathbf {X} )}
1059:
11163:
10415:. The matrix of covariances among various assets' returns is used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a
9307:
12145:
12064:
11117:
10657:
10598:
10576:
8926:
8293:
7576:
7546:
7261:
7239:
7217:
7170:
6537:
6515:
6420:
6398:
6373:
6351:
5905:
5883:
5805:
5783:
5577:
5555:
5445:
5397:
2321:
2226:
1898:
1161:
1135:
637:
615:
511:
8325:
153:
9962:
9936:
8978:
8952:
8872:
5423:
5375:
467:
345:
11650:
565:
12172:
12042:
2188:
857:
691:
664:
12168:
11608:
11154:
9149:
10384:, that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way (see
11873:
11708:
10677:
9280:
8896:
8846:
7596:
5013:
585:
441:
421:
319:
299:
7064:
6904:
9100:{\displaystyle \operatorname {var} (\mathbf {M} ^{1/2}\mathbf {X} )=\mathbf {M} ^{1/2}\,\operatorname {var} (\mathbf {X} )\,\mathbf {M} ^{1/2}=\mathbf {M} .}
11926:
6127:
6075:
702:
181:
2082:
8411:{\displaystyle \operatorname {var} (\mathbf {b} ^{\mathsf {T}}\mathbf {X} )=\mathbf {b} ^{\mathsf {T}}\operatorname {var} (\mathbf {X} )\mathbf {b} ,\,}
12069:
10509:
9158:
5241:{\displaystyle \mathbf {a} ^{T}\operatorname {K} _{\mathbf {X} \mathbf {X} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {R} ^{n}}
1795:
15504:
9419:
transposing and conjugating. In the following expression, the product of a vector with its conjugate transpose results in a square matrix called the
8155:
6254:{\displaystyle \operatorname {K} _{\mathbf {XY} }=\operatorname {K} _{\mathbf {YX} }^{\mathsf {T}}=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
9976:
rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices
12151:
shows, where interesting correlations of ion momenta are now clearly visible as straight lines centred on ionisation stages of atomic nitrogen.
3152:
Each element on the principal diagonal of a correlation matrix is the correlation of a random variable with itself, which always equals 1. Each
13996:
1875:
17:
6629:{\displaystyle \mathbf {Y} \mid \mathbf {X} \sim \ {\mathcal {N}}({\boldsymbol {\mu }}_{\mathbf {Y|X} },\operatorname {K} _{\mathbf {Y|X} }),}
14501:
5526:{\displaystyle \operatorname {var} (\mathbf {AX} +\mathbf {a} )=\mathbf {A} \,\operatorname {var} (\mathbf {X} )\,\mathbf {A} ^{\mathsf {T}}}
9115:
4876:
12175:. Mathematically, the former is expressed in terms of the sample covariance matrix and the technique is equivalent to covariance mapping.
14651:
3167:
212:
14275:
12916:
9315:
8768:
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11978:
10467:
5720:
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5126:
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14049:
2128:
1575:
1538:
1323:
1080:
796:
516:
174:
15809:
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11882:
273:
with a standard deviation of 3 in roughly the lower leftâupper right direction and of 1 in the orthogonal direction. Because the
143:
12199:
8199:
108:
7459:
12387:
12275:
12248:
9909:
7856:
6484:{\displaystyle \mathbf {X} ,\mathbf {Y} \sim \ {\mathcal {N}}({\boldsymbol {\mu }},\operatorname {\boldsymbol {\Sigma }} ),}
118:
15728:
15494:
12911:
12611:
12160:
8044:{\displaystyle \mathbf {c} ^{\mathsf {T}}\Sigma =\operatorname {cov} (\mathbf {c} ^{\mathsf {T}}\mathbf {X} ,\mathbf {X} )}
138:
103:
15882:
13515:
12663:
9646:{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }^{\mathsf {H}}=\operatorname {K} _{\mathbf {Z} \mathbf {Z} }}
6966:
5342:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }^{\mathsf {T}}=\operatorname {K} _{\mathbf {X} \mathbf {X} }}
4713:
This duality motivates a number of other dualities between marginalizing and conditioning for
Gaussian random variables.
167:
7497:
7007:
6298:
6266:
12502:
I Noda "Generalized two-dimensional correlation method applicable to infrared, Raman, and other types of spectroscopy"
11524:
113:
10452:
whose equidensity probability contours match the level sets of the landscape, and so they maximize the progress rate.
7612:
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14190:
12580:
12349:
12309:
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9841:
11527:
operator, which bypasses the requirement to invert a matrix and is available in some computational packages such as
14903:
14476:
14350:
11611:
7599:
6423:
1034:{\displaystyle \operatorname {K} _{X_{i}X_{j}}=\operatorname {cov} =\operatorname {E} )(X_{j}-\operatorname {E} )]}
11750:
10013:
9979:
3219:
Just as the covariance matrix can be written as the rescaling of a correlation matrix by the marginal variances:
1137:, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the
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270:
206:
10393:
9387:
7122:
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13604:
13493:
13451:
11574:
Fig. 1 illustrates how a partial covariance map is constructed on an example of an experiment performed at the
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14011:
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13792:
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10389:
9576:
9565:{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} =\operatorname {E} \left,}
8423:
7607:
5159:
9819:{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} =\operatorname {E} \left}
5819:
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15892:
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8828:
Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose
43:
12147:. Yet in practice it is often sufficient to overcompensate the partial covariance correction as panel
3206:, if it exists, is the inverse covariance matrix (or inverse concentration matrix), also known as the
15628:
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maps the partial covariance matrix that is corrected for the intensity fluctuations. Panel
11348:
8258:
7172:. In this form they correspond to the coefficients obtained by inverting the matrix of the
6777:
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2166:
830:
669:
642:
388:
11584:
11227:{\displaystyle \langle \mathbf {X} \rangle ={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {X} _{j}}
11130:
9134:
8:
15799:
15773:
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9152:
7962:
7112:{\displaystyle \operatorname {K} _{\mathbf {XX} }^{-1}\operatorname {K} _{\mathbf {XY} }}
6960:
6952:{\displaystyle \operatorname {K} _{\mathbf {YX} }\operatorname {K} _{\mathbf {XX} }^{-1}}
3719:
12241:
Fundamentals of
Probability and Stochastic Processes with Applications to Communications
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570:
426:
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360:
304:
284:
12552:
11964:{\displaystyle \langle \mathbf {X} \rangle \langle \mathbf {Y} ^{\mathsf {T}}\rangle }
1141:, because it is the matrix of covariances between the scalar components of the vector
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12345:
12305:
12271:
12244:
11615:
11235:
10388:
for a formal proof and additional properties of covariance matrices). This is called
10385:
6169:{\displaystyle \operatorname {K} _{\mathbf {YY} }=\operatorname {var} (\mathbf {Y} )}
6117:{\displaystyle \operatorname {K} _{\mathbf {XX} }=\operatorname {var} (\mathbf {X} )}
3722:, and partial variance, the inverse covariance matrix can be expressed analogously:
2118:{\displaystyle \operatorname {diag} (\operatorname {K} _{\mathbf {X} \mathbf {X} })}
15856:
15824:
15753:
15692:
15687:
15667:
15603:
15509:
15479:
15464:
15444:
15383:
15336:
15311:
15301:
15272:
15191:
15186:
15161:
15091:
15071:
14981:
14961:
14810:
14765:
14529:
14516:
14409:
14384:
14318:
14250:
14128:
13736:
13629:
13562:
13475:
13422:
13241:
13112:
12906:
12790:
12705:
12672:
12441:
9589:
9573:
9121:
8875:
7173:
7001:
6637:
5285:
3208:
477:
15449:
1320:
Both forms are quite standard, and there is no ambiguity between them. The matrix
15554:
15489:
15469:
15454:
15434:
15418:
15316:
15247:
15237:
15196:
15081:
15051:
14727:
14471:
14333:
14260:
13935:
13809:
13782:
13759:
13728:
13355:
13350:
13304:
13034:
12685:
12265:
12109:{\displaystyle \operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )}
10601:
10549:{\displaystyle \operatorname {pcov} (\mathbf {X} ,\mathbf {Y} \mid \mathbf {I} )}
2231:
Equivalently, the correlation matrix can be seen as the covariance matrix of the
2160:
782:
14217:
8874:
symmetric positive-semidefinite matrix. From the finite-dimensional case of the
1853:{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }=\operatorname {E} }
15814:
15758:
15738:
15723:
15682:
15559:
15519:
15484:
15408:
15347:
15326:
15267:
15257:
15242:
15176:
15121:
15111:
15106:
15016:
14676:
14671:
13134:
13064:
12710:
12194:
11156:. The expected values needed in the covariance formula are estimated using the
10444:
9125:
1070:
790:
469:
matrix would be necessary to fully characterize the two-dimensional variation.
12446:
12419:
8419:
15876:
15819:
15677:
15618:
15549:
15539:
15534:
15459:
15388:
15262:
15252:
15181:
15101:
15086:
15021:
14833:
14800:
14663:
14624:
14435:
14404:
13868:
13822:
13427:
13129:
12956:
12720:
12715:
11561:
maps common-mode correlations via intensity fluctuations of the laser. Panel
11343:
where the angular brackets denote sample averaging as before except that the
8186:{\displaystyle \mathbf {c} ^{\mathsf {T}}{\boldsymbol {\Sigma }}\mathbf {c} }
8052:
697:
396:
8196:
Similarly, the (pseudo-)inverse covariance matrix provides an inner product
7061:
The matrix of regression coefficients may often be given in transpose form,
347:
covariance matrix is needed; the directions of the arrows correspond to the
15702:
15659:
15564:
15277:
15216:
15126:
15006:
14775:
14708:
14685:
14600:
13930:
13226:
13124:
13059:
13001:
12986:
12923:
12878:
11351:. Using this estimation the partial covariance matrix can be calculated as
9579:, with real numbers in the main diagonal and complex numbers off-diagonal.
9116:
Complex random vector § Covariance matrix and pseudo-covariance matrix
10376:
The covariance matrix is a useful tool in many different areas. From it a
15544:
15514:
15282:
15116:
14986:
14818:
14780:
14463:
14364:
14226:
14039:
14006:
13498:
13415:
13410:
13054:
13011:
12991:
12971:
12961:
12730:
11157:
7928:
352:
348:
11543:
molecules undergoing
Coulomb explosion induced by a free-electron laser.
15595:
15056:
13664:
13144:
12844:
12775:
12725:
12700:
12620:
9665:
For complex random vectors, another kind of second central moment, the
7967:
Applied to one vector, the covariance matrix maps a linear combination
860:
473:
392:
364:
351:
of this covariance matrix and their lengths to the square roots of the
36:
12491:
12326:
11747:, which is shown in red at the bottom of Fig. 1. The average spectrum
7119:, suitable for post-multiplying a row vector of explanatory variables
4919:{\displaystyle {\boldsymbol {\mu }}_{\mathbf {X} }=\operatorname {E} }
15829:
15403:
13817:
13669:
13289:
13084:
12996:
12981:
12976:
12941:
12557:
8422:
of a real-valued random variable, so a covariance matrix is always a
7551:
774:{\displaystyle \mathbf {X} =(X_{1},X_{2},\dots ,X_{n})^{\mathsf {T}}}
265:
11553:
map the two terms of the covariance matrix, which is shown in panel
8954:
column vector-valued random variable whose covariance matrix is the
5015:-dimensional random variable, the following basic properties apply:
1874:
An entity closely related to the covariance matrix is the matrix of
1069:
Nomenclatures differ. Some statisticians, following the probabilist
15763:
13333:
12951:
12828:
12823:
12818:
12436:
9309:; thus the variance of a complex random variable is a real number.
786:
485:
3199:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }^{-1}}
255:{\displaystyle {\begin{bmatrix}1&0.5\\0.5&1\end{bmatrix}}}
51:
14915:
14838:
14539:
11575:
9381:
is a column vector of complex-valued random variables, then the
9374:{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{\mathsf {T}}}
9255:{\displaystyle \operatorname {var} (Z)=\operatorname {E} \left,}
8818:{\displaystyle w^{\mathsf {T}}(\mathbf {X} -\operatorname {E} )}
8055:, it yields the covariance between the two linear combinations:
4988:{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathsf {T}}}
14760:
13741:
13715:
13695:
12946:
12737:
12010:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
11528:
10499:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
5752:{\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}
5277:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }\,}
5151:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }\,}
201:
12589:
10448:
2152:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
1599:{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {X} }}
1562:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
1347:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
1104:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
820:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
540:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}
12420:"On the covariance-Hessian relation in evolution strategies"
12327:"Lectures on probability theory and mathematical statistics"
12301:
All of
Statistics: A Concise Course in Statistical Inference
8765:
where the last inequality follows from the observation that
12680:
11912:{\displaystyle \langle \mathbf {XY^{\mathsf {T}}} \rangle }
639:
are used to refer to random vectors, and Roman subscripted
12547:
12267:
An introduction to probability theory and its applications
11855:
are the same, except that the range of the time-of-flight
8268:
1878:
between each of the random variables in the random vector
1075:
1064:
8250:{\displaystyle \langle c-\mu |\Sigma ^{+}|c-\mu \rangle }
7956:
1530:
12154:
10556:
matrix are plotted as a 2-dimensional map. When vectors
9656:
The diagonal elements of the covariance matrix are real.
7487:{\displaystyle \operatorname {K} _{\mathbf {XY\mid I} }}
11539:
Figure 1: Construction of a partial covariance map of N
7888:{\displaystyle {\boldsymbol {\mu }}=\operatorname {E} }
443:
directions contain all of the necessary information; a
11477:
10752:
7770:
5983:
5927:
4487:
3985:
3764:
3612:
3350:
3251:
2355:
221:
29:
Measure of covariance of components of a random vector
12131:
12072:
12050:
12023:
11981:
11929:
11885:
11861:
11823:
11785:
11753:
11716:
11696:
11658:
11625:
11587:
11357:
11244:
11166:
11133:
11082:
10685:
10665:
10643:
10613:
10584:
10562:
10512:
10470:
10217:
10050:
10016:
9982:
9944:
9918:
9883:
9852:
9679:
9598:
9429:
9390:
9318:
9288:
9268:
9161:
9137:
8986:
8960:
8934:
8912:
8884:
8854:
8834:
8771:
8435:
8333:
8301:
8279:
8202:
8158:
8061:
7981:
7937:
7901:
7859:
7671:
7649:
7615:
7584:
7562:
7532:
7500:
7462:
7269:
7263:, the latter correlations are suppressed in a matrix
7247:
7225:
7203:
7156:
7125:
7067:
7042:
7010:
6969:
6907:
6785:
6645:
6545:
6523:
6501:
6432:
6406:
6384:
6359:
6337:
6301:
6269:
6182:
6130:
6078:
5913:
5891:
5869:
5847:
5822:
5791:
5769:
5723:
5585:
5563:
5541:
5453:
5431:
5405:
5383:
5357:
5294:
5255:
5168:
5129:
5024:
5001:
4932:
4879:
4727:
3728:
3225:
3170:
2329:
2291:
2240:
2196:
2169:
2131:
2085:
1906:
1884:
1863:
1798:
1612:
1578:
1541:
1374:
1326:
1169:
1147:
1121:
1083:
1047:
868:
833:
799:
705:
672:
645:
623:
601:
573:
553:
519:
497:
449:
429:
409:
327:
307:
287:
215:
14502:
Autoregressive conditional heteroskedasticity (ARCH)
5557:
is another random vector with the same dimension as
488:(i.e., the covariance of each element with itself).
209:
centered at (0, 0), with covariance matrix given by
7975:onto a vector of covariances with those variables:
7643:can be expressed in terms of the covariance matrix
6993:{\displaystyle \operatorname {K} _{\mathbf {Y|X} }}
3159:
1061:denotes the expected value (mean) of its argument.
13964:
12568:
12139:
12108:
12058:
12036:
12009:
11963:
11911:
11867:
11847:
11809:
11767:
11739:
11702:
11682:
11644:
11602:
11515:
11335:
11226:
11148:
11111:
11068:
10671:
10651:
10629:
10592:
10570:
10548:
10498:
10443:'s covariance matrix adapts to the inverse of the
10357:
10203:
10036:
10002:
9956:
9930:
9900:
9869:
9818:
9645:
9564:
9407:
9373:
9301:
9274:
9254:
9143:
9099:
8972:
8946:
8920:
8890:
8866:
8840:
8817:
8757:
8418:which must always be nonnegative, since it is the
8410:
8319:
8287:
8249:
8185:
8144:
8043:
7945:
7919:
7887:
7845:
7657:
7635:
7590:
7570:
7552:Covariance matrix as a parameter of a distribution
7540:
7519:{\displaystyle \operatorname {K} _{\mathbf {XY} }}
7518:
7486:
7448:
7255:
7233:
7211:
7164:
7142:
7111:
7050:
7029:{\displaystyle \operatorname {K} _{\mathbf {XX} }}
7028:
6992:
6951:
6891:
6768:
6628:
6531:
6509:
6483:
6414:
6392:
6367:
6345:
6327:can be identified as the variance matrices of the
6320:{\displaystyle \operatorname {K} _{\mathbf {YY} }}
6319:
6288:{\displaystyle \operatorname {K} _{\mathbf {XX} }}
6287:
6253:
6168:
6116:
6064:
5899:
5877:
5855:
5830:
5799:
5777:
5751:
5709:
5571:
5549:
5525:
5439:
5417:
5391:
5369:
5341:
5276:
5240:
5150:
5114:
5007:
4987:
4918:
4865:
4705:
3708:
3198:
3142:
2315:
2277:
2220:
2182:
2151:
2117:
2071:
1892:
1852:
1784:
1598:
1561:
1515:
1358:, since the diagonal terms are in fact variances.
1346:
1310:
1155:
1129:
1103:
1053:
1033:
851:
819:
773:
685:
658:
631:
609:
579:
559:
539:
505:
461:
435:
415:
339:
313:
293:
254:
12417:
7831:
7819:
7795:
7782:
6980:
6796:
6610:
6587:
595:Throughout this article, boldfaced unsubscripted
15874:
12544:", an easy way to visualize covariance matrices!
12342:Multivariate Statistics: a Vector Space Approach
9262:where the complex conjugate of a complex number
7636:{\displaystyle \operatorname {f} (\mathbf {X} )}
14050:Multivariate adaptive regression splines (MARS)
11740:{\displaystyle \langle \mathbf {X} (t)\rangle }
8152:. The variance of a linear combination is then
1876:Pearson product-moment correlation coefficients
1792:where the autocorrelation matrix is defined as
207:bivariate Gaussian probability density function
12460:L J Frasinski "Covariance mapping techniques"
12263:
12159:Two-dimensional infrared spectroscopy employs
11234:and the covariance matrix is estimated by the
8429:The above argument can be expanded as follows:
693:are used to refer to scalar random variables.
14931:
12605:
12571:Stochastic processes in physics and chemistry
8737:
8724:
8673:
8666:
2045:
2008:
1966:
1929:
175:
11958:
11941:
11938:
11930:
11906:
11886:
11768:{\displaystyle \langle \mathbf {X} \rangle }
11762:
11754:
11734:
11717:
11327:
11310:
11307:
11299:
11293:
11273:
11175:
11167:
10037:{\displaystyle \mathbf {Q} _{\mathbf {XY} }}
10003:{\displaystyle \mathbf {Q} _{\mathbf {XX} }}
8244:
8203:
7494:is effectively the simple covariance matrix
7183:
7150:rather than pre-multiplying a column vector
12363:
12361:
12234:
12232:
12230:
12167:. There are two versions of this analysis:
9901:{\displaystyle \mathbf {M} _{\mathbf {Y} }}
9870:{\displaystyle \mathbf {M} _{\mathbf {X} }}
9660:
15505:Fundamental (linear differential equation)
14938:
14924:
12650:
12612:
12598:
12566:
10399:The covariance matrix plays a key role in
10211:or, if the row means were known a priori,
9408:{\displaystyle \mathbf {Z} ^{\mathsf {H}}}
7143:{\displaystyle \mathbf {X} ^{\mathsf {T}}}
2125:is the matrix of the diagonal elements of
321:do not fully describe the distribution. A
182:
168:
13263:
12542:Covariance Matrix Explained With Pictures
12445:
12435:
12344:. John Wiley and Sons. pp. 116â117.
12297:
12257:
10630:{\displaystyle \mathbf {X} ,\mathbf {Y} }
9109:
9065:
9047:
8407:
7920:{\displaystyle |{\boldsymbol {\Sigma }}|}
7526:as if the uninteresting random variables
5508:
5490:
5273:
5228:
5147:
2005:
1987:
491:The covariance matrix of a random vector
395:between each pair of elements of a given
12377:
12358:
12227:
11533:
7598:possibly correlated random variables is
264:
200:
15810:Matrix representation of conic sections
12478:
12476:
12371:
12368:2003), 3rd ed., Chaps. 2.5.1 and 4.3.1.
10659:are acquired experimentally as rows of
9787:
9759:
9530:
9502:
8269:Which matrices are covariance matrices?
7946:{\displaystyle {\boldsymbol {\Sigma }}}
7861:
7658:{\displaystyle {\boldsymbol {\Sigma }}}
7051:{\displaystyle {\boldsymbol {\Sigma }}}
6749:
6675:
6648:
6577:
6463:
5948:
5932:
5915:
5856:{\displaystyle {\boldsymbol {\Sigma }}}
5824:
5093:
5079:
4882:
1065:Conflicting nomenclatures and notations
14:
15875:
14576:KaplanâMeier estimator (product limit)
12324:
12200:Lewandowski-Kurowicka-Joe distribution
11952:
11899:
11321:
11286:
10430:
10346:
10277:
10195:
10118:
9805:
9617:
9548:
9399:
9365:
8778:
8684:
8634:
8563:
8529:
8447:
8376:
8351:
8167:
8128:
8106:
8070:
8019:
7990:
7957:Covariance matrix as a linear operator
7800:
7134:
6963:coefficients, while in linear algebra
6217:
5517:
5313:
5106:
5064:
4979:
4852:
1841:
1776:
1707:
1531:Relation to the autocorrelation matrix
1499:
1294:
765:
14919:
14649:
14216:
13963:
13262:
13032:
12649:
12593:
12548:
12454:
12380:A Foundation in Digital Communication
12339:
12318:
12155:Two-dimensional infrared spectroscopy
10455:
8261:, a measure of the "unlikelihood" of
281:components co-vary, the variances of
14886:
14586:Accelerated failure time (AFT) model
12473:
12333:
12238:
12044:is recorded at every shot, put into
9131:random variable with expected value
5831:{\displaystyle {\boldsymbol {\mu }}}
2278:{\displaystyle X_{i}/\sigma (X_{i})}
1361:By comparison, the notation for the
14898:
14181:Analysis of variance (ANOVA, anova)
13033:
12402:
11848:{\displaystyle \mathbf {Y} _{j}(t)}
11810:{\displaystyle \mathbf {X} _{j}(t)}
11683:{\displaystyle \mathbf {X} _{j}(t)}
7197:If two vectors of random variables
5351:For any constant (i.e. non-random)
4908:
4716:
24:
14945:
14276:CochranâMantelâHaenszel statistics
12902:Pearson product-moment correlation
12516:
12418:Shir, O.M.; A. Yehudayoff (2020).
9735:
9681:
9627:
9600:
9478:
9431:
9180:
8795:
8701:
8658:
8611:
8580:
8546:
8506:
8475:
8453:
8273:From the identity just above, let
8221:
7996:
7868:
7672:
7616:
7502:
7464:
7271:
7241:are correlated via another vector
7095:
7069:
7012:
6971:
6927:
6909:
6872:
6846:
6828:
6810:
6787:
6709:
6691:
6601:
6567:
6454:
6303:
6271:
6202:
6184:
6132:
6080:
6040:
6023:
6004:
5987:
5323:
5296:
5257:
5182:
5131:
5045:
5026:
4897:
4828:
4792:
4768:
4729:
3172:
3006:
2888:
2740:
2612:
2489:
2366:
2133:
2096:
2023:
1989:
1944:
1864:Relation to the correlation matrix
1819:
1800:
1756:
1739:
1720:
1684:
1653:
1633:
1614:
1580:
1543:
1476:
1445:
1423:
1404:
1328:
1271:
1240:
1218:
1085:
1054:{\displaystyle \operatorname {E} }
1048:
1003:
962:
937:
870:
801:
554:
521:
25:
15904:
11975:shows their difference, which is
11779:In the example of Fig. 1 spectra
10423:) choose to hold in a context of
9842:Estimation of covariance matrices
5811:
15844:
14897:
14885:
14873:
14860:
14859:
14650:
12133:
12099:
12091:
12083:
12052:
12000:
11992:
11946:
11934:
11894:
11890:
11826:
11788:
11758:
11721:
11661:
11581:in Hamburg. The random function
11523:where the backslash denotes the
11498:
11490:
11470:
11462:
11440:
11432:
11412:
11404:
11384:
11376:
11368:
11315:
11303:
11281:
11277:
11263:
11255:
11214:
11171:
10729:
10708:
10693:
10645:
10623:
10615:
10586:
10564:
10539:
10531:
10523:
10489:
10481:
10339:
10333:
10325:
10319:
10298:
10295:
10289:
10270:
10264:
10256:
10250:
10229:
10226:
10220:
10188:
10182:
10174:
10168:
10139:
10136:
10130:
10111:
10105:
10097:
10091:
10062:
10059:
10053:
10028:
10025:
10019:
9994:
9991:
9985:
9892:
9886:
9861:
9855:
9793:
9778:
9765:
9750:
9720:
9710:
9691:
9686:
9637:
9632:
9610:
9605:
9536:
9521:
9508:
9493:
9468:
9460:
9441:
9436:
9393:
9320:
9151:is conventionally defined using
9090:
9068:
9058:
9029:
9017:
8998:
8914:
8805:
8788:
8711:
8694:
8621:
8604:
8590:
8573:
8516:
8499:
8485:
8468:
8400:
8392:
8370:
8358:
8345:
8281:
8179:
8174:
8161:
8135:
8122:
8113:
8100:
8082:
8077:
8064:
8034:
8026:
8013:
7984:
7939:
7908:
7878:
7825:
7822:
7814:
7811:
7807:
7788:
7785:
7728:
7682:
7651:
7626:
7564:
7534:
7510:
7507:
7478:
7472:
7469:
7436:
7428:
7401:
7393:
7376:
7368:
7348:
7340:
7320:
7312:
7304:
7285:
7279:
7276:
7249:
7227:
7205:
7158:
7128:
7103:
7100:
7077:
7074:
7044:
7020:
7017:
6984:
6976:
6935:
6932:
6917:
6914:
6880:
6877:
6854:
6851:
6836:
6833:
6818:
6815:
6800:
6792:
6755:
6740:
6717:
6714:
6699:
6696:
6681:
6664:
6654:
6614:
6606:
6591:
6583:
6555:
6547:
6525:
6503:
6471:
6442:
6434:
6408:
6386:
6361:
6339:
6311:
6308:
6279:
6276:
6244:
6236:
6210:
6207:
6192:
6189:
6159:
6140:
6137:
6107:
6088:
6085:
6048:
6045:
6031:
6028:
6012:
6009:
5995:
5992:
5971:
5893:
5871:
5849:
5793:
5771:
5742:
5734:
5700:
5680:
5672:
5652:
5644:
5624:
5604:
5596:
5565:
5543:
5511:
5501:
5486:
5475:
5467:
5464:
5433:
5385:
5333:
5328:
5306:
5301:
5267:
5262:
5219:
5202:
5192:
5187:
5171:
5141:
5136:
5099:
5085:
5059:
5055:
5036:
5031:
4934:
4888:
4838:
4821:
4802:
4785:
4758:
4739:
4734:
3739:
3236:
3182:
3177:
3160:Inverse of the covariance matrix
3156:is between â1 and +1 inclusive.
2340:
2143:
2138:
2106:
2101:
2033:
2028:
1999:
1994:
1954:
1949:
1917:
1886:
1835:
1829:
1810:
1805:
1766:
1749:
1730:
1725:
1694:
1677:
1663:
1646:
1624:
1619:
1590:
1585:
1553:
1548:
1486:
1469:
1455:
1438:
1414:
1409:
1393:
1385:
1338:
1333:
1281:
1264:
1250:
1233:
1208:
1200:
1180:
1149:
1123:
1095:
1090:
811:
806:
707:
625:
603:
531:
526:
499:
50:
15712:Used in science and engineering
14535:Least-squares spectral analysis
12496:
12485:J. Phys. B: At. Mol. Opt. Phys.
12462:J. Phys. B: At. Mol. Opt. Phys.
10607:In practice the column vectors
10371:
10286:
10127:
9572:The matrix so obtained will be
9302:{\displaystyle {\overline {z}}}
5969:
5212:
484:and its main diagonal contains
271:bivariate Gaussian distribution
14955:Explicitly constrained entries
13516:Mean-unbiased minimum-variance
12619:
12411:
12396:
12382:. Cambridge University Press.
12291:
12103:
12079:
12004:
11988:
11842:
11836:
11804:
11798:
11731:
11725:
11677:
11671:
11597:
11591:
11502:
11486:
11474:
11458:
11444:
11428:
11416:
11400:
11388:
11364:
11267:
11251:
11143:
11137:
11106:
11093:
11052:
11039:
11019:
11006:
10991:
10978:
10933:
10920:
10900:
10887:
10872:
10859:
10839:
10826:
10806:
10793:
10778:
10765:
10543:
10519:
10493:
10477:
10409:mutual fund separation theorem
9800:
9774:
9771:
9746:
9729:
9706:
9543:
9517:
9514:
9489:
9472:
9456:
9360:
9327:
9235:
9216:
9210:
9191:
9174:
9168:
9062:
9054:
9021:
8993:
8812:
8809:
8801:
8784:
8718:
8715:
8707:
8690:
8629:
8625:
8617:
8600:
8597:
8594:
8586:
8569:
8524:
8520:
8512:
8495:
8492:
8489:
8481:
8464:
8396:
8388:
8362:
8340:
8314:
8302:
8231:
8216:
8193:, its covariance with itself.
8139:
8095:
8038:
8008:
7913:
7903:
7882:
7874:
7734:
7723:
7702:
7692:
7686:
7678:
7630:
7622:
7456:The partial covariance matrix
7440:
7424:
7406:
7389:
7380:
7364:
7352:
7336:
7324:
7300:
6659:
6620:
6572:
6475:
6459:
6248:
6232:
6163:
6155:
6111:
6103:
5746:
5730:
5704:
5696:
5684:
5668:
5656:
5640:
5628:
5620:
5608:
5592:
5505:
5497:
5479:
5460:
5071:
5051:
4974:
4941:
4913:
4903:
4842:
4834:
4806:
4798:
4762:
4754:
4652:
4571:
4511:
3929:
3848:
3788:
3744:
3735:
3240:
3232:
3113:
3100:
3094:
3081:
3073:
3070:
3044:
3041:
3015:
3012:
2995:
2982:
2976:
2963:
2955:
2952:
2926:
2923:
2897:
2894:
2847:
2834:
2828:
2815:
2807:
2804:
2778:
2775:
2749:
2746:
2719:
2706:
2700:
2687:
2679:
2676:
2650:
2647:
2621:
2618:
2596:
2583:
2577:
2564:
2556:
2553:
2527:
2524:
2498:
2495:
2473:
2460:
2454:
2441:
2433:
2430:
2404:
2401:
2375:
2372:
2344:
2336:
2272:
2259:
2112:
2092:
2039:
2019:
1960:
1940:
1921:
1913:
1847:
1825:
1771:
1762:
1753:
1745:
1713:
1702:
1698:
1690:
1673:
1670:
1667:
1659:
1642:
1639:
1494:
1490:
1482:
1465:
1462:
1459:
1451:
1434:
1397:
1381:
1289:
1285:
1277:
1260:
1257:
1254:
1246:
1229:
1212:
1196:
1184:
1176:
1028:
1025:
1022:
1009:
987:
984:
981:
968:
946:
943:
931:
905:
846:
834:
760:
714:
13:
1:
15729:Fundamental (computer vision)
14829:Geographic information system
14045:Simultaneous equations models
12405:"The Matrix Reference Manual"
12220:
11123:-th discrete value in sample
9835:
5907:can be written in block form
2233:standardized random variables
1525:
793:, then the covariance matrix
590:
14012:Coefficient of determination
13623:Uniformly most powerful test
12424:Theoretical Computer Science
12163:to obtain 2D spectra of the
12140:{\displaystyle \mathbf {I} }
12059:{\displaystyle \mathbf {I} }
11112:{\displaystyle X_{j}(t_{i})}
10652:{\displaystyle \mathbf {I} }
10593:{\displaystyle \mathbf {Y} }
10571:{\displaystyle \mathbf {X} }
10419:) or are predicted to (in a
10390:principal component analysis
9724:
9294:
9239:
8921:{\displaystyle \mathbf {X} }
8898:has a nonnegative symmetric
8424:positive-semidefinite matrix
8288:{\displaystyle \mathbf {b} }
7608:probability density function
7600:jointly normally distributed
7571:{\displaystyle \mathbf {X} }
7541:{\displaystyle \mathbf {I} }
7256:{\displaystyle \mathbf {I} }
7234:{\displaystyle \mathbf {Y} }
7212:{\displaystyle \mathbf {X} }
7165:{\displaystyle \mathbf {X} }
6532:{\displaystyle \mathbf {X} }
6510:{\displaystyle \mathbf {Y} }
6424:jointly normally distributed
6415:{\displaystyle \mathbf {Y} }
6393:{\displaystyle \mathbf {X} }
6368:{\displaystyle \mathbf {Y} }
6346:{\displaystyle \mathbf {X} }
5900:{\displaystyle \mathbf {Y} }
5878:{\displaystyle \mathbf {X} }
5800:{\displaystyle \mathbf {Y} }
5778:{\displaystyle \mathbf {X} }
5572:{\displaystyle \mathbf {X} }
5550:{\displaystyle \mathbf {Y} }
5440:{\displaystyle \mathbf {a} }
5392:{\displaystyle \mathbf {A} }
3164:The inverse of this matrix,
2316:{\displaystyle i=1,\dots ,n}
2221:{\displaystyle i=1,\dots ,n}
1893:{\displaystyle \mathbf {X} }
1156:{\displaystyle \mathbf {X} }
1130:{\displaystyle \mathbf {X} }
632:{\displaystyle \mathbf {Y} }
610:{\displaystyle \mathbf {X} }
506:{\displaystyle \mathbf {X} }
7:
15495:Duplication and elimination
15294:eigenvalues or eigenvectors
14581:Proportional hazards models
14525:Spectral density estimation
14507:Vector autoregression (VAR)
13941:Maximum posterior estimator
13173:Randomized controlled trial
12575:. New York: North-Holland.
12529:Encyclopedia of Mathematics
12215:Quadratic form (statistics)
12178:
11710:produces a smooth spectrum
10413:capital asset pricing model
9968:columns of observations of
9588:The covariance matrix is a
8320:{\displaystyle (p\times 1)}
1535:The auto-covariance matrix
10:
15909:
15883:Covariance and correlation
15428:With specific applications
15057:Discrete Fourier Transform
14341:Multivariate distributions
12761:Average absolute deviation
12567:van Kampen, N. G. (1981).
11690:, and averaging them over
9839:
9113:
8902:, which can be denoted by
7960:
6959:is known as the matrix of
1900:, which can be written as
1867:
1356:variance-covariance matrix
385:varianceâcovariance matrix
193:
144:Cross-correlation function
109:Cross-correlation function
44:Correlation and covariance
18:Varianceâcovariance matrix
15838:
15787:
15719:CabibboâKobayashiâMaskawa
15711:
15657:
15593:
15427:
15346:Satisfying conditions on
15345:
15291:
15230:
14954:
14855:
14809:
14746:
14699:
14662:
14658:
14645:
14617:
14599:
14566:
14557:
14515:
14462:
14423:
14372:
14363:
14329:Structural equation model
14284:
14241:
14237:
14212:
14171:
14137:
14091:
14058:
14020:
13987:
13983:
13959:
13899:
13808:
13727:
13691:
13682:
13665:Score/Lagrange multiplier
13650:
13603:
13548:
13474:
13465:
13275:
13271:
13258:
13217:
13191:
13143:
13098:
13080:Sample size determination
13045:
13041:
13028:
12932:
12887:
12861:
12843:
12799:
12751:
12671:
12662:
12658:
12645:
12627:
12447:10.1016/j.tcs.2019.09.002
12340:Eaton, Morris L. (1983).
12298:Wasserman, Larry (2004).
10447:of the search landscape,
10380:can be derived, called a
9957:{\displaystyle q\times n}
9931:{\displaystyle p\times n}
9673:) is defined as follows:
8973:{\displaystyle p\times p}
8947:{\displaystyle p\times 1}
8867:{\displaystyle p\times p}
8327:real-valued vector, then
7184:Partial covariance matrix
5418:{\displaystyle m\times 1}
5370:{\displaystyle m\times n}
1354:is also often called the
462:{\displaystyle 2\times 2}
340:{\displaystyle 2\times 2}
154:Cross-covariance function
132:For deterministic signals
119:Cross-covariance function
14824:Environmental statistics
14346:Elliptical distributions
14139:Generalized linear model
14068:Simple linear regression
13838:HodgesâLehmann estimator
13295:Probability distribution
13204:Stochastic approximation
12766:Coefficient of variation
12190:Eigenvalue decomposition
12116:is calculated as panels
11645:{\displaystyle m=10^{4}}
11614:spectrum of ions from a
11347:should be made to avoid
10394:KarhunenâLoĂšve transform
10382:whitening transformation
9964:respectively, i.e. with
9830:complex symmetric matrix
9667:pseudo-covariance matrix
9661:Pseudo-covariance matrix
7971:of the random variables
7604:elliptically distributed
6493:conditional distribution
513:is typically denoted by
194:Not to be confused with
139:Autocorrelation function
104:Autocorrelation function
97:For stochastic processes
74:Cross-correlation matrix
15077:Generalized permutation
14484:Cross-correlation (XCF)
14092:Non-standard predictors
13526:LehmannâScheffĂ© theorem
13199:Adaptive clinical trial
12483:by partial covariance"
12378:Lapidoth, Amos (2009).
12264:William Feller (1971).
12205:Multivariate statistics
11127:of the random function
5840:joint covariance matrix
5761:cross-covariance matrix
1363:cross-covariance matrix
1073:in his two-volume book
560:{\displaystyle \Sigma }
196:Cross-covariance matrix
149:Autocovariance function
114:Autocovariance function
84:Cross-covariance matrix
15851:Mathematics portal
14880:Mathematics portal
14701:Engineering statistics
14609:NelsonâAalen estimator
14186:Analysis of covariance
14073:Ordinary least squares
13997:Pearson product-moment
13401:Statistical functional
13312:Empirical distribution
13145:Controlled experiments
12874:Frequency distribution
12652:Descriptive statistics
12325:Taboga, Marco (2010).
12141:
12110:
12060:
12038:
12011:
11965:
11913:
11869:
11849:
11811:
11769:
11741:
11704:
11684:
11646:
11604:
11571:
11517:
11337:
11228:
11211:
11150:
11113:
11070:
10673:
10653:
10631:
10594:
10572:
10550:
10500:
10359:
10205:
10038:
10004:
9958:
9932:
9902:
9871:
9820:
9647:
9566:
9423:, as its expectation:
9409:
9375:
9303:
9276:
9256:
9145:
9110:Complex random vectors
9101:
8980:identity matrix. Then
8974:
8948:
8922:
8892:
8868:
8842:
8819:
8759:
8412:
8321:
8289:
8251:
8187:
8146:
8045:
7947:
7921:
7889:
7847:
7659:
7637:
7592:
7572:
7542:
7520:
7488:
7450:
7257:
7235:
7213:
7178:ordinary least squares
7166:
7144:
7113:
7052:
7030:
6994:
6953:
6893:
6770:
6630:
6533:
6511:
6485:
6416:
6394:
6369:
6347:
6329:marginal distributions
6321:
6289:
6255:
6170:
6118:
6066:
5901:
5879:
5857:
5832:
5801:
5779:
5753:
5711:
5573:
5551:
5527:
5441:
5419:
5393:
5371:
5343:
5278:
5242:
5152:
5116:
5009:
4989:
4920:
4867:
4707:
3718:So, using the idea of
3710:
3200:
3144:
2317:
2279:
2222:
2184:
2153:
2119:
2073:
1894:
1854:
1786:
1600:
1571:autocorrelation matrix
1563:
1517:
1348:
1312:
1157:
1131:
1105:
1055:
1035:
853:
821:
775:
696:If the entries in the
687:
660:
633:
611:
581:
561:
541:
507:
482:positive semi-definite
463:
437:
417:
373:auto-covariance matrix
356:
341:
315:
295:
262:
256:
79:Auto-covariance matrix
69:Autocorrelation matrix
14796:Population statistics
14738:System identification
14472:Autocorrelation (ACF)
14400:Exponential smoothing
14314:Discriminant analysis
14309:Canonical correlation
14173:Partition of variance
14035:Regression validation
13879:(JonckheereâTerpstra)
13778:Likelihood-ratio test
13467:Frequentist inference
13379:Locationâscale family
13300:Sampling distribution
13265:Statistical inference
13232:Cross-sectional study
13219:Observational studies
13178:Randomized experiment
13007:Stem-and-leaf display
12809:Central limit theorem
12430:. Elsevier: 157â174.
12239:Park, Kun Il (2018).
12142:
12111:
12061:
12039:
12037:{\displaystyle I_{j}}
12012:
11966:
11914:
11870:
11850:
11812:
11770:
11742:
11705:
11685:
11647:
11605:
11537:
11518:
11338:
11229:
11191:
11151:
11114:
11071:
10674:
10654:
10632:
10595:
10573:
10551:
10501:
10378:transformation matrix
10360:
10206:
10044:can be defined to be
10039:
10005:
9959:
9933:
9903:
9872:
9821:
9648:
9577:positive-semidefinite
9567:
9410:
9376:
9304:
9277:
9257:
9146:
9114:Further information:
9102:
8975:
8949:
8923:
8893:
8869:
8843:
8820:
8760:
8413:
8322:
8290:
8252:
8188:
8147:
8046:
7948:
7922:
7890:
7848:
7660:
7638:
7593:
7573:
7543:
7521:
7489:
7451:
7258:
7236:
7214:
7167:
7145:
7114:
7053:
7031:
6995:
6954:
6894:
6771:
6631:
6534:
6512:
6486:
6417:
6395:
6370:
6348:
6322:
6290:
6256:
6171:
6119:
6067:
5902:
5880:
5858:
5833:
5802:
5780:
5754:
5712:
5574:
5552:
5528:
5442:
5420:
5394:
5372:
5344:
5279:
5243:
5160:positive-semidefinite
5153:
5117:
5010:
4990:
4921:
4868:
4708:
3711:
3201:
3145:
2318:
2280:
2223:
2185:
2183:{\displaystyle X_{i}}
2154:
2120:
2074:
1895:
1868:Further information:
1855:
1787:
1601:
1564:
1518:
1349:
1313:
1158:
1132:
1115:of the random vector
1106:
1056:
1036:
854:
852:{\displaystyle (i,j)}
822:
776:
688:
686:{\displaystyle Y_{i}}
661:
659:{\displaystyle X_{i}}
634:
612:
582:
562:
542:
508:
464:
438:
418:
342:
316:
296:
269:Sample points from a
268:
257:
204:
14719:Probabilistic design
14304:Principal components
14147:Exponential families
14099:Nonlinear regression
14078:General linear model
14040:Mixed effects models
14030:Errors and residuals
14007:Confounding variable
13909:Bayesian probability
13887:Van der Waerden test
13877:Ordered alternative
13642:Multiple comparisons
13521:RaoâBlackwellization
13484:Estimating equations
13440:Statistical distance
13158:Factorial experiment
12691:Arithmetic-Geometric
12210:Principal components
12161:correlation analysis
12129:
12070:
12048:
12021:
11979:
11927:
11883:
11859:
11821:
11783:
11751:
11714:
11694:
11656:
11623:
11603:{\displaystyle X(t)}
11585:
11525:left matrix division
11355:
11242:
11164:
11149:{\displaystyle X(t)}
11131:
11080:
10683:
10663:
10641:
10611:
10582:
10560:
10510:
10468:
10215:
10048:
10014:
9980:
9942:
9916:
9881:
9850:
9677:
9596:
9427:
9388:
9316:
9286:
9266:
9159:
9144:{\displaystyle \mu }
9135:
8984:
8958:
8932:
8910:
8882:
8852:
8832:
8769:
8433:
8331:
8299:
8277:
8259:Mahalanobis distance
8257:, which induces the
8200:
8156:
8059:
7979:
7935:
7899:
7857:
7669:
7647:
7613:
7602:, or more generally
7582:
7560:
7548:were held constant.
7530:
7498:
7460:
7267:
7245:
7223:
7201:
7154:
7123:
7065:
7040:
7008:
6967:
6905:
6783:
6778:conditional variance
6643:
6543:
6521:
6499:
6430:
6404:
6382:
6357:
6335:
6299:
6267:
6180:
6128:
6076:
5911:
5889:
5867:
5845:
5820:
5789:
5767:
5721:
5583:
5561:
5539:
5451:
5429:
5403:
5381:
5355:
5292:
5253:
5166:
5127:
5022:
4999:
4930:
4877:
4725:
3726:
3223:
3214:concentration matrix
3168:
3154:off-diagonal element
2327:
2289:
2238:
2194:
2167:
2163:of the variances of
2129:
2083:
1904:
1882:
1796:
1610:
1576:
1539:
1372:
1324:
1167:
1145:
1119:
1081:
1045:
866:
831:
827:is the matrix whose
797:
703:
670:
643:
621:
599:
571:
551:
517:
495:
447:
427:
407:
325:
305:
285:
213:
35:Part of a series on
15800:Linear independence
15047:Diagonally dominant
14791:Official statistics
14714:Methods engineering
14395:Seasonal adjustment
14163:Poisson regressions
14083:Bayesian regression
14022:Regression analysis
14002:Partial correlation
13974:Regression analysis
13573:Prediction interval
13568:Likelihood interval
13558:Confidence interval
13550:Interval estimation
13511:Unbiased estimators
13329:Model specification
13209:Up-and-down designs
12897:Partial correlation
12853:Index of dispersion
12771:Interquartile range
12553:"Covariance Matrix"
12524:"Covariance matrix"
12185:Covariance function
11579:free-electron laser
11345:Bessel's correction
10431:Use in optimization
10401:financial economics
10351:
10282:
10200:
10123:
9622:
9383:conjugate transpose
9153:complex conjugation
7963:Covariance operator
7556:If a column vector
7090:
6948:
6867:
6730:
6222:
5318:
5111:
3720:partial correlation
3195:
1077:, call the matrix
1041:where the operator
785:, each with finite
15893:Summary statistics
15805:Matrix exponential
15795:Jordan normal form
15629:Fisher information
15500:Euclidean distance
15414:Totally unimodular
14811:Spatial statistics
14691:Medical statistics
14591:First hitting time
14545:Whittle likelihood
14196:Degrees of freedom
14191:Multivariate ANOVA
14124:Heteroscedasticity
13936:Bayesian estimator
13901:Bayesian inference
13750:KolmogorovâSmirnov
13635:Randomization test
13605:Testing hypotheses
13578:Tolerance interval
13489:Maximum likelihood
13384:Exponential family
13317:Density estimation
13277:Statistical theory
13237:Natural experiment
13183:Scientific control
13100:Survey methodology
12786:Standard deviation
12550:Weisstein, Eric W.
12137:
12106:
12056:
12034:
12007:
11961:
11909:
11865:
11845:
11807:
11765:
11737:
11700:
11680:
11642:
11600:
11572:
11513:
11333:
11224:
11146:
11109:
11066:
11057:
10669:
10649:
10627:
10590:
10568:
10546:
10496:
10464:the values of the
10462:covariance mapping
10456:Covariance mapping
10441:evolution strategy
10437:evolution strategy
10417:normative analysis
10355:
10331:
10262:
10201:
10180:
10103:
10034:
10000:
9954:
9928:
9898:
9867:
9816:
9643:
9599:
9562:
9405:
9371:
9299:
9272:
9252:
9141:
9097:
8970:
8944:
8918:
8888:
8878:, it follows that
8864:
8838:
8815:
8755:
8753:
8408:
8317:
8285:
8247:
8183:
8142:
8041:
7943:
7917:
7885:
7843:
7779:
7655:
7633:
7588:
7568:
7538:
7516:
7484:
7446:
7253:
7231:
7209:
7162:
7140:
7109:
7068:
7048:
7026:
6990:
6949:
6926:
6889:
6845:
6766:
6708:
6626:
6529:
6507:
6481:
6412:
6390:
6365:
6343:
6317:
6285:
6251:
6201:
6166:
6114:
6062:
6056:
5960:
5897:
5875:
5853:
5828:
5797:
5775:
5749:
5707:
5569:
5547:
5523:
5437:
5415:
5389:
5367:
5339:
5295:
5274:
5238:
5148:
5112:
5091:
5005:
4985:
4916:
4863:
4703:
4697:
4476:
3974:
3706:
3700:
3601:
3339:
3196:
3171:
3140:
3131:
2313:
2275:
2218:
2180:
2149:
2115:
2069:
1890:
1870:Correlation matrix
1850:
1782:
1596:
1569:is related to the
1559:
1513:
1344:
1308:
1153:
1127:
1101:
1051:
1031:
849:
817:
771:
683:
656:
629:
607:
577:
557:
537:
503:
459:
433:
413:
361:probability theory
357:
337:
311:
291:
263:
252:
246:
62:For random vectors
15870:
15869:
15862:Category:Matrices
15734:Fuzzy associative
15624:Doubly stochastic
15332:Positive-definite
15012:Block tridiagonal
14913:
14912:
14851:
14850:
14847:
14846:
14786:National accounts
14756:Actuarial science
14748:Social statistics
14641:
14640:
14637:
14636:
14633:
14632:
14568:Survival function
14553:
14552:
14415:Granger causality
14256:Contingency table
14231:Survival analysis
14208:
14207:
14204:
14203:
14060:Linear regression
13955:
13954:
13951:
13950:
13926:Credible interval
13895:
13894:
13678:
13677:
13494:Method of moments
13363:Parametric family
13324:Statistical model
13254:
13253:
13250:
13249:
13168:Random assignment
13090:Statistical power
13024:
13023:
13020:
13019:
12869:Contingency table
12839:
12838:
12706:Generalized/power
12389:978-0-521-19395-5
12277:978-0-471-25709-7
12250:978-3-319-68074-3
11868:{\displaystyle t}
11703:{\displaystyle j}
11616:Coulomb explosion
11236:sample covariance
11189:
10672:{\displaystyle n}
10421:positive analysis
10386:Rayleigh quotient
10315:
10246:
10164:
10087:
9727:
9421:covariance matrix
9297:
9275:{\displaystyle z}
9242:
8891:{\displaystyle M}
8841:{\displaystyle M}
7778:
7591:{\displaystyle n}
6564:
6451:
5216:
5008:{\displaystyle n}
4910:
4693:
4609:
4536:
3970:
3886:
3813:
3117:
2999:
2851:
2723:
2600:
2477:
2062:
1983:
1139:covariance matrix
580:{\displaystyle S}
436:{\displaystyle y}
416:{\displaystyle x}
377:dispersion matrix
369:covariance matrix
314:{\displaystyle y}
294:{\displaystyle x}
192:
191:
16:(Redirected from
15900:
15857:List of matrices
15849:
15848:
15825:Row echelon form
15769:State transition
15698:Seidel adjacency
15580:Totally positive
15440:Alternating sign
15037:Complex Hadamard
14940:
14933:
14926:
14917:
14916:
14901:
14900:
14889:
14888:
14878:
14877:
14863:
14862:
14766:Crime statistics
14660:
14659:
14647:
14646:
14564:
14563:
14530:Fourier analysis
14517:Frequency domain
14497:
14444:
14410:Structural break
14370:
14369:
14319:Cluster analysis
14266:Log-linear model
14239:
14238:
14214:
14213:
14155:
14129:Homoscedasticity
13985:
13984:
13961:
13960:
13880:
13872:
13864:
13863:(KruskalâWallis)
13848:
13833:
13788:Cross validation
13773:
13755:AndersonâDarling
13702:
13689:
13688:
13660:Likelihood-ratio
13652:Parametric tests
13630:Permutation test
13613:1- & 2-tails
13504:Minimum distance
13476:Point estimation
13472:
13471:
13423:Optimal decision
13374:
13273:
13272:
13260:
13259:
13242:Quasi-experiment
13192:Adaptive designs
13043:
13042:
13030:
13029:
12907:Rank correlation
12669:
12668:
12660:
12659:
12647:
12646:
12614:
12607:
12600:
12591:
12590:
12586:
12574:
12563:
12562:
12537:
12510:
12504:Appl. Spectrosc.
12500:
12494:
12480:
12471:
12458:
12452:
12451:
12449:
12439:
12415:
12409:
12408:
12400:
12394:
12393:
12375:
12369:
12365:
12356:
12355:
12337:
12331:
12330:
12322:
12316:
12315:
12295:
12289:
12288:
12286:
12284:
12261:
12255:
12254:
12236:
12146:
12144:
12143:
12138:
12136:
12115:
12113:
12112:
12107:
12102:
12094:
12086:
12065:
12063:
12062:
12057:
12055:
12043:
12041:
12040:
12035:
12033:
12032:
12016:
12014:
12013:
12008:
12003:
11995:
11970:
11968:
11967:
11962:
11957:
11956:
11955:
11949:
11937:
11918:
11916:
11915:
11910:
11905:
11904:
11903:
11902:
11874:
11872:
11871:
11866:
11854:
11852:
11851:
11846:
11835:
11834:
11829:
11816:
11814:
11813:
11808:
11797:
11796:
11791:
11774:
11772:
11771:
11766:
11761:
11746:
11744:
11743:
11738:
11724:
11709:
11707:
11706:
11701:
11689:
11687:
11686:
11681:
11670:
11669:
11664:
11651:
11649:
11648:
11643:
11641:
11640:
11609:
11607:
11606:
11601:
11522:
11520:
11519:
11514:
11509:
11505:
11501:
11493:
11473:
11465:
11443:
11435:
11415:
11407:
11387:
11379:
11371:
11342:
11340:
11339:
11334:
11326:
11325:
11324:
11318:
11306:
11292:
11291:
11290:
11289:
11266:
11258:
11233:
11231:
11230:
11225:
11223:
11222:
11217:
11210:
11205:
11190:
11182:
11174:
11155:
11153:
11152:
11147:
11118:
11116:
11115:
11110:
11105:
11104:
11092:
11091:
11075:
11073:
11072:
11067:
11062:
11061:
11051:
11050:
11038:
11037:
11018:
11017:
11005:
11004:
10990:
10989:
10977:
10976:
10964:
10939:
10932:
10931:
10919:
10918:
10899:
10898:
10886:
10885:
10871:
10870:
10858:
10857:
10845:
10838:
10837:
10825:
10824:
10805:
10804:
10792:
10791:
10777:
10776:
10764:
10763:
10743:
10739:
10738:
10737:
10732:
10717:
10716:
10711:
10702:
10701:
10696:
10678:
10676:
10675:
10670:
10658:
10656:
10655:
10650:
10648:
10636:
10634:
10633:
10628:
10626:
10618:
10602:random functions
10599:
10597:
10596:
10591:
10589:
10577:
10575:
10574:
10569:
10567:
10555:
10553:
10552:
10547:
10542:
10534:
10526:
10505:
10503:
10502:
10497:
10492:
10484:
10405:portfolio theory
10403:, especially in
10396:(KL-transform).
10364:
10362:
10361:
10356:
10350:
10349:
10343:
10342:
10336:
10330:
10329:
10328:
10322:
10316:
10308:
10303:
10302:
10301:
10292:
10281:
10280:
10274:
10273:
10267:
10261:
10260:
10259:
10253:
10247:
10239:
10234:
10233:
10232:
10223:
10210:
10208:
10207:
10202:
10199:
10198:
10192:
10191:
10185:
10179:
10178:
10177:
10171:
10165:
10163:
10149:
10144:
10143:
10142:
10133:
10122:
10121:
10115:
10114:
10108:
10102:
10101:
10100:
10094:
10088:
10086:
10072:
10067:
10066:
10065:
10056:
10043:
10041:
10040:
10035:
10033:
10032:
10031:
10022:
10009:
10007:
10006:
10001:
9999:
9998:
9997:
9988:
9963:
9961:
9960:
9955:
9937:
9935:
9934:
9929:
9907:
9905:
9904:
9899:
9897:
9896:
9895:
9889:
9876:
9874:
9873:
9868:
9866:
9865:
9864:
9858:
9825:
9823:
9822:
9817:
9815:
9811:
9810:
9809:
9808:
9798:
9797:
9796:
9790:
9781:
9770:
9769:
9768:
9762:
9753:
9728:
9723:
9718:
9713:
9696:
9695:
9694:
9689:
9652:
9650:
9649:
9644:
9642:
9641:
9640:
9635:
9621:
9620:
9614:
9613:
9608:
9590:Hermitian matrix
9571:
9569:
9568:
9563:
9558:
9554:
9553:
9552:
9551:
9541:
9540:
9539:
9533:
9524:
9513:
9512:
9511:
9505:
9496:
9471:
9463:
9446:
9445:
9444:
9439:
9414:
9412:
9411:
9406:
9404:
9403:
9402:
9396:
9380:
9378:
9377:
9372:
9370:
9369:
9368:
9358:
9357:
9339:
9338:
9323:
9308:
9306:
9305:
9300:
9298:
9290:
9281:
9279:
9278:
9273:
9261:
9259:
9258:
9253:
9248:
9244:
9243:
9238:
9234:
9233:
9214:
9209:
9208:
9150:
9148:
9147:
9142:
9106:
9104:
9103:
9098:
9093:
9085:
9084:
9080:
9071:
9061:
9046:
9045:
9041:
9032:
9020:
9015:
9014:
9010:
9001:
8979:
8977:
8976:
8971:
8953:
8951:
8950:
8945:
8927:
8925:
8924:
8919:
8917:
8897:
8895:
8894:
8889:
8876:spectral theorem
8873:
8871:
8870:
8865:
8847:
8845:
8844:
8839:
8824:
8822:
8821:
8816:
8808:
8791:
8783:
8782:
8781:
8764:
8762:
8761:
8756:
8754:
8741:
8740:
8734:
8733:
8728:
8727:
8714:
8697:
8689:
8688:
8687:
8677:
8676:
8670:
8669:
8651:
8647:
8643:
8639:
8638:
8637:
8624:
8607:
8593:
8576:
8568:
8567:
8566:
8539:
8535:
8534:
8533:
8532:
8519:
8502:
8488:
8471:
8452:
8451:
8450:
8439:
8417:
8415:
8414:
8409:
8403:
8395:
8381:
8380:
8379:
8373:
8361:
8356:
8355:
8354:
8348:
8326:
8324:
8323:
8318:
8294:
8292:
8291:
8286:
8284:
8256:
8254:
8253:
8248:
8234:
8229:
8228:
8219:
8192:
8190:
8189:
8184:
8182:
8177:
8172:
8171:
8170:
8164:
8151:
8149:
8148:
8143:
8138:
8133:
8132:
8131:
8125:
8116:
8111:
8110:
8109:
8103:
8085:
8080:
8075:
8074:
8073:
8067:
8050:
8048:
8047:
8042:
8037:
8029:
8024:
8023:
8022:
8016:
7995:
7994:
7993:
7987:
7952:
7950:
7949:
7944:
7942:
7926:
7924:
7923:
7918:
7916:
7911:
7906:
7894:
7892:
7891:
7886:
7881:
7864:
7852:
7850:
7849:
7844:
7839:
7835:
7834:
7818:
7817:
7805:
7804:
7803:
7780:
7771:
7754:
7753:
7749:
7737:
7731:
7726:
7721:
7720:
7716:
7685:
7664:
7662:
7661:
7656:
7654:
7642:
7640:
7639:
7634:
7629:
7597:
7595:
7594:
7589:
7577:
7575:
7574:
7569:
7567:
7547:
7545:
7544:
7539:
7537:
7525:
7523:
7522:
7517:
7515:
7514:
7513:
7493:
7491:
7490:
7485:
7483:
7482:
7481:
7455:
7453:
7452:
7447:
7439:
7431:
7417:
7416:
7404:
7396:
7379:
7371:
7351:
7343:
7323:
7315:
7307:
7290:
7289:
7288:
7262:
7260:
7259:
7254:
7252:
7240:
7238:
7237:
7232:
7230:
7218:
7216:
7215:
7210:
7208:
7174:normal equations
7171:
7169:
7168:
7163:
7161:
7149:
7147:
7146:
7141:
7139:
7138:
7137:
7131:
7118:
7116:
7115:
7110:
7108:
7107:
7106:
7089:
7081:
7080:
7057:
7055:
7054:
7049:
7047:
7035:
7033:
7032:
7027:
7025:
7024:
7023:
7002:Schur complement
6999:
6997:
6996:
6991:
6989:
6988:
6987:
6983:
6958:
6956:
6955:
6950:
6947:
6939:
6938:
6922:
6921:
6920:
6898:
6896:
6895:
6890:
6885:
6884:
6883:
6866:
6858:
6857:
6841:
6840:
6839:
6823:
6822:
6821:
6805:
6804:
6803:
6799:
6775:
6773:
6772:
6767:
6765:
6761:
6760:
6759:
6758:
6752:
6743:
6729:
6721:
6720:
6704:
6703:
6702:
6686:
6685:
6684:
6678:
6669:
6668:
6667:
6662:
6657:
6651:
6638:conditional mean
6635:
6633:
6632:
6627:
6619:
6618:
6617:
6613:
6596:
6595:
6594:
6590:
6580:
6571:
6570:
6562:
6558:
6550:
6538:
6536:
6535:
6530:
6528:
6516:
6514:
6513:
6508:
6506:
6490:
6488:
6487:
6482:
6474:
6466:
6458:
6457:
6449:
6445:
6437:
6421:
6419:
6418:
6413:
6411:
6399:
6397:
6396:
6391:
6389:
6374:
6372:
6371:
6366:
6364:
6352:
6350:
6349:
6344:
6342:
6326:
6324:
6323:
6318:
6316:
6315:
6314:
6294:
6292:
6291:
6286:
6284:
6283:
6282:
6260:
6258:
6257:
6252:
6247:
6239:
6221:
6220:
6214:
6213:
6197:
6196:
6195:
6175:
6173:
6172:
6167:
6162:
6145:
6144:
6143:
6123:
6121:
6120:
6115:
6110:
6093:
6092:
6091:
6071:
6069:
6068:
6063:
6061:
6060:
6053:
6052:
6051:
6036:
6035:
6034:
6017:
6016:
6015:
6000:
5999:
5998:
5974:
5965:
5964:
5957:
5956:
5951:
5941:
5940:
5935:
5918:
5906:
5904:
5903:
5898:
5896:
5884:
5882:
5881:
5876:
5874:
5862:
5860:
5859:
5854:
5852:
5837:
5835:
5834:
5829:
5827:
5806:
5804:
5803:
5798:
5796:
5784:
5782:
5781:
5776:
5774:
5758:
5756:
5755:
5750:
5745:
5737:
5716:
5714:
5713:
5708:
5703:
5683:
5675:
5655:
5647:
5627:
5607:
5599:
5578:
5576:
5575:
5570:
5568:
5556:
5554:
5553:
5548:
5546:
5532:
5530:
5529:
5524:
5522:
5521:
5520:
5514:
5504:
5489:
5478:
5470:
5446:
5444:
5443:
5438:
5436:
5424:
5422:
5421:
5416:
5398:
5396:
5395:
5390:
5388:
5376:
5374:
5373:
5368:
5348:
5346:
5345:
5340:
5338:
5337:
5336:
5331:
5317:
5316:
5310:
5309:
5304:
5283:
5281:
5280:
5275:
5272:
5271:
5270:
5265:
5247:
5245:
5244:
5239:
5237:
5236:
5231:
5222:
5217:
5214:
5205:
5197:
5196:
5195:
5190:
5180:
5179:
5174:
5157:
5155:
5154:
5149:
5146:
5145:
5144:
5139:
5121:
5119:
5118:
5113:
5110:
5109:
5103:
5102:
5096:
5090:
5089:
5088:
5082:
5070:
5069:
5068:
5067:
5041:
5040:
5039:
5034:
5014:
5012:
5011:
5006:
4994:
4992:
4991:
4986:
4984:
4983:
4982:
4972:
4971:
4953:
4952:
4937:
4925:
4923:
4922:
4917:
4912:
4911:
4893:
4892:
4891:
4885:
4872:
4870:
4869:
4864:
4862:
4858:
4857:
4856:
4855:
4849:
4845:
4841:
4824:
4813:
4809:
4805:
4788:
4761:
4744:
4743:
4742:
4737:
4717:Basic properties
4712:
4710:
4709:
4704:
4702:
4701:
4694:
4692:
4691:
4690:
4689:
4665:
4664:
4655:
4650:
4649:
4632:
4629:
4628:
4615:
4614:
4610:
4608:
4607:
4597:
4596:
4584:
4583:
4574:
4569:
4568:
4551:
4548:
4540:
4539:
4537:
4535:
4534:
4524:
4523:
4514:
4509:
4508:
4491:
4481:
4480:
4463:
4462:
4461:
4460:
4436:
4435:
4423:
4422:
4410:
4409:
4397:
4396:
4377:
4376:
4375:
4374:
4350:
4349:
4337:
4336:
4324:
4323:
4280:
4279:
4278:
4277:
4253:
4252:
4240:
4239:
4227:
4226:
4214:
4213:
4184:
4183:
4173:
4172:
4160:
4159:
4147:
4146:
4125:
4124:
4123:
4122:
4098:
4097:
4085:
4084:
4072:
4071:
4047:
4046:
4036:
4035:
4023:
4022:
4010:
4009:
3979:
3978:
3971:
3969:
3968:
3967:
3966:
3942:
3941:
3932:
3927:
3926:
3909:
3906:
3905:
3892:
3891:
3887:
3885:
3884:
3874:
3873:
3861:
3860:
3851:
3846:
3845:
3828:
3825:
3817:
3816:
3814:
3812:
3811:
3801:
3800:
3791:
3786:
3785:
3768:
3755:
3754:
3742:
3715:
3713:
3712:
3707:
3705:
3704:
3697:
3696:
3695:
3694:
3679:
3678:
3665:
3664:
3660:
3659:
3658:
3657:
3642:
3634:
3633:
3631:
3630:
3629:
3628:
3606:
3605:
3588:
3587:
3586:
3585:
3573:
3572:
3556:
3555:
3554:
3553:
3541:
3540:
3500:
3499:
3498:
3497:
3485:
3484:
3458:
3457:
3456:
3455:
3443:
3442:
3424:
3423:
3422:
3421:
3409:
3408:
3387:
3386:
3385:
3384:
3372:
3371:
3344:
3343:
3336:
3335:
3334:
3333:
3318:
3317:
3304:
3303:
3299:
3298:
3297:
3296:
3281:
3273:
3272:
3270:
3269:
3268:
3267:
3239:
3209:precision matrix
3205:
3203:
3202:
3197:
3194:
3186:
3185:
3180:
3149:
3147:
3146:
3141:
3136:
3135:
3118:
3116:
3112:
3111:
3093:
3092:
3076:
3069:
3068:
3056:
3055:
3040:
3039:
3027:
3026:
3004:
3000:
2998:
2994:
2993:
2975:
2974:
2958:
2951:
2950:
2938:
2937:
2922:
2921:
2909:
2908:
2886:
2881:
2856:
2852:
2850:
2846:
2845:
2827:
2826:
2810:
2803:
2802:
2790:
2789:
2774:
2773:
2761:
2760:
2738:
2724:
2722:
2718:
2717:
2699:
2698:
2682:
2675:
2674:
2662:
2661:
2646:
2645:
2633:
2632:
2610:
2605:
2601:
2599:
2595:
2594:
2576:
2575:
2559:
2552:
2551:
2539:
2538:
2523:
2522:
2510:
2509:
2487:
2478:
2476:
2472:
2471:
2453:
2452:
2436:
2429:
2428:
2416:
2415:
2400:
2399:
2387:
2386:
2364:
2343:
2322:
2320:
2319:
2314:
2284:
2282:
2281:
2276:
2271:
2270:
2255:
2250:
2249:
2227:
2225:
2224:
2219:
2189:
2187:
2186:
2181:
2179:
2178:
2158:
2156:
2155:
2150:
2148:
2147:
2146:
2141:
2124:
2122:
2121:
2116:
2111:
2110:
2109:
2104:
2078:
2076:
2075:
2070:
2065:
2064:
2063:
2055:
2049:
2048:
2038:
2037:
2036:
2031:
2012:
2011:
2004:
2003:
2002:
1997:
1986:
1985:
1984:
1976:
1970:
1969:
1959:
1958:
1957:
1952:
1933:
1932:
1920:
1899:
1897:
1896:
1891:
1889:
1859:
1857:
1856:
1851:
1846:
1845:
1844:
1838:
1832:
1815:
1814:
1813:
1808:
1791:
1789:
1788:
1783:
1781:
1780:
1779:
1769:
1752:
1735:
1734:
1733:
1728:
1712:
1711:
1710:
1697:
1680:
1666:
1649:
1629:
1628:
1627:
1622:
1605:
1603:
1602:
1597:
1595:
1594:
1593:
1588:
1568:
1566:
1565:
1560:
1558:
1557:
1556:
1551:
1522:
1520:
1519:
1514:
1509:
1505:
1504:
1503:
1502:
1489:
1472:
1458:
1441:
1419:
1418:
1417:
1412:
1396:
1388:
1368:two vectors is
1353:
1351:
1350:
1345:
1343:
1342:
1341:
1336:
1317:
1315:
1314:
1309:
1304:
1300:
1299:
1298:
1297:
1284:
1267:
1253:
1236:
1211:
1203:
1183:
1162:
1160:
1159:
1154:
1152:
1136:
1134:
1133:
1128:
1126:
1110:
1108:
1107:
1102:
1100:
1099:
1098:
1093:
1060:
1058:
1057:
1052:
1040:
1038:
1037:
1032:
1021:
1020:
999:
998:
980:
979:
958:
957:
930:
929:
917:
916:
895:
894:
893:
892:
883:
882:
858:
856:
855:
850:
826:
824:
823:
818:
816:
815:
814:
809:
783:random variables
780:
778:
777:
772:
770:
769:
768:
758:
757:
739:
738:
726:
725:
710:
692:
690:
689:
684:
682:
681:
665:
663:
662:
657:
655:
654:
638:
636:
635:
630:
628:
616:
614:
613:
608:
606:
586:
584:
583:
578:
566:
564:
563:
558:
546:
544:
543:
538:
536:
535:
534:
529:
512:
510:
509:
504:
502:
468:
466:
465:
460:
442:
440:
439:
434:
422:
420:
419:
414:
346:
344:
343:
338:
320:
318:
317:
312:
300:
298:
297:
292:
261:
259:
258:
253:
251:
250:
184:
177:
170:
54:
32:
31:
21:
15908:
15907:
15903:
15902:
15901:
15899:
15898:
15897:
15873:
15872:
15871:
15866:
15843:
15834:
15783:
15707:
15653:
15589:
15423:
15341:
15287:
15226:
15027:Centrosymmetric
14950:
14944:
14914:
14909:
14872:
14843:
14805:
14742:
14728:quality control
14695:
14677:Clinical trials
14654:
14629:
14613:
14601:Hazard function
14595:
14549:
14511:
14495:
14458:
14454:BreuschâGodfrey
14442:
14419:
14359:
14334:Factor analysis
14280:
14261:Graphical model
14233:
14200:
14167:
14153:
14133:
14087:
14054:
14016:
13979:
13978:
13947:
13891:
13878:
13870:
13862:
13846:
13831:
13810:Rank statistics
13804:
13783:Model selection
13771:
13729:Goodness of fit
13723:
13700:
13674:
13646:
13599:
13544:
13533:Median unbiased
13461:
13372:
13305:Order statistic
13267:
13246:
13213:
13187:
13139:
13094:
13037:
13035:Data collection
13016:
12928:
12883:
12857:
12835:
12795:
12747:
12664:Continuous data
12654:
12641:
12623:
12618:
12583:
12522:
12519:
12517:Further reading
12514:
12513:
12501:
12497:
12490:164028 (2013),
12481:
12474:
12467:152004 (2016),
12459:
12455:
12416:
12412:
12403:Brookes, Mike.
12401:
12397:
12390:
12376:
12372:
12366:
12359:
12352:
12338:
12334:
12323:
12319:
12312:
12296:
12292:
12282:
12280:
12278:
12262:
12258:
12251:
12237:
12228:
12223:
12181:
12165:condensed phase
12157:
12132:
12130:
12127:
12126:
12098:
12090:
12082:
12071:
12068:
12067:
12051:
12049:
12046:
12045:
12028:
12024:
12022:
12019:
12018:
11999:
11991:
11980:
11977:
11976:
11951:
11950:
11945:
11944:
11933:
11928:
11925:
11924:
11898:
11897:
11893:
11889:
11884:
11881:
11880:
11875:differs. Panel
11860:
11857:
11856:
11830:
11825:
11824:
11822:
11819:
11818:
11792:
11787:
11786:
11784:
11781:
11780:
11757:
11752:
11749:
11748:
11720:
11715:
11712:
11711:
11695:
11692:
11691:
11665:
11660:
11659:
11657:
11654:
11653:
11636:
11632:
11624:
11621:
11620:
11586:
11583:
11582:
11542:
11497:
11489:
11469:
11461:
11451:
11447:
11439:
11431:
11411:
11403:
11383:
11375:
11367:
11356:
11353:
11352:
11320:
11319:
11314:
11313:
11302:
11285:
11284:
11280:
11276:
11262:
11254:
11243:
11240:
11239:
11218:
11213:
11212:
11206:
11195:
11181:
11170:
11165:
11162:
11161:
11132:
11129:
11128:
11100:
11096:
11087:
11083:
11081:
11078:
11077:
11056:
11055:
11046:
11042:
11033:
11029:
11027:
11022:
11013:
11009:
11000:
10996:
10994:
10985:
10981:
10972:
10968:
10965:
10962:
10961:
10956:
10951:
10946:
10940:
10937:
10936:
10927:
10923:
10914:
10910:
10908:
10903:
10894:
10890:
10881:
10877:
10875:
10866:
10862:
10853:
10849:
10846:
10843:
10842:
10833:
10829:
10820:
10816:
10814:
10809:
10800:
10796:
10787:
10783:
10781:
10772:
10768:
10759:
10755:
10748:
10747:
10733:
10728:
10727:
10712:
10707:
10706:
10697:
10692:
10691:
10690:
10686:
10684:
10681:
10680:
10664:
10661:
10660:
10644:
10642:
10639:
10638:
10622:
10614:
10612:
10609:
10608:
10585:
10583:
10580:
10579:
10563:
10561:
10558:
10557:
10538:
10530:
10522:
10511:
10508:
10507:
10488:
10480:
10469:
10466:
10465:
10458:
10433:
10425:diversification
10374:
10345:
10344:
10338:
10337:
10332:
10324:
10323:
10318:
10317:
10307:
10294:
10293:
10288:
10287:
10276:
10275:
10269:
10268:
10263:
10255:
10254:
10249:
10248:
10238:
10225:
10224:
10219:
10218:
10216:
10213:
10212:
10194:
10193:
10187:
10186:
10181:
10173:
10172:
10167:
10166:
10153:
10148:
10135:
10134:
10129:
10128:
10117:
10116:
10110:
10109:
10104:
10096:
10095:
10090:
10089:
10076:
10071:
10058:
10057:
10052:
10051:
10049:
10046:
10045:
10024:
10023:
10018:
10017:
10015:
10012:
10011:
9990:
9989:
9984:
9983:
9981:
9978:
9977:
9943:
9940:
9939:
9917:
9914:
9913:
9891:
9890:
9885:
9884:
9882:
9879:
9878:
9860:
9859:
9854:
9853:
9851:
9848:
9847:
9844:
9838:
9804:
9803:
9799:
9792:
9791:
9786:
9785:
9777:
9764:
9763:
9758:
9757:
9749:
9745:
9741:
9719:
9717:
9709:
9690:
9685:
9684:
9680:
9678:
9675:
9674:
9671:relation matrix
9663:
9636:
9631:
9630:
9626:
9616:
9615:
9609:
9604:
9603:
9597:
9594:
9593:
9547:
9546:
9542:
9535:
9534:
9529:
9528:
9520:
9507:
9506:
9501:
9500:
9492:
9488:
9484:
9467:
9459:
9440:
9435:
9434:
9430:
9428:
9425:
9424:
9398:
9397:
9392:
9391:
9389:
9386:
9385:
9364:
9363:
9359:
9353:
9349:
9334:
9330:
9319:
9317:
9314:
9313:
9289:
9287:
9284:
9283:
9267:
9264:
9263:
9229:
9225:
9215:
9213:
9204:
9200:
9190:
9186:
9160:
9157:
9156:
9136:
9133:
9132:
9118:
9112:
9089:
9076:
9072:
9067:
9066:
9057:
9037:
9033:
9028:
9027:
9016:
9006:
9002:
8997:
8996:
8985:
8982:
8981:
8959:
8956:
8955:
8933:
8930:
8929:
8913:
8911:
8908:
8907:
8883:
8880:
8879:
8853:
8850:
8849:
8833:
8830:
8829:
8804:
8787:
8777:
8776:
8772:
8770:
8767:
8766:
8752:
8751:
8736:
8735:
8729:
8723:
8722:
8721:
8710:
8693:
8683:
8682:
8678:
8672:
8671:
8665:
8664:
8649:
8648:
8633:
8632:
8628:
8620:
8603:
8589:
8572:
8562:
8561:
8557:
8556:
8552:
8528:
8527:
8523:
8515:
8498:
8484:
8467:
8463:
8459:
8446:
8445:
8441:
8436:
8434:
8431:
8430:
8399:
8391:
8375:
8374:
8369:
8368:
8357:
8350:
8349:
8344:
8343:
8332:
8329:
8328:
8300:
8297:
8296:
8280:
8278:
8275:
8274:
8271:
8230:
8224:
8220:
8215:
8201:
8198:
8197:
8178:
8173:
8166:
8165:
8160:
8159:
8157:
8154:
8153:
8134:
8127:
8126:
8121:
8120:
8112:
8105:
8104:
8099:
8098:
8081:
8076:
8069:
8068:
8063:
8062:
8060:
8057:
8056:
8051:. Treated as a
8033:
8025:
8018:
8017:
8012:
8011:
7989:
7988:
7983:
7982:
7980:
7977:
7976:
7965:
7959:
7938:
7936:
7933:
7932:
7912:
7907:
7902:
7900:
7897:
7896:
7877:
7860:
7858:
7855:
7854:
7810:
7806:
7799:
7798:
7794:
7781:
7769:
7765:
7761:
7745:
7738:
7733:
7732:
7727:
7722:
7712:
7705:
7701:
7681:
7670:
7667:
7666:
7650:
7648:
7645:
7644:
7625:
7614:
7611:
7610:
7583:
7580:
7579:
7563:
7561:
7558:
7557:
7554:
7533:
7531:
7528:
7527:
7506:
7505:
7501:
7499:
7496:
7495:
7468:
7467:
7463:
7461:
7458:
7457:
7435:
7427:
7409:
7405:
7400:
7392:
7375:
7367:
7347:
7339:
7319:
7311:
7303:
7275:
7274:
7270:
7268:
7265:
7264:
7248:
7246:
7243:
7242:
7226:
7224:
7221:
7220:
7204:
7202:
7199:
7198:
7186:
7157:
7155:
7152:
7151:
7133:
7132:
7127:
7126:
7124:
7121:
7120:
7099:
7098:
7094:
7082:
7073:
7072:
7066:
7063:
7062:
7043:
7041:
7038:
7037:
7016:
7015:
7011:
7009:
7006:
7005:
6979:
6975:
6974:
6970:
6968:
6965:
6964:
6940:
6931:
6930:
6913:
6912:
6908:
6906:
6903:
6902:
6876:
6875:
6871:
6859:
6850:
6849:
6832:
6831:
6827:
6814:
6813:
6809:
6795:
6791:
6790:
6786:
6784:
6781:
6780:
6754:
6753:
6748:
6747:
6739:
6738:
6734:
6722:
6713:
6712:
6695:
6694:
6690:
6680:
6679:
6674:
6673:
6663:
6658:
6653:
6652:
6647:
6646:
6644:
6641:
6640:
6609:
6605:
6604:
6600:
6586:
6582:
6581:
6576:
6575:
6566:
6565:
6554:
6546:
6544:
6541:
6540:
6524:
6522:
6519:
6518:
6502:
6500:
6497:
6496:
6470:
6462:
6453:
6452:
6441:
6433:
6431:
6428:
6427:
6407:
6405:
6402:
6401:
6385:
6383:
6380:
6379:
6360:
6358:
6355:
6354:
6338:
6336:
6333:
6332:
6307:
6306:
6302:
6300:
6297:
6296:
6275:
6274:
6270:
6268:
6265:
6264:
6243:
6235:
6216:
6215:
6206:
6205:
6188:
6187:
6183:
6181:
6178:
6177:
6158:
6136:
6135:
6131:
6129:
6126:
6125:
6106:
6084:
6083:
6079:
6077:
6074:
6073:
6055:
6054:
6044:
6043:
6039:
6037:
6027:
6026:
6022:
6019:
6018:
6008:
6007:
6003:
6001:
5991:
5990:
5986:
5979:
5978:
5970:
5959:
5958:
5952:
5947:
5946:
5943:
5942:
5936:
5931:
5930:
5923:
5922:
5914:
5912:
5909:
5908:
5892:
5890:
5887:
5886:
5870:
5868:
5865:
5864:
5848:
5846:
5843:
5842:
5823:
5821:
5818:
5817:
5816:The joint mean
5814:
5792:
5790:
5787:
5786:
5770:
5768:
5765:
5764:
5741:
5733:
5722:
5719:
5718:
5699:
5679:
5671:
5651:
5643:
5623:
5603:
5595:
5584:
5581:
5580:
5564:
5562:
5559:
5558:
5542:
5540:
5537:
5536:
5516:
5515:
5510:
5509:
5500:
5485:
5474:
5463:
5452:
5449:
5448:
5432:
5430:
5427:
5426:
5404:
5401:
5400:
5384:
5382:
5379:
5378:
5356:
5353:
5352:
5332:
5327:
5326:
5322:
5312:
5311:
5305:
5300:
5299:
5293:
5290:
5289:
5266:
5261:
5260:
5256:
5254:
5251:
5250:
5232:
5227:
5226:
5218:
5213:
5201:
5191:
5186:
5185:
5181:
5175:
5170:
5169:
5167:
5164:
5163:
5140:
5135:
5134:
5130:
5128:
5125:
5124:
5105:
5104:
5098:
5097:
5092:
5084:
5083:
5078:
5077:
5063:
5062:
5058:
5054:
5035:
5030:
5029:
5025:
5023:
5020:
5019:
5000:
4997:
4996:
4978:
4977:
4973:
4967:
4963:
4948:
4944:
4933:
4931:
4928:
4927:
4907:
4906:
4887:
4886:
4881:
4880:
4878:
4875:
4874:
4851:
4850:
4837:
4820:
4819:
4815:
4814:
4801:
4784:
4783:
4779:
4778:
4774:
4757:
4738:
4733:
4732:
4728:
4726:
4723:
4722:
4719:
4696:
4695:
4679:
4675:
4660:
4656:
4651:
4645:
4641:
4640:
4636:
4631:
4627:
4621:
4620:
4612:
4611:
4592:
4588:
4579:
4575:
4570:
4564:
4560:
4559:
4555:
4550:
4546:
4545:
4538:
4519:
4515:
4510:
4504:
4500:
4499:
4495:
4490:
4483:
4482:
4475:
4474:
4469:
4464:
4450:
4446:
4431:
4427:
4418:
4414:
4405:
4401:
4392:
4388:
4387:
4383:
4378:
4364:
4360:
4345:
4341:
4332:
4328:
4319:
4315:
4314:
4310:
4304:
4303:
4298:
4293:
4288:
4282:
4281:
4267:
4263:
4248:
4244:
4235:
4231:
4222:
4218:
4209:
4205:
4204:
4200:
4195:
4190:
4185:
4168:
4164:
4155:
4151:
4142:
4138:
4137:
4133:
4127:
4126:
4112:
4108:
4093:
4089:
4080:
4076:
4067:
4063:
4062:
4058:
4053:
4048:
4031:
4027:
4018:
4014:
4005:
4001:
4000:
3996:
3991:
3981:
3980:
3973:
3972:
3956:
3952:
3937:
3933:
3928:
3922:
3918:
3917:
3913:
3908:
3904:
3898:
3897:
3889:
3888:
3869:
3865:
3856:
3852:
3847:
3841:
3837:
3836:
3832:
3827:
3823:
3822:
3815:
3796:
3792:
3787:
3781:
3777:
3776:
3772:
3767:
3760:
3759:
3747:
3743:
3738:
3727:
3724:
3723:
3699:
3698:
3690:
3686:
3685:
3681:
3677:
3671:
3670:
3662:
3661:
3653:
3649:
3648:
3644:
3640:
3639:
3632:
3624:
3620:
3619:
3615:
3608:
3607:
3600:
3599:
3594:
3589:
3581:
3577:
3568:
3564:
3563:
3559:
3557:
3549:
3545:
3536:
3532:
3531:
3527:
3524:
3523:
3518:
3513:
3508:
3502:
3501:
3493:
3489:
3480:
3476:
3475:
3471:
3469:
3464:
3459:
3451:
3447:
3438:
3434:
3433:
3429:
3426:
3425:
3417:
3413:
3404:
3400:
3399:
3395:
3393:
3388:
3380:
3376:
3367:
3363:
3362:
3358:
3356:
3346:
3345:
3338:
3337:
3329:
3325:
3324:
3320:
3316:
3310:
3309:
3301:
3300:
3292:
3288:
3287:
3283:
3279:
3278:
3271:
3263:
3259:
3258:
3254:
3247:
3246:
3235:
3224:
3221:
3220:
3187:
3181:
3176:
3175:
3169:
3166:
3165:
3162:
3130:
3129:
3124:
3119:
3107:
3103:
3088:
3084:
3077:
3064:
3060:
3051:
3047:
3035:
3031:
3022:
3018:
3005:
3003:
3001:
2989:
2985:
2970:
2966:
2959:
2946:
2942:
2933:
2929:
2917:
2913:
2904:
2900:
2887:
2885:
2882:
2879:
2878:
2873:
2868:
2863:
2857:
2854:
2853:
2841:
2837:
2822:
2818:
2811:
2798:
2794:
2785:
2781:
2769:
2765:
2756:
2752:
2739:
2737:
2735:
2730:
2725:
2713:
2709:
2694:
2690:
2683:
2670:
2666:
2657:
2653:
2641:
2637:
2628:
2624:
2611:
2609:
2606:
2603:
2602:
2590:
2586:
2571:
2567:
2560:
2547:
2543:
2534:
2530:
2518:
2514:
2505:
2501:
2488:
2486:
2484:
2479:
2467:
2463:
2448:
2444:
2437:
2424:
2420:
2411:
2407:
2395:
2391:
2382:
2378:
2365:
2363:
2361:
2351:
2350:
2339:
2328:
2325:
2324:
2290:
2287:
2286:
2266:
2262:
2251:
2245:
2241:
2239:
2236:
2235:
2195:
2192:
2191:
2174:
2170:
2168:
2165:
2164:
2161:diagonal matrix
2142:
2137:
2136:
2132:
2130:
2127:
2126:
2105:
2100:
2099:
2095:
2084:
2081:
2080:
2054:
2050:
2044:
2043:
2042:
2032:
2027:
2026:
2022:
2007:
2006:
1998:
1993:
1992:
1988:
1975:
1971:
1965:
1964:
1963:
1953:
1948:
1947:
1943:
1928:
1927:
1916:
1905:
1902:
1901:
1885:
1883:
1880:
1879:
1872:
1866:
1840:
1839:
1834:
1833:
1828:
1809:
1804:
1803:
1799:
1797:
1794:
1793:
1775:
1774:
1770:
1765:
1748:
1729:
1724:
1723:
1719:
1706:
1705:
1701:
1693:
1676:
1662:
1645:
1623:
1618:
1617:
1613:
1611:
1608:
1607:
1589:
1584:
1583:
1579:
1577:
1574:
1573:
1552:
1547:
1546:
1542:
1540:
1537:
1536:
1533:
1528:
1498:
1497:
1493:
1485:
1468:
1454:
1437:
1433:
1429:
1413:
1408:
1407:
1403:
1392:
1384:
1373:
1370:
1369:
1337:
1332:
1331:
1327:
1325:
1322:
1321:
1293:
1292:
1288:
1280:
1263:
1249:
1232:
1228:
1224:
1207:
1199:
1179:
1168:
1165:
1164:
1148:
1146:
1143:
1142:
1122:
1120:
1117:
1116:
1094:
1089:
1088:
1084:
1082:
1079:
1078:
1067:
1046:
1043:
1042:
1016:
1012:
994:
990:
975:
971:
953:
949:
925:
921:
912:
908:
888:
884:
878:
874:
873:
869:
867:
864:
863:
832:
829:
828:
810:
805:
804:
800:
798:
795:
794:
764:
763:
759:
753:
749:
734:
730:
721:
717:
706:
704:
701:
700:
677:
673:
671:
668:
667:
650:
646:
644:
641:
640:
624:
622:
619:
618:
602:
600:
597:
596:
593:
572:
569:
568:
552:
549:
548:
530:
525:
524:
520:
518:
515:
514:
498:
496:
493:
492:
448:
445:
444:
428:
425:
424:
408:
405:
404:
381:variance matrix
371:(also known as
326:
323:
322:
306:
303:
302:
286:
283:
282:
245:
244:
239:
233:
232:
227:
217:
216:
214:
211:
210:
199:
188:
159:
158:
134:
124:
123:
99:
89:
88:
64:
30:
23:
22:
15:
12:
11:
5:
15906:
15896:
15895:
15890:
15885:
15868:
15867:
15865:
15864:
15859:
15854:
15839:
15836:
15835:
15833:
15832:
15827:
15822:
15817:
15815:Perfect matrix
15812:
15807:
15802:
15797:
15791:
15789:
15785:
15784:
15782:
15781:
15776:
15771:
15766:
15761:
15756:
15751:
15746:
15741:
15736:
15731:
15726:
15721:
15715:
15713:
15709:
15708:
15706:
15705:
15700:
15695:
15690:
15685:
15680:
15675:
15670:
15664:
15662:
15655:
15654:
15652:
15651:
15646:
15641:
15636:
15631:
15626:
15621:
15616:
15611:
15606:
15600:
15598:
15591:
15590:
15588:
15587:
15585:Transformation
15582:
15577:
15572:
15567:
15562:
15557:
15552:
15547:
15542:
15537:
15532:
15527:
15522:
15517:
15512:
15507:
15502:
15497:
15492:
15487:
15482:
15477:
15472:
15467:
15462:
15457:
15452:
15447:
15442:
15437:
15431:
15429:
15425:
15424:
15422:
15421:
15416:
15411:
15406:
15401:
15396:
15391:
15386:
15381:
15376:
15371:
15362:
15356:
15354:
15343:
15342:
15340:
15339:
15334:
15329:
15324:
15322:Diagonalizable
15319:
15314:
15309:
15304:
15298:
15296:
15292:Conditions on
15289:
15288:
15286:
15285:
15280:
15275:
15270:
15265:
15260:
15255:
15250:
15245:
15240:
15234:
15232:
15228:
15227:
15225:
15224:
15219:
15214:
15209:
15204:
15199:
15194:
15189:
15184:
15179:
15174:
15172:Skew-symmetric
15169:
15167:Skew-Hermitian
15164:
15159:
15154:
15149:
15144:
15139:
15134:
15129:
15124:
15119:
15114:
15109:
15104:
15099:
15094:
15089:
15084:
15079:
15074:
15069:
15064:
15059:
15054:
15049:
15044:
15039:
15034:
15029:
15024:
15019:
15014:
15009:
15004:
15002:Block-diagonal
14999:
14994:
14989:
14984:
14979:
14977:Anti-symmetric
14974:
14972:Anti-Hermitian
14969:
14964:
14958:
14956:
14952:
14951:
14943:
14942:
14935:
14928:
14920:
14911:
14910:
14908:
14907:
14895:
14883:
14869:
14856:
14853:
14852:
14849:
14848:
14845:
14844:
14842:
14841:
14836:
14831:
14826:
14821:
14815:
14813:
14807:
14806:
14804:
14803:
14798:
14793:
14788:
14783:
14778:
14773:
14768:
14763:
14758:
14752:
14750:
14744:
14743:
14741:
14740:
14735:
14730:
14721:
14716:
14711:
14705:
14703:
14697:
14696:
14694:
14693:
14688:
14683:
14674:
14672:Bioinformatics
14668:
14666:
14656:
14655:
14643:
14642:
14639:
14638:
14635:
14634:
14631:
14630:
14628:
14627:
14621:
14619:
14615:
14614:
14612:
14611:
14605:
14603:
14597:
14596:
14594:
14593:
14588:
14583:
14578:
14572:
14570:
14561:
14555:
14554:
14551:
14550:
14548:
14547:
14542:
14537:
14532:
14527:
14521:
14519:
14513:
14512:
14510:
14509:
14504:
14499:
14491:
14486:
14481:
14480:
14479:
14477:partial (PACF)
14468:
14466:
14460:
14459:
14457:
14456:
14451:
14446:
14438:
14433:
14427:
14425:
14424:Specific tests
14421:
14420:
14418:
14417:
14412:
14407:
14402:
14397:
14392:
14387:
14382:
14376:
14374:
14367:
14361:
14360:
14358:
14357:
14356:
14355:
14354:
14353:
14338:
14337:
14336:
14326:
14324:Classification
14321:
14316:
14311:
14306:
14301:
14296:
14290:
14288:
14282:
14281:
14279:
14278:
14273:
14271:McNemar's test
14268:
14263:
14258:
14253:
14247:
14245:
14235:
14234:
14210:
14209:
14206:
14205:
14202:
14201:
14199:
14198:
14193:
14188:
14183:
14177:
14175:
14169:
14168:
14166:
14165:
14149:
14143:
14141:
14135:
14134:
14132:
14131:
14126:
14121:
14116:
14111:
14109:Semiparametric
14106:
14101:
14095:
14093:
14089:
14088:
14086:
14085:
14080:
14075:
14070:
14064:
14062:
14056:
14055:
14053:
14052:
14047:
14042:
14037:
14032:
14026:
14024:
14018:
14017:
14015:
14014:
14009:
14004:
13999:
13993:
13991:
13981:
13980:
13977:
13976:
13971:
13965:
13957:
13956:
13953:
13952:
13949:
13948:
13946:
13945:
13944:
13943:
13933:
13928:
13923:
13922:
13921:
13916:
13905:
13903:
13897:
13896:
13893:
13892:
13890:
13889:
13884:
13883:
13882:
13874:
13866:
13850:
13847:(MannâWhitney)
13842:
13841:
13840:
13827:
13826:
13825:
13814:
13812:
13806:
13805:
13803:
13802:
13801:
13800:
13795:
13790:
13780:
13775:
13772:(ShapiroâWilk)
13767:
13762:
13757:
13752:
13747:
13739:
13733:
13731:
13725:
13724:
13722:
13721:
13713:
13704:
13692:
13686:
13684:Specific tests
13680:
13679:
13676:
13675:
13673:
13672:
13667:
13662:
13656:
13654:
13648:
13647:
13645:
13644:
13639:
13638:
13637:
13627:
13626:
13625:
13615:
13609:
13607:
13601:
13600:
13598:
13597:
13596:
13595:
13590:
13580:
13575:
13570:
13565:
13560:
13554:
13552:
13546:
13545:
13543:
13542:
13537:
13536:
13535:
13530:
13529:
13528:
13523:
13508:
13507:
13506:
13501:
13496:
13491:
13480:
13478:
13469:
13463:
13462:
13460:
13459:
13454:
13449:
13448:
13447:
13437:
13432:
13431:
13430:
13420:
13419:
13418:
13413:
13408:
13398:
13393:
13388:
13387:
13386:
13381:
13376:
13360:
13359:
13358:
13353:
13348:
13338:
13337:
13336:
13331:
13321:
13320:
13319:
13309:
13308:
13307:
13297:
13292:
13287:
13281:
13279:
13269:
13268:
13256:
13255:
13252:
13251:
13248:
13247:
13245:
13244:
13239:
13234:
13229:
13223:
13221:
13215:
13214:
13212:
13211:
13206:
13201:
13195:
13193:
13189:
13188:
13186:
13185:
13180:
13175:
13170:
13165:
13160:
13155:
13149:
13147:
13141:
13140:
13138:
13137:
13135:Standard error
13132:
13127:
13122:
13121:
13120:
13115:
13104:
13102:
13096:
13095:
13093:
13092:
13087:
13082:
13077:
13072:
13067:
13065:Optimal design
13062:
13057:
13051:
13049:
13039:
13038:
13026:
13025:
13022:
13021:
13018:
13017:
13015:
13014:
13009:
13004:
12999:
12994:
12989:
12984:
12979:
12974:
12969:
12964:
12959:
12954:
12949:
12944:
12938:
12936:
12930:
12929:
12927:
12926:
12921:
12920:
12919:
12914:
12904:
12899:
12893:
12891:
12885:
12884:
12882:
12881:
12876:
12871:
12865:
12863:
12862:Summary tables
12859:
12858:
12856:
12855:
12849:
12847:
12841:
12840:
12837:
12836:
12834:
12833:
12832:
12831:
12826:
12821:
12811:
12805:
12803:
12797:
12796:
12794:
12793:
12788:
12783:
12778:
12773:
12768:
12763:
12757:
12755:
12749:
12748:
12746:
12745:
12740:
12735:
12734:
12733:
12728:
12723:
12718:
12713:
12708:
12703:
12698:
12696:Contraharmonic
12693:
12688:
12677:
12675:
12666:
12656:
12655:
12643:
12642:
12640:
12639:
12634:
12628:
12625:
12624:
12617:
12616:
12609:
12602:
12594:
12588:
12587:
12581:
12564:
12545:
12538:
12518:
12515:
12512:
12511:
12509:1329â36 (1993)
12495:
12472:
12453:
12410:
12395:
12388:
12370:
12357:
12350:
12332:
12317:
12310:
12290:
12276:
12256:
12249:
12225:
12224:
12222:
12219:
12218:
12217:
12212:
12207:
12202:
12197:
12195:Gramian matrix
12192:
12187:
12180:
12177:
12156:
12153:
12135:
12105:
12101:
12097:
12093:
12089:
12085:
12081:
12078:
12075:
12054:
12031:
12027:
12006:
12002:
11998:
11994:
11990:
11987:
11984:
11960:
11954:
11948:
11943:
11940:
11936:
11932:
11908:
11901:
11896:
11892:
11888:
11864:
11844:
11841:
11838:
11833:
11828:
11806:
11803:
11800:
11795:
11790:
11764:
11760:
11756:
11736:
11733:
11730:
11727:
11723:
11719:
11699:
11679:
11676:
11673:
11668:
11663:
11652:such spectra,
11639:
11635:
11631:
11628:
11612:time-of-flight
11599:
11596:
11593:
11590:
11540:
11512:
11508:
11504:
11500:
11496:
11492:
11488:
11485:
11482:
11479:
11476:
11472:
11468:
11464:
11460:
11457:
11454:
11450:
11446:
11442:
11438:
11434:
11430:
11427:
11424:
11421:
11418:
11414:
11410:
11406:
11402:
11399:
11396:
11393:
11390:
11386:
11382:
11378:
11374:
11370:
11366:
11363:
11360:
11332:
11329:
11323:
11317:
11312:
11309:
11305:
11301:
11298:
11295:
11288:
11283:
11279:
11275:
11272:
11269:
11265:
11261:
11257:
11253:
11250:
11247:
11221:
11216:
11209:
11204:
11201:
11198:
11194:
11188:
11185:
11180:
11177:
11173:
11169:
11145:
11142:
11139:
11136:
11108:
11103:
11099:
11095:
11090:
11086:
11065:
11060:
11054:
11049:
11045:
11041:
11036:
11032:
11028:
11026:
11023:
11021:
11016:
11012:
11008:
11003:
10999:
10995:
10993:
10988:
10984:
10980:
10975:
10971:
10967:
10966:
10963:
10960:
10957:
10955:
10952:
10950:
10947:
10945:
10942:
10941:
10938:
10935:
10930:
10926:
10922:
10917:
10913:
10909:
10907:
10904:
10902:
10897:
10893:
10889:
10884:
10880:
10876:
10874:
10869:
10865:
10861:
10856:
10852:
10848:
10847:
10844:
10841:
10836:
10832:
10828:
10823:
10819:
10815:
10813:
10810:
10808:
10803:
10799:
10795:
10790:
10786:
10782:
10780:
10775:
10771:
10767:
10762:
10758:
10754:
10753:
10751:
10746:
10742:
10736:
10731:
10726:
10723:
10720:
10715:
10710:
10705:
10700:
10695:
10689:
10679:samples, e.g.
10668:
10647:
10625:
10621:
10617:
10588:
10566:
10545:
10541:
10537:
10533:
10529:
10525:
10521:
10518:
10515:
10495:
10491:
10487:
10483:
10479:
10476:
10473:
10457:
10454:
10445:Hessian matrix
10432:
10429:
10392:(PCA) and the
10373:
10370:
10354:
10348:
10341:
10335:
10327:
10321:
10314:
10311:
10306:
10300:
10297:
10291:
10285:
10279:
10272:
10266:
10258:
10252:
10245:
10242:
10237:
10231:
10228:
10222:
10197:
10190:
10184:
10176:
10170:
10162:
10159:
10156:
10152:
10147:
10141:
10138:
10132:
10126:
10120:
10113:
10107:
10099:
10093:
10085:
10082:
10079:
10075:
10070:
10064:
10061:
10055:
10030:
10027:
10021:
9996:
9993:
9987:
9953:
9950:
9947:
9927:
9924:
9921:
9894:
9888:
9863:
9857:
9840:Main article:
9837:
9834:
9814:
9807:
9802:
9795:
9789:
9784:
9780:
9776:
9773:
9767:
9761:
9756:
9752:
9748:
9744:
9740:
9737:
9734:
9731:
9726:
9722:
9716:
9712:
9708:
9705:
9702:
9699:
9693:
9688:
9683:
9662:
9659:
9658:
9657:
9654:
9639:
9634:
9629:
9625:
9619:
9612:
9607:
9602:
9585:
9584:
9561:
9557:
9550:
9545:
9538:
9532:
9527:
9523:
9519:
9516:
9510:
9504:
9499:
9495:
9491:
9487:
9483:
9480:
9477:
9474:
9470:
9466:
9462:
9458:
9455:
9452:
9449:
9443:
9438:
9433:
9401:
9395:
9367:
9362:
9356:
9352:
9348:
9345:
9342:
9337:
9333:
9329:
9326:
9322:
9296:
9293:
9271:
9251:
9247:
9241:
9237:
9232:
9228:
9224:
9221:
9218:
9212:
9207:
9203:
9199:
9196:
9193:
9189:
9185:
9182:
9179:
9176:
9173:
9170:
9167:
9164:
9140:
9111:
9108:
9096:
9092:
9088:
9083:
9079:
9075:
9070:
9064:
9060:
9056:
9053:
9050:
9044:
9040:
9036:
9031:
9026:
9023:
9019:
9013:
9009:
9005:
9000:
8995:
8992:
8989:
8969:
8966:
8963:
8943:
8940:
8937:
8916:
8887:
8863:
8860:
8857:
8837:
8814:
8811:
8807:
8803:
8800:
8797:
8794:
8790:
8786:
8780:
8775:
8750:
8747:
8744:
8739:
8732:
8726:
8720:
8717:
8713:
8709:
8706:
8703:
8700:
8696:
8692:
8686:
8681:
8675:
8668:
8663:
8660:
8657:
8654:
8652:
8650:
8646:
8642:
8636:
8631:
8627:
8623:
8619:
8616:
8613:
8610:
8606:
8602:
8599:
8596:
8592:
8588:
8585:
8582:
8579:
8575:
8571:
8565:
8560:
8555:
8551:
8548:
8545:
8542:
8538:
8531:
8526:
8522:
8518:
8514:
8511:
8508:
8505:
8501:
8497:
8494:
8491:
8487:
8483:
8480:
8477:
8474:
8470:
8466:
8462:
8458:
8455:
8449:
8444:
8440:
8438:
8406:
8402:
8398:
8394:
8390:
8387:
8384:
8378:
8372:
8367:
8364:
8360:
8353:
8347:
8342:
8339:
8336:
8316:
8313:
8310:
8307:
8304:
8283:
8270:
8267:
8246:
8243:
8240:
8237:
8233:
8227:
8223:
8218:
8214:
8211:
8208:
8205:
8181:
8176:
8169:
8163:
8141:
8137:
8130:
8124:
8119:
8115:
8108:
8102:
8097:
8094:
8091:
8088:
8084:
8079:
8072:
8066:
8040:
8036:
8032:
8028:
8021:
8015:
8010:
8007:
8004:
8001:
7998:
7992:
7986:
7961:Main article:
7958:
7955:
7941:
7915:
7910:
7905:
7884:
7880:
7876:
7873:
7870:
7867:
7863:
7842:
7838:
7833:
7830:
7827:
7824:
7821:
7816:
7813:
7809:
7802:
7797:
7793:
7790:
7787:
7784:
7777:
7774:
7768:
7764:
7760:
7757:
7752:
7748:
7744:
7741:
7736:
7730:
7725:
7719:
7715:
7711:
7708:
7704:
7700:
7697:
7694:
7691:
7688:
7684:
7680:
7677:
7674:
7653:
7632:
7628:
7624:
7621:
7618:
7587:
7566:
7553:
7550:
7536:
7512:
7509:
7504:
7480:
7477:
7474:
7471:
7466:
7445:
7442:
7438:
7434:
7430:
7426:
7423:
7420:
7415:
7412:
7408:
7403:
7399:
7395:
7391:
7388:
7385:
7382:
7378:
7374:
7370:
7366:
7363:
7360:
7357:
7354:
7350:
7346:
7342:
7338:
7335:
7332:
7329:
7326:
7322:
7318:
7314:
7310:
7306:
7302:
7299:
7296:
7293:
7287:
7284:
7281:
7278:
7273:
7251:
7229:
7207:
7185:
7182:
7160:
7136:
7130:
7105:
7102:
7097:
7093:
7088:
7085:
7079:
7076:
7071:
7046:
7022:
7019:
7014:
6986:
6982:
6978:
6973:
6946:
6943:
6937:
6934:
6929:
6925:
6919:
6916:
6911:
6888:
6882:
6879:
6874:
6870:
6865:
6862:
6856:
6853:
6848:
6844:
6838:
6835:
6830:
6826:
6820:
6817:
6812:
6808:
6802:
6798:
6794:
6789:
6764:
6757:
6751:
6746:
6742:
6737:
6733:
6728:
6725:
6719:
6716:
6711:
6707:
6701:
6698:
6693:
6689:
6683:
6677:
6672:
6666:
6661:
6656:
6650:
6625:
6622:
6616:
6612:
6608:
6603:
6599:
6593:
6589:
6585:
6579:
6574:
6569:
6561:
6557:
6553:
6549:
6527:
6505:
6480:
6477:
6473:
6469:
6465:
6461:
6456:
6448:
6444:
6440:
6436:
6410:
6388:
6375:respectively.
6363:
6341:
6313:
6310:
6305:
6281:
6278:
6273:
6250:
6246:
6242:
6238:
6234:
6231:
6228:
6225:
6219:
6212:
6209:
6204:
6200:
6194:
6191:
6186:
6165:
6161:
6157:
6154:
6151:
6148:
6142:
6139:
6134:
6113:
6109:
6105:
6102:
6099:
6096:
6090:
6087:
6082:
6059:
6050:
6047:
6042:
6038:
6033:
6030:
6025:
6021:
6020:
6014:
6011:
6006:
6002:
5997:
5994:
5989:
5985:
5984:
5982:
5977:
5973:
5968:
5963:
5955:
5950:
5945:
5944:
5939:
5934:
5929:
5928:
5926:
5921:
5917:
5895:
5873:
5851:
5826:
5813:
5812:Block matrices
5810:
5809:
5808:
5795:
5773:
5748:
5744:
5740:
5736:
5732:
5729:
5726:
5706:
5702:
5698:
5695:
5692:
5689:
5686:
5682:
5678:
5674:
5670:
5667:
5664:
5661:
5658:
5654:
5650:
5646:
5642:
5639:
5636:
5633:
5630:
5626:
5622:
5619:
5616:
5613:
5610:
5606:
5602:
5598:
5594:
5591:
5588:
5567:
5545:
5533:
5519:
5513:
5507:
5503:
5499:
5496:
5493:
5488:
5484:
5481:
5477:
5473:
5469:
5466:
5462:
5459:
5456:
5435:
5414:
5411:
5408:
5387:
5366:
5363:
5360:
5349:
5335:
5330:
5325:
5321:
5315:
5308:
5303:
5298:
5269:
5264:
5259:
5248:
5235:
5230:
5225:
5221:
5211:
5208:
5204:
5200:
5194:
5189:
5184:
5178:
5173:
5143:
5138:
5133:
5122:
5108:
5101:
5095:
5087:
5081:
5076:
5073:
5066:
5061:
5057:
5053:
5050:
5047:
5044:
5038:
5033:
5028:
5004:
4981:
4976:
4970:
4966:
4962:
4959:
4956:
4951:
4947:
4943:
4940:
4936:
4915:
4905:
4902:
4899:
4896:
4890:
4884:
4861:
4854:
4848:
4844:
4840:
4836:
4833:
4830:
4827:
4823:
4818:
4812:
4808:
4804:
4800:
4797:
4794:
4791:
4787:
4782:
4777:
4773:
4770:
4767:
4764:
4760:
4756:
4753:
4750:
4747:
4741:
4736:
4731:
4718:
4715:
4700:
4688:
4685:
4682:
4678:
4674:
4671:
4668:
4663:
4659:
4654:
4648:
4644:
4639:
4635:
4630:
4626:
4623:
4622:
4619:
4616:
4613:
4606:
4603:
4600:
4595:
4591:
4587:
4582:
4578:
4573:
4567:
4563:
4558:
4554:
4549:
4547:
4544:
4541:
4533:
4530:
4527:
4522:
4518:
4513:
4507:
4503:
4498:
4494:
4489:
4488:
4486:
4479:
4473:
4470:
4468:
4465:
4459:
4456:
4453:
4449:
4445:
4442:
4439:
4434:
4430:
4426:
4421:
4417:
4413:
4408:
4404:
4400:
4395:
4391:
4386:
4382:
4379:
4373:
4370:
4367:
4363:
4359:
4356:
4353:
4348:
4344:
4340:
4335:
4331:
4327:
4322:
4318:
4313:
4309:
4306:
4305:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4283:
4276:
4273:
4270:
4266:
4262:
4259:
4256:
4251:
4247:
4243:
4238:
4234:
4230:
4225:
4221:
4217:
4212:
4208:
4203:
4199:
4196:
4194:
4191:
4189:
4186:
4182:
4179:
4176:
4171:
4167:
4163:
4158:
4154:
4150:
4145:
4141:
4136:
4132:
4129:
4128:
4121:
4118:
4115:
4111:
4107:
4104:
4101:
4096:
4092:
4088:
4083:
4079:
4075:
4070:
4066:
4061:
4057:
4054:
4052:
4049:
4045:
4042:
4039:
4034:
4030:
4026:
4021:
4017:
4013:
4008:
4004:
3999:
3995:
3992:
3990:
3987:
3986:
3984:
3977:
3965:
3962:
3959:
3955:
3951:
3948:
3945:
3940:
3936:
3931:
3925:
3921:
3916:
3912:
3907:
3903:
3900:
3899:
3896:
3893:
3890:
3883:
3880:
3877:
3872:
3868:
3864:
3859:
3855:
3850:
3844:
3840:
3835:
3831:
3826:
3824:
3821:
3818:
3810:
3807:
3804:
3799:
3795:
3790:
3784:
3780:
3775:
3771:
3766:
3765:
3763:
3758:
3753:
3750:
3746:
3741:
3737:
3734:
3731:
3703:
3693:
3689:
3684:
3680:
3676:
3673:
3672:
3669:
3666:
3663:
3656:
3652:
3647:
3643:
3641:
3638:
3635:
3627:
3623:
3618:
3614:
3613:
3611:
3604:
3598:
3595:
3593:
3590:
3584:
3580:
3576:
3571:
3567:
3562:
3558:
3552:
3548:
3544:
3539:
3535:
3530:
3526:
3525:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3503:
3496:
3492:
3488:
3483:
3479:
3474:
3470:
3468:
3465:
3463:
3460:
3454:
3450:
3446:
3441:
3437:
3432:
3428:
3427:
3420:
3416:
3412:
3407:
3403:
3398:
3394:
3392:
3389:
3383:
3379:
3375:
3370:
3366:
3361:
3357:
3355:
3352:
3351:
3349:
3342:
3332:
3328:
3323:
3319:
3315:
3312:
3311:
3308:
3305:
3302:
3295:
3291:
3286:
3282:
3280:
3277:
3274:
3266:
3262:
3257:
3253:
3252:
3250:
3245:
3242:
3238:
3234:
3231:
3228:
3193:
3190:
3184:
3179:
3174:
3161:
3158:
3139:
3134:
3128:
3125:
3123:
3120:
3115:
3110:
3106:
3102:
3099:
3096:
3091:
3087:
3083:
3080:
3075:
3072:
3067:
3063:
3059:
3054:
3050:
3046:
3043:
3038:
3034:
3030:
3025:
3021:
3017:
3014:
3011:
3008:
3002:
2997:
2992:
2988:
2984:
2981:
2978:
2973:
2969:
2965:
2962:
2957:
2954:
2949:
2945:
2941:
2936:
2932:
2928:
2925:
2920:
2916:
2912:
2907:
2903:
2899:
2896:
2893:
2890:
2884:
2883:
2880:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2858:
2855:
2849:
2844:
2840:
2836:
2833:
2830:
2825:
2821:
2817:
2814:
2809:
2806:
2801:
2797:
2793:
2788:
2784:
2780:
2777:
2772:
2768:
2764:
2759:
2755:
2751:
2748:
2745:
2742:
2736:
2734:
2731:
2729:
2726:
2721:
2716:
2712:
2708:
2705:
2702:
2697:
2693:
2689:
2686:
2681:
2678:
2673:
2669:
2665:
2660:
2656:
2652:
2649:
2644:
2640:
2636:
2631:
2627:
2623:
2620:
2617:
2614:
2608:
2607:
2604:
2598:
2593:
2589:
2585:
2582:
2579:
2574:
2570:
2566:
2563:
2558:
2555:
2550:
2546:
2542:
2537:
2533:
2529:
2526:
2521:
2517:
2513:
2508:
2504:
2500:
2497:
2494:
2491:
2485:
2483:
2480:
2475:
2470:
2466:
2462:
2459:
2456:
2451:
2447:
2443:
2440:
2435:
2432:
2427:
2423:
2419:
2414:
2410:
2406:
2403:
2398:
2394:
2390:
2385:
2381:
2377:
2374:
2371:
2368:
2362:
2360:
2357:
2356:
2354:
2349:
2346:
2342:
2338:
2335:
2332:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2274:
2269:
2265:
2261:
2258:
2254:
2248:
2244:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2177:
2173:
2145:
2140:
2135:
2114:
2108:
2103:
2098:
2094:
2091:
2088:
2068:
2061:
2058:
2053:
2047:
2041:
2035:
2030:
2025:
2021:
2018:
2015:
2010:
2001:
1996:
1991:
1982:
1979:
1974:
1968:
1962:
1956:
1951:
1946:
1942:
1939:
1936:
1931:
1926:
1923:
1919:
1915:
1912:
1909:
1888:
1865:
1862:
1849:
1843:
1837:
1831:
1827:
1824:
1821:
1818:
1812:
1807:
1802:
1778:
1773:
1768:
1764:
1761:
1758:
1755:
1751:
1747:
1744:
1741:
1738:
1732:
1727:
1722:
1718:
1715:
1709:
1704:
1700:
1696:
1692:
1689:
1686:
1683:
1679:
1675:
1672:
1669:
1665:
1661:
1658:
1655:
1652:
1648:
1644:
1641:
1638:
1635:
1632:
1626:
1621:
1616:
1592:
1587:
1582:
1555:
1550:
1545:
1532:
1529:
1527:
1524:
1512:
1508:
1501:
1496:
1492:
1488:
1484:
1481:
1478:
1475:
1471:
1467:
1464:
1461:
1457:
1453:
1450:
1447:
1444:
1440:
1436:
1432:
1428:
1425:
1422:
1416:
1411:
1406:
1402:
1399:
1395:
1391:
1387:
1383:
1380:
1377:
1340:
1335:
1330:
1307:
1303:
1296:
1291:
1287:
1283:
1279:
1276:
1273:
1270:
1266:
1262:
1259:
1256:
1252:
1248:
1245:
1242:
1239:
1235:
1231:
1227:
1223:
1220:
1217:
1214:
1210:
1206:
1202:
1198:
1195:
1192:
1189:
1186:
1182:
1178:
1175:
1172:
1151:
1125:
1097:
1092:
1087:
1071:William Feller
1066:
1063:
1050:
1030:
1027:
1024:
1019:
1015:
1011:
1008:
1005:
1002:
997:
993:
989:
986:
983:
978:
974:
970:
967:
964:
961:
956:
952:
948:
945:
942:
939:
936:
933:
928:
924:
920:
915:
911:
907:
904:
901:
898:
891:
887:
881:
877:
872:
848:
845:
842:
839:
836:
813:
808:
803:
791:expected value
767:
762:
756:
752:
748:
745:
742:
737:
733:
729:
724:
720:
716:
713:
709:
680:
676:
653:
649:
627:
605:
592:
589:
576:
556:
533:
528:
523:
501:
458:
455:
452:
432:
412:
387:) is a square
336:
333:
330:
310:
290:
249:
243:
240:
238:
235:
234:
231:
228:
226:
223:
222:
220:
190:
189:
187:
186:
179:
172:
164:
161:
160:
157:
156:
151:
146:
141:
135:
130:
129:
126:
125:
122:
121:
116:
111:
106:
100:
95:
94:
91:
90:
87:
86:
81:
76:
71:
65:
60:
59:
56:
55:
47:
46:
40:
39:
28:
9:
6:
4:
3:
2:
15905:
15894:
15891:
15889:
15886:
15884:
15881:
15880:
15878:
15863:
15860:
15858:
15855:
15853:
15852:
15847:
15841:
15840:
15837:
15831:
15828:
15826:
15823:
15821:
15820:Pseudoinverse
15818:
15816:
15813:
15811:
15808:
15806:
15803:
15801:
15798:
15796:
15793:
15792:
15790:
15788:Related terms
15786:
15780:
15779:Z (chemistry)
15777:
15775:
15772:
15770:
15767:
15765:
15762:
15760:
15757:
15755:
15752:
15750:
15747:
15745:
15742:
15740:
15737:
15735:
15732:
15730:
15727:
15725:
15722:
15720:
15717:
15716:
15714:
15710:
15704:
15701:
15699:
15696:
15694:
15691:
15689:
15686:
15684:
15681:
15679:
15676:
15674:
15671:
15669:
15666:
15665:
15663:
15661:
15656:
15650:
15647:
15645:
15642:
15640:
15637:
15635:
15632:
15630:
15627:
15625:
15622:
15620:
15617:
15615:
15612:
15610:
15607:
15605:
15602:
15601:
15599:
15597:
15592:
15586:
15583:
15581:
15578:
15576:
15573:
15571:
15568:
15566:
15563:
15561:
15558:
15556:
15553:
15551:
15548:
15546:
15543:
15541:
15538:
15536:
15533:
15531:
15528:
15526:
15523:
15521:
15518:
15516:
15513:
15511:
15508:
15506:
15503:
15501:
15498:
15496:
15493:
15491:
15488:
15486:
15483:
15481:
15478:
15476:
15473:
15471:
15468:
15466:
15463:
15461:
15458:
15456:
15453:
15451:
15448:
15446:
15443:
15441:
15438:
15436:
15433:
15432:
15430:
15426:
15420:
15417:
15415:
15412:
15410:
15407:
15405:
15402:
15400:
15397:
15395:
15392:
15390:
15387:
15385:
15382:
15380:
15377:
15375:
15372:
15370:
15366:
15363:
15361:
15358:
15357:
15355:
15353:
15349:
15344:
15338:
15335:
15333:
15330:
15328:
15325:
15323:
15320:
15318:
15315:
15313:
15310:
15308:
15305:
15303:
15300:
15299:
15297:
15295:
15290:
15284:
15281:
15279:
15276:
15274:
15271:
15269:
15266:
15264:
15261:
15259:
15256:
15254:
15251:
15249:
15246:
15244:
15241:
15239:
15236:
15235:
15233:
15229:
15223:
15220:
15218:
15215:
15213:
15210:
15208:
15205:
15203:
15200:
15198:
15195:
15193:
15190:
15188:
15185:
15183:
15180:
15178:
15175:
15173:
15170:
15168:
15165:
15163:
15160:
15158:
15155:
15153:
15150:
15148:
15145:
15143:
15140:
15138:
15137:Pentadiagonal
15135:
15133:
15130:
15128:
15125:
15123:
15120:
15118:
15115:
15113:
15110:
15108:
15105:
15103:
15100:
15098:
15095:
15093:
15090:
15088:
15085:
15083:
15080:
15078:
15075:
15073:
15070:
15068:
15065:
15063:
15060:
15058:
15055:
15053:
15050:
15048:
15045:
15043:
15040:
15038:
15035:
15033:
15030:
15028:
15025:
15023:
15020:
15018:
15015:
15013:
15010:
15008:
15005:
15003:
15000:
14998:
14995:
14993:
14990:
14988:
14985:
14983:
14980:
14978:
14975:
14973:
14970:
14968:
14967:Anti-diagonal
14965:
14963:
14960:
14959:
14957:
14953:
14948:
14941:
14936:
14934:
14929:
14927:
14922:
14921:
14918:
14906:
14905:
14896:
14894:
14893:
14884:
14882:
14881:
14876:
14870:
14868:
14867:
14858:
14857:
14854:
14840:
14837:
14835:
14834:Geostatistics
14832:
14830:
14827:
14825:
14822:
14820:
14817:
14816:
14814:
14812:
14808:
14802:
14801:Psychometrics
14799:
14797:
14794:
14792:
14789:
14787:
14784:
14782:
14779:
14777:
14774:
14772:
14769:
14767:
14764:
14762:
14759:
14757:
14754:
14753:
14751:
14749:
14745:
14739:
14736:
14734:
14731:
14729:
14725:
14722:
14720:
14717:
14715:
14712:
14710:
14707:
14706:
14704:
14702:
14698:
14692:
14689:
14687:
14684:
14682:
14678:
14675:
14673:
14670:
14669:
14667:
14665:
14664:Biostatistics
14661:
14657:
14653:
14648:
14644:
14626:
14625:Log-rank test
14623:
14622:
14620:
14616:
14610:
14607:
14606:
14604:
14602:
14598:
14592:
14589:
14587:
14584:
14582:
14579:
14577:
14574:
14573:
14571:
14569:
14565:
14562:
14560:
14556:
14546:
14543:
14541:
14538:
14536:
14533:
14531:
14528:
14526:
14523:
14522:
14520:
14518:
14514:
14508:
14505:
14503:
14500:
14498:
14496:(BoxâJenkins)
14492:
14490:
14487:
14485:
14482:
14478:
14475:
14474:
14473:
14470:
14469:
14467:
14465:
14461:
14455:
14452:
14450:
14449:DurbinâWatson
14447:
14445:
14439:
14437:
14434:
14432:
14431:DickeyâFuller
14429:
14428:
14426:
14422:
14416:
14413:
14411:
14408:
14406:
14405:Cointegration
14403:
14401:
14398:
14396:
14393:
14391:
14388:
14386:
14383:
14381:
14380:Decomposition
14378:
14377:
14375:
14371:
14368:
14366:
14362:
14352:
14349:
14348:
14347:
14344:
14343:
14342:
14339:
14335:
14332:
14331:
14330:
14327:
14325:
14322:
14320:
14317:
14315:
14312:
14310:
14307:
14305:
14302:
14300:
14297:
14295:
14292:
14291:
14289:
14287:
14283:
14277:
14274:
14272:
14269:
14267:
14264:
14262:
14259:
14257:
14254:
14252:
14251:Cohen's kappa
14249:
14248:
14246:
14244:
14240:
14236:
14232:
14228:
14224:
14220:
14215:
14211:
14197:
14194:
14192:
14189:
14187:
14184:
14182:
14179:
14178:
14176:
14174:
14170:
14164:
14160:
14156:
14150:
14148:
14145:
14144:
14142:
14140:
14136:
14130:
14127:
14125:
14122:
14120:
14117:
14115:
14112:
14110:
14107:
14105:
14104:Nonparametric
14102:
14100:
14097:
14096:
14094:
14090:
14084:
14081:
14079:
14076:
14074:
14071:
14069:
14066:
14065:
14063:
14061:
14057:
14051:
14048:
14046:
14043:
14041:
14038:
14036:
14033:
14031:
14028:
14027:
14025:
14023:
14019:
14013:
14010:
14008:
14005:
14003:
14000:
13998:
13995:
13994:
13992:
13990:
13986:
13982:
13975:
13972:
13970:
13967:
13966:
13962:
13958:
13942:
13939:
13938:
13937:
13934:
13932:
13929:
13927:
13924:
13920:
13917:
13915:
13912:
13911:
13910:
13907:
13906:
13904:
13902:
13898:
13888:
13885:
13881:
13875:
13873:
13867:
13865:
13859:
13858:
13857:
13854:
13853:Nonparametric
13851:
13849:
13843:
13839:
13836:
13835:
13834:
13828:
13824:
13823:Sample median
13821:
13820:
13819:
13816:
13815:
13813:
13811:
13807:
13799:
13796:
13794:
13791:
13789:
13786:
13785:
13784:
13781:
13779:
13776:
13774:
13768:
13766:
13763:
13761:
13758:
13756:
13753:
13751:
13748:
13746:
13744:
13740:
13738:
13735:
13734:
13732:
13730:
13726:
13720:
13718:
13714:
13712:
13710:
13705:
13703:
13698:
13694:
13693:
13690:
13687:
13685:
13681:
13671:
13668:
13666:
13663:
13661:
13658:
13657:
13655:
13653:
13649:
13643:
13640:
13636:
13633:
13632:
13631:
13628:
13624:
13621:
13620:
13619:
13616:
13614:
13611:
13610:
13608:
13606:
13602:
13594:
13591:
13589:
13586:
13585:
13584:
13581:
13579:
13576:
13574:
13571:
13569:
13566:
13564:
13561:
13559:
13556:
13555:
13553:
13551:
13547:
13541:
13538:
13534:
13531:
13527:
13524:
13522:
13519:
13518:
13517:
13514:
13513:
13512:
13509:
13505:
13502:
13500:
13497:
13495:
13492:
13490:
13487:
13486:
13485:
13482:
13481:
13479:
13477:
13473:
13470:
13468:
13464:
13458:
13455:
13453:
13450:
13446:
13443:
13442:
13441:
13438:
13436:
13433:
13429:
13428:loss function
13426:
13425:
13424:
13421:
13417:
13414:
13412:
13409:
13407:
13404:
13403:
13402:
13399:
13397:
13394:
13392:
13389:
13385:
13382:
13380:
13377:
13375:
13369:
13366:
13365:
13364:
13361:
13357:
13354:
13352:
13349:
13347:
13344:
13343:
13342:
13339:
13335:
13332:
13330:
13327:
13326:
13325:
13322:
13318:
13315:
13314:
13313:
13310:
13306:
13303:
13302:
13301:
13298:
13296:
13293:
13291:
13288:
13286:
13283:
13282:
13280:
13278:
13274:
13270:
13266:
13261:
13257:
13243:
13240:
13238:
13235:
13233:
13230:
13228:
13225:
13224:
13222:
13220:
13216:
13210:
13207:
13205:
13202:
13200:
13197:
13196:
13194:
13190:
13184:
13181:
13179:
13176:
13174:
13171:
13169:
13166:
13164:
13161:
13159:
13156:
13154:
13151:
13150:
13148:
13146:
13142:
13136:
13133:
13131:
13130:Questionnaire
13128:
13126:
13123:
13119:
13116:
13114:
13111:
13110:
13109:
13106:
13105:
13103:
13101:
13097:
13091:
13088:
13086:
13083:
13081:
13078:
13076:
13073:
13071:
13068:
13066:
13063:
13061:
13058:
13056:
13053:
13052:
13050:
13048:
13044:
13040:
13036:
13031:
13027:
13013:
13010:
13008:
13005:
13003:
13000:
12998:
12995:
12993:
12990:
12988:
12985:
12983:
12980:
12978:
12975:
12973:
12970:
12968:
12965:
12963:
12960:
12958:
12957:Control chart
12955:
12953:
12950:
12948:
12945:
12943:
12940:
12939:
12937:
12935:
12931:
12925:
12922:
12918:
12915:
12913:
12910:
12909:
12908:
12905:
12903:
12900:
12898:
12895:
12894:
12892:
12890:
12886:
12880:
12877:
12875:
12872:
12870:
12867:
12866:
12864:
12860:
12854:
12851:
12850:
12848:
12846:
12842:
12830:
12827:
12825:
12822:
12820:
12817:
12816:
12815:
12812:
12810:
12807:
12806:
12804:
12802:
12798:
12792:
12789:
12787:
12784:
12782:
12779:
12777:
12774:
12772:
12769:
12767:
12764:
12762:
12759:
12758:
12756:
12754:
12750:
12744:
12741:
12739:
12736:
12732:
12729:
12727:
12724:
12722:
12719:
12717:
12714:
12712:
12709:
12707:
12704:
12702:
12699:
12697:
12694:
12692:
12689:
12687:
12684:
12683:
12682:
12679:
12678:
12676:
12674:
12670:
12667:
12665:
12661:
12657:
12653:
12648:
12644:
12638:
12635:
12633:
12630:
12629:
12626:
12622:
12615:
12610:
12608:
12603:
12601:
12596:
12595:
12592:
12584:
12582:0-444-86200-5
12578:
12573:
12572:
12565:
12560:
12559:
12554:
12551:
12546:
12543:
12539:
12535:
12531:
12530:
12525:
12521:
12520:
12508:
12505:
12499:
12493:
12489:
12486:
12479:
12477:
12470:
12466:
12463:
12457:
12448:
12443:
12438:
12433:
12429:
12425:
12421:
12414:
12406:
12399:
12391:
12385:
12381:
12374:
12364:
12362:
12353:
12351:0-471-02776-6
12347:
12343:
12336:
12328:
12321:
12313:
12311:0-387-40272-1
12307:
12303:
12302:
12294:
12279:
12273:
12269:
12268:
12260:
12252:
12246:
12242:
12235:
12233:
12231:
12226:
12216:
12213:
12211:
12208:
12206:
12203:
12201:
12198:
12196:
12193:
12191:
12188:
12186:
12183:
12182:
12176:
12174:
12170:
12166:
12162:
12152:
12150:
12123:
12119:
12095:
12087:
12076:
12073:
12029:
12025:
11996:
11985:
11982:
11974:
11922:
11878:
11862:
11839:
11831:
11801:
11793:
11777:
11728:
11697:
11674:
11666:
11637:
11633:
11629:
11626:
11617:
11613:
11594:
11588:
11580:
11577:
11568:
11564:
11560:
11556:
11552:
11548:
11544:
11536:
11532:
11530:
11526:
11510:
11506:
11494:
11483:
11480:
11466:
11455:
11452:
11448:
11436:
11425:
11422:
11419:
11408:
11397:
11394:
11391:
11380:
11372:
11361:
11358:
11350:
11346:
11330:
11296:
11270:
11259:
11248:
11245:
11237:
11219:
11207:
11202:
11199:
11196:
11192:
11186:
11183:
11178:
11159:
11140:
11134:
11126:
11122:
11101:
11097:
11088:
11084:
11063:
11058:
11047:
11043:
11034:
11030:
11024:
11014:
11010:
11001:
10997:
10986:
10982:
10973:
10969:
10958:
10953:
10948:
10943:
10928:
10924:
10915:
10911:
10905:
10895:
10891:
10882:
10878:
10867:
10863:
10854:
10850:
10834:
10830:
10821:
10817:
10811:
10801:
10797:
10788:
10784:
10773:
10769:
10760:
10756:
10749:
10744:
10740:
10734:
10724:
10721:
10718:
10713:
10703:
10698:
10687:
10666:
10619:
10605:
10603:
10600:are discrete
10535:
10527:
10516:
10513:
10485:
10474:
10471:
10463:
10453:
10450:
10446:
10442:
10438:
10428:
10426:
10422:
10418:
10414:
10410:
10406:
10402:
10397:
10395:
10391:
10387:
10383:
10379:
10369:
10365:
10352:
10312:
10309:
10304:
10283:
10243:
10240:
10235:
10160:
10157:
10154:
10150:
10145:
10124:
10083:
10080:
10077:
10073:
10068:
9975:
9971:
9967:
9951:
9948:
9945:
9925:
9922:
9919:
9912:of dimension
9911:
9910:data matrices
9908:are centered
9843:
9833:
9831:
9826:
9812:
9782:
9754:
9742:
9738:
9732:
9714:
9703:
9700:
9697:
9672:
9669:(also called
9668:
9655:
9623:
9591:
9587:
9586:
9582:
9581:
9580:
9578:
9575:
9559:
9555:
9525:
9497:
9485:
9481:
9475:
9464:
9453:
9450:
9447:
9422:
9418:
9415:is formed by
9384:
9354:
9350:
9346:
9343:
9340:
9335:
9331:
9324:
9310:
9291:
9269:
9249:
9245:
9230:
9226:
9222:
9219:
9205:
9201:
9197:
9194:
9187:
9183:
9177:
9171:
9165:
9162:
9154:
9138:
9130:
9129:scalar-valued
9127:
9123:
9117:
9107:
9094:
9086:
9081:
9077:
9073:
9051:
9048:
9042:
9038:
9034:
9024:
9011:
9007:
9003:
8990:
8987:
8967:
8964:
8961:
8941:
8938:
8935:
8905:
8901:
8885:
8877:
8861:
8858:
8855:
8835:
8826:
8825:is a scalar.
8798:
8792:
8773:
8748:
8745:
8742:
8730:
8704:
8698:
8679:
8661:
8655:
8653:
8644:
8640:
8614:
8608:
8583:
8577:
8558:
8553:
8549:
8543:
8540:
8536:
8509:
8503:
8478:
8472:
8460:
8456:
8442:
8427:
8425:
8421:
8404:
8385:
8382:
8365:
8337:
8334:
8311:
8308:
8305:
8266:
8264:
8260:
8241:
8238:
8235:
8225:
8212:
8209:
8206:
8194:
8117:
8092:
8089:
8086:
8054:
8053:bilinear form
8030:
8005:
8002:
7999:
7974:
7970:
7964:
7954:
7930:
7871:
7865:
7840:
7836:
7828:
7791:
7775:
7772:
7766:
7762:
7758:
7755:
7750:
7746:
7742:
7739:
7717:
7713:
7709:
7706:
7698:
7695:
7689:
7675:
7619:
7609:
7605:
7601:
7585:
7549:
7475:
7443:
7432:
7421:
7418:
7413:
7410:
7397:
7386:
7383:
7372:
7361:
7358:
7355:
7344:
7333:
7330:
7327:
7316:
7308:
7297:
7294:
7291:
7282:
7195:
7192:
7181:
7179:
7175:
7091:
7086:
7083:
7059:
7003:
6962:
6944:
6941:
6923:
6899:
6886:
6868:
6863:
6860:
6842:
6824:
6806:
6779:
6762:
6744:
6735:
6731:
6726:
6723:
6705:
6687:
6670:
6639:
6623:
6597:
6559:
6551:
6494:
6478:
6467:
6446:
6438:
6425:
6376:
6330:
6262:
6240:
6229:
6226:
6223:
6198:
6152:
6149:
6146:
6100:
6097:
6094:
6057:
5980:
5975:
5966:
5961:
5953:
5937:
5924:
5919:
5841:
5762:
5738:
5727:
5724:
5693:
5690:
5687:
5676:
5665:
5662:
5659:
5648:
5637:
5634:
5631:
5617:
5614:
5611:
5600:
5589:
5586:
5534:
5494:
5491:
5482:
5471:
5457:
5454:
5412:
5409:
5406:
5399:and constant
5364:
5361:
5358:
5350:
5319:
5287:
5249:
5233:
5223:
5215:for all
5209:
5206:
5198:
5176:
5161:
5123:
5074:
5048:
5042:
5018:
5017:
5016:
5002:
4968:
4964:
4960:
4957:
4954:
4949:
4945:
4938:
4900:
4894:
4859:
4846:
4831:
4825:
4816:
4810:
4795:
4789:
4780:
4775:
4771:
4765:
4751:
4748:
4745:
4714:
4698:
4686:
4683:
4680:
4676:
4672:
4669:
4666:
4661:
4657:
4646:
4642:
4637:
4633:
4624:
4617:
4604:
4601:
4598:
4593:
4589:
4585:
4580:
4576:
4565:
4561:
4556:
4552:
4542:
4531:
4528:
4525:
4520:
4516:
4505:
4501:
4496:
4492:
4484:
4477:
4471:
4466:
4457:
4454:
4451:
4447:
4443:
4440:
4437:
4432:
4428:
4424:
4419:
4415:
4411:
4406:
4402:
4398:
4393:
4389:
4384:
4380:
4371:
4368:
4365:
4361:
4357:
4354:
4351:
4346:
4342:
4338:
4333:
4329:
4325:
4320:
4316:
4311:
4307:
4300:
4295:
4290:
4285:
4274:
4271:
4268:
4264:
4260:
4257:
4254:
4249:
4245:
4241:
4236:
4232:
4228:
4223:
4219:
4215:
4210:
4206:
4201:
4197:
4192:
4187:
4180:
4177:
4174:
4169:
4165:
4161:
4156:
4152:
4148:
4143:
4139:
4134:
4130:
4119:
4116:
4113:
4109:
4105:
4102:
4099:
4094:
4090:
4086:
4081:
4077:
4073:
4068:
4064:
4059:
4055:
4050:
4043:
4040:
4037:
4032:
4028:
4024:
4019:
4015:
4011:
4006:
4002:
3997:
3993:
3988:
3982:
3975:
3963:
3960:
3957:
3953:
3949:
3946:
3943:
3938:
3934:
3923:
3919:
3914:
3910:
3901:
3894:
3881:
3878:
3875:
3870:
3866:
3862:
3857:
3853:
3842:
3838:
3833:
3829:
3819:
3808:
3805:
3802:
3797:
3793:
3782:
3778:
3773:
3769:
3761:
3756:
3751:
3748:
3732:
3729:
3721:
3716:
3701:
3691:
3687:
3682:
3674:
3667:
3654:
3650:
3645:
3636:
3625:
3621:
3616:
3609:
3602:
3596:
3591:
3582:
3578:
3574:
3569:
3565:
3560:
3550:
3546:
3542:
3537:
3533:
3528:
3520:
3515:
3510:
3505:
3494:
3490:
3486:
3481:
3477:
3472:
3466:
3461:
3452:
3448:
3444:
3439:
3435:
3430:
3418:
3414:
3410:
3405:
3401:
3396:
3390:
3381:
3377:
3373:
3368:
3364:
3359:
3353:
3347:
3340:
3330:
3326:
3321:
3313:
3306:
3293:
3289:
3284:
3275:
3264:
3260:
3255:
3248:
3243:
3229:
3226:
3217:
3215:
3211:
3210:
3191:
3188:
3157:
3155:
3150:
3137:
3132:
3126:
3121:
3108:
3104:
3097:
3089:
3085:
3078:
3065:
3061:
3057:
3052:
3048:
3036:
3032:
3028:
3023:
3019:
3009:
2990:
2986:
2979:
2971:
2967:
2960:
2947:
2943:
2939:
2934:
2930:
2918:
2914:
2910:
2905:
2901:
2891:
2875:
2870:
2865:
2860:
2842:
2838:
2831:
2823:
2819:
2812:
2799:
2795:
2791:
2786:
2782:
2770:
2766:
2762:
2757:
2753:
2743:
2732:
2727:
2714:
2710:
2703:
2695:
2691:
2684:
2671:
2667:
2663:
2658:
2654:
2642:
2638:
2634:
2629:
2625:
2615:
2591:
2587:
2580:
2572:
2568:
2561:
2548:
2544:
2540:
2535:
2531:
2519:
2515:
2511:
2506:
2502:
2492:
2481:
2468:
2464:
2457:
2449:
2445:
2438:
2425:
2421:
2417:
2412:
2408:
2396:
2392:
2388:
2383:
2379:
2369:
2358:
2352:
2347:
2333:
2330:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2267:
2263:
2256:
2252:
2246:
2242:
2234:
2229:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2175:
2171:
2162:
2089:
2086:
2066:
2059:
2056:
2051:
2016:
2013:
1980:
1977:
1972:
1937:
1934:
1924:
1910:
1907:
1877:
1871:
1861:
1822:
1816:
1759:
1742:
1736:
1716:
1687:
1681:
1656:
1650:
1636:
1630:
1572:
1523:
1510:
1506:
1479:
1473:
1448:
1442:
1430:
1426:
1420:
1400:
1389:
1378:
1375:
1367:
1364:
1359:
1357:
1318:
1305:
1301:
1274:
1268:
1243:
1237:
1225:
1221:
1215:
1204:
1193:
1190:
1187:
1173:
1170:
1140:
1114:
1076:
1072:
1062:
1017:
1013:
1006:
1000:
995:
991:
976:
972:
965:
959:
954:
950:
940:
934:
926:
922:
918:
913:
909:
902:
899:
896:
889:
885:
879:
875:
862:
859:entry is the
843:
840:
837:
792:
788:
784:
754:
750:
746:
743:
740:
735:
731:
727:
722:
718:
711:
699:
698:column vector
694:
678:
674:
651:
647:
588:
574:
489:
487:
483:
479:
475:
470:
456:
453:
450:
430:
410:
400:
398:
397:random vector
394:
390:
386:
382:
378:
374:
370:
366:
362:
354:
350:
334:
331:
328:
308:
288:
280:
276:
272:
267:
247:
241:
236:
229:
224:
218:
208:
203:
197:
185:
180:
178:
173:
171:
166:
165:
163:
162:
155:
152:
150:
147:
145:
142:
140:
137:
136:
133:
128:
127:
120:
117:
115:
112:
110:
107:
105:
102:
101:
98:
93:
92:
85:
82:
80:
77:
75:
72:
70:
67:
66:
63:
58:
57:
53:
49:
48:
45:
42:
41:
38:
34:
33:
27:
19:
15842:
15774:Substitution
15660:graph theory
15613:
15157:Quaternionic
15147:Persymmetric
14902:
14890:
14871:
14864:
14776:Econometrics
14726: /
14709:Chemometrics
14686:Epidemiology
14679: /
14652:Applications
14494:ARIMA model
14441:Q-statistic
14390:Stationarity
14286:Multivariate
14229: /
14225: /
14223:Multivariate
14221: /
14161: /
14157: /
13931:Bayes factor
13830:Signed rank
13742:
13716:
13708:
13696:
13391:Completeness
13227:Cohort study
13125:Opinion poll
13060:Missing data
13047:Study design
13002:Scatter plot
12924:Scatter plot
12917:Spearman's Ï
12879:Grouped data
12570:
12556:
12527:
12506:
12503:
12498:
12487:
12484:
12464:
12461:
12456:
12427:
12423:
12413:
12398:
12379:
12373:
12341:
12335:
12320:
12304:. Springer.
12300:
12293:
12281:. Retrieved
12266:
12259:
12243:. Springer.
12240:
12173:asynchronous
12158:
12148:
12121:
12117:
11972:
11920:
11876:
11778:
11573:
11566:
11562:
11558:
11554:
11550:
11546:
11538:
11124:
11120:
10606:
10461:
10459:
10434:
10398:
10375:
10372:Applications
10366:
9973:
9969:
9965:
9845:
9827:
9670:
9666:
9664:
9420:
9416:
9311:
9128:
9119:
8903:
8827:
8428:
8272:
8262:
8195:
7972:
7968:
7966:
7555:
7196:
7187:
7060:
6900:
6539:is given by
6377:
6263:
5815:
4720:
3717:
3218:
3213:
3207:
3163:
3151:
2230:
1873:
1534:
1365:
1360:
1355:
1319:
1138:
1112:
1074:
1068:
695:
594:
490:
471:
401:
384:
380:
376:
372:
368:
358:
349:eigenvectors
278:
274:
131:
96:
78:
61:
26:
15749:Hamiltonian
15673:Biadjacency
15609:Correlation
15525:Householder
15475:Commutation
15212:Vandermonde
15207:Tridiagonal
15142:Permutation
15132:Nonnegative
15117:Matrix unit
14997:Bisymmetric
14904:WikiProject
14819:Cartography
14781:Jurimetrics
14733:Reliability
14464:Time domain
14443:(LjungâBox)
14365:Time-series
14243:Categorical
14227:Time-series
14219:Categorical
14154:(Bernoulli)
13989:Correlation
13969:Correlation
13765:JarqueâBera
13737:Chi-squared
13499:M-estimator
13452:Asymptotics
13396:Sufficiency
13163:Interaction
13075:Replication
13055:Effect size
13012:Violin plot
12992:Radar chart
12972:Forest plot
12962:Correlogram
12912:Kendall's Ï
12492:open access
12469:open access
12169:synchronous
11158:sample mean
10411:and in the
9282:is denoted
8900:square root
7929:determinant
7665:as follows
7606:, then its
7191:common-mode
6901:The matrix
6636:defined by
391:giving the
353:eigenvalues
15877:Categories
15649:Transition
15644:Stochastic
15614:Covariance
15596:statistics
15575:Symplectic
15570:Similarity
15399:Unimodular
15394:Orthogonal
15379:Involutory
15374:Invertible
15369:Projection
15365:Idempotent
15307:Convergent
15202:Triangular
15152:Polynomial
15097:Hessenberg
15067:Equivalent
15062:Elementary
15042:Copositive
15032:Conference
14992:Bidiagonal
14771:Demography
14489:ARMA model
14294:Regression
13871:(Friedman)
13832:(Wilcoxon)
13770:Normality
13760:Lilliefors
13707:Student's
13583:Resampling
13457:Robustness
13445:divergence
13435:Efficiency
13373:(monotone)
13368:Likelihood
13285:Population
13118:Stratified
13070:Population
12889:Dependence
12845:Count data
12776:Percentile
12753:Dispersion
12686:Arithmetic
12621:Statistics
12437:1806.03674
12221:References
11971:and panel
9836:Estimation
9583:Properties
6961:regression
5447:, one has
1526:Properties
861:covariance
591:Definition
476:matrix is
474:covariance
393:covariance
365:statistics
37:Statistics
15830:Wronskian
15754:Irregular
15744:Gell-Mann
15693:Laplacian
15688:Incidence
15668:Adjacency
15639:Precision
15604:Centering
15510:Generator
15480:Confusion
15465:Circulant
15445:Augmented
15404:Unipotent
15384:Nilpotent
15360:Congruent
15337:Stieltjes
15312:Defective
15302:Companion
15273:Redheffer
15192:Symmetric
15187:Sylvester
15162:Signature
15092:Hermitian
15072:Frobenius
14982:Arrowhead
14962:Alternant
14152:Logistic
13919:posterior
13845:Rank sum
13593:Jackknife
13588:Bootstrap
13406:Bootstrap
13341:Parameter
13290:Statistic
13085:Statistic
12997:Run chart
12982:Pie chart
12977:Histogram
12967:Fan chart
12942:Bar chart
12824:L-moments
12711:Geometric
12558:MathWorld
12534:EMS Press
12283:10 August
12270:. Wiley.
12096:∣
12077:
11986:
11959:⟩
11942:⟨
11939:⟩
11931:⟨
11907:⟩
11887:⟨
11763:⟩
11755:⟨
11735:⟩
11718:⟨
11570:ringing).
11484:
11478:∖
11456:
11426:
11420:−
11398:
11381:∣
11362:
11328:⟩
11311:⟨
11308:⟩
11300:⟨
11297:−
11294:⟩
11274:⟨
11271:≈
11249:
11193:∑
11176:⟩
11168:⟨
11025:⋯
10959:⋮
10954:⋱
10949:⋮
10944:⋮
10906:⋯
10812:⋯
10722:…
10536:∣
10517:
10475:
10158:−
10081:−
9949:×
9923:×
9788:μ
9783:−
9760:μ
9755:−
9739:
9725:¯
9704:
9574:Hermitian
9531:μ
9526:−
9503:μ
9498:−
9482:
9454:
9344:…
9295:¯
9240:¯
9227:μ
9223:−
9202:μ
9198:−
9184:
9166:
9139:μ
9052:
8991:
8965:×
8939:×
8859:×
8799:
8793:−
8743:≥
8705:
8699:−
8662:
8615:
8609:−
8584:
8578:−
8550:
8510:
8504:−
8479:
8473:−
8457:
8386:
8338:
8309:×
8245:⟩
8242:μ
8239:−
8222:Σ
8213:μ
8210:−
8204:⟨
8175:Σ
8093:
8078:Σ
8006:
7997:Σ
7940:Σ
7909:Σ
7872:
7862:μ
7829:μ
7826:−
7812:−
7808:Σ
7792:μ
7789:−
7767:−
7759:
7740:−
7729:Σ
7707:−
7699:π
7676:
7652:Σ
7620:
7476:∣
7422:
7411:−
7387:
7362:
7356:−
7334:
7317:∣
7298:
7283:∣
7092:
7084:−
7045:Σ
6942:−
6924:
6869:
6861:−
6843:
6825:−
6750:μ
6745:−
6732:
6724:−
6706:
6676:μ
6649:μ
6578:μ
6560:∼
6552:∣
6491:then the
6472:Σ
6464:μ
6447:∼
6230:
6153:
6101:
5972:Σ
5949:μ
5933:μ
5916:μ
5850:Σ
5825:μ
5728:
5694:
5666:
5638:
5618:
5590:
5495:
5458:
5410:×
5362:×
5286:symmetric
5224:∈
5207:≥
5199:
5094:μ
5080:μ
5075:−
5049:
4958:…
4901:
4883:μ
4832:
4826:−
4796:
4790:−
4772:
4752:
4684:−
4638:σ
4618:⋱
4557:σ
4497:σ
4467:⋯
4455:−
4412:∣
4385:ρ
4381:−
4369:−
4339:∣
4312:ρ
4308:−
4301:⋮
4296:⋱
4291:⋮
4286:⋮
4272:−
4229:∣
4202:ρ
4198:−
4193:⋯
4162:∣
4135:ρ
4131:−
4117:−
4087:∣
4060:ρ
4056:−
4051:⋯
4025:∣
3998:ρ
3994:−
3961:−
3915:σ
3895:⋱
3834:σ
3774:σ
3749:−
3733:
3683:σ
3668:⋱
3646:σ
3617:σ
3592:⋯
3561:ρ
3529:ρ
3521:⋮
3516:⋱
3511:⋮
3506:⋮
3473:ρ
3467:⋯
3431:ρ
3397:ρ
3391:⋯
3360:ρ
3322:σ
3307:⋱
3285:σ
3256:σ
3230:
3189:−
3122:⋯
3098:σ
3079:σ
3062:μ
3058:−
3033:μ
3029:−
3010:
2980:σ
2961:σ
2944:μ
2940:−
2915:μ
2911:−
2892:
2876:⋮
2871:⋱
2866:⋮
2861:⋮
2832:σ
2813:σ
2796:μ
2792:−
2767:μ
2763:−
2744:
2733:⋯
2704:σ
2685:σ
2668:μ
2664:−
2639:μ
2635:−
2616:
2581:σ
2562:σ
2545:μ
2541:−
2516:μ
2512:−
2493:
2482:⋯
2458:σ
2439:σ
2422:μ
2418:−
2393:μ
2389:−
2370:
2334:
2305:…
2257:σ
2210:…
2159:(i.e., a
2090:
2052:−
2017:
1973:−
1938:
1911:
1823:
1760:
1743:
1737:−
1688:
1682:−
1657:
1651:−
1637:
1480:
1474:−
1449:
1443:−
1427:
1379:
1275:
1269:−
1244:
1238:−
1222:
1194:
1174:
1007:
1001:−
966:
960:−
941:
903:
744:…
555:Σ
486:variances
478:symmetric
454:×
332:×
15888:Matrices
15658:Used in
15594:Used in
15555:Rotation
15530:Jacobian
15490:Distance
15470:Cofactor
15455:Carleman
15435:Adjugate
15419:Weighing
15352:inverses
15348:products
15317:Definite
15248:Identity
15238:Exchange
15231:Constant
15197:Toeplitz
15082:Hadamard
15052:Diagonal
14866:Category
14559:Survival
14436:Johansen
14159:Binomial
14114:Isotonic
13701:(normal)
13346:location
13153:Blocking
13108:Sampling
12987:QâQ plot
12952:Box plot
12934:Graphics
12829:Skewness
12819:Kurtosis
12791:Variance
12721:Heronian
12716:Harmonic
12179:See also
11919:, panel
11557:. Panel
10407:and its
9122:variance
8420:variance
4926:, where
1113:variance
787:variance
15759:Overlap
15724:Density
15683:Edmonds
15560:Seifert
15520:Hessian
15485:Coxeter
15409:Unitary
15327:Hurwitz
15258:Of ones
15243:Hilbert
15177:Skyline
15122:Metzler
15112:Logical
15107:Integer
15017:Boolean
14949:classes
14892:Commons
14839:Kriging
14724:Process
14681:studies
14540:Wavelet
14373:General
13540:Plug-in
13334:L space
13113:Cluster
12814:Moments
12632:Outline
12536:, 2001
11610:is the
11545:Panels
11238:matrix
11160:, e.g.
11119:is the
9592:, i.e.
9126:complex
8928:be any
7927:is the
7180:(OLS).
7000:is the
5759:is the
5579:, then
5425:vector
5377:matrix
5288:, i.e.
5162:, i.e.
2079:where
1366:between
15678:Degree
15619:Design
15550:Random
15540:Payoff
15535:Moment
15460:Cartan
15450:BĂ©zout
15389:Normal
15263:Pascal
15253:Lehmer
15182:Sparse
15102:Hollow
15087:Hankel
15022:Cauchy
14947:Matrix
14761:Census
14351:Normal
14299:Manova
14119:Robust
13869:2-way
13861:1-way
13699:-test
13370:
12947:Biplot
12738:Median
12731:Lehmer
12673:Center
12579:
12386:
12348:
12308:
12274:
12247:
11923:shows
11879:shows
11529:Matlab
11076:where
10637:, and
8906:. Let
7853:where
6563:
6517:given
6450:
6072:where
5717:where
389:matrix
15739:Gamma
15703:Tutte
15565:Shear
15278:Shift
15268:Pauli
15217:Walsh
15127:Moore
15007:Block
14385:Trend
13914:prior
13856:anova
13745:-test
13719:-test
13711:-test
13618:Power
13563:Pivot
13356:shape
13351:scale
12801:Shape
12781:Range
12726:Heinz
12701:Cubic
12637:Index
12432:arXiv
11576:FLASH
10449:up to
9124:of a
8848:is a
8295:be a
4995:is a
383:, or
15545:Pick
15515:Gram
15283:Zero
14987:Band
14618:Test
13818:Sign
13670:Wald
12743:Mode
12681:Mean
12577:ISBN
12384:ISBN
12346:ISBN
12306:ISBN
12285:2012
12272:ISBN
12245:ISBN
12171:and
12120:and
12074:pcov
12066:and
11817:and
11549:and
11359:pcov
11349:bias
10578:and
10514:pcov
10435:The
10010:and
9972:and
9938:and
9877:and
9417:both
9120:The
7895:and
7295:pcov
7219:and
6776:and
6495:for
6422:are
6400:and
6353:and
6331:for
6295:and
6176:and
5885:and
5838:and
5785:and
4873:and
4721:For
3212:(or
2331:corr
2285:for
2190:for
2087:diag
2014:diag
1935:diag
1908:corr
1111:the
789:and
781:are
666:and
617:and
480:and
472:Any
423:and
367:, a
363:and
301:and
277:and
15634:Hat
15367:or
15350:or
13798:BIC
13793:AIC
12442:doi
12428:801
11983:cov
11481:cov
11453:cov
11423:cov
11395:cov
11246:cov
10506:or
10472:cov
10460:In
9846:If
9701:cov
9451:cov
9312:If
9163:var
9049:var
8988:var
8383:var
8335:var
8090:cov
8003:cov
7931:of
7756:exp
7578:of
7419:cov
7384:cov
7359:cov
7331:cov
7176:of
7036:in
7004:of
6378:If
6227:cov
6150:var
6098:var
5863:of
5763:of
5725:cov
5691:var
5663:cov
5635:cov
5615:var
5587:var
5535:If
5492:var
5455:var
5284:is
5158:is
4749:var
3730:cov
3227:cov
3216:).
2228:).
1606:by
1376:cov
1191:cov
1171:var
900:cov
567:or
359:In
237:0.5
230:0.5
15879::
12555:.
12532:,
12526:,
12507:47
12488:46
12475:^
12465:49
12440:.
12426:.
12422:.
12360:^
12229:^
11634:10
11531:.
10427:.
9832:.
9155::
8426:.
8265:.
7953:.
7058:.
6426:,
6261:.
6124:,
2323:.
1860:.
1163:.
587:.
547:,
399:.
379:,
375:,
205:A
15764:S
15222:Z
14939:e
14932:t
14925:v
13743:G
13717:F
13709:t
13697:Z
13416:V
13411:U
12613:e
12606:t
12599:v
12585:.
12561:.
12540:"
12450:.
12444::
12434::
12407:.
12392:.
12354:.
12329:.
12314:.
12287:.
12253:.
12149:f
12134:I
12122:e
12118:d
12104:)
12100:I
12092:Y
12088:,
12084:X
12080:(
12053:I
12030:j
12026:I
12005:)
12001:Y
11997:,
11993:X
11989:(
11973:c
11953:T
11947:Y
11935:X
11921:b
11900:T
11895:Y
11891:X
11877:a
11863:t
11843:)
11840:t
11837:(
11832:j
11827:Y
11805:)
11802:t
11799:(
11794:j
11789:X
11759:X
11732:)
11729:t
11726:(
11722:X
11698:j
11678:)
11675:t
11672:(
11667:j
11662:X
11638:4
11630:=
11627:m
11598:)
11595:t
11592:(
11589:X
11567:f
11563:e
11559:d
11555:c
11551:b
11547:a
11541:2
11511:,
11507:)
11503:)
11499:Y
11495:,
11491:I
11487:(
11475:)
11471:I
11467:,
11463:I
11459:(
11449:(
11445:)
11441:I
11437:,
11433:X
11429:(
11417:)
11413:Y
11409:,
11405:X
11401:(
11392:=
11389:)
11385:I
11377:Y
11373:,
11369:X
11365:(
11331:,
11322:T
11316:Y
11304:X
11287:T
11282:Y
11278:X
11268:)
11264:Y
11260:,
11256:X
11252:(
11220:j
11215:X
11208:n
11203:1
11200:=
11197:j
11187:n
11184:1
11179:=
11172:X
11144:)
11141:t
11138:(
11135:X
11125:j
11121:i
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11102:i
11098:t
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11064:,
11059:]
11053:)
11048:m
11044:t
11040:(
11035:n
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10916:n
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10896:2
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10888:(
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