For a Gaussian Bivariate Distribution, the distribution of correlation Coefficients is given by
where
is the population correlation Coefficient,
is a Hypergeometric Function, and
is the Gamma Function (Kenney and Keeping 1951, pp. 217-221). The Moments are
where
. If the variates are uncorrelated, then
and
so
But from the Legendre Duplication Formula,
![\begin{displaymath}
\sqrt{\pi}\,\Gamma(N-1)=2^{N-2}\Gamma({\textstyle{N\over 2}})\Gamma({\textstyle{N-1\over 2}}),
\end{displaymath}](c3_274.gif) |
(7) |
so
The uncorrelated case can be derived more simply by letting
be the true slope, so that
. Then
![\begin{displaymath}
t\equiv (b-\beta) {{\rm s}_x\over {\rm s}_y} \sqrt{N-2\over 1-r^2}
= {(b-\beta)r\over b} \sqrt{N-2\over 1-r^2}
\end{displaymath}](c3_281.gif) |
(9) |
is distributed as Student's t-Distribution with
Degrees of
Freedom. Let the population regression Coefficient
be 0, then
, so
![\begin{displaymath}
t=r\sqrt{\nu\over 1-r^2},
\end{displaymath}](c3_284.gif) |
(10) |
and the distribution is
![\begin{displaymath}
P(t)\,dt = {1\over\sqrt{\nu\pi}} {\Gamma({\textstyle{\nu+1\o...
...u\over 2}})
\left({1+{t^2\over\nu}}\right)^{(\nu+1)/2}}\,dt.
\end{displaymath}](c3_285.gif) |
(11) |
Plugging in for
and using
gives
so
![\begin{displaymath}
P(r)={1\over\sqrt{\pi}} {\Gamma\left({\nu+1\over 2}\right)\over\Gamma\left({\nu\over 2}\right)}(1-r^2)^{(\nu-2)/2}
\end{displaymath}](c3_295.gif) |
(14) |
as before. See Bevington (1969, pp. 122-123) or Pugh and Winslow (1966, §12-8). If we are interested instead in the
probability that a correlation Coefficient would be obtained
, where
is the observed Coefficient, then
Let
. For Even
, the exponent
is an Integer so, by the Binomial
Theorem,
![\begin{displaymath}
(1-r^2)^I = \sum_{k=0}^I {I\choose k} (-r^2)^k
\end{displaymath}](c3_304.gif) |
(16) |
and
For Odd
, the integral is
Let
so
, then
But
is Odd, so
is Even. Therefore
Combining with the result from the Cosine Integral gives
![\begin{displaymath}
P_c(r)=1-{2\over \pi}{(2n)!!(2n-1)!!\over (2n-1)!!(2n)!!}\le...
...er (2k+1)!!}\cos^{2k+1}x+ x}\right]_0^{\sin^{-1}\vert r\vert}.
\end{displaymath}](c3_318.gif) |
(21) |
Use
![\begin{displaymath}
\cos^{2k-1} x=(1-r^2)^{(2k-1)/2} = (1-r^2)^{(k-1/2)},
\end{displaymath}](c3_319.gif) |
(22) |
and define
, then
![\begin{displaymath}
P_c(r)= 1-{2\over \pi} \left[{\sin^{-1}\vert r\vert+\vert r\vert\sum_{k=0}^J {(2k)!!\over (2k+1)!!} (1-r^2)^{k+1/2}}\right].
\end{displaymath}](c3_321.gif) |
(23) |
(In Bevington 1969, this is given incorrectly.) Combining the correct solutions
![\begin{displaymath}
P_c(r) = \cases{
1-{2\over\sqrt{\pi}} {\Gamma[(\nu+1)/2]\ove...
...}(1-r^2)^{k+1/2}}\right]\cr
\quad {\rm for\ }\nu{\rm\ odd}\cr}
\end{displaymath}](c3_322.gif) |
(24) |
If
, a skew distribution is obtained, but the variable
defined by
![\begin{displaymath}
z\equiv \tanh^{-1} r
\end{displaymath}](c3_325.gif) |
(25) |
is approximately normal with
(Kenney and Keeping 1962, p. 266).
Let
be the slope of a best-fit line, then the multiple correlation Coefficient is
![\begin{displaymath}
R^2\equiv \sum_{j=1}^n\left({b_j {{s_{jy}}^2\over {s_y}^2}}\right)= \sum_{j=1}^n \left({b_j {s_j\over s_y} r_{jy}}\right),
\end{displaymath}](c3_331.gif) |
(28) |
where
is the sample Variance.
On the surface of a Sphere,
![\begin{displaymath}
r\equiv {\int fg\,d\Omega\over \int f\,d\Omega \int g\,d\Omega},
\end{displaymath}](c3_333.gif) |
(29) |
where
is a differential Solid Angle.
This definition guarantees that
. If
and
are expanded in Real Spherical Harmonics,
Then
![\begin{displaymath}
r_l ={\sum_{m=0}^l (C_l^mA_l^m+S_l^mB_l^m)\over \sqrt{\sum_{...
...^m}^2+{S_l^m}^2)}
\sqrt{\sum_{m=0}^l ({A_l^m}^2+{B_l^m}^2)}}.
\end{displaymath}](c3_340.gif) |
(32) |
The confidence levels are then given by
where
![\begin{displaymath}
s\equiv \sqrt{1-r^2}
\end{displaymath}](c3_350.gif) |
(33) |
(Eckhardt 1984).
See also Fisher's z'-Transformation, Spearman Rank Correlation Coefficient,
Spherical Harmonic
References
Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969.
Eckhardt, D. H. ``Correlations Between Global Features of Terrestrial Fields.'' Math. Geology 16,
155-171, 1984.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Pugh, E. M. and Winslow, G. H. The Analysis of Physical Measurements. Reading, MA: Addison-Wesley, 1966.
© 1996-9 Eric W. Weisstein
1999-05-25