Correlation Coefficient--Gaussian Bivariate Distribution

For a Gaussian Bivariate Distribution, the distribution of correlation Coefficients is given by

$$P(r) = {1\over \pi} (N-2)(1-r^2)^{(N-4)/2}(1-\rho^2)^{(N-1)/2}\int_0^\infty {d\beta\over (\cosh\beta -\rho r)^{N-1}}$$

$$= {1\over \pi} (N-2)(1-r^2)^{(N-4)/2}(1-\rho^2)^{(N-1)/2} \sqrt{\pi\over 2}\, {\Gamma(N-1)\over \Gamma(N-{\textstyle{1\over 2}})}$$

$$\phantom{=} \mathop{\times} (1-\rho r)^{-(N-3/2)}{}_2F_1({\textstyle{1\over 2... ...textstyle{1\over 2}}, {\textstyle{2N-1\over 2}}; {\textstyle{\rho r+1\over 2}})$$

$$= {(N-2)\Gamma(N-1)(1-\rho^2)^{(N-1)/2}(1-r^2)^{(N-4)/2}\over\sqrt{2\pi}\,\Gamma(N-{\textstyle{1\over 2}})(1-\rho r)^{N-3/2}}$$

$$\phantom{=} \mathop{\times} \left[{1+{1\over 4} {\rho r+1\over 2N-1} +{9\over 16} {(\rho r+1)^2\over (2N-1)(2N+1)} +\cdots}\right],$$ (1)

where $\rho$ is the population correlation Coefficient, ${}_2F_1(a,b;c;x)$ is a Hypergeometric Function, and $\Gamma(z)$ is the Gamma Function (Kenney and Keeping 1951, pp. 217-221). The Moments are

$$\left\langle{r}\right\rangle{} = \rho-{\rho(1-\rho^2)\over 2n}$$ (2)

$$\mathop{\rm var}\nolimits (r) = {(1-\rho^2)^2\over n} \left({1+{11\rho^2\over 2n}+\cdots}\right)$$ (3)

$$\gamma_1 = {6\rho\over \sqrt{n}}\left({1+{77\rho^2-30\over 12 n}+\cdots}\right)$$

$$\gamma_2 = {6\over n}(12\rho^2-1)+\ldots,$$ (4)

where $n\equiv N-1$. If the variates are uncorrelated, then $\rho=0$ and

$${}_2F_1({\textstyle{1\over 2}}, {\textstyle{1\over 2}}, {\textstyle{2N-1\over 2}}; {\textstyle{\rho r+1\over 2}}) = {}_2F_1({\textstyle{1\over 2}}, {\textstyle{1\over 2}}, {\textstyle{2N-1\over 2}}; {\textstyle{1\over 2}})$$

$$= {\Gamma(N-{\textstyle{1\over 2}})2^{3/2-N}\sqrt{\pi}\over [\Gamma({\textstyle{N\over 2}})]^2},$$ (5)

so

$$P(r) = {(N-2)\Gamma(N-1)\over \sqrt{2\pi}\,\Gamma(N-{\textstyle{1\over 2... ...tstyle{1\over 2}}) 2^{3/2-N}\sqrt{\pi}\over [\Gamma({\textstyle{N\over 2}})]^2}$$

$$= {2^{1-N}(N-2)\Gamma(N-1)\over [\Gamma({\textstyle{N\over 2}})]^2} (1-r^2)^{(N-4/2)}.$$ (6)

But from the Legendre Duplication Formula,

$$\sqrt{\pi}\,\Gamma(N-1)=2^{N-2}\Gamma({\textstyle{N\over 2}})\Gamma({\textstyle{N-1\over 2}}),$$ (7)

so

$$P(r) = {(2^{1-N})(2^{N-2})(N-2)\Gamma({\textstyle{N\over 2}})\Gamma({\te... ...over 2}})\over\sqrt{\pi}\,[\Gamma({\textstyle{N\over 2}})]^2} (1-r^2)^{(N-4)/2}$$

$$= {(N-2)\Gamma({\textstyle{N-1\over 2}})\over 2\sqrt{\pi}\,\Gamma({\textstyle{N\over 2}})}(1-r^2)^{(N-4)/2}$$

$$= {1\over\sqrt{\pi}} {{\nu\over 2} \Gamma({\textstyle{\nu+1\over 2}})\over\Gamma({\textstyle{\nu\over 2}}+1)} (1-r^2)^{(\nu-2)/2}$$

$$= {1\over\sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over \Gamma({\textstyle{\nu\over 2}})}(1-r^2)^{(\nu-2)/2}.$$ (8)

The uncorrelated case can be derived more simply by letting $\beta$ be the true slope, so that $\eta=\alpha+\beta x$. Then

$$t\equiv (b-\beta) {{\rm s}_x\over {\rm s}_y} \sqrt{N-2\over 1-r^2} = {(b-\beta)r\over b} \sqrt{N-2\over 1-r^2}$$ (9)

is distributed as Student's t-Distribution with $\nu\equiv N-2$ Degrees of Freedom. Let the population regression Coefficient $\rho$ be 0, then $\beta=0$, so

$$t=r\sqrt{\nu\over 1-r^2},$$ (10)

and the distribution is

$$P(t)\,dt = {1\over\sqrt{\nu\pi}} {\Gamma({\textstyle{\nu+1\o... ...u\over 2}}) \left({1+{t^2\over\nu}}\right)^{(\nu+1)/2}}\,dt.$$ (11)

Plugging in for $t$ and using

$$dt = \sqrt{\nu} \left[{\sqrt{1-r^2}-r({\textstyle{1\over 2}})(-2r)(1-r^2)^{-1/2}\over 1-r^2}\right]\,dr$$

$$= \sqrt{\nu\over 1-r^2} \left({1-r^2+r^2\over 1-r^2}\right)\,dr$$

$$= \sqrt{\nu\over (1-r)^3}\,dr$$ (12)

gives

$$P(t) \,dt = {1\over \sqrt{\nu\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over \... ...left[{1+{r^2\nu\over(1-r^2)\nu}}\right]^{(\nu+1)/2}}\sqrt{\nu\over (1-r)^3}\,dr$$

$$= {(1-r^2)^{-3/2}\over\sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}... ...r \Gamma({\textstyle{\nu\over 2}}) \left({1\over 1-r^2}\right)^{(\nu+1)/2}}\,dr$$

$$= {1\over\sqrt{\pi}}{\Gamma({\textstyle{\nu+1\over 2}})\over \Gamma({\textstyle{\nu\over 2}})}(1-r^2)^{-3/2}(1-r^2)^{(\nu+1)/2}\,dr$$

$$= {1\over\sqrt{\pi}}{\Gamma({\textstyle{\nu+1\over 2}})\over\Gamma({\textstyle{\nu\over 2}})}(1-r^2)^{(\nu-2)/2}\,dr,$$ (13)

so

$$P(r)={1\over\sqrt{\pi}} {\Gamma\left({\nu+1\over 2}\right)\over\Gamma\left({\nu\over 2}\right)}(1-r^2)^{(\nu-2)/2}$$ (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 $\geq \vert r\vert$, where $r$ is the observed Coefficient, then

$$P_c(r,N) = 2\int_{\vert r\vert}^1 P(r',N)\,dr' = 1-2\int_0^{\vert r\vert} P(r',N)\,dr'$$

$$= 1-{2\over\sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over\Gamma({\textstyle{\nu\over 2}})}\int_0^{\vert r\vert}(1-r^2)^{(\nu-2)/2}\,dr.$$ (15)

Let $I\equiv {1\over 2}(\nu-2)$. For Even $\nu$, the exponent $I$ is an Integer so, by the Binomial Theorem,

$$(1-r^2)^I = \sum_{k=0}^I {I\choose k} (-r^2)^k$$ (16)

and

$$P_c(r) = 1-{2\over\sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over \Ga... ... 2}})} (-1)^k {I!\over(I-k)!k!} \int_0^{\vert r\vert} \sum_{k=0}^I r'^{2k}\,dr'$$

$$= 1-{2\over\sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over \Ga... ...k=0}^I \left[{(-1)^k{I!\over (I-k)!k!} {\vert r\vert^{2k+1}\over 2k+1}}\right].$$ (17)

For Odd $\nu$, the integral is

$$P_c(r) = 1-2\int_0^{\vert r\vert} P(r')\,dr'$$

$$= 1-{2\over\sqrt{\pi}}{\Gamma({\textstyle{\nu+1\over 2}})\over\Gamma({\textstyle{\nu\over 2}})}\int_0^{\vert r\vert}(\sqrt{1-r^2}\,)^{\nu-2}\,dr.$$ (18)

Let $r\equiv\sin x$ so $dr=\cos x \,dx$, then

$$P_c(r) = 1-{2\over\sqrt{\pi}} {\Gamma[({\textstyle{\nu+1\over 2}})]\over \... ...tstyle{\nu\over 2}})} \int_0^{\sin^{-1} \vert r\vert} \cos^{\nu-2}x \cos x \,dx$$

$$= 1-{2\over\sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over \Gamma({\textstyle{\nu\over 2}})}+ \int_0^{\sin^{-1} \vert r\vert} \cos^{\nu-1}x\,dx.$$ (19)

But $\nu$ is Odd, so $\nu-1\equiv 2n$ is Even. Therefore

$${2\over \sqrt{\pi}} {\Gamma({\textstyle{\nu+1\over 2}})\over \Gamma({\textstyle{\nu\over 2}})} = {2\over \sqrt{\pi}} {\Gamma(n+1)\over \Gamma(n+{\textstyle{1\over 2}})} = {2\over \sqrt{\pi}} {n!\over {(2n-1)!!\sqrt{\pi}\over 2^n}}$$

$$= {2\over \pi} {2^n n!\over (2n-1)!!} = {2\over \pi}{(2n)!!\over (2n-1)!!}.$$ (20)

Combining with the result from the Cosine Integral gives

$$P_c(r)=1-{2\over \pi}{(2n)!!(2n-1)!!\over (2n-1)!!(2n)!!}\le... ...er (2k+1)!!}\cos^{2k+1}x+ x}\right]_0^{\sin^{-1}\vert r\vert}.$$ (21)

Use

$$\cos^{2k-1} x=(1-r^2)^{(2k-1)/2} = (1-r^2)^{(k-1/2)},$$ (22)

and define $J\equiv n-1=(\nu-3)/2$, then

$$P_c(r)= 1-{2\over \pi} \left[{\sin^{-1}\vert r\vert+\vert r\vert\sum_{k=0}^J {(2k)!!\over (2k+1)!!} (1-r^2)^{k+1/2}}\right].$$ (23)

(In Bevington 1969, this is given incorrectly.) Combining the correct solutions

$$P_c(r) = \cases{ 1-{2\over\sqrt{\pi}} {\Gamma[(\nu+1)/2]\ove... ...}(1-r^2)^{k+1/2}}\right]\cr \quad {\rm for\ }\nu{\rm\ odd}\cr}$$ (24)

If $\rho\not=0$, a skew distribution is obtained, but the variable $z$ defined by

$$z\equiv \tanh^{-1} r$$ (25)

is approximately normal with

$$\mu_z = \tanh^{-1}\rho$$ (26)

$${\sigma_z}^2 = {1\over N-3}$$ (27)

(Kenney and Keeping 1962, p. 266).

Let $b_j$ be the slope of a best-fit line, then the multiple correlation Coefficient is

$$R^2\equiv \sum_{j=1}^n\left({b_j {{s_{jy}}^2\over {s_y}^2}}\right)= \sum_{j=1}^n \left({b_j {s_j\over s_y} r_{jy}}\right),$$ (28)

where $s_{jy}$ is the sample Variance.

On the surface of a Sphere,

$$r\equiv {\int fg\,d\Omega\over \int f\,d\Omega \int g\,d\Omega},$$ (29)

where $d\Omega$ is a differential Solid Angle. This definition guarantees that $-1<r<1$. If $f$ and $g$ are expanded in Real Spherical Harmonics,

$$f(\theta,\phi) \equiv \sum_{l=0}^\infty \sum_{m=0}^l [C_l^m {Y_l^m}^c(\theta,\phi)\sin(m\phi)+S_l^m {Y_l^m}^s(\theta, \phi)]$$ (30)

$$g(\theta,\phi) \equiv \sum_{l=0}^\infty \sum_{m=0}^l [A_l^m {Y_l^m}^c(\theta,\phi)\sin(m\phi)+B_l^m {Y_l^m}^s(\theta, \phi)].$$ (31)

Then

$$r_l ={\sum_{m=0}^l (C_l^mA_l^m+S_l^mB_l^m)\over \sqrt{\sum_{... ...^m}^2+{S_l^m}^2)} \sqrt{\sum_{m=0}^l ({A_l^m}^2+{B_l^m}^2)}}.$$ (32)

The confidence levels are then given by

$$G_1(r) = r$$

$$G_2(r) = r(1+{\textstyle{1\over 2}}s^2)={\textstyle{1\over 2}}r(3-r^2)$$

$$G_3(r) = r[1+{\textstyle{1\over 2}}s^2(1+{\textstyle{3\over 4}} s^2)]={\textstyle{1\over 8}} r(15-10r^2+3r^4)$$

$$G_4(r) = r\{1+{\textstyle{1\over 2}}s^2[1+{\textstyle{3\over 4}}s^2(1+{\textstyle{5\over 6}}s^2)]\}$$

$$= {\textstyle{1\over 16}}r(35-35r^2+21r^4-5r^6),$$

where

$$s\equiv \sqrt{1-r^2}$$ (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.