# min ChiSquare to fit dipole axis

I am trying to fit a dipole. As I am not sure about some aspects (and not sure whether I understand it at all) I would like to ask this question here.

Basically I have a distribution of $n$ sample points on a sphere. Each point is defined by two coordinate $\theta$ (latitude) and $\phi$ (longitude). Each of the point can take a value $\mu$ between -1 and 1. I have a set of $k$ dipole axises that I test. ChiSqure is defined as

$$\chi^2 = \sum\limits_{i=0}^n \frac{\left(o_i - e_i \right)^2}{\sigma_i} = \sum\limits_{i=0}^n \frac{\left(\mu_i - a \cos \gamma_i\right)^2}{\sigma_i}$$

Where $\gamma_i$ is the angle between two points given by

$$\gamma =\arccos\left(\sin(\phi_A) \cdot \sin(\phi_B) + \cos(\phi_A) \cdot \cos(\phi_B) \cdot \cos(\theta_B - \theta_A) \right)$$

Now for every possible axis dipole axis I calculate the $\chi^2$ while variating $a$ to find the minimal chiSquare $\chi^2_{min}$ for the each candidate axis. At the end my best fit dipole axis should be the one with the smallest $\chi^2_{min}$. The significance of the result would then be calculated with making statistics with the best fit $a_j$'s of the candidate axises (with $a_j$ being the value $a$ yielding the minimal $\chi^2$ for the $j$ candidate axis). So I if my best fit axis would be $l$ 'th candidate axis I would look at the position of $a_l$ in the histogram.

Now my questions:

1. Is this approach correct at all?

2. How is this minimal minimal ChiSquare value labeled? ${\chi^2_{min}}_{min}$ would be weird.

3. How do I know $\sigma_i$. If I assume the same uncertainty for all samples. Can I just set $\sigma_i=1$?

4. What is $dof$ in this case? Simply $n-1$?

• The $\chi^2$ statistic is not appropriate for these data: that formula applies to counts $o_i$. By "fit a dipole" are you trying to find a mean axis for a set of directions or are you actually assuming the distribution on the sphere is given by a spherical harmonic function? – whuber Mar 26 '15 at 15:00
• I try to find a mean axis. There are only data points for a part of the sphere. I am not assuming a spherical harmonic function (though it would maybe also be interesting to assume a spherical harmonics). – Sjoerd222888 Mar 26 '15 at 17:51