Online material

Appendix A: Statistics in Thiele-Innes space

In Appendix A of L14, formulae are derived for , the least squares Thiele-Innes constants at given φ = (log P,e,τ). The resulting value of χ² then determines the profile likelihood ℒ^† used in the approximation of posterior means – Eq. (7) of L14. Now we wish to sample points displaced from . Let such a displacement be δψ = (a,b,f,g).

Appendix A.1: Probability density function p(ψ | φ, D)

In Sect. A.4 of L14, the PDF at δψ is shown to be the product of two independent PDF’s, each a bivariate normal distribution, one for (a,f), the other for (b,g). Formulae for the variances and the covariances cov(a,f),cov(b,g) that define these PDF’s are given in Eqs. (A.9) and (A.10) of L14.

The PDF for (a,f) is (A.1)where ρ_af = cov(a,f)/(σ_aσ_f) and (A.2)The point (0,0) corresponds to the minimum- solution , where is the x-coordinate contribution to χ² – see Eq. (2). The displacement (a,f) therefore results in a positive increment given by (A.3)Now, for displacement (a,f), the predicted (A.4)where X,Y are given by Eqs. (A.3) in L14. Substitution in Eq. (A.3) then gives (A.5)where terms linear in a and f vanish because the minimum is at (0,0). The summations in Eq. (A.5) can be eliminated in favour of and cov(a,f) with the formulae given in Eqs. (A.6), (A.9) and (A.10) of L14. After lengthy algebra, we find that (A.6)and so (A.7)Exactly the same analysis applies to the independent pair (b,g), so that (A.8)Combining these formulae, we find that the PDF at is (A.9)where and η(φ | D) is given by (A.10)

Appendix A.2: Modified Thiele-Innes constants

The familiar device of “completing the square” applied to Eq. (A.2) suggests new variables defined by (A.11)Substitution in Eq. (A.6) then gives (A.12)The Jacobian of this transformation is (A.13)so that, by conservation of probability, the PDF of is (A.14)Now, exactly the same analysis applies to the independent pair (b,g). Thus, if we define new variables by the equations (A.15)then the PDF of is , and (A.16)If ζ denotes the vector , then Π(ζ), the PDF in ζ-space, is – i.e., (A.17)Accordingly, the distribution of probability in ζ-space is simply the product of four independent normal distributions, each with zero mean and unit variance.

The increment in χ² is given by Eqs. (A.12) and (A.16) as (A.18)

Appendix A.3: Approximate PDFs

A distribution of probability can be represented by a sum of δ functions in such a way that the probability attached to any finite element of space is approximated with arbitrary accuracy. Thus, the PDF giving the distribution of probability in ζ-space is given approximately by (A.19)where each ζ_ℓ is an independent random vector sampling the PDF Π(ζ) given by Eq. (A.17). The integral of p(ζ) over a finite element in ζ-space converges to the exact value as .

Equations (A.11) and (A.15) transform the point ζ_ℓ into the displacement . The corresponding approximate PDF in ψ-space is therefore (A.20)Note that the Jacobian of this transformation from ζ- to ψ- space is implicit in the changes in number densities of the delta functions in the respective spaces.

Similarly, the PDF for the Campbell elements ϑ corresponding to the Thiele-Innes elements ψ is (A.21)where ϑ_ℓ = ϑ(ψ_ℓ) is derived as described in Sect. A.4 of L14.

Appendix A.4: Random sampling in ψ-space

According to Eq. (A.17), a random point in ζ-space is (z₁,z₂,z₃,z₄), where the z_i are independent random Gaussian variates drawn from . This point corresponds to the displacement (a,b,f,g) given by Eqs. (A.11) and (A.15) and therefore to the point in ψ-space. Thus, a point randomly selected from the exact PDF p(ψ | φ,D) can be derived from four independent Gaussian variates, and the resulting increment in χ² is given by Eq. (A.18) as (A.22)

Appendix A.5: Random sampling at fixed δχ²

A random point in ψ-space subject to a constraint on δχ² can be found by first selecting a random point on the 4D sphere in ζ-space defined by Eq. (A.18). This is achieved as follows: If z_i again denotes a Gaussian variate from , then a random point on this hypersphere is (z₁,z₂,z₃,z₄) /Z, where (A.23)(Muller 1979). The corresponding point (A,B,F,G) in ψ-space is then derived from Eqs. (A.11) and (A.15).

The random sampling procedures of Sects. A.4 and A.5 predict χ² without the need to compute an orbit. This is achieved by exploiting the linearity of the ψ-elements and is the basis of the computational efficiency of the techniques of Sects. 3.3 and 4.1. However, during code development, this prediction should be tested by actually computing the orbit and independently evaluating χ² from Eq. (2).

© ESO, 2014