Appendix A: Regularity periodograms for cubes

Regularity periodograms for constant density cubes can be found analytically, counting regions of different density generated by partially filled period cells at the boundaries. This is effectively a description of edge effects.

We found these expressions useful when debugging our code. Moreover, we discovered that our original estimator was biased, by comparing the results of our program with the analytic formula.

As this method can be used in a similar way also for 2-D and 1-D cubes (squares and line segments), we give formulae for these cases, too.

We shall use two auxiliary functions below:

$$\begin{eqnarray*}m(x)=\lfloor 1/x\rfloor, \end{eqnarray*}$$

where the floor function $\lfloor\cdot\rfloor$ is defined as the largest integer smaller than or equal to its argument, and v(x)=1-xm(x).

We use the normalized period y=d/L, where d is the test period and L is the size of the cube.

Figure A.1: Periodograms for uniform density cubes (using the full estimator), for 1D, 2D and 2D cases. The full period range is shown.

Figure A.2: Periodograms for uniform density cubes (using the estimator that discards local correlations), for 1D, 2D and 2D cases.

The formulae:

3-D, all points:

$$\begin{eqnarray*}\kappa_{3D}(y)&=&y^3\big\{[y-v(y)]^3 m^6(y)\\ &&+{}3v(y)[y-v(... ...t]^2\\ &&+{}v^3(y)\left[m^3(y)+3m^2(y)+3m(y)+1\right]^2\big\}; \end{eqnarray*}$$

3-D, no local correlations:

$$\begin{eqnarray*}\kappa'_{3D}(y)&=&y^3\big\{[y-v(y)]^3 m^3(y)\left[m^3(y)-1\righ... ...+1\right]\\ &&\times{}\left[m^3(y)+3m^2(y)+3m(y)\right]\big\}; \end{eqnarray*}$$

[2]

2-D, all points:

$$\begin{eqnarray*}\kappa_{2D}(y)&=&y^2\big\{[y-v(y)]^2 m^4(y)\\ &&+{}2v(y)[y-v(... ...y)\right]^2\\ &&+{}2v^2(y)\left[m^2(y)+2m(y)+1\right]^2\big\}; \end{eqnarray*}$$

2-D, no local correlations:

$$\begin{eqnarray*}\kappa'_{2D}(y)&=&y^2\big\{[y-v(y)]^2 m^2(y)\left[m^2(y)-1\righ... ...y)+2m(y)+1\right]\\ &&\times{}\left[m^2(y)+2m(y)\right]\big\}; \end{eqnarray*}$$

1-D, all points:

$$\begin{eqnarray*}\kappa_{1D}(y)&=&y\{[y-v(y)]m^2(y)\\ &&+{}v(y)[m(y)+1]^2\}; \end{eqnarray*}$$

1-D, no local correlations:

$$\begin{eqnarray*}\kappa'_{1D}(y)&=&y\{[y-v(y)]m(y)[m(y)-1]\\ &&+{}v(y)[m(y)+1]m(y)\}. \end{eqnarray*}$$

We show all these periodograms in Figs. A.1 and A.2, which show the behaviour of these functions over the entire range of periods. In applications, of course, the periods d>L/2 do not make much sense.

The amplitudes of both estimators (edge effects) grow with dimension. Nevertheless the second estimator that discards local correlations, also reduces the edge effects considerably.