The Variance and Covariance of Counts-in-Cells Probabilities

We now turn to the problem of determining the error on the measured CIC probability in a way that accounts for correlations between neighboring cells. Given a bin $B$ of counts, and writing $\mathcal{S}$ for $\mathcal{S}_{B}$, we see from Equation 2 that the probability of that bin $\mathcal{P}(B)=\langle\mathcal{S}\rangle$. Furthermore, the variance of $\mathcal{S}$ is $\sigma^{2}_{\mathcal{S}}=\langle\mathcal{S}^{2}\rangle-\langle\mathcal{S}\rangle^{2}=\mathcal{P}(B)\left(1-\mathcal{P}(B)\right)$.

If the survey cells are uncorrelated, it follows that the variance of $\mathcal{P}(B)$ is given by

(Note that this expression appears in Colombi et al. (1995) for $B=\{0\}$ with no correlation.) Equation 4 treats each cell as an independent measurement of the probability; correlations between cells will decrease the effective number of independent measurements, requiring modification of the expression.

Thus, to handle the case of correlated cells, we first consider the artificial case of an $\mathcal{S}$-field consisting of $n$ survey cells such that the correlation between any two of them is a fixed value $\xi$. Explicitly, if $i\neq j$, we have $\langle s_{i}s_{j}\rangle=(1+\xi)P^{2}$, where $P=\mathcal{P}(B)$ is the probability that $\mathcal{S}=1$ in any given cell. (Note that $\xi$ is the correlation of $\mathcal{S}$, not of the counts-in-cells $N$.) If $s_{i}$ is the value of the $\mathcal{S}$ in the $i$th cell, we know that

In Appendix A.1 we show that the variance $\sigma^{2}_{\mathcal{P}(B)}$ of this quantity is given by

However, this expression presupposes an equal degree of correlation $\xi$ between all $n$ cells. Obtaining a similar expression for the general case of varying $\xi$ is analytically intractable. However, we note that, in the inductive proof of Equation 6 (Appendix A.1), the term through which new instances of $\xi$ accumulate into the expression is $\langle s_{i}s_{n+1}\rangle$ (in Equation 22). This term has precisely the form expected for a Monte Carlo volume average of a spatially-varying $\xi(r)$. With this motivation, we approximate $\xi$ in Equation 6 with the average slice-field correlation $\overline{\xi}_{\mathcal{S}}$, where the average is taken over all pairs $(\mathbf{r}_{i},\mathbf{r}_{j})$ of survey cells ($i\neq j$). (In Section 3.3 we verify the validity of this approximation numerically.) Performing this substitution and writing $N_{c}$ (the number of cells) for $n$, we obtain for any counts-in-cells bin $B$,

which reduces to Equation 4 in the uncorrelated case.

In order to test Equation 7, we need multiple measurements of the probability $\mathcal{P}(B)$. It is possible to obtain such measurements from a single simulation by dividing it into multiple subvolumes, calculating $\mathcal{P}(B)$ in each subvolume, and then observing the variance of those measured $\mathcal{P}(B)$-values. However, a meaningful comparison between measurement and theory requires an estimate of the possible scatter in these observations of the variance. To this we now turn.

Suppose we have a set $\left\{P_{i}\right\}$ of $N_{m}$ measurements of a probability $\mathcal{P}(B)$. Let us denote the measured variance of this set as $s_{P}^{2}$. Then according to a standard result (e.g., Kendall and Stuart 1958, eq. 12.35, converting cumulants to moments),

where $\mu_{j}$ denotes the $j$th central moment of the distribution of $P_{i}$-values. We can use the measured variance $s_{P}^{2}$ for $\mu_{2}$; however, we must perform an additional estimation of $\mu_{4}$. For simplicity, in calculating $\mu_{4}$ we will ignore the correlation between neighboring cells; we shall however mitigate the effect of this simplification by expressing our results in terms of $s_{p}^{2}$, which do include the effects of correlation.

Proceding to determine $\mu_{4}$, and employing $P$ as shorthand for $\mathcal{P}(B)$, it is straightforward to obtain the moment-generating function for the slice-field $\mathcal{S}$ corresponding to bin $B$:

Now, if each measurement $P_{i}$ was obtained by averaging the $\mathcal{S}$-values in $N_{c}$ cells (see Equation 5), then the moment-generating function for the probability is

From this function (see Appendix A.2 for details) we obtain the following expression for $\mu_{4}$:

inserting this result into Equation 8 and simplifying, we obtain

where $N_{c}$ is the number of survey cells in each measurement of $s_{P}^{2}$, and $N_{m}$ is the total number of measurements. Equation 12 thus gives us the uncertainty on the measured value of $s_{p}^{2}$.

Note also that each probability measurement $P_{i}$ is a mean of $N_{c}$ values of the $\mathcal{S}$-field. Thus for reasonably large values of $N_{c}$, we can invoke the central limit theorem and treat the distribution of $P_{i}$-values as Gaussian, in which case $\mathrm{Var}\left(s_{p}^{2}\right)=2\left(s_{p}^{2}\right)^{2}\!/(N_{m}-1)$. This result is, of course, the limit of Equation 12 for large $N_{c}$.

We can now compare the predictions of Equation 7 with measured variances. To do so, we make use of the L-galaxies catalog¹¹1From the repository at http://gavo.mpa-garching.mpg.de/
Millennium/ (Bertone et al. 2007) from the Millennium Simulation (Springel et al. 2005), imposing a stellar mass cut of $M_{\star}\geq 10^{9}\mathrm{M}_{\odot}$ to obtain a mock galaxy survey. We perform two tests, one with the survey volume divided into $256^{3}$ cubical cells (with side length $1.95h^{-1}$Mpc) and a second with the volume divided into $16^{3}$ cubical cells (with side length $31.25h^{-1}$Mpc).

We next split the survey into $N_{m}$ subvolumes, each consisting of $N_{c}$ survey cells (so that $N_{m}N_{c}=N_{\mathrm{tot}}$, the total number of cells in the survey). Given a CIC bin $B$, we determine the probability $\mathcal{P}(B)$ within each subvolume, thus obtaining an ensemble of $N_{m}$ measured probabilities. The variance of this ensemble gives us a measured value for $\sigma^{2}_{\mathcal{P}}(B)$, and this value will depend on the number of cells $N_{c}$ used in the measurement of each probability. These empirical variances appear as thick curves in Fig. 1. In this figure, the bottom axis shows $N_{c}$, the number of cells in each subvolume, and the top axis shows $N_{m}$, the number of subvolumes. (Note also that we choose unit bin widths for the $1.95h^{-1}$-Mpc cells and varying bin widths for the $31.25h^{-1}$-Mpc cells.) Equation 12 gives us the error bars on these measurements, where we use for $s^{2}_{p}$ and $P$ the measured variances and probabilities.

The thin curves in Fig. 1 show the predictions of Equation 7. This prediction is not entirely a priori, since it requires the (measured) probability values $\mathcal{P}(B)$ and the measured volume-averaged correlation $\overline{\xi}_{\mathcal{S}}$ of the corresponding slice field. To obtain the latter, we first measure the two-point correlation function $\xi_{\mathcal{S}}(r)$ of the appropriate slice field using a standard fast Fourier transform method; we then use Monte Carlo sampling of the slice field to obtain random pairs, using $\xi(r)$ to calculate their correlation and folding the result into the average; we continue the sampling process until the variation in the running average has subpercent effect on the predicted $\sigma^{2}_{\mathcal{P}}(B)$.

The endpoints of the curves illustrate two extremes. The left-hand endpoints represent the situation in which each survey cell constitutes its own subvolume ($N_{c}=1$, and $N_{m}=256^{3}$ or $16^{3}$). In this case, each cell provides an estimate of $\mathcal{P}(B)$, and these estimates are either 0 or 1 (depending on whether the cell falls into that probability bin). In this case we expect the variance of these estimates to be large.

The right-hand endpoint of each curve represents the opposite situation of high $N_{c}$ and low $N_{m}$. At this endpoint, the survey is divided into 8 subvolumes ($N_{c}=256^{3}\!/8$ or $16^{3}\!/8$, and in both cases $N_{m}=8$). Here we have 8 measurements of $\mathcal{P}(B)$, and the variance of these measurements is small due to the large number of cells $N_{c}$ involved in the calculation of each one. On the other hand, since we have only 8 measurements of $\mathcal{P}(B)$, our estimate $s^{2}$ of the variance is less certain.

In both cases ($1.95h^{-1}$- and $31.25h^{-1}$-Mpc cells), we see from this figure that the predicted variance is in excellent agreement with the measured values. Furthermore, we see the expected trends: as the number of cells $N_{c}$ used for the measurement of $\mathcal{P}(B)$ increases, the variance in those measurements decreases; however, as the number of measurements of $\mathcal{P}(B)$ decreases, the uncertainty in the variance increases.

The top panels of Fig. 1 also show, as thin dotted lines, the variance predicted under the assumption of no correlation between survey cells (Equation 4), which is proportional to $1/N_{c}$; the bottom panels show the ratio between the correlated and uncorrelated cases. It is evident that inter-cell correlations can have a significant effect on the variance of $\mathcal{P}(B)$; in the case of $\mathcal{P}(0)$ for the $1.95h^{-1}$-Mpc cells, the difference is more than two orders of magnitude.

The second issue one must consider in fitting models to CIC results is the covariance between different bins of counts. (It is clear that such covariance must exist, given that a survey cell falling into one bin is thereby excluded from all other bins.) Thus we here derive an expression for the covariance $\sigma_{\mathcal{P}(B_{1})\mathcal{P}(B_{2})}$ of counts-in-cells in two (disjoint) bins $B_{1}$ and $B_{2}$. To simplify notation, we write $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ for the slice fields of the two bins, and $P_{1}$, $P_{2}$ for $\mathcal{P}(B_{1})$, $\mathcal{P}(B_{2})$.

We begin by considering the case of two slice fields, both drawn from the same survey consisting of $n$ cells, and, as before, we initially assume that the cross-correlation between the two slice fields is a fixed, constant value $\xi_{12}$. In particular, for $i\neq j$ we let $s_{1i}=\mathcal{S}_{1}(\mathrm{r}_{i})$ and $s_{2j}=\mathcal{S}_{2}(\mathrm{r}_{j})$; then given this cross-correlation, we can say that the joint probability of $(s_{1i}=1,s_{2j}=1)$ is $\mathcal{P}(1,1)=P_{1}P_{2}(1+\xi_{12})$. Furthermore, since $\mathcal{S}_{1}(\mathrm{r}_{i})\mathcal{S}_{2}(\mathrm{r}_{j})$ vanishes unless $\mathcal{S}_{1}(\mathrm{r}_{i})=\mathcal{S}_{2}(\mathrm{r}_{j})=1$, the expected value $\langle\mathcal{S}_{1}(\mathrm{r}_{i})\mathcal{S}_{2}(\mathrm{r}_{j})\rangle=\mathcal{P}(1,1)$.

Now $P_{1}$ is simply $(1/n)\sum s_{1i}$, and $P_{2}$ is $(1/n)\sum s_{2j}$. Given these relationships, we prove in Appendix A.3 the following statement (analogous to Equation 6) concerning the covariance of the probabilities $P_{1}$ and $P_{2}$:

At this point we make an approximation analogous to that in Section 3.1 by replacing the fixed $\xi_{12}$ with the average cross-correlation $\overline{\xi}_{\mathcal{S}_{1}\mathcal{S}_{2}}$ of the two slice fields. Again writing $N_{c}$ for $n$, we obtain, for any disjoint counts-in-cells bins $B_{1}$ and $B_{2}$,

We note that in the absence of cross-correlation, the covariance is negative since the bins are mutually exclusive.

As in Section 3, we now wish to compare Equation 14 with results measured from simulations, and thus we require an estimate for the uncertainty of the measured covariances.

Let us begin with two disjoint CIC bins $B_{1}$ and $B_{2}$; let us also suppose that we have two sets $\left\{P_{1i}\right\}$ and $\left\{P_{2i}\right\}$ of measurements of probabilities $\mathcal{P}(B_{1})$ and $\mathcal{P}(B_{2})$, each set consisting of $N_{m}$ measurements. We denote the covariance of these two sets of probability measurements as $S_{P_{1}P_{2}}$.

The following expression (Kendall and Stuart 1958, p. 322) gives the variance of $S_{P_{1}P_{2}}$ (where our $S_{P_{1}P_{2}}$ corresponds to $k_{11}$ of Kendall and Stuart):

where we have converted cumulants into moments, with $\mu_{rs}$ denoting the $r$th, $s$th product moments about the means of the random variables $P_{1}$, $P_{2}$. In calculating these moments we will ignore cross-correlations between nearby cells (as in Section 3.2), but we shall again seek to reduce the impact of this simplification by expressing our result in terms of $S_{P_{1}P_{2}}$.

Let us employ $P_{1}$, $P_{2}$ as shorthand for $\mathcal{P}(B_{1})$, $\mathcal{P}(B_{2})$. Then the joint moment-generating function for the corresponding slice fields $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ is

Furthermore (as in Section 3.2), each of the measurements $P_{1i}$, $P_{2i}$ is an average over the $\mathcal{S}_{1}$, $\mathcal{S}_{2}$-values in $N_{c}$ cells. Thus the joint moment-generating function for the measured probabilities is

From this function, we calculate in Appendix A.4 the required joint cental moments of the $\mathcal{P}_{1},\mathcal{P}_{2}$ distribution. We then apply these results to Equation 15 and, using $S_{P_{1}P_{2}}=-P_{1}P_{2}/N_{c}$ (Equation 14 with no cross-correlation), it eventually follows that

As with the variance in Section 3.2, we can note that for large values of $N_{c}$, the $P_{1i}$, $P_{2i}$ values quickly approach a joint Gaussian distribution; in this case we can take the limit of Equation 18 to obtain $\mathrm{Var}\left(S_{P_{1}P_{2}}\right)=2\left(S_{P_{1}P_{2}}\right)^{2}/(N_{m}-1)$.

As in Section 3.3, we proceed to compare the predictions of Equation 7 with measured covariances; we use the same mock survey, with the same cell sizes ($1.95h^{-1}$ and $31.25h^{-1}$Mpc), as in Section 3.3. Again we split each survey into $N_{m}$ subvolumes, with each subvolume consisting of $N_{c}$ survey cells.

In this case we choose two CIC bins $B_{1}$ and $B_{2}$ and empirically determine the probabilities $\mathcal{P}(B_{1})$ and $\mathcal{P}(B_{1})$ within each subvolume; the result is an ensemble of $N_{m}$ measured probabilities $P_{1i}$ in $B_{1}$ and a corresponding ensemble of $N_{m}$ measured probabilities $P_{2i}$ within $B_{2}$. The covariance $S_{P_{1}P_{2}}$ of these two sets of measurements is our estimate for $\mathrm{Cov}(\mathcal{P}(B_{1})\mathcal{P}(B_{2}))$, and this value will depend on the number of cells $N_{c}$ used in the measurement of the probabilities. As before, we use unit bin widths with the $1.95h^{-1}$-Mpc cells and varying bin widths with the $31.25h^{-1}$-Mpc cells.

These empirical covariances appear as thick curves in Fig. 2. Since in these cases the covariances are typically negative, we plot $-\mathrm{Cov}$ with solid lines (using dashed lines to indicate positive covariances). Equation 18 provides the error bars for these measurements, where we use for $S_{P_{1}P_{2}}$, $P_{1}$, and $P_{2}$ the measured covariances and probabilities.

The thin curves in Fig. 2 show the predictions of Equation 14. For the volume-averaged correlation $\overline{\xi}_{\mathcal{S}_{1}\mathcal{S}_{2}}$ of the slice fields for the two bins, we first empirically calculate the two-point cross-correlation function $\xi_{\mathcal{S}_{1}\mathcal{S}_{2}}(r)$ of the two slice fields with FFT methods. We then, as in Section 3.3, sample the slice fields to obtain random pairs of positions within the survey volume, determine the cross-correlation of those positions from $\xi_{\mathcal{S}_{1}\mathcal{S}_{2}}(r)$, and fold the result into a running average, terminating the sampling process once the variation in the running average has subpercent effect on the predicted $\sigma_{\mathcal{P}(B_{1})\mathcal{P}(B_{2})}$.

Once again, the agreement between prediction and measurement is excellent, although the large error bars at $31h^{-1}$-Mpc cells mean that most of the measurements yield only upper limits of the absolute value. We also observe the same (expected) trends as in Fig. 1: the covariance of the CIC measurements decreases as the number of cells $N_{c}$ in each measurement increases; likewise the error bars on the covariance increase as the number $N_{m}$ of measurements decreases.

The thin dotted lines in Fig. 2 show the predicted covariance in the case of no cross-correlation, and the lower panels show the ratio between the cross-correlated and non-cross-correlated results. It is again clear that cross-correlations among survey cells in different probability bins can increase the covariance by multiple orders of magnitude.