5 Properties

In this section we show a number of interesting properties of C(w)and $\bigl\langle \tilde f(\vec\theta) \bigr\rangle$.

   
5.1 Normalization

As already shown [see above Eq. (13)], $\bigl\langle \tilde f(\vec\theta) \bigr\rangle$ is a simple convolution of $f(\vec\theta)$with $w_{\rm eff}$. The normalization I of this convolution can be obtained from the expression

$$I = \int_\Omega w_{\rm eff}(\vec\theta) \, \rm d^2 \theta =... ...(\vec\theta) C\bigl( w(\vec\theta) \bigr) \, \rm d^2 \theta .$$ (35)

From Eq. (34) we find

$$I = \frac{\rho}{1 - P_0} \int_\Omega \rm d^2 \theta \int_0^\infty w(\vec\theta) \rm e^{-s w(\vec\theta)} Y(s) \, \rm ds .$$ (36)

We first note that, in the general case, Y(s) depends on $w(\vec\theta)$. On the other hand, since the second term of Y(s)is proportional to $\Delta(w)$ [see Eq. (33)], it does not contribute to the integral. We can then evaluate first the integral over  $\vec\theta$, obtaining

$$\int_\Omega w(\vec\theta) \rm e^{-s w(\vec\theta)} \, \rm d^2 \theta = - Q'(s) ,$$ (37)

and thus we have

$$I = - \frac{\rho}{1 - P_0} \int_0^\infty Q'(s) \rm e^{\rho ... ...frac{1}{1 - P_0} \rm e^{\rho Q(s)} \right\vert _0^\infty = 1 .$$ (38)

Hence we conclude that the estimator (5) is correctly normalized.

   
5.2 Scaling

It is easily verified that a simple scaling property holds for the effective weight. Suppose that we evaluate $w_{\rm eff}(\vec\theta)$ for a given weight function $w(\vec\theta)$ and density $\rho$. If we rescale the weight function into $k^2 w(k \vec\theta)$, and the density into $k^2 \rho$, where kis a positive factor, the corresponding effective weight is also rescaled similarly to w, i.e. $k^2 w_{\rm eff}( k \vec\theta)$.

We also recall here that, although a normalized weight function has been assumed [see Eq. (6)], all results are clearly independent of the normalization of w. Hence, we can also consider a trivial scaling property: the weight function $k w(\vec\theta)$ has effective weight $w_{\rm eff}(\vec\theta)$ independent of k.

We anticipate that the effective weight is very close to the original weight w for large densities $\rho$ (see below Sect. 5.4). The scaling properties discussed above suggest that the shape of the effective weight is actually controlled by the expected number of objects for which the weight is significantly different from zero. This justifies the definition of the weight area $\mathcal{A}$ of w:

$$\mathcal{A} = \biggl[ \int_\Omega w(\vec\theta) \, \rm d^2... ...ega \bigl[ w(\vec\theta) \bigr]^2 \, \rm d^2 \theta \biggr] .$$ (39)

The first factor in this definition ensures that $\mathcal{A}$ does not depend on the normalization of $w(\vec\theta)$. It is easily verified that $\mathcal{A} = \pi_w$ for a top-hat weight function. Correspondingly, we define the weight number of objects as $\mathcal{N} = \rho \mathcal{A}$. Clearly, this quantity is left unchanged by the scalings considered above. Similar definitions can be provided for the effective weight $w_{\rm eff}$. Explicitely, we have

$$\mathcal{A}_{\rm eff} = \biggl[ \int_\Omega w_{\rm eff}(\v... ... w_{\rm eff}(\vec\theta) \bigr]^2 \, \rm d^2 \theta \biggr] ,$$ (40)

and $\mathcal{N}_{\rm eff} = \rho \mathcal{A}_{\rm eff}$.

Numerical calculations for $w_{\rm eff}$ show clearly that $\mathcal{N}$, rather than $\rho$, is the key factor that controls how close the effective weight $w_{\rm eff}(\vec\theta)$ is to $w(\vec\theta)$ (cf. Figs. 1, 3, and 4).

   
5.3 Behavior of w C(w)

We can easily study the general behavior of $w_{\rm eff}$. We first consider the case of infinite-support weights (P₀ = 0).

We first note that Y(s) > 0 for every s, and thus C(w) decreases as w increases. On the other hand, we have

$$w_{\rm eff} = w C(w) = \rho Y(0) + \rho \mathcal L[Y'](w) .$$ (41)

Since $Y'(s) = \rho Q'(s) Y(s) < 0$ is negative, we have that w C(w)increases with w. This result shows that the effective weight $w_{\rm eff}$ follows the general shape of the weight w, as expected. Moreover, the fact that C(w) is monotonically decreasing implies that the effective weight $w_{\rm eff}$ is "broader'' than w. For instance, we can say that there is a value w₁ for the weight such that $w_{\rm eff} = w C(w)$ is not larger than w for w > w₁, and not smaller than w for w < w₁. In fact, since C(w) is monotonic, the equation C(w) = 1 can have at most one solution. On the other hand, if C(w) < 1 (respectively, if C(w) > 1) for all w, then $w_{\rm eff}(\vec\theta) < w(\vec\theta)$( $w_{\rm eff}(\vec\theta) > w(\vec\theta)$) for all $\vec\theta$. This inequality, however, cannot be true since both w and $w_{\rm eff}$ are normalized.

Figure 1: The effective weight $w_{\rm eff}$ never exceeds $\rho$. The original weight $w(\vec\theta)$ is the combination of two Gaussians with different widths ( $\sigma _1 = 1$ and $\sigma _2 = 0.1$). The central peak is severely depressed for relatively low densities. Note that the weight area is $\mathcal{A} \simeq 12.2$, so that a density of $\rho = 0.2$ corresponds to $\mathcal{N} \simeq 2.4$.

The property just shown can be used to derive a relation between the weight area $\mathcal{A}$ and the effective weight area $\mathcal{A}_{\rm eff}$. Let us evaluate the integral

$$D = \int_\Omega \bigl[ w(\vec\theta) + w_{\rm eff}(\vec\the... ...c\theta) - w_{\rm eff}(\vec\theta) \bigr] \, \rm d^2 \theta .$$ (42)

This quantity is positive, since the integrand is positive (this is easily verified by distinguishing the two cases $w(\vec\theta) > w_1$ and $w(\vec\theta) < w_1$, and by noting that both factors in the integrand have the same sign). On the other hand, if we expand the product we obtain

 

$$0 < D =\int_\Omega \bigl[ w(\vec\theta) \bigr]^2 \, \rm d^2 \thet... ...eta) \bigr] \, \rm d^2 \theta = \mathcal{A}^{-1} - \mathcal{A}_{\rm eff}^{-1} .$$ (43)

The last relation holds because of the normalization of $w(\vec\theta)$ and $w_{\rm eff}(\vec\theta)$. We thus have shown that the effective weight area $\mathcal{A}_{\rm eff}$ is always larger than the original weight area $\mathcal{A}$; analogously we have $\mathcal{N}_{\rm eff} > \mathcal{N}$. This, clearly, is another indication that $w_{\rm eff}$ is "broader'' than w.

It is also interesting to evaluate the limits of w C(w) for small and large values of w. We have

$$\lim_{w \rightarrow \infty} w C(w) = \rho Y(0) = \rho .$$ (44)

Since we know that w C(w) is monotonic, $\rho$ is also a superior limit for the effective weight function. In other words, even if whas high peaks, the effective weight $w_{\rm eff}$ (that, we recall, is normalized) will never exceed the value $\rho$ (see Fig. 1). We stress here that, since $w_{\rm eff}$ is normalized, its maximum value [which, in virtue of Eq. (44) does not exceed $\rho$] is a significant parameter. For example, using the relation $w_{\rm eff}(\vec\theta) < \rho$ in the definition of $\mathcal{A}_{\rm eff}$ we find immediately

$$\mathcal{A}_{\rm eff}^{-1} = \int_\Omega \bigl[ w_{\rm eff... ...\int_\Omega w_{\rm eff}(\vec\theta) \, \rm d^2 \theta = \rho ,$$ (45)

or, equivalently, $\mathcal{N}_{\rm eff} > 1$. In other words, no matter how small $\mathcal{A}$ is, the effective weight will always "force'' us to use at least one object.

Equation (44) suggests also a local order-of-magnitude check for the effective weight: assuming $w(\vec\theta)$ normalized, we expect the effective weight to be significantly different from w for points where $w(\vec\theta)$ is of the order of $\rho$ or larger. Equation (45), instead, provides a global criterion: the effective weight will be significantly broader than w if $\mathcal{N}$ is of the order of unity or smaller. As anticipated above, thus, $\mathcal{N}$ is the real key factor that controls the shape of $w_{\rm eff}$ with respect to w.

The other limit for the effective weight is

$$\lim_{w \rightarrow 0^+} w C(w) = \rho \lim_{s \rightarrow \infty} Y(s) = 0 .$$ (46)

Thus, as expected, the effective weight $w_{\rm eff}$ vanishes as w vanishes.

If w has finite support, then the situation is slightly different. Given the definition of Y(s), we expect for C(w) a discontinuity for w = 0. Apart from this difference, the behavior of C(w) and of w C(w) is similar to the case considered above, namely C(w) is monotonically decreasing and w C(w) is increasing. We also have

$$\lim_{w \rightarrow \infty} w C(w) = \frac{\rho}{1 - P_0} ,$$ (47)

which is similar to Eq. (44). Note that, as expected from the normalization of $w_{\rm eff}$, $\pi_w \rho / (1 - P_0) \ge 1$for all densities. We also find

$$\lim_{w \rightarrow 0^{+}} w C(w) = \frac{\rho P_0}{1 - P_0} \cdot$$ (48)

In other words, the effective weight w C(w) does not vanish for small w. If, instead, we take w = 0, then we have w C(w) = 0. This discontinuity, related to the term $\Delta(w)$ in Eq. (33), is a consequence of a number of properties for the effective weight: (i) $w_{\rm eff}$ is normalized; (ii) $w_{\rm eff}$ is broader than w; (iii) $w_{\rm eff}$ has the same support as w. We thus are forced to have a discontinuity for the effective weight. The result obtained above is also convenient for simplifying equations for finite-support weight functions. In fact, for w > 0 we can clearly drop the last term of Eq. (33), thus recovering Eq. (19); if, instead, w = 0, we can directly set w C(w) = 0, without further calculations.

   
5.4 Limit of high and low densities

The final result considered in this section is the behavior of the correcting factor C(w) in the limit $\rho \rightarrow \infty$ and $\rho \rightarrow 0$.

We observe that $Q(s) \leq 0$ and moreover $\left\vert Q(s) \right\vert$ increases with s. Thus, if s is large, $\rm e^{\rho Q(s)}$ vanishes quickly when $\rho \rightarrow \infty$. As a result, in the limit of large densities we are mainly interested in Q(s) with s small. Expanding Q(s) around s = 0⁺, we find

$$Q(s) = \sum_{k=1}^\infty \frac{(-1)^k s^k S_k}{k!} ,$$ (49)

where S_k is the kth moment of w:

$$S_k = \int_\Omega \bigl[ w(\vec\theta) \bigr]^k \, \rm d^2 \theta .$$ (50)

The normalization of w clearly implies S₁ = 1. Assuming w > 0, to first order we have then $Y(s) \simeq \rm e^{-s \rho}$, so that

$$C(w) \simeq \frac{1}{1 - P_0} \int_0^\infty \rho \rm e^{-s ... ...ho} \, \rm ds = \frac{1}{1 - P_0} \frac{\rho}{\rho + w} \cdot$$ (51)

This expression gives the correcting factor at the first order in $1/\rho$. If w has infinite support the expression reduces to $C(w) = 1 / \bigl[ 1 + w/\rho\bigr]$. Note that the quantity $w/\rho \sim 1/\mathcal{N}$ is related the weight number, i.e. the expected number of objects which contribute significantly to the signal, for which the weight is not exceedingly small. Finally, at the zero order in the limit $\rho \rightarrow \infty$, C(w)converges to unity. In this case the map $\tilde f(\vec\theta)$ is expected to be a smoothing of $f(\vec\theta)$ with the same weight function w.

In the limit $\rho \rightarrow 0$, we have $Y(s) \rightarrow 1$, and thus

$$C(w) \simeq \frac{\rho}{1 - P_0} \frac{1}{w} \cdot$$ (52)

In the same limit, $P_0 \simeq 1 - \pi_w \rho$, so that we find

$$w_{\rm eff} = w C(w) \simeq \frac{1}{\pi_w} \cdot$$ (53)

Recalling that w C(w) vanishes for w = 0, we conclude that the effective weight converges to a top-hat function with support $\pi_w$ if $w(\vec\theta)$ has finite-support (see Fig. 4 for an example).