1 Introduction
We start by analyzing a procedure that is used in predictive security applications to identify crime hot spots, [2, 8, 4, 9]. This procedure is known as the Prediction Accuracy Index (PAI) and it is used to compare among possible hot spots regions. In this context, a higher PAI is preferred to a lower one. In particular, in some applications it is used (along side other measures such as distances and divergences [9, 4]
) as an indicator of similarity between two densities of random variables (a density from a prediction and the other from an observation). Roughly speaking, the PAI of a region is defined as the ratio between the hit rate and the volume proportion (see
[2, 8, 4] and the next section). When the PAI is used as a similarity measure between two densities, say and , this indicator is computed as follows
Identify several regions of high probability according to
. 
Evaluate the PAI on each one of the regions identified before. Here the idea is that higher values of PAI will indicate certain similarity between and .
In case several regions of highprobability are selected in 1. and 2. above, one can add to this procedure the following:

Compute the average among the PAI indicators obtained above. See Section 2.
Our preliminary analysis readily shows that PAI measures (even average ones) shall not be used as a similarity measure between two densities and it have to be used cautiously to identify hot spots. See also
[8, 4, 9] for some other issues related to the PAI indicator.
We believe this is an important take home message in this note. See
Remark 1.
Anyhow, the procedure described above (1., 2. and 3.) led us to the definition of the integral average transform. See Section 3 and Remark 2. The integral average transform is defined for a given function , assumed regular enough just to fix ideas. The transform is the result of integrating plain unweighted averages of over a family of neighborhoods of a point , say that is indexed by the real parameter . Note that we are talking about plain averages and not weighted averages such as . See (12) for a precise definition. We show that this procedure is equivalent to computing integrals against a two variables kernel with a possible singularity at , that is, computing,
We also show that the integral above can be interpreted as an integral average transform, that is, an integral of regular averages of over a parametrized family of regions. As an example of application we show that integrating against the fundamental solution of the Laplace equation, that is, solving the Poisson equation, can be reinterpreted as an integral of averages of the forcing terms over balls. See Section 3 and Remark 5.
The rest of the paper is organized as follows. In Section 2 we present the mathematical context related to the PAI including some modification know as the penalized PAI. We introduce the average PAI that motivates our main definition. In Section 3 we present the definition of integral average transform and, as an application of this novel interpretation of kernels, we show how to write the fundamental solution of the Laplace operator as an integral average transform. In Section 4 we present some final comments.
2 Averaging prediction accuracy
In this section we present the motivation that lead us to study integrals of average of function on parametrized regions.
Let us consider the real random variables and . Denote by and
be probability density functions associated with
and , respectively. Given a measurable set we write for the Lebesgue measure of . Denote by the set with probability density level higher (greater) than , that is,(1) 
Let us define . In some practical applications a threshold value is determined and the region is refereed to as hot spot area (and in case of being the union of disjoint small regions these regions are know as hot spots), see [2] and references therein. In fact, note that if is close to the set will contain the most probable occurrences of .
It is common in applications to use the hot spots of the random variable to predict the hot spots of another a random variable . A common way to measure the performance of such predictions is by using the socalled Prediction Accuracy Index (PAI) defined next (see [2]). In some cases, when several possible predictions of are available, it is used the PAI measure to decide which model to use to report hots spots predictions of the target . See [2, 8].
In order to fix ideas and simplify the presentation we assume that and are continuous functions but we can as well consider discretized version up to some given spatial resolution as it is most common in applications.
2.1 PAI of a subregion
Assume that we want to analyze a measurable study region with positive volume measure. Region corresponds to the study region where the random variable occur. The region is a possible hot spot region for in . Define the hit rate by ([2])
If define the average of on by
Note that in our notation for we do not make explicit reference to the region since we assume is fixed. A commonly used indicator ([2]) of the quality of using to approximate the hot spots of in is given by the ratio of the hit rate with the percentage of volume , that is,
Note that the can also be written as the ratio between the averages of on
and which gives another interpretation of the that readily reveals some possible issues with this indicator such as the preferences for small subregions with highvalue of .
In order to better understand this indicator we present some simple examples. Note that if we get . If for instance we select with volume percentage
and hit rate
we still have since we can predict half of occurrences of in half of the study area. Assume now that we have
We then have since we can predict 80% of occurrences of in 20% of the study area.
In practical applications we are interested in finding regions with a high value of PAI for a given density . We make the following observations:

Since we always have we see that

Let . If is such that then and all its possitive measure subsets maximizes the PAI indicator. In fact, for all such that we have
We then see that, in practical applications the PAI indicator will favor small area regions around the maximum values of the density . Here small will depend on the spatial resolution at which subregions are computed. This might not be convenient in some applications as pointed out in [8].

If defined above is such that then, given a subregion with and , there always exists with and such that
Therefore we see that searching for Borel subregions
with as high PAI as possible is not a well posed problem under general considerations. Other possible optimization problems may be needed for practical applications. See [8, 4, 9] for more details.
Due to the fact that the PAI indicator prefers small area regions some modifications have been introduced. Among them, in [8] was introduced a penalized PAI where the area ratio is penalized, that is,
Here there was introduced the penalization . The exponent may depend on , e.g., (and in this case, for small hit rate, the indicator is penalized multiplicatively by the volume proportion of the region while for large hit rate close to 1 we do not have that penalization, [8]). Many other penalization alternatives can also be consider at the light of practical applications. For instance, a penalization of the form
where is the surface volume of with regular enough. See (17) below.
One additional observation is the following. One can think to compute an average of several PAI values over subregions. Let
and consider
Define the piecewise constant layered function
, where . We have that
(2) 
Note that the function depends on the sets and the weight but we not make this dependence explicit in our notation. We conclude that the average value (of PAI indicators) over some regions corresponds to the inner product between and a piecewise constant function that weights the regions by the inverse their areas penalized by . The layered function can be taught as an approximation of a kernel that has singularities in the region where it takes maximum value.
2.2 PAI of a random variable
Recall that is the density of the random variable that we want to use to predict the hot spots of the random variable . See [2]. Assume that is continuous and that . For any define by
(3) 
and the subregion by
(4) 
Define the prediction accuracy index at level as the PPAI of the subregion .
(5) 
As before, the level
prediction accuracy index is computed by
dividing the hit rate by the volume percentage using the region
and multiplying by a penalization.
From the comments on the previous subsection we have for the case (no penalization),



If is such that then .

If defined above is such that and there exists such that with and , then for small enough we have and

If (with finite volume) we have .
As mentioned before, in order to have an overall quantity, the performance may be measured by an average of the prediction accuracy index at different levels. More precisely, chose an integer and define
(6) 
Note that, under appropriate assumptions we shall have a limiting value when . We define
(7) 
2.3 Average PAI and kernels
Note that, by using Fubini’s theorem (see also (2)),
Here we have introduced the “layered” function
In case is continuous then and have the same level sets. Indeed, if we put defined such that where is defined in (3) we have that
Then,
Summarizing,
(8) 
The is a positive function with the same level curves as with maximum value (and possible singularities) at . Appearance of singularities depends on the integrability of the function
Therefore, the value will give higher weight to the regions containing . To further clarify our point we present the next example.
Example 1
Consider , and defined by
We have and and . Additionally if we take we have and it holds that the mass of the region is given by
Therefore, given , we can find such that to obtain,
Then, the measure of the region is the lengh of the interval that is,
Given , we can find by
which gives
Thus we obtain for the case and for
(9)  
(10)  
(11) 
See Figure 1 for an illustration.
From (8) we have , that is, an inner product with the function in (11). Note that puts a very high weight around the value and therefore ignoring other possible regions with hot spots. For instance if then the possible hot spots of at and wont be detected but the value of will be very high (as far as we select high enough).
Remark 1
We then conclude that not the PAI at a particular level, not the average of several PAIs at different levels, are good measures or indicators in order to select one among different possible predictions of the random variable , in particular, hots spots reported after selecting among several models using PAI as a main indicator are not adequate. See [8, 2].
3 The integral average transform
In the previous section we presented an application where integral of averages of a function on subregions are computed. We summarize this procedure as follows: consider a function and family of regions neighboring , say such that: for all and . Define the following integral average transform of by
(12) 
That is, a weighed integral of plain average values of over the regions , . Here is a nonnegative weight function.
This procedure is equivalent to integrating against a possible singular kernel. We formalize this interpretation in the next statement.
Theorem 3.1
As in the previous subsections we can choose for instance, subsets related to levels curves of another functions , say,
(13) 
Remark 2
In Section 2 the average PPAI, , defined in (7), was an integral average transform of where the family of sets were given by levels sets of . In particular, in Section 2 is continuous and has a maximum value , then corresponds to the integral average transform of evaluated at . The definition of subregions in (4) are related to superlevel sets of while for this section, and the rest of the paper, we use as sublevel sets; See (13). In the case of super level set the possible singularity of the associated kernel will be related to the minimum value of that, in order to fix ideas we are assuming to be a singleton. Recall that we are assuming that for all and .
Let us now consider a nonnegative kernel given by a with a possible singularity at and smooth for . We can then take, for instance, for some . Then
(14) 
Note that corresponds to the level curve of . In this case if we define
and therefore we have
We conclude that there is several ways to reinterpret an inner product against a kernel as a integral average transform.
Due to the importance of kernels and its ubiquity in pure and applied mathematics it is always useful to have several interpretations and equivalent formulations for the results that are written as integration against kernels. In the next section we present some well know examples.
3.1 Fundamental solution of Laplace equation as an integral average transform
As a particular example let us consider the application of singular kernels related to the Poisson problem ([7]). In order to fix ideas we do not considered the more general setting regarding minimal regularity of functions involved. Instead we assume that all functions are sufficiently regular in order to show the usage of the integral average transform (12).
Let . Here we need to find a function such that
(15) 
Consider In this case
(16) 
Denote the volume of the unit ball in and define the weight
(17) 
Then, in order to compute the corresponding kernel note that, for and ,
(19)  
(20) 
where the constant is given by
and is the fundamental solution of the Laplace equation. See [7, 5]. By taking we see that, for ,
We have the following result.
Theorem 3.2
Proof. Using (20) we have, for , the truncated integral,
(22)  
(23) 
By taking large enough, we see that
The results follows from the fact that is the fundamental solution of the Laplace equation and the other term vanishes when for and .
Remark 3
Consider now the case , , and . For all large enough we have
for all . It is enough to take where contains the support of . We then see that .
We have also the following result concerning a generalization of the mean value property. See [3].
Theorem 3.3
Assume that and are sufficiently smooth and satisfy
Then the following mean value property holds,
(24) 
Proof. From [3] we have
where is the Green function on the ball given by
From (19) and the Fubini’s theorem it follows
(25)  
This finished the proof.
We now turn to show how to rewrite the solution of the Poisson problem on the positive semispace as an integral average transform. Here we need to find a function such that
(26) 
with on
Theorem 3.4
Assume that , is regular enough and with compact support contained in and considered the integral average transform defined by
(27) 
Then
and on
(28)  
(29) 
Here we have used (21). The result follows from recalling that the support of is contained in and by noting that
Remark 4
For we have, for large enough
and therefore
(30)  
(31) 
We conclude the same result as in (27) by taking .
Remark 5
In the previous results we write the solution of the Poisson equation in the half space as an integral of averages of the forcing term over balls centered at . In the case the ball , the function is cut to . See Figure 2.
Finally we mention that formula (21) give us additional interpretation of the solution of the Poisson equations in domain with boundaries. Instead of subtracting two solution as in formula (28), we could extend the forcing term in such a way that averages centered at the boundary vanish. For instance, let us consider the domain and . In order to use (21) to obtain the solution of the Poisson equation in we could extend to the whole , say such that and with for all . We then have the following results as an alternative the Theorem 3.4.
Theorem 3.5
Assume that , is regular enough and with compact support contained in and considered
and defined by
(33) 
Then
and on
Similar argument can be applied to other domains with simple boundaries such as strips and cubes.
4 Final comments
In this short note we introduced the integral average transform. Our motivation comes from the need of a better mathematical understanding of some practical measures such as the prediction accuracy index that is popular in problems related to predictive security. In this paper we have explained the mathematical and practical context of this application in order to motivate our main definition. The integral average transform is defined for a given function and it is the result of integrating plain averages of over an family of sets (containing a point ) indexed by the integration argument. See (12) for a precise definition. We show that this procedure is equivalent to computing integrals against a two variable kernel with a possible singularity at . We also show that any kernel integral of the form can also be interpreted as an integral average transform. Given the ubiquity of kernels in the solution of many problems, we believe that this novel interpretation may be worth of further investigation. For instance, other kernels can be considered such as Poisson’s kernels. We can also associate some dynamics to the parameter .
References
 [1] (2007) Comprehensive survey on distance/similarity measures between probability density functions. City 1 (2), pp. 1. Cited by: Remark 1.
 [2] (2008) The utility of hotspot mapping for predicting spatial patterns of crime. Security Journal 21 (1), pp. 4–28. External Links: Document, Link, https://doi.org/10.1057/palgrave.sj.8350066 Cited by: §1, §2.1, §2.2, §2, §2, Remark 1.
 [3] (1990) A monte carlo method for poisson’s equation. Journal of Computational Physics 90 (1), pp. 123–140. Cited by: §3.1, §3.1.
 [4] (2016) A metric comparison of predictive hot spot techniques and rtm. Justice Quarterly 33 (3), pp. 369–397. Cited by: §1, §1, §2.1.
 [5] (1998) Partial differential equations. Graduate studies in mathematics 19 (4), pp. 7. Cited by: §3.1, §3.1.
 [6] (2002) On choosing and bounding probability metrics. International statistical review 70 (3), pp. 419–435. Cited by: Remark 1.
 [7] (2015) Elliptic partial differential equations of second order. Vol. 224, springer. Cited by: §3.1, §3.1, §3.1.
 [8] (2021) Considerations for developing predictive spatial models of crime and new methods for measuring their accuracy. ISPRS International Journal of GeoInformation 10 (9), pp. 597. Cited by: §1, §1, 2nd item, §2.1, §2.1, §2, Remark 1.
 [9] (2020) Learning to rank spatiotemporal event hotspots. Crime Science 9 (1), pp. 1–12. Cited by: §1, §1, §2.1.