The general motivation and strategy behind tomographic mapping is the reduction of dimensionality [31]. Many one-dimensional methods are difficult to extend to higher dimensions. Multi-dimensional data can always be reduced in dimension via projection to a hyperspace, but this reduction comes at the price of loss of information, determined by the particular hyperspace chosen. However, several projections can be considered simultaneously, and thus we can describe a dimension reduction strategy schematically as using the following steps.

**Step 1**:Select a projection of the data to a chosen one-dimensional subspace.

**Step 2**:Make a one-dimensional comparison.

**Step 3**:Repeat steps 1 and 2 for a representative sample of subspaces.

**Step 4**:Aggregate the one-dimensional comparisons for a composite multivariate comparison.

We follow this strategy to compare *F* and *F*_{0} at each point . As each of the four steps requires prior decisions, such as the choice of a projection in Step 1, we describe them in detail in the following four sections. We assume for now that is known, but consider dropping this assumption later.

To illustrate the different steps, we provide several pictures in Figure 1 and refer to them when necessary.

Figure 1 (a) Project the data in one dimension. (b) Make a comparison between two one-dimensional distributions. (c) Several one-dimensional comparisons. (d) Averaging over all five circle points.

**Step 1: Select a projection of the data in one dimension** To reduce the multi-dimensionality of the data to one dimension, we consider the distribution of distances to a chosen fixed point. More specifically, let be one of several chosen fixed points, indexed by *i*. Then, as one-dimensional counterparts of *F* and *F*_{0}, consider the CDFs of distances from

where is true, and 0 otherwise. This first step is illustrated in the top left panel of Figure 1. The study region *R* is shaped as a disk. The point is where we want to make the local comparison of and . The point *c*_{i} is chosen outside of *R*. The next step focuses on comparing and at .

**Step 2: Make a one-dimensional comparison** As a comparison we define the function:

where the function and is a subset of the real line containing . The function*ψ* is a comparison measure, for example, a difference or a ratio, while is a neighborhood of on which to make the comparison. To make this common support unique with regards to , we choose it of the form , where the choice of is described in the following step.

The top right panel of Figure 1 illustrates this second step. The segment joining *c*_{i} to *y* is of length . The annulus delimited by the two dashed lines illustrates the interval . Points whose distance from *c*_{i} fall in are represented by the shaded intersection of the annulus and *R*. We call that intersection .

**Step 3: Repeat steps 1 and 2 for a representative sample of subspaces** The projection described in Step 1 simplifies the data to one dimension which results in a loss of information. Our one-dimensional comparison cannot discern changes in the two distributions for points within the same distance of *c*_{i}. To compensate we reproduce steps 1 and 2 using different locations for *c*_{i}. We choose to place these points on a circle surrounding the region *R*. Let *C* denote the circle circumscribed around *R*, centered at with radius . Choose *x*_{C} as the center of gravity of *R* under *F*_{0}. We divide the circumference of *C* into *N* equal arcs described by the points on the circumference: .

To ensure all projections are given equal role, we can either fix the length of or fix the proportion of the reference population it covers. We choose the second option, that is, given a fixed proportion *p*_{0}, we determine the unique that guaranties

[2]By keeping the parameter *p*_{0} fixed across all projections, the value *h**(*t, i) will vary with *i* and *t*. Each comparison from Step 2 can then be rewritten as

[3]The parameter *p*_{0} plays the role of a ‘bandwidth’ parameter defining the area covered by each around . We usually use

Finally, we introduce a further notation to simplify expression [3] to
where denotes the expected value relative to *F*. We also refer to the points as “circle points”. For each of them we can then calculate the corresponding . In the bottom left panel of Figure 1, four other circle points are placed around *R*, and their corresponding are delimited by dotted lines.

**Step 4: Composite multivariate comparison** In the final step, we define the distance-based mapping (DBM) as a real-valued function for any point
as the average of the one-dimensional comparisons. To interpret this quantity further, we consider the following property for .

**Definition 2.3**. Let be a real-valued function. We say *ψ* is *semi-linear* if and only if for any , any , and any such that , the following holds:

If *ψ* is semi-linear, for example, when it is a ratio or a difference, using for all simplifies to:

[4]From this expression [4], that is when is semi-linear, we can interpret as a function making a local comparison of transformations of *F* and *F*_{0}. More precisely, given a function and a point , the transformation is mapping to . When , the transformation reduces to the single scalar *p*_{0}. A good feature of Γ is that, similarly to the log-risk ratio from eq. [1], if the two functions to be compared are equal (i.e. ), then is a scalar, equal to 0 or 1 when is defined as a difference or a ratio, respectively.

The bottom right panel of Figure 1 depicts graphically the calculation of the composite score Γ. One hundred observations from which *F* can be estimated have also been plotted. All the intersections overlap around *y*, the point of interest. Observations falling in that overlapping area will contribute the most to the composite score. This contribution decreases in a discrete fashion as the observations get further away from *y*. Yet if there are more observations than expected in some of the , the corresponding one-dimensional comparison should be large, leading to a high value for .

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