Combining local and global smoothing in multivariate density estimation

10/07/2016
by   Adelchi Azzalini, et al.
0

Non-parametric estimation of a multivariate density estimation is tackled via a method which combines traditional local smoothing with a form of global smoothing but without imposing a rigid structure. Simulation work delivers encouraging indications on the effectiveness of the method. An application to density-based clustering illustrates a possible usage.

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1 Local and global smoothing

Consider estimation of the probability density function

of a continuous random variable in cases when a parametric formulation for

is not considered appropriate. Given a random sample drawn form , a variety of non-parametric estimation methods are available. Most of these methods share the common feature of being ‘fully non-parametric’, meaning that the set of competing alternatives from which an estimate must be selected is constituted by the entire set of possible densities, except for some conditions of mathematical regularity.

Limited work has been dedicated to methods which allow inclusion of some qualitative requirement about . One problem which has attracted a fair amount of attention is estimation, in the univariate case, of a unimodal density or, more generally, of a density with a pre-assigned number of modes, like in Hall & Huang (2002). Other qualitative requirements on seem to have received less consideration.

We shall be dealing with estimation of a density on , or possibly a subset of it, with . For reasons which will become clear shortly, the case is technically possible, but both uninteresting and nearly degenerate in our framework; therefore is the situation really considered.

It is well-known that, as

increases, non-parametric methods, and in particular those for density estimation, degrade in performance, eventually running into the so-called problem of ‘curse of dimensionality’ when

is large. On the other hand, there is the widespread perception that, in many real situations, the dependence structure of a multivariate distribution is largely governed by the dependence among a smaller number of components. An explicit statement of this view has been expressed by Scott (1992, p. 195): “Multivariate data in are almost never -dimensional. That is, the underlying structure of data in is almost always of dimension lower than ”.

The present contribution examines an estimation method motivated by these considerations. Broadly speaking, we impose a ‘light structure’ on the density , moving away from a fully non-parametric construction, but without imposing a detailed structure, such as parametric form. A bit more specifically, it is assumed that, at least in an approximate sense, the dependence is regulated by a structure based on -dimensional subsets of the variables, with . The introduction of this constraint leads to a form of global smoothing of the estimated distribution which can improve upon existing methods, in appropriate situations, by reducing variability connected to estimation of fine details of regulating high-order interactions among variables. It is plausible that, even if these high-order interactions are not exactly null, the reduction of variability of the estimate overcomes the bias so introduced; an assumption of this sort is ubiquitous in any modelling operation.

Clearly, the success of this scheme relies on the suitability of the imposed structure in a given situation. To exemplify by what here represents an extreme case, application of the stated criterion when

would entail to introduce a joint distribution constituted only by marginals of dimension

, that is, assuming independence of the two component variables. In the majority of situations, the more interesting ones in fact, this extreme simplification would not be appropriate; this explains why earlier we have effectively restricted ourselves to the case .

Estimation of will still be carried out via a classical local smoothing method, such as the kernel estimator, but in a way which reflects the global smoothing imposed by the assumed structure of . Therefore the final outcome of the procedure will reflect both the local and the global smoothing operations.

In the next two sections, we transfer this broad criterion into a specific operational formulation. This is then followed by numerical exploration to evaluate its practical working with simulated data and by its utilization within a density-based clustering process of some real data.

2 Global smoothing via a log-linear model

The criterion described only qualitatively so far can be translated into an operational procedure. Given the broad nature of the above formulation, there is not a unique prescribed way to define such a procedure. The route to be presented here is driven by simplicity and flexibility, since it can be used in conjunction with any local smoother which allows weighted observations with only simple adjustments of an existing method.

Assume that a sample of observations drawn from is available, where for . Denote by the parent random variable from which the ’s are drawn, all independently from each other. We introduce subdivisions of the coordinates axes into disjoint intervals, creating a partition of the sample space into hyper-rectangles or cells. Correspondingly, there are probabilities associated to the cells. The th element of this partition, denoted , can be associated to a multidimensional subscript

, but this is not of relevance at the moment.

Denote by the number of sample elements falling into , for , so that . The basic estimate of is and, correspondingly, for a given point , a crude estimate of is

(1)

where is the geometric volume of .

A key weakness of this scheme is that it implies distinct probabilities to be estimated, up to a constraint on their sum. If is not small and the subdivisions are not coarse, can be large. To reduce the number of free parameters to be estimated, we introduce a log-linear model for the cells probabilities, ’s, where interaction terms involving more than component variables are set to zero. For instance, if and we choose , terms of the log-linear representation of the ’s are retained up to pairwise interactions while all three-factor interaction terms are eliminated, reducing the number of underlying parameters by with respect to the saturated model. See Section 9.2.2 of Agresti (2013) for a detailed discussion of the pertaining log-linear model; there is only the difference that those expressions refer to the expected values of the frequencies instead of the probabilities, but this is irrelevant since the two sets of quantities are proportional to each other.

After the log-linear model has been fit to the observations, a set of expected frequencies is obtained, denoted , and corresponding estimated probabilities . Replacing by in (1) provides a revised estimate.

The constraints enforced by the log-linear model refer to the cell probabilities, , and so to the expected frequencies , but not to the density . However, provided the values are not too disparate, at least in the region where most of distribution is located, the originally intended dependence structure will hold approximately. Furthermore, the local smoothing step to be presented shortly introduces an additional perturbation in this sense. Since the imposed dependence structure is motivated by practical considerations of improved estimation performance rather than exact model compliance, we are not concerned about these approximations.

3 Non-parametric local smoothing

In the second step of the procedure, the expected frequencies obtained in the first step are used to assign weights to the observations, , so that the estimate produced by the subsequent local smoothing respects, approximately, the estimated cell probabilities .

To illustrate the procedure, we use the following simple form of the kernel density estimate. Take the kernel function to be the

-dimensional circular normal density with standardized components, denoted

, and choose a vector

of smoothing parameters; then the classical kernel estimate at point is

(2)

where . This classical estimate is modified by weighting an observation in the -th cell with , so that the overall weight of the observations in the th cell is instead of . The new estimate takes the form

(3)

where is the weight of the cell to which belongs. The type of perturbation of estimator (2) is denoted ‘tilting’ by Doosti & Hall (2016).

Clearly several variants forms can be considered, such as replacing the normal kernel in (3) by some other multivariate kernel or using smoothing parameters which vary with the observations. Not only these variants are immediately accommodated, but we are not restricted to kernel-based methods. For instance, if we use instead a projection method based on an orthogonal series decomposition of , the weight is assigned to observation when the coefficients of the projection are estimated as sample averages of suitable data-dependent functions.

A complication arises with empty cells, where , since the corresponding weights are not well defined. While for cells with the method works by suitably increasing or decreasing the weight of the observations belonging to that cell, no such adjustment is possible if the cell is empty.

The simplest approach to the problem is to just use (3), only replacing in the denominator by ; we are then effectively ignoring the ‘empty cells’ problem. This variant form is denoted ‘Plain’ (or P) later on. When corresponding to is of non-negligible magnitude, possibly so for several cells, the above solution may appear too crude, hence supporting the alternative approach of introducing some fictitious data to fill the empty cells. Note that this implies that the overall number of data points exceeds , although the sum of their weights remains

. However, while appealing in principle, it is hard to say how to pursue this route in a theoretically-motivated procedure. An heuristic method has been constructed, described in Appendix A. This variant form is denoted ‘Fill’ (or F) later on.

4 Practical and computational aspects

The practical implementation of the method requires to specify, in the first place, the subdivisions of the axes which identify the cells. For each axis, we have started by applying the ‘normal reference rule’ proposed by Scott (1992, p. 82) for choosing the histogram bins, assuming joint normality of the multivariate distribution, that is, the th bin-width is initially taken equal to

where

is the standard deviation of the

-th component variable, ; in practice, must be replaced by its sample value. Division of the range of observations by , rounded to the nearest integer, lends the number of bins for the -th component, . The sample range of the -th variable has then been subdivided into

intervals, constructed as follows: first the quantiles of level

of the distribution have been computed and then the sample quantiles with level equal to these Beta quantiles have been used as the end-points of the intervals on the -th axis. The underlying idea is to have the central intervals shorter than those near the margins of the sample range. This process is repeated for .

This procedure for choosing the end-points of the intervals appears somewhat arbitrary if examined from a formal viewpoint. The scheme must rather be regarded as a way of mimicking the non-automatic process followed when intervals are chosen by subjective judgement.

To compensate the possible effect of the choice of the number of subdivisions , a variant form of the procedure involved using three choices of the subdivisions: one as described above, one decreasing each by 1 and the third one increasing each by 1. For each of these three grids, the log-linear fitting and computation of (3) were applied, followed by averaging of the three estimates. This variant is denoted ‘Average’ (or A) later on.

Once the grid of the space has been fixed, the sample frequencies of the cells are identified. For a given value of , we must fit a log-linear model as described in Section 2; we temporarily leave aside the choice of , to which we return later. In our problem the interest is only in the fitted frequencies, , not in the log-linear parameters. In this case the recommendation of Agresti (2013, § 9.7.3) is to adopt the iterative proportional fitting algorithm, since it “converges to the ML [maximum likelihood] fit even when the likelihood is poorly behaved”. A Fortran implementation of this algorithm has been provided by Haberman (1972), subsequently ported to the R computing environment with name loglin.

The final step is application of the the weighted estimator (3). In most of our numerical work, the diagonal smoothing matrix has been chosen by the multivariate version of the plug-in method of Wand and Jones (1994) available in the R package ks (Duong, 2015).

An illustration of the working of the procedure is provided in Figure 1, which refers to the simplest possible case, that is, with and . As already explained, this situation is not of practical relevance, but it is appropriate for simple illustration. Specifically, the

plotted points constitute a sample drawn from a circular bivariate normal distribution with standardized marginals. A rectangular area slightly wider than the range of the observed points has been selected and, using the above-described rule, a

grid has been identified; for all cells of this grid, was observed. The rectangles have been shaded using a -level grey scale which discretizes the values of the crude estimate (1). The dashed red lines represent the contour level curves of the classical kernel estimate (2) while the continuous black curves refer to the weighted estimate (3); this estimate appears somewhat smoother than the unweighted one, with more limited departures from convexity, especially so in the central region.

Figure 1: Illustrative example of the estimate in the simple case with and

5 Simulation work

The performance of the proposed method has been examined in a number of cases, using simulated data from a range of distributions: normal, skew-normal, Student’s

and its skew version, and two-component mixtures of these distributions. The general expression of the distribution in use is

(4)

where and are of skew-normal (SN) or skew- (ST) type, which include the classical normal and distributions as special cases; is the mixing proportion. The distributions and are specified by the following parameters: a -vector location , a symmetric positive-definite scale matrix , a -vector slant and a positive real number . The component exists only for the ST distribution; when , or equivalently when it is not present, the distribution is of SN type. A detailed treatment of the multivariate SN and ST distributions is provided by Azzalini & Capitanio (2014). When , there is effectively no mixture mechanism and only the parameters are required. The parameters considered have been selected among the following options.

  • If , the location parameter is always . If , the location of is and the one of is , where denotes the -vector of all 1’s.

  • The scale matrix has been chosen among the following options:

    • the identity matrix

      ;

    • a Toeplitz-type matrix with -th entry where , or equivalently with AR(1) correlation structure;

    • an ARMA(2,1) correlation structure;

    • a matrix with elements specified individually, in some instances with .

  • The components of the slant parameter have been specified individually. When and , the distribution is a regular (symmetric) Gaussian or Student’s distribution.

  • The degrees of freedom

    was given a value among the following: (corresponding to the SN distribution), 5 or 2.

Distributions with dimension from to have been considered. The value employed in the log-linear model was and , with the constraint . In most cases, the sample size was ; a few experiments used either or . For each combination of parameter values and sample size, replicates have been considered and the following estimation methods have been tested:

  1. the classical kernel method in (2), denoted ‘kde’;

  2. the weighted kernel method in (3), in the Plain variant form described in Section 3;

  3. the Average variant which averages three estimates computed from three grid subdivisions of the sample space;

  4. the Fill variant which places constructed points in empty cells, as described at the end of Section 3 and more in detail in an Appendix.

The vector of smoothing parameters for the kernel method and its variants was obtained by the function Hpi.diag of the R package ks described earlier.

The possible number of factor combinations so obtainable is enormous even if one selects only a few possible choices for each of the above-described parameters. Moreover the computation burden with certain parameters, especially for the Average variant, was appreciable. This prevents any attempt of running a full factorial experiment. Only a selection of factor combinations has been considered, driven by subjective judgement on the outcome of earlier experiments, paying more attentions to situations which appeared more interesting in some sense. For instance, for a certain combination of parameters, the value of could have been decreased to or increased to to examine the effect of sample size alone, when this appeared to be an interesting case.

To evaluate the performance of the proposed estimate (3), in their variant forms, with respect to the classical estimate (2), the estimation error at has been expressed by

(5)

where has been used in the outcomes presented below. Initial numerical work had also considered and , but the general qualitative indication which emerged was not very different and may represent a reasonable compromise between absolute error and relative error. Two sets of points have been considered for evaluating (5): (i) a non-random grid of points spanning the area of non-negligible density of and (ii) the sample values. The second option is relevant in certain applications like the one of Section 6. A detailed description of the non-random grid of points is provided in Appendix B. For the (ii) case, only the real observations have been considered in the Fill variant, ignoring the fictitious observations which it involves.

Direct consideration of (5) for all the evaluation points, in either of the two considered sets, is not feasible. The quantiles of such sets of estimation errors have examined instead, at probability levels . Even with this reduction, the amount of tabular material so produced is considerable; the full set of such tables is provided in Appendix B. A more compact summary exhibit of the overall outcome is provided by Figure 2. The values on the vertical axis represent

(6)

where is the -level quantile of the relative error (5), evaluated over a given set of points, for a the proposed method (in one of its variants) and is the similar quantity for the standard kernel density estimation. Therefore, represents a measure of reduction of the estimation error with respect to the classical estimate, or a measure of its increase in case this quantity is negative. Figure 2 reports only the more noteworthy aspects of the full outcome, as described next.

Figure 2: Relative reduction of estimation error quantiles of the proposed estimate versus the classical kernel estimate; see text for a detailed description.

Only variants Plain and Fill have been considered in Figure 2 since the Average form was essentially equivalent to the Plain one, with extra computing effort; it is however reassuring to know that the specific choice of the grid size is not critical. For each of the three panels of the figure, the left portion refers to the choice for the log-linear model, the right portion to . Three values of entering (6) have been reported, namely , from the full six values in the complete outcome. For each pair of and , there are two vertical stripes of numbers; the left blue stripe refers to distributions which are mixtures, while the right red stripe refers to single-component distributions, that is, those having in (4). In all cases, the digit plotted at ordinate (6) denotes . Of the three panels, two refer to the Plain variant of the method, with evaluation is performed either at a fixed grid of points or at the sample points. The third panel refers to the Fill variant, but only with evaluation at the sample points; evaluation at the grid points was markedly unsatisfactory.

The first message emerging from inspection of Figure 2 is that an improvement of the weighted kernel estimate over the classical one occurs in the majority of cases, often with an appreciable magnitude; the negative values are limited in number and in magnitude. This consideration is substantially reinforced if we confine attention to , irrespectively of ; this explain why has not been considered in the simulations. Another indication is that the method, in all variants, performs better with a single component distribution than with a non-generate mixture.

Operationally, the following recommendations for use of the method can be extracted: (i) set in all cases; (ii) the Plain variant is preferable when the whole density surface must be estimated, while the Fill variant is preferable for evaluation at the observed data points; (iii) expect more improvement in case of a unimodal distribution than a multimodal one. These recommendations refer to the kernel estimate and smoothing matrix described above, and they may not necessarily hold for other forms of non-parametric estimation.

6 Application to density-based clustering

The proposed density estimate has been used in conjunction with the clustering method presented by Azzalini and Torelli (2007), implemented in the R package pdfCluster (Azzalini and Menardi, 2014). Since this clustering technique is firmly based on estimation of the density of the underlying -dimensional random variable, it represents an ideal framework for application of the present proposal.

The real-data application presented by Azzalini and Torelli (2007, Section 4.3) concerned eight chemical components of specimens of olive-oil originating from various regions of Italy. We re-examine their clustering exercise whose aim was the reconstruction the production area of the specimens from the values of their chemical components. The data themselves are available in the pdfCluster package. A more detailed description of the data and of their preliminary transformations, which we also apply here, is provided by Azzalini and Torelli (2007). We only specify the undocumented detail that, in the additive log-ratio transform applied to the compositional data (), namely for , the choice was made because the values of are well separated from 0; this is also the choice of the original article. The first principal components of the ’s constitute the variables used for the actual clustering step.

The pdfCluster package was applied to the five principal components just described both in its current public version (1.0-2, as available at the time of writing) and a modified version which replaces the classical kernel estimate (2) and the weighted kernel estimate (3) with ; all other ingredients have been kept at the default specification of the package. Table 1 displays the cross-classification table of the true geographical areas and the groups formed by clustering for the classical estimate (2) in the first three columns and the new proposed estimate in the last three columns refer. The latter estimate has actually been computed using both the Fill and the Plain variant, but the outcome was the same. The ARI values underneath each sub-table denote the ‘adjusted Rand index’ which constitutes a measure of agreement between the true and the reconstructed classification (Huber and Arabie, 1985). There is a clear improvement in using the new estimate, from consideration both of direct inspection of the table and by the ARI values. The left portion of Table 1 is slightly different from the table originally obtained by Azzalini and Torelli (2007), but the essential traits are the same and the ARI value was even smaller there, namely 0.792.

classical estimate proposed estimate
1 2 3 1 2 3
South 321 0 2 323 0 0
Sardinia 0 98 0 0 98 0
Centre-North 0 45 106 0 22 129
ARI 0.873 0.937
Table 1: Clustering of olive-oil data: true versus reconstructed groups using the current package pdfCluster and its modification with density estimation replaced by the new method.

The values in Table 1 have been obtained using the default smoothing parameter of pdfCluster, which is the asymptotically optimal bandwidth under normality, multiplied by a shrinkage factor. For completeness, we considered also the choice of produced by Hpi.diag, already used in the simulation work. In this case the shrinkage factor usually introduced by pdfCluster has been to the neutral value of 1, since that shrinkage loses meaning with another choice of . The ARI value of the new groupings decreases slightly to 0.910 for the proposed estimate, while where was a much worse degrade for the classical estimate, which lead to four groups instead of three, with an ARI of 0.817.

At first sight, it may look surprising that the use of estimate (3) produces such a noticeable improvement over (2), considering that Figure 2 indicates a limited improvement in connection with multimodal densities, which is the typical situation in a clustering context. One must however bear in mind that the procedure underlying pdfCluster involves two main stages: in the first stage, the density of the overall population is estimated, to locate the cluster cores associated to the subpopulations, while, in the second stage, the distribution of each identified cluster core is estimated separately. The densities of these sub-populations are naturally of unimodal type, where Figure 2 indicates a better performance. It is then reasonable to link the successful effect of the new estimate mainly to its role in the second stage of the procedure.

7 Final remarks

The numerical outcome, both from the simulation work and from the clustering application, provides quite clear evidence in support of the proposed method. However, there is still much room for improvement. For instance, a better motivated method for filling empty cells would be welcome. Even more importantly, some mathematically-argumented understanding of why the method works is lacking. Moreover, the global smoothing technique of Section 2 represents one possible route to implement the qualitative criterion stated in Section 1, but other routes may be considered.

An implementation of the proposed method will be made publicly available in the R package pdfCluster.

Acknowledgements

The development of this work has much benefited from stimulating discussions with Giuliana Regoli.

References

Agresti, A. (2013). Categorical Data Analysis, 3rd edition. Wiley, New York.

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS monographs.

Azzalini, A. and Menardi, G. (2014). Clustering via nonparametric density estimation: The R package pdfCluster. J. Stat. Software, 57(11), 1–26. http://www.jstatsoft.org/v57/i11/

Azzalini, A and Torelli, N. (2007). Clustering via nonparametric density estimation. Stat. Comput., 17, 71–80.

Doosti, H. and Hall, P. (2016) Making a non-parametric density estimator more attractive, and more accurate, by data perturbation. J. R. Stat. Soc, series B, 78, 445–462.

Haberman, S. J. (1972). Algorithm AS 51: Log-linear fit for contingency tables.

Appl. Stat., 21, 218–225.

Hall, P. and Huang, L. S. (2002). Unimodal density estimation using kernel methods. Stat. Sinica, 12, 965–990.

Scott D. (1992). Multivariate Density Estimation. John Wiley & Sons, New York.

Duong, T. (2015). ks: Kernel Smoothing. R package version 1.10.0. https://CRAN.R-project.org/package=ks

Wand, M. P. and Jones, M. C. (1994). Multivariate plugin bandwidth selection. Comp. Stat. 9, 97–116.

Appendix

A. Filling empty cells

As explained in Section 3, empty cells having are problematic. One approach is to fill them with some fictitious data before applying the weighted kernel estimate (3). Unfortunately, the construction of such data by some theoretically-supported procedure appears to be a challenging problem. We describe instead a fairly simple heuristic procedure.

Consider a given cell with but and denote by the smallest integer value larger than or equal to . The aim is to choose fictitious points in ; recall that the points will be suitably weighted so that the overall weights of the cell will be .

An instinctive idea is to consider a component-wise average of the coordinates of some neighbouring observations falling in adjacent cells, but this may easily produce points outside the cell . To avoid this problem, we consider instead an average of the coordinates of the corners of , giving more weight to the corners closer to nearby observations. The specific procedure is as follows:

  • the set of Euclidean distances of each observations from the centre of are computed and sorted in increasing order;

  • construct a fictitious observation from the following two steps:

    1. compute the Euclidean distances between the first element of the available observations (in the sorted list just constructed) and the corner points of , and assign to each corner point a weight inversely proportional to the square root of its distance;

    2. the above step is repeat times (in our work has been used), each time discarding the already employed observation from the sorted list, adding up the weights of the corner points; finally, retain the weighted average of the corner points as a new constructed observation;

  • step 1 and 2 are repeated for each of the points to be constructed.

B. Output of the simulation study

The definition of the distributions considered in the simulation study and the description of their parameters have been provided in Section 5. It remains to describe the non-random grid of points on which the estimation error has been evaluated. Consider the hypercube where when is a mixture with and is . On the interval , select equally spaced points, where is the smallest integer larger than or equal to and usually was used. The Cartesian product of these coordinates for coordinated axes produces a grid of evaluation points.

The following pages provided a summary for each simulation run followed by an overall summary.

Case No.  1

Niter : 2500
n : 500
d : 4
m : 3
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 5
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 5
25% 50% 75% 90% 95% 99%
e1 2.9E-05 9.79E-05 0.000348 0.00115 0.0028 0.0344
e2 2.9E-05 9.79E-05 0.000348 0.00115 0.0028 0.034
e3 2.89E-05 9.79E-05 0.000348 0.00115 0.0028 0.0341
e4 2.89E-05 9.79E-05 0.00035 0.00118 0.00299 0.0371
Table 2: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
e1 0.0274 0.06 0.114 0.183 0.232 0.636
e2 0.0272 0.0596 0.114 0.182 0.232 0.634
e3 0.0273 0.0597 0.114 0.182 0.232 0.634
e4 0.0272 0.0596 0.114 0.182 0.232 0.63
Table 3: Error quantiles evaluating at the observed sample points

Case No.  2

Niter : 2500
n : 500
d : 4
m : 2
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 5
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 5
25% 50% 75% 90% 95% 99%
e1 2.89E-05 9.79E-05 0.000348 0.00115 0.00279 0.0346
e2 2.9E-05 9.79E-05 0.000348 0.00115 0.00272 0.0303
e3 2.9E-05 9.79E-05 0.000348 0.00115 0.00273 0.0306
e4 2.9E-05 9.98E-05 0.000371 0.00167 0.00581 0.0577
Table 4: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
e1 0.0274 0.06 0.114 0.182 0.232 0.622
e2 0.0255 0.0563 0.108 0.175 0.22 0.497
e3 0.0261 0.0568 0.108 0.175 0.22 0.497
e4 0.0252 0.0561 0.109 0.176 0.221 0.471
Table 5: Error quantiles evaluating at the observed sample points

Case No.  3

Niter : 2500
n : 500
d : 4
m : 3
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
25% 50% 75% 90% 95% 99%
e1 5.13E-22 3.77E-12 6.97E-05 3.57E+03 1.81E+13 2.11E+36
e2 5.12E-22 3.76E-12 6.94E-05 3.58E+03 1.8E+13 2.15E+36
e3 5.11E-22 3.76E-12 6.94E-05 3.56E+03 1.81E+13 1.88E+36
e4 5.11E-20 5.64E-11 0.000709 1.25E+06 1.19E+20 1.2E+51
Table 6: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
e1 0.0267 0.0569 0.0986 0.141 0.167 0.233
e2 0.0266 0.0568 0.0985 0.141 0.167 0.233
e3 0.0266 0.0568 0.0985 0.141 0.167 0.233
e4 0.0266 0.0569 0.0986 0.141 0.167 0.233
Table 7: Error quantiles evaluating at the observed sample points

Case No.  4

Niter : 2500
n : 500
d : 4
m : 2
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
25% 50% 75% 90% 95% 99%
kde 5.14E-22 3.79E-12 7.13E-05 4.21E+03 3.22E+13 1.29E+36
wkde 5.01E-22 3.68E-12 6.75E-05 4.02E+03 3.03E+13 1.25E+36
wkdeA 5E-22 3.7E-12 6.84E-05 4.06E+03 2.92E+13 1.19E+36
fill+wkde 4.55E-16 4.4E-08 0.277 8.34E+12 9.88E+27 1.56E+61
Table 8: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0266 0.057 0.0986 0.141 0.167 0.234
wkde 0.0264 0.0565 0.098 0.14 0.166 0.232
wkdeA 0.0263 0.0563 0.0976 0.14 0.165 0.23
fill+wkde 0.0265 0.0568 0.0985 0.141 0.166 0.231
Table 9: Error quantiles evaluating at the observed sample points

Case No.  5

Niter : 2500
n : 500
d : 4
m : 3
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 2
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 0.000202 0.000462 0.00115 0.00321 0.00813 0.0485
wkde 0.000202 0.000462 0.00116 0.00322 0.00818 0.0486
wkdeA 0.000202 0.000462 0.00116 0.00322 0.00818 0.0486
fill+wkde 0.000202 0.000462 0.00115 0.00321 0.00816 0.0486
Table 10: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.025 0.0586 0.136 0.246 0.343 2.63
wkde 0.025 0.0587 0.136 0.247 0.345 2.67
wkdeA 0.025 0.0587 0.136 0.246 0.345 2.66
fill+wkde 0.0249 0.0585 0.136 0.246 0.343 2.62
Table 11: Error quantiles evaluating at the observed sample points

Case No.  6

Niter : 2500
n : 500
d : 4
m : 2
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 2
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 0.000208 0.000481 0.00121 0.00331 0.00827 0.0487
wkde 0.00021 0.000486 0.00122 0.00319 0.00689 0.0398
wkdeA 0.00021 0.000486 0.00121 0.00319 0.00694 0.0396
fill+wkde 0.000205 0.000478 0.00121 0.00329 0.00775 0.0443
Table 12: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.025 0.0587 0.135 0.245 0.341 2.6
wkde 0.0182 0.0494 0.125 0.234 0.323 2.95
wkdeA 0.0188 0.0496 0.125 0.234 0.323 2.95
fill+wkde 0.0181 0.0485 0.124 0.228 0.302 1.73
Table 13: Error quantiles evaluating at the observed sample points

Case No.  7

Niter : 2500
n : 500
d : 4
m : 3
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 2
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 0.000182 0.000429 0.00112 0.00323 0.00812 0.0477
wkde 0.000182 0.000429 0.00112 0.00324 0.00815 0.0479
wkdeA 0.000182 0.000429 0.00112 0.00324 0.00815 0.0479
fill+wkde 0.000182 0.000429 0.00112 0.00324 0.00814 0.0477
Table 14: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.025 0.0614 0.141 0.254 0.355 2.65
wkde 0.0251 0.0616 0.142 0.254 0.357 2.69
wkdeA 0.0251 0.0616 0.142 0.254 0.357 2.69
fill+wkde 0.025 0.0614 0.141 0.254 0.355 2.65
Table 15: Error quantiles evaluating at the observed sample points

Case No.  8

Niter : 2500
n : 500
d : 4
m : 2
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 2
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 0.000182 0.000433 0.00113 0.00321 0.00806 0.0478
wkde 0.000184 0.000439 0.00114 0.00304 0.00702 0.0392
wkdeA 0.000184 0.000439 0.00114 0.00305 0.00707 0.039
fill+wkde 0.000179 0.000426 0.00113 0.00324 0.00769 0.0417
Table 16: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0251 0.0616 0.141 0.254 0.355 2.73
wkde 0.0211 0.0552 0.131 0.243 0.337 3.06
wkdeA 0.0218 0.0554 0.131 0.243 0.338 3.07
fill+wkde 0.0184 0.0514 0.13 0.235 0.314 1.71
Table 17: Error quantiles evaluating at the observed sample points

Case No.  9

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha : 0 0 0 0
dp1$nu : 2
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00304 0.00495 0.00858 0.0182 0.0268 0.0472
wkde 0.00304 0.00496 0.00861 0.0183 0.027 0.0475
wkdeA 0.00304 0.00496 0.00861 0.0183 0.027 0.0475
fill+wkde 0.00304 0.00495 0.00857 0.0182 0.0268 0.0471
Table 18: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0158 0.0326 0.0707 0.185 0.506 5.55
wkde 0.0158 0.0328 0.0712 0.186 0.509 5.57
wkdeA 0.0158 0.0328 0.0713 0.187 0.509 5.57
fill+wkde 0.0157 0.0325 0.0706 0.185 0.504 5.52
Table 19: Error quantiles evaluating at the observed sample points

Case No.  10

Niter : 2500
n : 500
d : 4
m : 2
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00071 0.00213 0.00624 0.0197 0.0349 0.0938
wkde 0.000712 0.00215 0.00614 0.018 0.0306 0.0815
wkdeA 0.000712 0.00213 0.00615 0.018 0.0308 0.0812
fill+wkde 0.000976 0.00403 0.0142 0.0364 0.0585 0.125
Table 20: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0152 0.0334 0.0643 0.113 0.164 0.388
wkde 0.0149 0.0329 0.0617 0.103 0.146 0.335
wkdeA 0.0151 0.0327 0.0601 0.1 0.143 0.333
fill+wkde 0.0108 0.0228 0.0406 0.0715 0.106 0.239
Table 21: Error quantiles evaluating at the observed sample points

Case No.  11

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00071 0.00214 0.00625 0.0198 0.0349 0.0942
wkde 0.000712 0.00218 0.00622 0.0195 0.0342 0.0937
wkdeA 0.000711 0.00215 0.00623 0.0196 0.0344 0.0931
fill+wkde 0.000646 0.00257 0.00921 0.026 0.0435 0.105
Table 22: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0152 0.0334 0.0643 0.113 0.164 0.389
wkde 0.0153 0.0338 0.0651 0.114 0.164 0.387
wkdeA 0.0153 0.0337 0.0648 0.113 0.162 0.382
fill+wkde 0.0126 0.0275 0.0532 0.0979 0.143 0.336
Table 23: Error quantiles evaluating at the observed sample points

Case No.  12

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00143 0.0127 3.02E+12 8.48E+53 3.35E+90 Inf
wkde 0.00144 0.0127 2.94E+12 7.96E+53 2.92E+90 Inf
wkdeA 0.00143 0.0127 3.05E+12 8.25E+53 3.1E+90 Inf
fill+wkde 0.00165 0.016 1.55E+14 1.08E+60 4.45E+100 Inf
Table 24: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0198 0.0428 0.0796 0.14 0.206 0.49
wkde 0.0197 0.0428 0.0805 0.141 0.207 0.488
wkdeA 0.0198 0.043 0.0801 0.141 0.205 0.485
fill+wkde 0.0177 0.0385 0.0718 0.126 0.187 0.439
Table 25: Error quantiles evaluating at the observed sample points

Case No.  13

Niter : 2500
n : 500
d : 4
m : 2
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00143 0.0127 2.91E+12 8.23E+53 3.23E+90 Inf
wkde 0.00141 0.0121 1.94E+12 5.96E+53 2.23E+90 Inf
wkdeA 0.00141 0.0121 2.17E+12 6.34E+53 2.42E+90 Inf
fill+wkde 0.00219 0.023 2.33E+16 1.55E+63 3.62E+105 Inf
Table 26: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0197 0.0426 0.0794 0.14 0.205 0.487
wkde 0.0189 0.0414 0.0778 0.135 0.195 0.458
wkdeA 0.019 0.0413 0.0761 0.131 0.19 0.448
fill+wkde 0.0157 0.0341 0.0621 0.0997 0.144 0.336
Table 27: Error quantiles evaluating at the observed sample points

Case No.  14

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 2.01E-05 0.00195 0.033 0.283 1.31 122
wkde 1.98E-05 0.00193 0.0325 0.269 1.2 104
wkdeA 1.99E-05 0.00194 0.0327 0.273 1.23 107
fill+wkde 8.63E-05 0.00794 0.429 5.43E+03 3.73E+06 1.02E+12
Table 28: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0265 0.0561 0.0977 0.158 0.228 0.5
wkde 0.0265 0.0562 0.0977 0.157 0.227 0.498
wkdeA 0.0265 0.0562 0.0978 0.157 0.227 0.498
fill+wkde 0.0266 0.0562 0.0973 0.155 0.223 0.49
Table 29: Error quantiles evaluating at the observed sample points

Case No.  15

Niter : 2500
n : 500
d : 4
m : 2
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 2E-05 0.00194 0.0328 0.274 1.2 203
wkde 1.55E-05 0.00159 0.0267 0.202 0.742 28.1
wkdeA 1.66E-05 0.00169 0.0284 0.218 0.831 38.7
fill+wkde 0.00882 0.391 9.27E+03 3.55E+10 1.78E+15 6.86E+25
Table 30: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0265 0.0561 0.0976 0.158 0.229 0.503
wkde 0.0249 0.053 0.0928 0.149 0.215 0.472
wkdeA 0.0249 0.0527 0.092 0.148 0.211 0.458
fill+wkde 0.0257 0.0543 0.0926 0.138 0.191 0.423
Table 31: Error quantiles evaluating at the observed sample points

Case No.  16

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 3.11E-07 0.000968 4.54E+07 3.54E+48 3.09E+82 Inf
wkde 3.1E-07 0.000953 4.43E+07 3.39E+48 3.02E+82 Inf
wkdeA 3.11E-07 0.00096 4.48E+07 3.48E+48 3.03E+82 Inf
fill+wkde 7.65E-07 0.00272 7.55E+08 1.14E+53 1.31E+88 Inf
Table 32: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0312 0.0665 0.117 0.184 0.257 0.58
wkde 0.0306 0.0653 0.115 0.182 0.256 0.58
wkdeA 0.0306 0.0654 0.116 0.182 0.255 0.579
fill+wkde 0.0306 0.0653 0.115 0.179 0.25 0.568
Table 33: Error quantiles evaluating at the observed sample points

Case No.  17

Niter : 2500
n : 500
d : 4
m : 2
qN : 3
Npts : 1296
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 3.1E-07 0.00096 4.36E+07 3.15E+48 1.37E+82 Inf
wkde 2.85E-07 0.000778 2.73E+07 1.89E+48 8.26E+81 Inf
wkdeA 2.9E-07 0.000807 3.02E+07 2.14E+48 8.73E+81 Inf
fill+wkde 0.000213 0.0343 9.94E+13 7.86E+59 3.93E+99 Inf
Table 34: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0312 0.0665 0.117 0.184 0.257 0.574
wkde 0.0274 0.0592 0.106 0.168 0.235 0.531
wkdeA 0.0273 0.0588 0.105 0.165 0.229 0.518
fill+wkde 0.0274 0.059 0.105 0.16 0.211 0.475
Table 35: Error quantiles evaluating at the observed sample points

Case No.  18

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
     [,1] [,2] [,3]
[1,] 1.00  0.8 0.32
[2,] 0.80  1.0 0.40
[3,] 0.32  0.4 1.00
dp1$alpha : 6 3 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 5.5E-06 0.00257 0.0482 5.2E+03 3.79E+08 4.83E+14
wkde 5.23E-06 0.00243 0.0436 5.26E+03 2.59E+08 3.08E+14
wkdeA 5.32E-06 0.00244 0.0438 4.1E+03 2.66E+08 3.18E+14
fill+wkde 2.73E-05 0.0091 0.149 6.56E+07 3.94E+13 1.33E+22
Table 36: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0256 0.0541 0.0941 0.144 0.189 0.357
wkde 0.0238 0.0505 0.0882 0.135 0.177 0.335
wkdeA 0.0234 0.0497 0.0866 0.133 0.173 0.327
fill+wkde 0.0239 0.0504 0.0872 0.132 0.17 0.324
Table 37: Error quantiles evaluating at the observed sample points

Case No.  19

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
     [,1] [,2] [,3]
[1,] 1.00  0.8 0.32
[2,] 0.80  1.0 0.40
[3,] 0.32  0.4 1.00
dp1$alpha : 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 8.01E-05 0.00341 0.0216 0.0572 0.0879 0.187
wkde 7.23E-05 0.00298 0.0193 0.0501 0.0767 0.156
wkdeA 7.29E-05 0.00303 0.0195 0.0501 0.0762 0.153
fill+wkde 0.00148 0.0128 0.0468 0.146 0.475 33.2
Table 38: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0198 0.0415 0.0713 0.108 0.141 0.258
wkde 0.0186 0.039 0.0669 0.101 0.13 0.237
wkdeA 0.0184 0.0386 0.0662 0.0992 0.128 0.232
fill+wkde 0.019 0.0397 0.0674 0.0991 0.126 0.232
Table 39: Error quantiles evaluating at the observed sample points

Case No.  20

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
     [,1] [,2] [,3]
[1,]  1.0  0.8 -0.1
[2,]  0.8  1.0  0.4
[3,] -0.1  0.4  1.0
dp1$alpha : 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 3.96E-05 0.00577 0.0462 0.143 0.248 1.12
wkde 3.3E-05 0.00534 0.0436 0.131 0.219 0.777
wkdeA 3.47E-05 0.00544 0.0441 0.133 0.221 0.785
fill+wkde 0.0117 0.105 23.3 1.39E+05 1.16E+08 2.87E+15
Table 40: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0308 0.0642 0.108 0.156 0.202 0.367
wkde 0.0304 0.0634 0.106 0.154 0.197 0.347
wkdeA 0.0303 0.0632 0.106 0.153 0.196 0.342
fill+wkde 0.0311 0.0647 0.108 0.154 0.194 0.343
Table 41: Error quantiles evaluating at the observed sample points

Case No.  21

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
       [,1] [,2]   [,3]
[1,] 1.0000 0.75 0.5625
[2,] 0.7500 1.00 0.7500
[3,] 0.5625 0.75 1.0000
dp1$alpha : 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 9E-06 0.00113 0.0174 0.0557 0.0893 0.194
wkde 8.21E-06 0.000996 0.0153 0.0489 0.0784 0.164
wkdeA 8.28E-06 0.00101 0.0155 0.049 0.078 0.161
fill+wkde 0.000244 0.00794 0.0425 0.138 0.354 10.8
Table 42: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0224 0.047 0.0803 0.119 0.154 0.278
wkde 0.0209 0.0439 0.0752 0.112 0.143 0.255
wkdeA 0.0207 0.0435 0.0744 0.11 0.14 0.25
fill+wkde 0.0214 0.0449 0.0761 0.111 0.139 0.25
Table 43: Error quantiles evaluating at the observed sample points

Case No.  22

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1
dp1$alpha : 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00828 0.0143 0.0246 0.0434 0.0588 0.0937
wkde 0.00872 0.0147 0.0239 0.0399 0.0559 0.1
wkdeA 0.00853 0.0144 0.0235 0.0388 0.0538 0.0949
fill+wkde 0.0081 0.014 0.0232 0.0366 0.05 0.0867
Table 44: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0157 0.0329 0.0573 0.0914 0.139 0.474
wkde 0.0134 0.0288 0.0524 0.0858 0.125 0.454
wkdeA 0.0133 0.0285 0.0513 0.0832 0.122 0.448
fill+wkde 0.0107 0.0234 0.0434 0.07 0.0988 0.318
Table 45: Error quantiles evaluating at the observed sample points

Case No.  23

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
       [,1] [,2]   [,3]
[1,] 1.0000 0.75 0.5625
[2,] 0.7500 1.00 0.7500
[3,] 0.5625 0.75 1.0000
dp1$alpha : 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00369 0.00857 0.0224 0.0532 0.0811 0.155
wkde 0.00373 0.00845 0.0194 0.0419 0.0652 0.126
wkdeA 0.00371 0.00834 0.0192 0.0415 0.064 0.122
fill+wkde 0.00354 0.00836 0.0221 0.0496 0.075 0.144
Table 46: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0275 0.0576 0.098 0.144 0.197 0.639
wkde 0.0229 0.0492 0.0863 0.131 0.181 0.61
wkdeA 0.0232 0.0492 0.0854 0.129 0.178 0.607
fill+wkde 0.024 0.0512 0.0894 0.132 0.167 0.517
Table 47: Error quantiles evaluating at the observed sample points

Case No.  24

Niter : 2500
n : 500
d : 3
m : 2
qN : 3
Npts : 1000
dp1$xi : 0 0 0
dp1$Omega :
       [,1] [,2]   [,3]
[1,] 1.0000 0.75 0.5625
[2,] 0.7500 1.00 0.7500
[3,] 0.5625 0.75 1.0000
dp1$alpha :  3  6 -6
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000145 0.00468 0.0195 0.0651 0.126 0.422
wkde 0.000145 0.00462 0.0184 0.0546 0.109 0.388
wkdeA 0.000145 0.00461 0.0183 0.0543 0.108 0.386
fill+wkde 0.000524 0.00638 0.027 0.101 0.264 1.78
Table 48: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0363 0.0771 0.137 0.21 0.274 0.81
wkde 0.0311 0.0685 0.126 0.198 0.26 0.784
wkdeA 0.0311 0.0681 0.125 0.196 0.258 0.782
fill+wkde 0.0318 0.0702 0.131 0.205 0.256 0.689
Table 49: Error quantiles evaluating at the observed sample points

Case No.  25

Niter : 2500
n : 500
d : 5
m : 2
qN : 3
Npts : 3125
dp1$xi : 0 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3  0 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 4.95E-10 1.9E-05 1.29E+07 1.84E+53 2.87E+84 Inf
wkde 4.16E-10 1.39E-05 6.14E+06 5.23E+52 1.83E+84 Inf
wkdeA 4.35E-10 1.52E-05 7.75E+06 8.53E+52 2.11E+84 Inf
fill+wkde 0.000446 0.143 6.02E+15 2.72E+70 5.5E+102 Inf
Table 50: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.031 0.0671 0.125 0.23 0.353 0.901
wkde 0.0253 0.0561 0.108 0.205 0.32 0.837
wkdeA 0.0262 0.0573 0.109 0.204 0.317 0.833
fill+wkde 0.0238 0.052 0.0959 0.16 0.245 0.65
Table 51: Error quantiles evaluating at the observed sample points

Case No.  26

Niter : 2500
n : 500
d : 5
m : 3
qN : 3
Npts : 3125
dp1$xi : 0 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3  0 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 4.73E-10 1.78E-05 9.03E+06 6.8E+52 1.45E+84 Inf
wkde 4.6E-10 1.69E-05 8.12E+06 6.19E+52 1.27E+84 Inf
wkdeA 4.66E-10 1.74E-05 8.65E+06 6.6E+52 1.38E+84 Inf
fill+wkde 8.95E-08 0.000937 5.11E+11 8.65E+60 1.23E+95 Inf
Table 52: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.031 0.0672 0.125 0.231 0.353 0.898
wkde 0.0296 0.0647 0.122 0.229 0.352 0.91
wkdeA 0.0302 0.0657 0.123 0.229 0.353 0.907
fill+wkde 0.0286 0.062 0.115 0.208 0.323 0.834
Table 53: Error quantiles evaluating at the observed sample points

Case No.  27

Niter : 2500
n : 500
d : 5
m : 2
qN : 6
Npts : 3125
dp1$xi : 2 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,]  1.00000  0.60714 -0.04464 -0.33705 -0.23047
[2,]  0.60714  1.00000  0.60714 -0.04464 -0.33705
[3,] -0.04464  0.60714  1.00000  0.60714 -0.04464
[4,] -0.33705 -0.04464  0.60714  1.00000  0.60714
[5,] -0.23047 -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3  0 -6 -3
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  0  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 3.31E-06 3.14E-05 0.000159 0.00115 0.00416 0.0373
wkde 3.72E-06 3.37E-05 0.000163 0.00113 0.00361 0.0272
wkdeA 3.7E-06 3.39E-05 0.000163 0.00113 0.00359 0.0273
fill+wkde 3.4E-06 3.24E-05 0.000162 0.0012 0.0042 0.0742
Table 54: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0274 0.0619 0.115 0.176 0.249 3.42
wkde 0.0288 0.0638 0.116 0.174 0.223 3.57
wkdeA 0.0295 0.0643 0.116 0.174 0.226 3.66
fill+wkde 0.026 0.0619 0.115 0.17 0.208 1.74
Table 55: Error quantiles evaluating at the observed sample points

Case No.  28

Niter : 2500
n : 500
d : 5
m : 3
qN : 6
Npts : 3125
dp1$xi : 2 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,]  1.00000  0.60714 -0.04464 -0.33705 -0.23047
[2,]  0.60714  1.00000  0.60714 -0.04464 -0.33705
[3,] -0.04464  0.60714  1.00000  0.60714 -0.04464
[4,] -0.33705 -0.04464  0.60714  1.00000  0.60714
[5,] -0.23047 -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3  0 -6 -3
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  0  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 3.44E-06 3.15E-05 0.000149 0.00099 0.00317 0.0243
wkde 3.43E-06 3.15E-05 0.000149 0.000987 0.00316 0.0245
wkdeA 3.43E-06 3.15E-05 0.000149 0.000987 0.00316 0.0245
fill+wkde 3.47E-06 3.19E-05 0.00015 0.000996 0.00319 0.0269
Table 56: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0275 0.0621 0.115 0.176 0.25 3.34
wkde 0.0277 0.0625 0.116 0.177 0.253 3.46
wkdeA 0.0277 0.0625 0.116 0.177 0.253 3.46
fill+wkde 0.0275 0.0624 0.116 0.177 0.247 3.32
Table 57: Error quantiles evaluating at the observed sample points

Case No.  29

Niter : 2500
n : 500
d : 5
m : 2
qN : 6
Npts : 7776
dp1$xi : 2 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,]  1.00000  0.60714 -0.04464 -0.33705 -0.23047
[2,]  0.60714  1.00000  0.60714 -0.04464 -0.33705
[3,] -0.04464  0.60714  1.00000  0.60714 -0.04464
[4,] -0.33705 -0.04464  0.60714  1.00000  0.60714
[5,] -0.23047 -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3  0 -6 -3
dp1$nu : 5
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  0  6  3
dp2$nu : 5
25% 50% 75% 90% 95% 99%
kde 3.61E-06 1.26E-05 4.88E-05 0.000173 0.000381 0.00405
wkde 3.61E-06 1.26E-05 4.88E-05 0.000173 0.000376 0.00349
wkdeA 3.61E-06 1.26E-05 4.88E-05 0.000173 0.000377 0.00359
fill+wkde 3.68E-06 1.32E-05 5.59E-05 0.000281 0.00115 0.0201
Table 58: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0231 0.0531 0.107 0.184 0.248 0.889
wkde 0.0215 0.05 0.102 0.174 0.23 0.616
wkdeA 0.0222 0.0506 0.102 0.174 0.231 0.629
fill+wkde 0.021 0.0494 0.102 0.173 0.228 0.548
Table 59: Error quantiles evaluating at the observed sample points

Case No.  30

Niter : 2500
n : 500
d : 5
m : 3
qN : 6
Npts : 7776
dp1$xi : 2 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,]  1.00000  0.60714 -0.04464 -0.33705 -0.23047
[2,]  0.60714  1.00000  0.60714 -0.04464 -0.33705
[3,] -0.04464  0.60714  1.00000  0.60714 -0.04464
[4,] -0.33705 -0.04464  0.60714  1.00000  0.60714
[5,] -0.23047 -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3  0 -6 -3
dp1$nu : 5
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  0  6  3
dp2$nu : 5
25% 50% 75% 90% 95% 99%
kde 3.61E-06 1.26E-05 4.88E-05 0.000173 0.000381 0.00403
wkde 3.61E-06 1.26E-05 4.88E-05 0.000173 0.000381 0.004
wkdeA 3.61E-06 1.26E-05 4.88E-05 0.000173 0.000381 0.00401
fill+wkde 3.61E-06 1.27E-05 5E-05 0.00019 0.000471 0.00742
Table 60: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0232 0.0532 0.108 0.184 0.248 0.869
wkde 0.0231 0.0529 0.107 0.183 0.247 0.865
wkdeA 0.0232 0.0531 0.107 0.183 0.247 0.868
fill+wkde 0.0229 0.0528 0.107 0.183 0.246 0.844
Table 61: Error quantiles evaluating at the observed sample points

Case No.  31

Niter : 2500
n : 500
d : 5
m : 2
qN : 6
Npts : 7776
dp1$xi : 2 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,]  1.00000  0.60714 -0.04464 -0.33705 -0.23047
[2,]  0.60714  1.00000  0.60714 -0.04464 -0.33705
[3,] -0.04464  0.60714  1.00000  0.60714 -0.04464
[4,] -0.33705 -0.04464  0.60714  1.00000  0.60714
[5,] -0.23047 -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3  0 -6 -3
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  0  6  3
25% 50% 75% 90% 95% 99%
kde 3.83E-21 1.02E-11 0.00701 5.73E+18 8.16E+41 Inf
wkde 3.54E-21 9.36E-12 0.0064 5.27E+18 6.29E+41 Inf
wkdeA 3.69E-21 9.87E-12 0.0067 5.8E+18 1E+42 Inf
fill+wkde 5.42E-11 0.0174 1.67E+11 1.76E+37 1.27E+65 Inf
Table 62: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0229 0.0501 0.0902 0.135 0.164 0.226
wkde 0.0226 0.0497 0.0896 0.135 0.163 0.222
wkdeA 0.0228 0.05 0.0901 0.135 0.164 0.227
fill+wkde 0.0227 0.05 0.0901 0.135 0.164 0.221
Table 63: Error quantiles evaluating at the observed sample points

Case No.  32

Niter : 2500
n : 500
d : 5
m : 3
qN : 6
Npts : 7776
dp1$xi : 2 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,]  1.00000  0.60714 -0.04464 -0.33705 -0.23047
[2,]  0.60714  1.00000  0.60714 -0.04464 -0.33705
[3,] -0.04464  0.60714  1.00000  0.60714 -0.04464
[4,] -0.33705 -0.04464  0.60714  1.00000  0.60714
[5,] -0.23047 -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3  0 -6 -3
mix.p : 0.6667
dp2$xi : -2 -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  0  6  3
25% 50% 75% 90% 95% 99%
kde 2.94E-21 9.36E-12 0.00714 4.98E+18 5.1E+41 Inf
wkde 2.89E-21 9.29E-12 0.00707 4.93E+18 4.93E+41 Inf
wkdeA 2.93E-21 9.33E-12 0.00711 4.89E+18 4.98E+41 Inf
fill+wkde 3.05E-16 5.17E-07 1.99E+04 1.01E+28 1.47E+55 Inf
Table 64: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0229 0.0501 0.0903 0.135 0.164 0.227
wkde 0.0229 0.0501 0.0903 0.135 0.164 0.227
wkdeA 0.0229 0.05 0.0903 0.135 0.164 0.227
fill+wkde 0.0229 0.0501 0.0904 0.136 0.164 0.226
Table 65: Error quantiles evaluating at the observed sample points

Case No.  33

Niter : 2500
n : 1000
d : 5
m : 3
qN : 3
Npts : 3125
dp1$xi : 0 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3  0 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 1.79E-10 5.7E-06 1.02E+04 3.89E+46 3.37E+78 Inf
wkde 1.73E-10 5.59E-06 9.04E+03 3.35E+46 3.01E+78 Inf
wkdeA 1.76E-10 5.64E-06 9.62E+03 3.51E+46 3.11E+78 Inf
fill+wkde 1.37E-08 0.000521 7.61E+09 2.95E+56 2.3E+92 Inf
Table 66: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0267 0.0578 0.106 0.188 0.287 0.728
wkde 0.0245 0.0536 0.101 0.185 0.285 0.744
wkdeA 0.0251 0.0545 0.102 0.184 0.284 0.738
fill+wkde 0.0236 0.0514 0.0944 0.162 0.251 0.657
Table 67: Error quantiles evaluating at the observed sample points

Case No.  34

Niter : 2500
n : 250
d : 5
m : 3
qN : 3
Npts : 3125
dp1$xi : 0 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3  0 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 3.71E-09 8.1E-05 1.58E+10 4.49E+58 1.18E+89 Inf
wkde 3.58E-09 7.9E-05 1.51E+10 4.04E+58 1.14E+89 Inf
wkdeA 3.65E-09 8.01E-05 1.53E+10 4.28E+58 1.14E+89 Inf
fill+wkde 5.88E-07 0.00162 1.1E+13 5.75E+64 1.33E+98 Inf
Table 68: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0361 0.0783 0.149 0.284 0.435 1.09
wkde 0.0355 0.0773 0.148 0.285 0.438 1.11
wkdeA 0.036 0.0779 0.149 0.285 0.437 1.11
fill+wkde 0.0345 0.0748 0.141 0.268 0.414 1.05
Table 69: Error quantiles evaluating at the observed sample points

Case No.  35

Niter : 2500
n : 250
d : 4
m : 3
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000683 0.00247 0.00828 0.0239 0.0424 0.119
wkde 0.000686 0.00248 0.00826 0.0236 0.0417 0.119
wkdeA 0.000684 0.00247 0.00828 0.0238 0.0422 0.119
fill+wkde 0.000698 0.00282 0.0098 0.0288 0.0502 0.134
Table 70: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0222 0.049 0.0943 0.165 0.243 0.579
wkde 0.022 0.049 0.0951 0.167 0.245 0.582
wkdeA 0.0222 0.0494 0.0949 0.166 0.243 0.578
fill+wkde 0.0184 0.041 0.0825 0.15 0.221 0.522
Table 71: Error quantiles evaluating at the observed sample points

Case No.  36

Niter : 2500
n : 250
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000685 0.00247 0.00826 0.0238 0.0423 0.119
wkde 0.000691 0.00247 0.00807 0.0224 0.0388 0.11
wkdeA 0.000687 0.00245 0.00808 0.0225 0.0391 0.109
fill+wkde 0.000857 0.00372 0.014 0.0398 0.0674 0.176
Table 72: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0225 0.0493 0.0941 0.165 0.243 0.586
wkde 0.0216 0.0478 0.0909 0.157 0.229 0.542
wkdeA 0.0226 0.0486 0.0899 0.154 0.225 0.535
fill+wkde 0.0145 0.0312 0.0595 0.112 0.168 0.39
Table 73: Error quantiles evaluating at the observed sample points

Case No.  37

Niter : 2500
n : 1000
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha : 0 0 0 0
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000652 0.00225 0.00714 0.019 0.0321 0.0787
wkde 0.000654 0.00224 0.00676 0.0164 0.0268 0.0644
wkdeA 0.000653 0.00224 0.00673 0.0164 0.0268 0.0637
fill+wkde 0.00225 0.00766 0.0214 0.0467 0.0718 0.149
Table 74: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0111 0.024 0.045 0.0779 0.112 0.262
wkde 0.0108 0.0235 0.0433 0.0699 0.095 0.209
wkdeA 0.0107 0.023 0.0414 0.0667 0.0922 0.208
fill+wkde 0.00844 0.0177 0.0303 0.0491 0.0729 0.156
Table 75: Error quantiles evaluating at the observed sample points

Case No.  38

Niter : 2500
n : 250
d : 4
m : 2
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 5
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 5
25% 50% 75% 90% 95% 99%
kde 2.89E-05 9.71E-05 0.000349 0.00119 0.00318 0.0473
wkde 2.89E-05 9.71E-05 0.000349 0.00118 0.00307 0.0445
wkdeA 2.89E-05 9.71E-05 0.000349 0.00118 0.00309 0.0446
fill+wkde 2.9E-05 9.93E-05 0.000372 0.00168 0.00613 0.0651
Table 76: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0309 0.0677 0.127 0.199 0.25 0.708
wkde 0.0297 0.0654 0.124 0.194 0.243 0.595
wkdeA 0.0301 0.0658 0.124 0.194 0.243 0.605
fill+wkde 0.0295 0.0652 0.124 0.194 0.242 0.569
Table 77: Error quantiles evaluating at the observed sample points

Case No.  39

Niter : 2500
n : 250
d : 4
m : 2
qN : 6
Npts : 1296
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 2
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
dp2$nu : 2
25% 50% 75% 90% 95% 99%
kde 0.000209 0.00049 0.00127 0.00378 0.0102 0.0589
wkde 0.000209 0.000491 0.00126 0.00352 0.00854 0.0527
wkdeA 0.000209 0.00049 0.00126 0.00354 0.00866 0.0526
fill+wkde 0.000205 0.00049 0.0013 0.00416 0.0115 0.0606
Table 78: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0282 0.0663 0.148 0.259 0.366 2.95
wkde 0.0233 0.0589 0.139 0.248 0.339 2.99
wkdeA 0.0242 0.0593 0.139 0.249 0.341 3
fill+wkde 0.0225 0.0577 0.138 0.244 0.323 2.16
Table 79: Error quantiles evaluating at the observed sample points

Case No.  40

Niter : 2500
n : 250
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00198 0.016 4.96E+11 3.54E+51 3.73E+82 Inf
wkde 0.00198 0.0156 4.06E+11 2.84E+51 2.83E+82 Inf
wkdeA 0.00198 0.0156 4.34E+11 3.07E+51 3.15E+82 Inf
fill+wkde 0.00243 0.0278 9.96E+13 4.85E+58 4.97E+93 Inf
Table 80: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0264 0.0573 0.11 0.197 0.292 0.696
wkde 0.0251 0.0554 0.107 0.193 0.284 0.679
wkdeA 0.0257 0.0559 0.106 0.189 0.278 0.663
fill+wkde 0.0201 0.0435 0.0803 0.142 0.214 0.509
Table 81: Error quantiles evaluating at the observed sample points

Case No.  41

Niter : 2500
n : 1000
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00189 0.0135 3.41E+10 2.95E+47 1.92E+76 Inf
wkde 0.0019 0.013 2.43E+10 2.13E+47 1.38E+76 Inf
wkdeA 0.00189 0.013 2.58E+10 2.27E+47 1.45E+76 Inf
fill+wkde 0.00348 0.0329 1.26E+14 5.02E+58 4.16E+93 Inf
Table 82: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0154 0.0332 0.0604 0.102 0.148 0.351
wkde 0.015 0.0327 0.0599 0.0983 0.138 0.316
wkdeA 0.015 0.0324 0.0581 0.0946 0.134 0.311
fill+wkde 0.0128 0.028 0.0517 0.0801 0.105 0.235
Table 83: Error quantiles evaluating at the observed sample points

Case No.  42

Niter : 2500
n : 250
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 1.51E-06 0.00317 6.56E+09 2.19E+49 9.61E+79 Inf
wkde 1.3E-06 0.00267 4.83E+09 1.74E+49 7.73E+79 Inf
wkdeA 1.34E-06 0.00279 5.23E+09 1.84E+49 7.78E+79 Inf
fill+wkde 0.0011 0.096 3.54E+13 5.81E+57 1.99E+93 Inf
Table 84: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0366 0.078 0.138 0.224 0.323 0.739
wkde 0.0334 0.0718 0.129 0.21 0.302 0.693
wkdeA 0.0335 0.0717 0.128 0.207 0.297 0.68
fill+wkde 0.033 0.0707 0.125 0.194 0.271 0.627
Table 85: Error quantiles evaluating at the observed sample points

Case No.  43

Niter : 2500
n : 1000
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 2.78E-07 0.000777 5.58E+04 1.12E+40 1.22E+67 Inf
wkde 2.53E-07 0.000672 3E+04 6.66E+39 1.09E+67 Inf
wkdeA 2.57E-07 0.000686 3.3E+04 7.05E+39 1.12E+67 Inf
fill+wkde 0.000119 0.027 1.64E+11 4.85E+53 1.44E+87 Inf
Table 86: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0264 0.0563 0.0991 0.153 0.207 0.462
wkde 0.0225 0.0487 0.0877 0.137 0.185 0.417
wkdeA 0.0222 0.048 0.0858 0.134 0.181 0.41
fill+wkde 0.0228 0.0494 0.0887 0.135 0.172 0.373
Table 87: Error quantiles evaluating at the observed sample points

Case No.  44

Niter : 2500
n : 1000
d : 5
m : 2
qN : 3
Npts : 3125
dp1$xi : 0 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3  0 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 1.79E-10 5.72E-06 1.17E+04 4.81E+46 3.23E+78 Inf
wkde 1.46E-10 5.02E-06 4.71E+03 1.16E+46 1.74E+78 Inf
wkdeA 1.53E-10 5.17E-06 5.95E+03 1.59E+46 1.96E+78 Inf
fill+wkde 0.000183 0.0602 3.59E+14 3.77E+67 6.75E+99 Inf
Table 88: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0267 0.0577 0.106 0.188 0.287 0.73
wkde 0.0203 0.0455 0.0883 0.163 0.254 0.667
wkdeA 0.0208 0.0459 0.0875 0.161 0.252 0.66
fill+wkde 0.0196 0.0433 0.08 0.13 0.187 0.501
Table 89: Error quantiles evaluating at the observed sample points

Case No.  45

Niter : 5000
n : 250
d : 5
m : 2
qN : 3
Npts : 3125
dp1$xi : 0 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 1.0000 0.7500 0.5625 0.4219 0.3164
[2,] 0.7500 1.0000 0.7500 0.5625 0.4219
[3,] 0.5625 0.7500 1.0000 0.7500 0.5625
[4,] 0.4219 0.5625 0.7500 1.0000 0.7500
[5,] 0.3164 0.4219 0.5625 0.7500 1.0000
dp1$alpha :  6  3  0 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 3.71E-09 8.17E-05 1.11E+10 2.47E+58 1.55E+89 Inf
wkde 2.76E-09 6.56E-05 6.99E+09 1.34E+58 1.15E+89 Inf
wkdeA 3.07E-09 7.12E-05 8.5E+09 1.7E+58 1.22E+89 Inf
fill+wkde 0.0014 0.695 1.19E+17 6.06E+71 1.77E+105 Inf
Table 90: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0361 0.0784 0.15 0.284 0.436 1.11
wkde 0.0314 0.0692 0.135 0.263 0.41 1.06
wkdeA 0.0326 0.0713 0.137 0.263 0.408 1.05
fill+wkde 0.0291 0.0632 0.116 0.206 0.324 0.838
Table 91: Error quantiles evaluating at the observed sample points

Case No.  46

Niter : 10000
n : 250
d : 4
m : 2
qN : 6
Npts : 2401
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
25% 50% 75% 90% 95% 99%
kde 5.63E-14 4.35E-07 0.112 3.97E+11 1.62E+25 1.01E+55
wkde 5.57E-14 4.32E-07 0.11 3.82E+11 1.33E+25 1.01E+55
wkdeA 5.55E-14 4.29E-07 0.11 3.82E+11 1.51E+25 9.41E+54
fill+wkde 6.93E-11 0.000103 3.38E+03 1.09E+19 2.37E+37 2.28E+73
Table 92: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0307 0.0652 0.112 0.157 0.185 0.26
wkde 0.0307 0.0651 0.112 0.157 0.185 0.261
wkdeA 0.0305 0.0648 0.111 0.156 0.184 0.258
fill+wkde 0.0308 0.0654 0.112 0.158 0.185 0.26
Table 93: Error quantiles evaluating at the observed sample points

Case No.  47

Niter : 2500
n : 1000
d : 4
m : 2
qN : 6
Npts : 2401
dp1$xi : 2 2 2 2
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
mix.p : 0.3333
dp2$xi : -2 -2 -2 -2
dp2$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp2$alpha : -6 -3  6  3
25% 50% 75% 90% 95% 99%
kde 1.73E-25 6.07E-15 1.58E-07 0.0214 160 2.42E+15
wkde 1.66E-25 5.72E-15 1.5E-07 0.0201 145 8.08E+14
wkdeA 1.68E-25 5.84E-15 1.53E-07 0.0206 150 1.44E+15
fill+wkde 7.64E-20 7.15E-11 0.00147 2.77E+08 5.19E+21 1.24E+49
Table 94: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0229 0.0492 0.0857 0.124 0.148 0.209
wkde 0.0225 0.0483 0.0844 0.122 0.146 0.207
wkdeA 0.0226 0.0485 0.0846 0.122 0.146 0.205
fill+wkde 0.0226 0.0486 0.085 0.123 0.147 0.206
Table 95: Error quantiles evaluating at the observed sample points

Case No.  48

Niter : 2500
n : 250
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha : 0 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000881 0.00218 0.00624 0.02 0.0424 0.123
wkde 0.000897 0.00221 0.00587 0.016 0.0319 0.1
wkdeA 0.000893 0.0022 0.00586 0.0163 0.0325 0.0983
fill+wkde 0.000826 0.00265 0.0106 0.037 0.0715 0.199
Table 96: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0358 0.0763 0.137 0.25 0.461 2.11
wkde 0.0286 0.0629 0.118 0.221 0.42 2.09
wkdeA 0.0303 0.0647 0.118 0.22 0.421 2.09
fill+wkde 0.0259 0.0575 0.109 0.181 0.295 1.37
Table 97: Error quantiles evaluating at the observed sample points

Case No.  49

Niter : 2500
n : 500
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha : 0 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00089 0.00221 0.00612 0.018 0.0374 0.108
wkde 0.000908 0.00227 0.00588 0.0141 0.0257 0.0839
wkdeA 0.000906 0.00226 0.00583 0.0141 0.0257 0.0815
fill+wkde 0.000833 0.00229 0.00728 0.0231 0.0462 0.135
Table 98: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0305 0.0651 0.116 0.204 0.375 1.62
wkde 0.0217 0.0495 0.0956 0.178 0.342 1.73
wkdeA 0.0226 0.0501 0.0947 0.175 0.34 1.72
fill+wkde 0.02 0.0452 0.0878 0.148 0.215 0.992
Table 99: Error quantiles evaluating at the observed sample points

Case No.  50

Niter : 2500
n : 1000
d : 4
m : 2
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha : 0 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000894 0.00223 0.00603 0.0166 0.0333 0.0951
wkde 0.000914 0.00232 0.00601 0.0134 0.0224 0.0737
wkdeA 0.000912 0.0023 0.00594 0.0132 0.0222 0.0713
fill+wkde 0.000863 0.00212 0.00569 0.014 0.0235 0.059
Table 100: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0259 0.0549 0.0978 0.167 0.3 1.33
wkde 0.0166 0.0392 0.0786 0.148 0.285 1.52
wkdeA 0.0168 0.0391 0.0773 0.145 0.282 1.5
fill+wkde 0.0157 0.0361 0.0714 0.123 0.166 0.733
Table 101: Error quantiles evaluating at the observed sample points

Case No.  51

Niter : 2500
n : 250
d : 4
m : 3
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha : 0 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000882 0.00219 0.00623 0.0199 0.0421 0.123
wkde 0.000882 0.00219 0.00623 0.0199 0.0422 0.124
wkdeA 0.000882 0.00219 0.00623 0.0199 0.0421 0.124
fill+wkde 0.000866 0.00215 0.00634 0.021 0.0436 0.125
Table 102: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0358 0.0763 0.137 0.251 0.46 2.08
wkde 0.0357 0.0761 0.137 0.254 0.467 2.13
wkdeA 0.0358 0.0762 0.137 0.253 0.465 2.12
fill+wkde 0.0353 0.0753 0.136 0.248 0.455 2.07
Table 103: Error quantiles evaluating at the observed sample points

Case No.  52

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
       [,1]   [,2]   [,3]   [,4]
[1,] 1.0000 0.7500 0.5625 0.4219
[2,] 0.7500 1.0000 0.7500 0.5625
[3,] 0.5625 0.7500 1.0000 0.7500
[4,] 0.4219 0.5625 0.7500 1.0000
dp1$alpha : 0 0 0 0
dp1$nu : 5
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00089 0.00221 0.00612 0.0181 0.0374 0.108
wkde 0.00089 0.00221 0.00612 0.0181 0.0378 0.111
wkdeA 0.00089 0.00221 0.00612 0.0181 0.0377 0.11
fill+wkde 0.000883 0.00219 0.00613 0.0186 0.0382 0.11
Table 104: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0307 0.0652 0.116 0.204 0.372 1.64
wkde 0.0305 0.0652 0.117 0.209 0.384 1.71
wkdeA 0.0306 0.0651 0.117 0.208 0.382 1.7
fill+wkde 0.03 0.0641 0.115 0.201 0.367 1.63
Table 105: Error quantiles evaluating at the observed sample points

Case No.  53

Niter : 2500
n : 250
d : 4
m : 3
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1
dp1$alpha :  6  3 -6 -3
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.00198 0.0159 4.86E+11 3.75E+51 3.97E+82 Inf
wkde 0.00198 0.0159 4.72E+11 3.57E+51 3.75E+82 Inf
wkdeA 0.00198 0.016 4.82E+11 3.66E+51 3.84E+82 Inf
fill+wkde 0.00204 0.0188 3.69E+12 1.61E+56 4.42E+90 Inf
Table 106: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0265 0.0576 0.11 0.198 0.292 0.701
wkde 0.0265 0.0578 0.111 0.2 0.294 0.704
wkdeA 0.0266 0.0579 0.111 0.199 0.293 0.701
fill+wkde 0.0241 0.0524 0.101 0.184 0.273 0.652
Table 107: Error quantiles evaluating at the observed sample points

Case No.  54

Niter : 2500
n : 500
d : 4
m : 3
qN : 3
Npts : 2401
dp1$xi : 0 0 0 0
dp1$Omega :
         [,1]     [,2]     [,3]     [,4]
[1,]  1.00000  0.60714 -0.04464 -0.33705
[2,]  0.60714  1.00000  0.60714 -0.04464
[3,] -0.04464  0.60714  1.00000  0.60714
[4,] -0.33705 -0.04464  0.60714  1.00000
dp1$alpha :  6  3 -6 -3
dp1$nu : 2
mix.p : 1
25% 50% 75% 90% 95% 99%
kde 0.000443 0.00242 0.0128 0.0505 0.105 0.374
wkde 0.000443 0.00243 0.0128 0.0506 0.106 0.373
wkdeA 0.000443 0.00243 0.0128 0.0506 0.106 0.373
fill+wkde 0.000444 0.00243 0.0128 0.0507 0.106 0.374
Table 108: Error quantiles for a fixed grid of points
25% 50% 75% 90% 95% 99%
kde 0.0348 0.085 0.189 0.344 0.556 4.94
wkde 0.035 0.0854 0.19 0.346 0.56 4.99
wkdeA 0.035 0.0854 0.19 0.346 0.561 4.99
fill+wkde 0.0348 0.0849 0.189 0.344 0.555 4.94
Table 109: Error quantiles evaluating at the observed sample points
Summary: relative improvement of error quantiles with Plain variant
rel.improvP :
   d m  mix.p    n    grid.50     obs.50    grid.75     obs.75    grid.95     obs.95
1  4 3 0.3333  500 -7.024e-06  0.0062091  6.057e-05  0.0035047  0.0014075  0.0023555
2  4 2 0.3333  500 -4.816e-04  0.0632107  4.517e-04  0.0496820  0.0281589  0.0487456
3  4 3 0.3333  500  8.854e-04  0.0010320  4.857e-03  0.0007320  0.0064476  0.0001832
4  4 2 0.3333  500  3.086e-02  0.0075780  5.299e-02  0.0066434  0.0596355  0.0062920
5  4 3 0.3333  500  9.247e-05 -0.0019726 -5.787e-04 -0.0034900 -0.0056127 -0.0049934
6  4 2 0.3333  500 -1.073e-02  0.1587915 -8.131e-03  0.0766432  0.1658389  0.0535351
7  4 3 0.6667  500  9.290e-05 -0.0025718 -4.604e-04 -0.0024352 -0.0039162 -0.0043010
8  4 2 0.6667  500 -1.272e-02  0.1035375 -6.344e-03  0.0723508  0.1285367  0.0483129
9  4 3 1.0000  500 -5.496e-04 -0.0061320 -2.988e-03 -0.0078146 -0.0067139 -0.0068770
10 4 2 1.0000  500 -7.959e-03  0.0138169  1.543e-02  0.0400199  0.1232558  0.1119632
11 4 3 1.0000  500 -1.523e-02 -0.0115403  4.932e-03 -0.0129228  0.0199083  0.0013560
12 4 3 1.0000  500  4.819e-04  0.0001732  2.682e-02 -0.0115994  0.1283942 -0.0057434
13 4 2 1.0000  500  4.079e-02  0.0287409  3.354e-01  0.0208001  0.3082178  0.0502174
14 4 3 1.0000  500  1.392e-02 -0.0005857  1.794e-02 -0.0003149  0.0857899  0.0052793
15 4 2 1.0000  500  1.780e-01  0.0549584  1.853e-01  0.0490660  0.3811008  0.0629187
16 4 3 1.0000  500  1.549e-02  0.0183119  2.561e-02  0.0132881  0.0247188  0.0042106
17 4 2 1.0000  500  1.892e-01  0.1097304  3.741e-01  0.0922566  0.3969550  0.0870600
18 3 2 1.0000  500  5.556e-02  0.0653200  9.482e-02  0.0619619  0.3163063  0.0641892
19 3 2 1.0000  500  1.278e-01  0.0616721  1.092e-01  0.0616103  0.1263927  0.0748406
20 3 2 1.0000  500  7.405e-02  0.0116158  5.613e-02  0.0130387  0.1171125  0.0287566
21 3 2 1.0000  500  1.212e-01  0.0649599  1.197e-01  0.0632092  0.1219351  0.0703953
22 3 2 1.0000  500 -3.033e-02  0.1249595  2.755e-02  0.0842165  0.0500413  0.0997428
23 3 2 1.0000  500  1.408e-02  0.1458633  1.316e-01  0.1193372  0.1957737  0.0798755
24 3 2 1.0000  500  1.280e-02  0.1119781  5.674e-02  0.0825847  0.1378309  0.0490871
25 5 2 1.0000  500  2.696e-01  0.1640928  5.246e-01  0.1346124  0.3636144  0.0941541
26 5 3 1.0000  500  4.909e-02  0.0370564  1.004e-01  0.0243688  0.1197060  0.0004601
27 5 2 0.6667  500 -7.520e-02 -0.0304363 -2.542e-02 -0.0044766  0.1318902  0.1014228
28 5 3 0.6667  500  1.255e-03 -0.0060955  6.055e-04 -0.0053435  0.0031362 -0.0110032
29 5 2 0.6667  500 -6.223e-04  0.0587803 -5.015e-05  0.0519677  0.0112995  0.0727365
30 5 3 0.6667  500 -6.200e-05  0.0058907 -1.652e-05  0.0041479  0.0007639  0.0044376
31 5 2 0.6667  500  7.981e-02  0.0089196  8.742e-02  0.0057118  0.2295232  0.0057754
32 5 3 0.6667  500  8.174e-03  0.0005249  9.317e-03  0.0005979  0.0327445 -0.0004954
33 5 3 1.0000 1000  1.913e-02  0.0719979  1.118e-01  0.0487230  0.1075626  0.0061596
34 5 3 1.0000  250  2.444e-02  0.0129532  4.412e-02  0.0070907  0.0300844 -0.0081575
35 4 3 1.0000  250 -4.661e-03 -0.0003045  2.770e-03 -0.0085057  0.0155507 -0.0066655
36 4 2 1.0000  250  1.988e-03  0.0306898  2.359e-02  0.0336224  0.0848010  0.0593612
37 4 2 1.0000 1000  1.177e-03  0.0211555  5.332e-02  0.0386698  0.1650018  0.1494835
38 4 2 0.3333  250 -2.769e-04  0.0349276  6.568e-04  0.0278809  0.0361631  0.0305796
39 4 2 0.3333  250 -1.565e-03  0.1122842  6.472e-03  0.0603346  0.1604286  0.0743937
40 4 2 1.0000  250  2.481e-02  0.0342828  1.827e-01  0.0243983  0.2413600  0.0253359
41 4 2 1.0000 1000  3.731e-02  0.0142255  2.865e-01  0.0088516  0.2804137  0.0682228
42 4 2 1.0000  250  1.553e-01  0.0793144  2.639e-01  0.0659200  0.1952579  0.0646610
43 4 2 1.0000 1000  1.352e-01  0.1350389  4.624e-01  0.1144852  0.1098095  0.1037914
44 5 2 1.0000 1000  1.213e-01  0.2117341  5.973e-01  0.1692204  0.4616741  0.1138866
45 5 2 1.0000  250  1.976e-01  0.1174394  3.706e-01  0.0972674  0.2576898  0.0604635
46 4 2 0.3333  250  7.136e-03  0.0008641  2.205e-02  0.0008915  0.1833463 -0.0002455
47 4 2 0.3333 1000  5.672e-02  0.0173900  5.416e-02  0.0152593  0.0917034  0.0135010
48 4 2 1.0000  250 -1.282e-02  0.1762382  5.927e-02  0.1358727  0.2485566  0.0901304
49 4 2 1.0000  500 -2.638e-02  0.2399888  3.831e-02  0.1777767  0.3134105  0.0873002
50 4 2 1.0000 1000 -3.982e-02  0.2860935  3.176e-03  0.1967130  0.3255958  0.0511283
51 4 3 1.0000  250 -5.343e-04  0.0023868 -1.210e-04 -0.0022015 -0.0012113 -0.0157336
52 4 3 1.0000  500 -5.069e-04  0.0008678 -1.077e-03 -0.0087687 -0.0101262 -0.0329717
53 4 3 1.0000  250  7.377e-05 -0.0036595  2.863e-02 -0.0080380  0.0559315 -0.0069071
54 4 3 1.0000  500 -8.766e-04 -0.0047209 -1.407e-03 -0.0039813 -0.0012095 -0.0084011
Summary: relative improvement of error quantiles with Fill variant
rel.improvF :
   d m  mix.p    n    grid.50     obs.50    grid.75     obs.75    grid.95     obs.95
1  4 3 0.3333  500 -4.600e-04  0.0065579 -6.411e-03  2.335e-03 -6.859e-02  0.0022439
2  4 2 0.3333  500 -1.959e-02  0.0649588 -6.643e-02  3.934e-02 -1.079e+00  0.0440172
3  4 3 0.3333  500 -1.399e+01  0.0004061 -9.173e+00  6.291e-05 -6.550e+06 -0.0002471
4  4 2 0.3333  500 -1.160e+04  0.0027459 -3.888e+03  1.721e-03 -3.067e+14  0.0031734
5  4 3 0.3333  500  9.827e-04  0.0007644  1.951e-05 -2.568e-04 -2.826e-03  0.0008446
6  4 2 0.3333  500  6.021e-03  0.1741547  7.287e-04  8.196e-02  6.284e-02  0.1145357
7  4 3 0.6667  500  8.073e-04  0.0008974  1.189e-03 -2.243e-04 -2.716e-03  0.0003218
8  4 2 0.6667  500  1.550e-02  0.1657329  2.504e-03  8.309e-02  4.552e-02  0.1147171
9  4 3 1.0000  500  1.076e-03  0.0022720  1.565e-03  1.678e-03  2.173e-03  0.0033365
10 4 2 1.0000  500 -8.867e-01  0.3168617 -1.270e+00  3.677e-01 -6.757e-01  0.3515219
11 4 3 1.0000  500 -1.976e-01  0.1761395 -4.734e-01  1.718e-01 -2.449e-01  0.1259928
12 4 3 1.0000  500 -2.617e-01  0.1002917 -5.022e+01  9.682e-02 -1.329e+10  0.0910333
13 4 2 1.0000  500 -8.144e-01  0.1998363 -8.004e+03  2.178e-01 -1.122e+15  0.2994834
14 4 3 1.0000  500 -3.064e+00 -0.0013903 -1.199e+01  3.579e-03 -2.836e+06  0.0224118
15 4 2 1.0000  500 -2.003e+02  0.0315370 -2.827e+05  5.117e-02 -1.488e+15  0.1669600
16 4 3 1.0000  500 -1.805e+00  0.0184925 -1.562e+01  1.698e-02 -4.247e+05  0.0248494
17 4 2 1.0000  500 -3.474e+01  0.1123845 -2.280e+06  1.026e-01 -2.870e+17  0.1781653
18 3 2 1.0000  500 -2.543e+00  0.0673693 -2.094e+00  7.298e-02 -1.039e+05  0.1014109
19 3 2 1.0000  500 -2.745e+00  0.0444310 -1.163e+00  5.524e-02 -4.409e+00  0.1023358
20 3 2 1.0000  500 -1.717e+01 -0.0085571 -5.043e+02 -7.551e-04 -4.679e+08  0.0422174
21 3 2 1.0000  500 -6.003e+00  0.0447325 -1.435e+00  5.213e-02 -2.967e+00  0.0926267
22 3 2 1.0000  500  1.853e-02  0.2906052  5.808e-02  2.419e-01  1.503e-01  0.2865112
23 3 2 1.0000  500  2.427e-02  0.1115106  1.054e-02  8.790e-02  7.459e-02  0.1522683
24 3 2 1.0000  500 -3.636e-01  0.0895953 -3.814e-01  4.749e-02 -1.092e+00  0.0648791
25 5 2 1.0000  500 -7.552e+03  0.2255596 -4.661e+08  2.350e-01 -1.918e+18  0.3047575
26 5 3 1.0000  500 -5.164e+01  0.0770386 -5.662e+04  8.528e-02 -8.533e+10  0.0845638
27 5 2 0.6667  500 -3.261e-02  0.0011853 -2.148e-02  3.002e-03 -9.525e-03  0.1642382
28 5 3 0.6667  500 -1.229e-02 -0.0049056 -4.794e-03 -5.097e-03 -6.760e-03  0.0112358
29 5 2 0.6667  500 -4.957e-02  0.0695890 -1.461e-01  5.281e-02 -2.013e+00  0.0838223
30 5 3 0.6667  500 -7.283e-03  0.0087676 -2.375e-02  4.969e-03 -2.364e-01  0.0088703
31 5 2 0.6667  500 -1.716e+09  0.0031929 -2.377e+13  7.005e-04 -1.561e+23  0.0033200
32 5 3 0.6667  500 -5.526e+04 -0.0007819 -2.782e+06 -6.267e-04 -2.883e+13 -0.0011976
33 5 3 1.0000 1000 -9.046e+01  0.1110201 -7.476e+05  1.119e-01 -6.829e+13  0.1251793
34 5 3 1.0000  250 -1.906e+01  0.0447503 -6.994e+02  5.600e-02 -1.127e+09  0.0475500
35 4 3 1.0000  250 -1.403e-01  0.1631826 -1.835e-01  1.248e-01 -1.843e-01  0.0918364
36 4 2 1.0000  250 -5.069e-01  0.3671602 -6.925e-01  3.680e-01 -5.919e-01  0.3108264
37 4 2 1.0000 1000 -2.412e+00  0.2654077 -1.998e+00  3.256e-01 -1.236e+00  0.3468042
38 4 2 0.3333  250 -2.260e-02  0.0378355 -6.597e-02  2.369e-02 -9.244e-01  0.0312826
39 4 2 0.3333  250  3.576e-04  0.1308397 -2.349e-02  6.312e-02 -1.320e-01  0.1185749
40 4 2 1.0000  250 -7.377e-01  0.2406346 -1.997e+02  2.695e-01 -1.333e+11  0.2652797
41 4 2 1.0000 1000 -1.427e+00  0.1570527 -3.692e+03  1.433e-01 -2.168e+17  0.2897565
42 4 2 1.0000  250 -2.933e+01  0.0927426 -5.400e+03  9.286e-02 -2.071e+13  0.1600587
43 4 2 1.0000 1000 -3.373e+01  0.1223061 -2.937e+06  1.047e-01 -1.181e+20  0.1685462
44 5 2 1.0000 1000 -1.053e+04  0.2498730 -3.070e+10  2.472e-01 -2.087e+21  0.3463910
45 5 2 1.0000  250 -8.504e+03  0.1943648 -1.067e+07  2.217e-01 -1.141e+16  0.2580643
46 4 2 0.3333  250 -2.347e+02 -0.0037129 -3.010e+04 -3.090e-03 -1.462e+12 -0.0025253
47 4 2 0.3333 1000 -1.179e+04  0.0110207 -9.264e+03  8.527e-03 -3.248e+19  0.0089931
48 4 2 1.0000  250 -2.107e-01  0.2459490 -6.981e-01  2.060e-01 -6.854e-01  0.3602301
49 4 2 1.0000  500 -3.605e-02  0.3055664 -1.901e-01  2.454e-01 -2.352e-01  0.4245999
50 4 2 1.0000 1000  4.666e-02  0.3433847  5.629e-02  2.699e-01  2.940e-01  0.4479900
51 4 3 1.0000  250  1.637e-02  0.0127077 -1.676e-02  9.533e-03 -3.566e-02  0.0099689
52 4 3 1.0000  500  1.214e-02  0.0172478 -1.972e-03  1.108e-02 -2.081e-02  0.0128858
53 4 3 1.0000  250 -1.813e-01  0.0893326 -6.598e+00  8.483e-02 -1.112e+08  0.0658840
54 4 3 1.0000  500 -2.634e-03  0.0006553 -5.151e-03  3.109e-04 -1.835e-03  0.0011444