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 nonparametric estimation methods are available. Most of these methods share the common feature of being ‘fully nonparametric’, 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 preassigned 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 wellknown that, as
increases, nonparametric methods, and in particular those for density estimation, degrade in performance, eventually running into the socalled 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 nonparametric 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 highorder interactions among variables. It is plausible that, even if these highorder 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 densitybased clustering process of some real data.
2 Global smoothing via a loglinear 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 hyperrectangles 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 loglinear 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 loglinear representation of the ’s are retained up to pairwise interactions while all threefactor 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 loglinear 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 loglinear 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 loglinear 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 Nonparametric 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 kernelbased 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 datadependent 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 nonnegligible 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 theoreticallymotivated 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 binwidth 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 intointervals, 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 endpoints 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 endpoints of the intervals appears somewhat arbitrary if examined from a formal viewpoint. The scheme must rather be regarded as a way of mimicking the nonautomatic 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 loglinear 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 loglinear 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 loglinear 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 plugin 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 abovedescribed 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.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, skewnormal, Student’s
and its skew version, and twocomponent mixtures of these distributions. The general expression of the distribution in use is(4) 
where and are of skewnormal (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 positivedefinite 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 Toeplitztype 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.

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 loglinear 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:

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

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

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 abovedescribed 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 nonrandom grid of points spanning the area of nonnegligible 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 nonrandom 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.
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 loglinear 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 singlecomponent 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 nongenerate 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 nonparametric estimation.
6 Application to densitybased 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 realdata application presented by Azzalini and Torelli (2007, Section 4.3) concerned eight chemical components of specimens of oliveoil originating from various regions of Italy. We reexamine 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 logratio 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 (1.02, 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 crossclassification 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 subtable 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 
CentreNorth  0  45  106  0  22  129 
ARI  0.873  0.937 
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 subpopulations 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 mathematicallyargumented 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 SkewNormal 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 nonparametric 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: Loglinear 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.Rproject.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 theoreticallysupported 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 componentwise 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:

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;

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 nonrandom 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  2.9E05  9.79E05  0.000348  0.00115  0.0028  0.0344 
e2  2.9E05  9.79E05  0.000348  0.00115  0.0028  0.034 
e3  2.89E05  9.79E05  0.000348  0.00115  0.0028  0.0341 
e4  2.89E05  9.79E05  0.00035  0.00118  0.00299  0.0371 
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 
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  2.89E05  9.79E05  0.000348  0.00115  0.00279  0.0346 
e2  2.9E05  9.79E05  0.000348  0.00115  0.00272  0.0303 
e3  2.9E05  9.79E05  0.000348  0.00115  0.00273  0.0306 
e4  2.9E05  9.98E05  0.000371  0.00167  0.00581  0.0577 
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 
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  5.13E22  3.77E12  6.97E05  3.57E+03  1.81E+13  2.11E+36 
e2  5.12E22  3.76E12  6.94E05  3.58E+03  1.8E+13  2.15E+36 
e3  5.11E22  3.76E12  6.94E05  3.56E+03  1.81E+13  1.88E+36 
e4  5.11E20  5.64E11  0.000709  1.25E+06  1.19E+20  1.2E+51 
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 
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  5.14E22  3.79E12  7.13E05  4.21E+03  3.22E+13  1.29E+36 
wkde  5.01E22  3.68E12  6.75E05  4.02E+03  3.03E+13  1.25E+36 
wkdeA  5E22  3.7E12  6.84E05  4.06E+03  2.92E+13  1.19E+36 
fill+wkde  4.55E16  4.4E08  0.277  8.34E+12  9.88E+27  1.56E+61 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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  2.01E05  0.00195  0.033  0.283  1.31  122 
wkde  1.98E05  0.00193  0.0325  0.269  1.2  104 
wkdeA  1.99E05  0.00194  0.0327  0.273  1.23  107 
fill+wkde  8.63E05  0.00794  0.429  5.43E+03  3.73E+06  1.02E+12 
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 
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  2E05  0.00194  0.0328  0.274  1.2  203 
wkde  1.55E05  0.00159  0.0267  0.202  0.742  28.1 
wkdeA  1.66E05  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 
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 
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  3.11E07  0.000968  4.54E+07  3.54E+48  3.09E+82  Inf 
wkde  3.1E07  0.000953  4.43E+07  3.39E+48  3.02E+82  Inf 
wkdeA  3.11E07  0.00096  4.48E+07  3.48E+48  3.03E+82  Inf 
fill+wkde  7.65E07  0.00272  7.55E+08  1.14E+53  1.31E+88  Inf 
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 
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  3.1E07  0.00096  4.36E+07  3.15E+48  1.37E+82  Inf 
wkde  2.85E07  0.000778  2.73E+07  1.89E+48  8.26E+81  Inf 
wkdeA  2.9E07  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 
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 
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  5.5E06  0.00257  0.0482  5.2E+03  3.79E+08  4.83E+14 
wkde  5.23E06  0.00243  0.0436  5.26E+03  2.59E+08  3.08E+14 
wkdeA  5.32E06  0.00244  0.0438  4.1E+03  2.66E+08  3.18E+14 
fill+wkde  2.73E05  0.0091  0.149  6.56E+07  3.94E+13  1.33E+22 
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 
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  8.01E05  0.00341  0.0216  0.0572  0.0879  0.187 
wkde  7.23E05  0.00298  0.0193  0.0501  0.0767  0.156 
wkdeA  7.29E05  0.00303  0.0195  0.0501  0.0762  0.153 
fill+wkde  0.00148  0.0128  0.0468  0.146  0.475  33.2 
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 
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  3.96E05  0.00577  0.0462  0.143  0.248  1.12 
wkde  3.3E05  0.00534  0.0436  0.131  0.219  0.777 
wkdeA  3.47E05  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 
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 
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  9E06  0.00113  0.0174  0.0557  0.0893  0.194 
wkde  8.21E06  0.000996  0.0153  0.0489  0.0784  0.164 
wkdeA  8.28E06  0.00101  0.0155  0.049  0.078  0.161 
fill+wkde  0.000244  0.00794  0.0425  0.138  0.354  10.8 
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 
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 
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 
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 
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 
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 
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 
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  4.95E10  1.9E05  1.29E+07  1.84E+53  2.87E+84  Inf 
wkde  4.16E10  1.39E05  6.14E+06  5.23E+52  1.83E+84  Inf 
wkdeA  4.35E10  1.52E05  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 
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 
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  4.73E10  1.78E05  9.03E+06  6.8E+52  1.45E+84  Inf 
wkde  4.6E10  1.69E05  8.12E+06  6.19E+52  1.27E+84  Inf 
wkdeA  4.66E10  1.74E05  8.65E+06  6.6E+52  1.38E+84  Inf 
fill+wkde  8.95E08  0.000937  5.11E+11  8.65E+60  1.23E+95  Inf 
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 
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  3.31E06  3.14E05  0.000159  0.00115  0.00416  0.0373 
wkde  3.72E06  3.37E05  0.000163  0.00113  0.00361  0.0272 
wkdeA  3.7E06  3.39E05  0.000163  0.00113  0.00359  0.0273 
fill+wkde  3.4E06  3.24E05  0.000162  0.0012  0.0042  0.0742 
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 
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  3.44E06  3.15E05  0.000149  0.00099  0.00317  0.0243 
wkde  3.43E06  3.15E05  0.000149  0.000987  0.00316  0.0245 
wkdeA  3.43E06  3.15E05  0.000149  0.000987  0.00316  0.0245 
fill+wkde  3.47E06  3.19E05  0.00015  0.000996  0.00319  0.0269 
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 
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  3.61E06  1.26E05  4.88E05  0.000173  0.000381  0.00405 
wkde  3.61E06  1.26E05  4.88E05  0.000173  0.000376  0.00349 
wkdeA  3.61E06  1.26E05  4.88E05  0.000173  0.000377  0.00359 
fill+wkde  3.68E06  1.32E05  5.59E05  0.000281  0.00115  0.0201 
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 
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  3.61E06  1.26E05  4.88E05  0.000173  0.000381  0.00403 
wkde  3.61E06  1.26E05  4.88E05  0.000173  0.000381  0.004 
wkdeA  3.61E06  1.26E05  4.88E05  0.000173  0.000381  0.00401 
fill+wkde  3.61E06  1.27E05  5E05  0.00019  0.000471  0.00742 
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 
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  3.83E21  1.02E11  0.00701  5.73E+18  8.16E+41  Inf 
wkde  3.54E21  9.36E12  0.0064  5.27E+18  6.29E+41  Inf 
wkdeA  3.69E21  9.87E12  0.0067  5.8E+18  1E+42  Inf 
fill+wkde  5.42E11  0.0174  1.67E+11  1.76E+37  1.27E+65  Inf 
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 
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  2.94E21  9.36E12  0.00714  4.98E+18  5.1E+41  Inf 
wkde  2.89E21  9.29E12  0.00707  4.93E+18  4.93E+41  Inf 
wkdeA  2.93E21  9.33E12  0.00711  4.89E+18  4.98E+41  Inf 
fill+wkde  3.05E16  5.17E07  1.99E+04  1.01E+28  1.47E+55  Inf 
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 
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  1.79E10  5.7E06  1.02E+04  3.89E+46  3.37E+78  Inf 
wkde  1.73E10  5.59E06  9.04E+03  3.35E+46  3.01E+78  Inf 
wkdeA  1.76E10  5.64E06  9.62E+03  3.51E+46  3.11E+78  Inf 
fill+wkde  1.37E08  0.000521  7.61E+09  2.95E+56  2.3E+92  Inf 
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 
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  3.71E09  8.1E05  1.58E+10  4.49E+58  1.18E+89  Inf 
wkde  3.58E09  7.9E05  1.51E+10  4.04E+58  1.14E+89  Inf 
wkdeA  3.65E09  8.01E05  1.53E+10  4.28E+58  1.14E+89  Inf 
fill+wkde  5.88E07  0.00162  1.1E+13  5.75E+64  1.33E+98  Inf 
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 
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 
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 
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 
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 
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 
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 
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  2.89E05  9.71E05  0.000349  0.00119  0.00318  0.0473 
wkde  2.89E05  9.71E05  0.000349  0.00118  0.00307  0.0445 
wkdeA  2.89E05  9.71E05  0.000349  0.00118  0.00309  0.0446 
fill+wkde  2.9E05  9.93E05  0.000372  0.00168  0.00613  0.0651 
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 
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 
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 
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 
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 
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 
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 
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  1.51E06  0.00317  6.56E+09  2.19E+49  9.61E+79  Inf 
wkde  1.3E06  0.00267  4.83E+09  1.74E+49  7.73E+79  Inf 
wkdeA  1.34E06  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 
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 
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  2.78E07  0.000777  5.58E+04  1.12E+40  1.22E+67  Inf 
wkde  2.53E07  0.000672  3E+04  6.66E+39  1.09E+67  Inf 
wkdeA  2.57E07  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 
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 
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  1.79E10  5.72E06  1.17E+04  4.81E+46  3.23E+78  Inf 
wkde  1.46E10  5.02E06  4.71E+03  1.16E+46  1.74E+78  Inf 
wkdeA  1.53E10  5.17E06  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 
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 
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  3.71E09  8.17E05  1.11E+10  2.47E+58  1.55E+89  Inf 
wkde  2.76E09  6.56E05  6.99E+09  1.34E+58  1.15E+89  Inf 
wkdeA  3.07E09  7.12E05  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 
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 
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  5.63E14  4.35E07  0.112  3.97E+11  1.62E+25  1.01E+55 
wkde  5.57E14  4.32E07  0.11  3.82E+11  1.33E+25  1.01E+55 
wkdeA  5.55E14  4.29E07  0.11  3.82E+11  1.51E+25  9.41E+54 
fill+wkde  6.93E11  0.000103  3.38E+03  1.09E+19  2.37E+37  2.28E+73 
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 
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  1.73E25  6.07E15  1.58E07  0.0214  160  2.42E+15 
wkde  1.66E25  5.72E15  1.5E07  0.0201  145  8.08E+14 
wkdeA  1.68E25  5.84E15  1.53E07  0.0206  150  1.44E+15 
fill+wkde  7.64E20  7.15E11  0.00147  2.77E+08  5.19E+21  1.24E+49 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 
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 7.024e06 0.0062091 6.057e05 0.0035047 0.0014075 0.0023555 2 4 2 0.3333 500 4.816e04 0.0632107 4.517e04 0.0496820 0.0281589 0.0487456 3 4 3 0.3333 500 8.854e04 0.0010320 4.857e03 0.0007320 0.0064476 0.0001832 4 4 2 0.3333 500 3.086e02 0.0075780 5.299e02 0.0066434 0.0596355 0.0062920 5 4 3 0.3333 500 9.247e05 0.0019726 5.787e04 0.0034900 0.0056127 0.0049934 6 4 2 0.3333 500 1.073e02 0.1587915 8.131e03 0.0766432 0.1658389 0.0535351 7 4 3 0.6667 500 9.290e05 0.0025718 4.604e04 0.0024352 0.0039162 0.0043010 8 4 2 0.6667 500 1.272e02 0.1035375 6.344e03 0.0723508 0.1285367 0.0483129 9 4 3 1.0000 500 5.496e04 0.0061320 2.988e03 0.0078146 0.0067139 0.0068770 10 4 2 1.0000 500 7.959e03 0.0138169 1.543e02 0.0400199 0.1232558 0.1119632 11 4 3 1.0000 500 1.523e02 0.0115403 4.932e03 0.0129228 0.0199083 0.0013560 12 4 3 1.0000 500 4.819e04 0.0001732 2.682e02 0.0115994 0.1283942 0.0057434 13 4 2 1.0000 500 4.079e02 0.0287409 3.354e01 0.0208001 0.3082178 0.0502174 14 4 3 1.0000 500 1.392e02 0.0005857 1.794e02 0.0003149 0.0857899 0.0052793 15 4 2 1.0000 500 1.780e01 0.0549584 1.853e01 0.0490660 0.3811008 0.0629187 16 4 3 1.0000 500 1.549e02 0.0183119 2.561e02 0.0132881 0.0247188 0.0042106 17 4 2 1.0000 500 1.892e01 0.1097304 3.741e01 0.0922566 0.3969550 0.0870600 18 3 2 1.0000 500 5.556e02 0.0653200 9.482e02 0.0619619 0.3163063 0.0641892 19 3 2 1.0000 500 1.278e01 0.0616721 1.092e01 0.0616103 0.1263927 0.0748406 20 3 2 1.0000 500 7.405e02 0.0116158 5.613e02 0.0130387 0.1171125 0.0287566 21 3 2 1.0000 500 1.212e01 0.0649599 1.197e01 0.0632092 0.1219351 0.0703953 22 3 2 1.0000 500 3.033e02 0.1249595 2.755e02 0.0842165 0.0500413 0.0997428 23 3 2 1.0000 500 1.408e02 0.1458633 1.316e01 0.1193372 0.1957737 0.0798755 24 3 2 1.0000 500 1.280e02 0.1119781 5.674e02 0.0825847 0.1378309 0.0490871 25 5 2 1.0000 500 2.696e01 0.1640928 5.246e01 0.1346124 0.3636144 0.0941541 26 5 3 1.0000 500 4.909e02 0.0370564 1.004e01 0.0243688 0.1197060 0.0004601 27 5 2 0.6667 500 7.520e02 0.0304363 2.542e02 0.0044766 0.1318902 0.1014228 28 5 3 0.6667 500 1.255e03 0.0060955 6.055e04 0.0053435 0.0031362 0.0110032 29 5 2 0.6667 500 6.223e04 0.0587803 5.015e05 0.0519677 0.0112995 0.0727365 30 5 3 0.6667 500 6.200e05 0.0058907 1.652e05 0.0041479 0.0007639 0.0044376 31 5 2 0.6667 500 7.981e02 0.0089196 8.742e02 0.0057118 0.2295232 0.0057754 32 5 3 0.6667 500 8.174e03 0.0005249 9.317e03 0.0005979 0.0327445 0.0004954 33 5 3 1.0000 1000 1.913e02 0.0719979 1.118e01 0.0487230 0.1075626 0.0061596 34 5 3 1.0000 250 2.444e02 0.0129532 4.412e02 0.0070907 0.0300844 0.0081575 35 4 3 1.0000 250 4.661e03 0.0003045 2.770e03 0.0085057 0.0155507 0.0066655 36 4 2 1.0000 250 1.988e03 0.0306898 2.359e02 0.0336224 0.0848010 0.0593612 37 4 2 1.0000 1000 1.177e03 0.0211555 5.332e02 0.0386698 0.1650018 0.1494835 38 4 2 0.3333 250 2.769e04 0.0349276 6.568e04 0.0278809 0.0361631 0.0305796 39 4 2 0.3333 250 1.565e03 0.1122842 6.472e03 0.0603346 0.1604286 0.0743937 40 4 2 1.0000 250 2.481e02 0.0342828 1.827e01 0.0243983 0.2413600 0.0253359 41 4 2 1.0000 1000 3.731e02 0.0142255 2.865e01 0.0088516 0.2804137 0.0682228 42 4 2 1.0000 250 1.553e01 0.0793144 2.639e01 0.0659200 0.1952579 0.0646610 43 4 2 1.0000 1000 1.352e01 0.1350389 4.624e01 0.1144852 0.1098095 0.1037914 44 5 2 1.0000 1000 1.213e01 0.2117341 5.973e01 0.1692204 0.4616741 0.1138866 45 5 2 1.0000 250 1.976e01 0.1174394 3.706e01 0.0972674 0.2576898 0.0604635 46 4 2 0.3333 250 7.136e03 0.0008641 2.205e02 0.0008915 0.1833463 0.0002455 47 4 2 0.3333 1000 5.672e02 0.0173900 5.416e02 0.0152593 0.0917034 0.0135010 48 4 2 1.0000 250 1.282e02 0.1762382 5.927e02 0.1358727 0.2485566 0.0901304 49 4 2 1.0000 500 2.638e02 0.2399888 3.831e02 0.1777767 0.3134105 0.0873002 50 4 2 1.0000 1000 3.982e02 0.2860935 3.176e03 0.1967130 0.3255958 0.0511283 51 4 3 1.0000 250 5.343e04 0.0023868 1.210e04 0.0022015 0.0012113 0.0157336 52 4 3 1.0000 500 5.069e04 0.0008678 1.077e03 0.0087687 0.0101262 0.0329717 53 4 3 1.0000 250 7.377e05 0.0036595 2.863e02 0.0080380 0.0559315 0.0069071 54 4 3 1.0000 500 8.766e04 0.0047209 1.407e03 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 4.600e04 0.0065579 6.411e03 2.335e03 6.859e02 0.0022439 2 4 2 0.3333 500 1.959e02 0.0649588 6.643e02 3.934e02 1.079e+00 0.0440172 3 4 3 0.3333 500 1.399e+01 0.0004061 9.173e+00 6.291e05 6.550e+06 0.0002471 4 4 2 0.3333 500 1.160e+04 0.0027459 3.888e+03 1.721e03 3.067e+14 0.0031734 5 4 3 0.3333 500 9.827e04 0.0007644 1.951e05 2.568e04 2.826e03 0.0008446 6 4 2 0.3333 500 6.021e03 0.1741547 7.287e04 8.196e02 6.284e02 0.1145357 7 4 3 0.6667 500 8.073e04 0.0008974 1.189e03 2.243e04 2.716e03 0.0003218 8 4 2 0.6667 500 1.550e02 0.1657329 2.504e03 8.309e02 4.552e02 0.1147171 9 4 3 1.0000 500 1.076e03 0.0022720 1.565e03 1.678e03 2.173e03 0.0033365 10 4 2 1.0000 500 8.867e01 0.3168617 1.270e+00 3.677e01 6.757e01 0.3515219 11 4 3 1.0000 500 1.976e01 0.1761395 4.734e01 1.718e01 2.449e01 0.1259928 12 4 3 1.0000 500 2.617e01 0.1002917 5.022e+01 9.682e02 1.329e+10 0.0910333 13 4 2 1.0000 500 8.144e01 0.1998363 8.004e+03 2.178e01 1.122e+15 0.2994834 14 4 3 1.0000 500 3.064e+00 0.0013903 1.199e+01 3.579e03 2.836e+06 0.0224118 15 4 2 1.0000 500 2.003e+02 0.0315370 2.827e+05 5.117e02 1.488e+15 0.1669600 16 4 3 1.0000 500 1.805e+00 0.0184925 1.562e+01 1.698e02 4.247e+05 0.0248494 17 4 2 1.0000 500 3.474e+01 0.1123845 2.280e+06 1.026e01 2.870e+17 0.1781653 18 3 2 1.0000 500 2.543e+00 0.0673693 2.094e+00 7.298e02 1.039e+05 0.1014109 19 3 2 1.0000 500 2.745e+00 0.0444310 1.163e+00 5.524e02 4.409e+00 0.1023358 20 3 2 1.0000 500 1.717e+01 0.0085571 5.043e+02 7.551e04 4.679e+08 0.0422174 21 3 2 1.0000 500 6.003e+00 0.0447325 1.435e+00 5.213e02 2.967e+00 0.0926267 22 3 2 1.0000 500 1.853e02 0.2906052 5.808e02 2.419e01 1.503e01 0.2865112 23 3 2 1.0000 500 2.427e02 0.1115106 1.054e02 8.790e02 7.459e02 0.1522683 24 3 2 1.0000 500 3.636e01 0.0895953 3.814e01 4.749e02 1.092e+00 0.0648791 25 5 2 1.0000 500 7.552e+03 0.2255596 4.661e+08 2.350e01 1.918e+18 0.3047575 26 5 3 1.0000 500 5.164e+01 0.0770386 5.662e+04 8.528e02 8.533e+10 0.0845638 27 5 2 0.6667 500 3.261e02 0.0011853 2.148e02 3.002e03 9.525e03 0.1642382 28 5 3 0.6667 500 1.229e02 0.0049056 4.794e03 5.097e03 6.760e03 0.0112358 29 5 2 0.6667 500 4.957e02 0.0695890 1.461e01 5.281e02 2.013e+00 0.0838223 30 5 3 0.6667 500 7.283e03 0.0087676 2.375e02 4.969e03 2.364e01 0.0088703 31 5 2 0.6667 500 1.716e+09 0.0031929 2.377e+13 7.005e04 1.561e+23 0.0033200 32 5 3 0.6667 500 5.526e+04 0.0007819 2.782e+06 6.267e04 2.883e+13 0.0011976 33 5 3 1.0000 1000 9.046e+01 0.1110201 7.476e+05 1.119e01 6.829e+13 0.1251793 34 5 3 1.0000 250 1.906e+01 0.0447503 6.994e+02 5.600e02 1.127e+09 0.0475500 35 4 3 1.0000 250 1.403e01 0.1631826 1.835e01 1.248e01 1.843e01 0.0918364 36 4 2 1.0000 250 5.069e01 0.3671602 6.925e01 3.680e01 5.919e01 0.3108264 37 4 2 1.0000 1000 2.412e+00 0.2654077 1.998e+00 3.256e01 1.236e+00 0.3468042 38 4 2 0.3333 250 2.260e02 0.0378355 6.597e02 2.369e02 9.244e01 0.0312826 39 4 2 0.3333 250 3.576e04 0.1308397 2.349e02 6.312e02 1.320e01 0.1185749 40 4 2 1.0000 250 7.377e01 0.2406346 1.997e+02 2.695e01 1.333e+11 0.2652797 41 4 2 1.0000 1000 1.427e+00 0.1570527 3.692e+03 1.433e01 2.168e+17 0.2897565 42 4 2 1.0000 250 2.933e+01 0.0927426 5.400e+03 9.286e02 2.071e+13 0.1600587 43 4 2 1.0000 1000 3.373e+01 0.1223061 2.937e+06 1.047e01 1.181e+20 0.1685462 44 5 2 1.0000 1000 1.053e+04 0.2498730 3.070e+10 2.472e01 2.087e+21 0.3463910 45 5 2 1.0000 250 8.504e+03 0.1943648 1.067e+07 2.217e01 1.141e+16 0.2580643 46 4 2 0.3333 250 2.347e+02 0.0037129 3.010e+04 3.090e03 1.462e+12 0.0025253 47 4 2 0.3333 1000 1.179e+04 0.0110207 9.264e+03 8.527e03 3.248e+19 0.0089931 48 4 2 1.0000 250 2.107e01 0.2459490 6.981e01 2.060e01 6.854e01 0.3602301 49 4 2 1.0000 500 3.605e02 0.3055664 1.901e01 2.454e01 2.352e01 0.4245999 50 4 2 1.0000 1000 4.666e02 0.3433847 5.629e02 2.699e01 2.940e01 0.4479900 51 4 3 1.0000 250 1.637e02 0.0127077 1.676e02 9.533e03 3.566e02 0.0099689 52 4 3 1.0000 500 1.214e02 0.0172478 1.972e03 1.108e02 2.081e02 0.0128858 53 4 3 1.0000 250 1.813e01 0.0893326 6.598e+00 8.483e02 1.112e+08 0.0658840 54 4 3 1.0000 500 2.634e03 0.0006553 5.151e03 3.109e04 1.835e03 0.0011444