Normalization of the Spectral Test in High Dimensions. Karl Entacher1, Gerold Laimer2, Harald Röck2, and Andreas Uhl2. 1School of Telecommunications ...
Monte Carlo Methods and Appl., Vol. 10, No. 3 – 4, pp. 341 – 366 (2004) c VSP 2004
Normalization of the Spectral Test in High Dimensions Karl Entacher1 , Gerold Laimer2 , Harald R¨ ock2 , and Andreas Uhl2 1 School
of Telecommunications Engineering, Salzburg Univ. of Applied Sciences and Technology 2 Department of Scientific Computing, Salzburg University, AUSTRIA
Abstracts — The spectral test provides a reliable measure for lattice assessment and can be computed very efficiently. It has extensively been applied to find good lattices for several MC and QMC applications. In order to enable comparisons across dimensions, a normalized spectral test is widely used. We empirically demonstrate significant shortcomings of this normalization in high dimensions, discuss the empirical distribution of the normalized spectral test values, and propose a new normalization strategy. The new normalization is shown to give more reliable results, especially concerning the comparability of the values accross dimensions. Keywords: Monte Carlo Simulation, pseudo-random numbers, LCG, spectral test, parameter search, lattice rules, quasi Monte Carlo Methods
1
Introduction
Quality assessements of integer lattice rules play an important role in the development of efficient node sets for the approximative calculation of high dimensional integrals using Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods. Integer lattices with (in a certain sense) optimal resolution or distribution properties are classical node sets of QMC integration. Lattice assessements are also used to get reliable linear random number generators to produce node sets for MC integration, the counterpart of QMC. This is due to the fact that different vectors from linear random numbers are contained in lattice structures. An analysis of the underlying lattices provides generators with optimal distribution and correlation quality. Linear congruential pseudo-random number generators (LCGs) are currently the best analyzed and most widely used pseudorandom number generators (PRNGs). They have recently attained special interest due to the fact that some state of the art generation methods are equivalent or approximately equivalent to big size LCGs [12]. A LCG(m, a, b, y0 ), a, b, y0 ∈ Zm := {0, . . . , m − 1} (the least residue system modulo m), is defined by the linear recurrence yn+1 ≡ ayn + b (mod m), n ≥ 0, and by an initial seed y0 . Here, m represents the modulus and a represents the multiplier. Standard parametrization schemes which guarantee maximal period [12], [17] and corresponding parameter tables can be found in the literature [8], [14]. For recent surveys on PRNGs see [10], [12], [17]. The empirical properties of pseudo-random numbers (PRNs) generated by LCGs strongly depend on the choice of parameters of the underlying generation method. The quality of the normalized PRNs xn := yn /m, n ≥ 0 is usually determined by the coarseness of the lattice {Xn(s) , n ≥ 0} created by all overlapping s-dimensional vectors Xn(s) := (xn , . . . , xn+s−1 ). This coarseness can change dramatically if either s or a are varied. Various approaches to assess the lattice structure have been proposed in order to find “optimal” LCG-parameters: among them, the most popular measure is the spectral test, which provides a figure of merit ds for the quality of the corresponding lattice [3, 8, 11, 15]. LCG parameters with good spectral
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Karl Entacher, Gerold Laimer, Harald R¨ock, and Andreas Uhl
tests may also be applied for QMC integration since the vectors Xn generated over the full period of the LCG produce rank-1 lattice rules which are classical QMC node sets [5]. Widely used is a normalized spectral test Ss := d∗s /ds , 2 ≤ s ≤ 8, for which 0 ≤ Ss ≤ 1 (values near 1 imply a good lattice structure). Whereas the normalization constants d∗s are absolute lower bounds on ds for s ≤ 8, also certain lower bounds d∗s for dimensions s > 8 are used in order to compute Ss for arbitrary dimensions. However, these bounds are suboptimal. In this work we show some problems arising with the normalization constants in higher dimensions and propose a novel type of normalization to overcome some of the shown problems for parameter searches. The normalization is derived from large scale experimental data and is fitted according to statistical principles. In Section 2 we review the spectral test in some detail and in Section 3 we demonstrate significant shortcomings of the presently used normalization. In particular we empirically analyse the variation of the empirical distribution of normalized spectral test values across dimensions. In Section 4 we propose the new normalization strategy and show examples where the improved behaviour is demonstrated. Section 5 concludes the paper.
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The Spectral Test
The spectral test gives the maximal distance ds between adjacent parallel hyperplanes, the maximum being taken over all families of parallel hyperplanes that cover all points {Xn(s) } [3, 8, 11, 15]. Low values ds imply a fine lattice structure in dimension s. To compute the spectral test, the shortest vector of the dual lattice is calculated [4] (recent implementations use the Fincke-Pohst algorithm [7]). One over the euclidean length of this shortest vector yields the spectral test ds 1 . To enable the comparison of spectral test results obtained in different dimensions, a normalized spectral test Ss := d∗s /ds for which 0 ≤ Ss ≤ 1 was introduced [8, 11]. Here, high values of Ss imply good lattice structures. The constants d∗s = γs−1 · m−1/s are absolute lower bounds on ds based on Hermite constants γs for s ≤ 8 [11]. Lower bounds for dimension s > 8 have been proposed as well in order to compute Ss for arbitrary dimensions [14]. In this context, an upper bound ρs for γs is used: ρs = 2eR(s)/s For s ≤ 24, R(s) is known from the theory of sphere packings, for s > 24 R(s) itself can only be approximated. Consequently, for s > 8 all these bounds are not optimal. A typical function measuring the “quality” of a multiplier a in terms of the spectral test across dimensions is: Mk := min Ss . 1≤s≤k
The measure Mk is maximized in the context of large scale parameter searches [14, 13, 16]. In order to considerably speed up the computations, the LLL basis reduction algorithm [2] may be applied instead of the Fincke-Pohst algorithm as an efficient and reliable approximation to the spectral test. Our experiments use an approximation of this type. The high quality of the LLL-approximation and the speedup with respect to the “original” spectral test have recently been shown [6]. 1
The spectral test is a geometrical measure for lattice assessment. The name spectral test originates from [3] where this measure was defined using discrete Fourier analysis for LCG tests. For generalized spectral test concepts, see [9].
Normalization of the Spectral Test in High Dimensions
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One reason for the introduction of the normalized spectral test was to be able to compare the values across dimensions and to be able to introduce quality measures like Mk . However, it turns out that there are problems associated with that, especially with respect to taking minima across dimensions. For a motivation, we randomly generate 200000 LCG multipliers a where each a needs to be a primitive root for m = 261 − 1. Each multiplier is evaluated by applying the spectral test in dimensions s = 2, . . . , 8, 16, 24 and each entry in the table corresponds to the minimal spectral test value found for a specific dimension s. s=2 2.3 · 10−8
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4 5 6 7 8 16 24 0.0008 0.0064 0.0249 0.0655 0.1029 0.3071 0.4157
Table 1: Minimal spectral test values found for each dimension s. Interestingly, the values increase with increasing dimension. The lowest values found for dimensions 16 and 24 would be of no significance in dimensions 2 - 8. In particular, only low dimensions contribute small values to Mk and the large values in high dimensions do not influence the result at all. This raises the question whether it makes sense at all to compare normalized spectral test values across dimensions or to search for parameters resulting from a large minimum taken over a set of different dimensions as currently done in many large scale parameter searches. 23
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Figure 1: Comparsion of normalized and not normalized spectral test histograms in dimension s = 24. In order to study this phenonemon in more detail, we have conducted computer experiments targeting the distribution of the spectral test values. As a first step, we compare the distributions of normalized and not-normalized values. In Figure 1.a and 1.b we display all spectral test values for LCG primitive root multipliers a for the prime modulus m = 223 − 15 in dimension s = 24. The x-axis of the diagram related to the not normalized values is scaled corresponding to maximal and minimal spectral test values attained. It is interesting to note that the normalized histogram is shifted significantly to the right half of the interval [0,1] as compared to the histogram of the not-normalized values in the interval [min,max]. The same behaviour is found for modulus m = 264 − 59 in Fig. 1.c and 1.d, of course we have not done an exhaustive analysis due to exceeding computational demand. This distribution behaviour illustrates the values in Table 1. In higher dimensions, we do not find low values since the distribution is shifted to the right half of the interval, which is obviously caused by the normalization. As a second step in the empirical investigation of the distribution of the spectral test values, we computed the distributions of the normalized spectral test values for dimensions 2 ≤ s ≤
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Karl Entacher, Gerold Laimer, Harald R¨ock, and Andreas Uhl
24. The data has been generated for moduli m = 217 −1, 219 −1, 221 −9, 231 −1, 264 −59, 2127 −1, exhaustive (i.e. considering all primitive roots with respect to the corresponding modulus) for small moduli up to 227 − 1 and randomly sampled (1300000 samples) for larger moduli. In Figure 2 we vizualize the results for m = 264 − 59, however, for all moduli considered the results are almost identical. Modul: 264 − 59
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Normalization of the Spectral Test in High Dimensions
5
Two facts are shown very clearly. First, for dimensions s = 2 we face a triangle-type distribution, for increasing dimension, the histograms get more and more symmetric. Second, the entire histograms are shifted to the right half of the interval [0, 1] by an increasing amount for increasing dimensions. The latter fact is also the reason for the phenomenon shown in Table 1. The regularity of the historams suggest the existence of an underlying classical density funtion. Although several attempts have been made to verify this (e.g. [1]), the distributions of the spectral test values remain an open question. Nevertheless, we have tried to find known density functions which approximate the empirical distributions very well with computational methods. Among several distribution fits (Weibull, Chi-square, Lognormal, ...) the Gamma distributions turned out show the best approximations to the distribution of 1 − Ss with surprising good quality, for dimensions s ≥ 3 and all moduli considered. With increasing dimensions the quality of approximation increased as well. The distribution fits were realized in MATLAB using maximum likelihood estimation methods for the corresponding distribution parameters. Figure 3 exhibits the quality of approximation. The left graphics show the histograms and the corresponding density, and the right graphics the empirical distribution function versus the Gamma distribution function. 3.5
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4 4.1
Karl Entacher, Gerold Laimer, Harald R¨ock, and Andreas Uhl
A Proposal for Improved Normalization Distribution Dilation
Since the distributions of the spectral values are almost invariant with respect to increasing size of the modulus it might be sufficient to apply a distribution dilation to get improved normalizations across dimensions. To cope with empirical outliers we determined estimates for 0.1% and 99.9% quantiles using regression. Normalized spectral test values between the inter-quantile ranges are linearly transformed to [0,1], and values outside of the quantiles are mapped to zero and one respectively. Fig. 4 illustrates the normalization adjustment.
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Figure 4: Normalization adjustment using distribution dilation. For the estimation of the 0.1% and 99.1% quantiles we applied randomly chosen moduli mi in the intervals [2i , 2i+1 ], i = 25, . . . , 35 using a random sample of 1300000 multipliers for each modulus. The data from the moduli from Sect. 3 were used to control the results and the regression quality. From the empirical quantiles for all moduli and all dimensions s = 2, . . . , 24, we generated a polynomial fit to get single estimates for the quantiles per dimension. Figure 5 exhibit the regression results, the quality of the residuals and the coefficients of determination. The regression functions for the 0.1% quantile q1(s) and the 99.9% quantile q2(s) are: q1(s) := 0.000042s3 − 0.0027s2 + 0.067s − 0.097 q2(s) := −0.000058s3 + 0.0036s2 − 0.059s + 1.09 Using these functions we performed the linear transformation for the normalized spectral Ss test in dimension 2 ≤ s ≤ 24: Ss − q1(s) q2(s) − q1(s) for the experiments in the following section.
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4.2
Examples for Improved Behaviour
In order to give concrete examples for improved behaviour in realistic applications, we conduct random searches for good primitive root multipliers for a set of moduli. The multipliers are ranked according to the criteria M24 . For m = 217 − 1 an exhaustive analysis have been performed (all primitive root multipliers have been assessed). Figure 6.a shows the spectral test values of multiplier 70357 employing the classical normalization strategy which is ranked on position 484 even though the spectral test value for dimension 24 lies below the 1% quantile of all values in this dimension. In case the proposed normalization strategy is used, this multiplier is ranked on position 26835 according to M24 which corresponds well to the poor behaviour in dimension 24 (see Figure 6.b). A similar example is provided for m = 264 − 59 where 1300000 primitive root multipliers have been assessed. Figure 6.a shows the spectral test values of a multiplier which is ranked on position 32711 even though its value in dimension 20 is below 0.55. Relating this absolute value (which does not seem to be particularly bad) to the corresponding histogram in Figure 2 it turns out that again this value lies below the 1% quantile of all values in this dimension. The proposed normalization corrects the spectral test value to 0.1 and the multiplier is subsequently ranked on position 1016895 according to M24 .
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The classical normalization constants of the spectral test in high dimensions have been found 0.3 to0.3be weak and may give misleading results when comparing spectral test values across di0.2 0.2 mensions. We propose a new normalization scheme based on the empirical distribution of the constants from quantiles of the empirical 0.1 spectral test, where we derive the normalization 0.1 distribution. Practical examples from the area0 of searching high quality LCG parameters 0 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24 Dimension Dimension suggest the new normalization strategy to deliver more reliable results. In future work we will apply the proposed normalization scheme to large scale parameter searches to identify LCG parameters with equally stable behaviour across low and high dimensions. 0
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12 14 Dimension
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0
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12 14 Dimension
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Acknowledgements This work was supported by the Austrian Science Fund, Grants S8303, S8311 and P13732.
Normalization of the Spectral Test in High Dimensions
9
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