Astroparticle Physics 34 (2011) 871–877

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Astroparticle Physics

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Haar wavelets as a tool for the statistical characterization of variability

Ryan Price, Stephane Vincent, Stephan LeBohec ⇑Department of Physics and Astronomy, University of Utah, Salt-Lake-City, UT 84112-0830, USA

a r t i c l e i n f o

Article history:Received 12 October 2010Received in revised form 17 February 2011Accepted 23 March 2011Available online 29 March 2011

Keywords:WaveletLight curveAnalysis

0927-6505/$ - see front matter Published by Elsevierdoi:10.1016/j.astropartphys.2011.03.006

⇑ Corresponding author.E-mail addresses: ryan.g.price@utah.edu (R. Price

(S. LeBohec).

a b s t r a c t

In the field of gamma-ray astronomy, irregular and noisy datasets make difficult the characterization oflight-curve features in terms of statistical significance while properly accounting for trial factors associ-ated with the search for variability at different times and over different timescales. In order to addressthese difficulties, we propose a method based on the Haar wavelet decomposition of the data. It allowsstatistical characterization of possible variability, embedded in a white noise background, in terms of aconfidence level. The method is applied to artificially generated data for characterization as well as tothe very high energy M87 light curve recorded with VERITAS in 2008 which serves here as a realisticapplication example.

Published by Elsevier B.V.

1. Introduction

In high energy astrophysics and possibly in other fields ofresearch, problems associated with the analysis of data with largerstatistical errors (with rarely more than a few standard deviationsper data point) lead to the development of very systematic andcareful practices when determining the detection of a new source.As an example, for a discovery to be claimed, because of alwayspossible systematic effects, unrealistically high statistical signifi-cance thresholds of �5 standard deviations above the background[12] are typically self-imposed [15,6]. Less systematic and also lessstringent practices are however often applied when the analysisinvestigates higher order features such as time variability orenergy spectrum breaks (e.g. [5,7,8]). We are here investigatingthe potential of using a wavelet analysis in these situations.

In this paper, the data will be considered as a function of time,but the analysis method can easily be used on data that is a func-tion of any variable. (In fact, this method could be generalized todata recorded as a function of more than one parameter such aswith significance maps, but this is not to be investigated here.)The variability may concentrate on individual data points or overbroad intervals. In either case, the aim is to establish a confidencelevel (CL) with which variability can be reported. An approachsometimes taken for this is to determine if a data model, such asa constant value, fits the data with a satisfying v2 (e.g. [1–3]). Ifit does, then variability is generally not further investigated withthe same data. However, this can be misleading, and does not make

B.V.

), lebohec@physics.utah.edu

full use of the data as these statistical characterizations do notdepend on the order in which the data points occurred.

Wavelet analysis should then be a better tool to use in this typeof situations. The approach consists of describing the dataset by alinear combination of wavelets. A wavelet family is a complete andorthonormal functional basis whose members reach their largestmagnitudes over compact domains in both their time and fre-quency representations [13]. Wavelet families are generally orga-nized in subsets corresponding to different scales of variability.Within each subset, each wavelet corresponds to a different posi-tion in the dataset. When errors are associated with the originaldata, they can be propagated through the wavelet coefficientscalculation. The ratio between the wavelet coefficient value andits error can be regarded as the statistical significance of the contri-bution of that wavelet to the original data. It is related to the CLwith which variability at the corresponding position and scale inthe data sequence is observed. Generally, as is the case with themost commonly used Daubechies wavelets [9], different waveletswithin one timescale overlap and their coefficients are not statisti-cally independent. For this reason, this paper presents an analysis,which utilizes the most compact of the Daubechies wavelet familyknown as D2 and usually referred to as the Haar wavelets [10],which do not suffer from this inconvenience and which have re-cently seen a regain of interest in the domains of signal processingand optimal control of linear time varying systems [11].

In Section 2 we describe the Haar wavelet basis and some of itsproperties. Section 3 presents a few examples and applicationsbased on simulated data to characterize the sensitivity of the wave-let analysis relative to a standardv2 test. Finally, Section 4 presents afew caveats concerning the implementation and interpretation of aHaar wavelet analysis when applied to real data, and includes anapplication to real data from the VERITAS gamma ray observatory.

872 R. Price et al. / Astroparticle Physics 34 (2011) 871–877

2. Analysis with the Haar wavelets

Consider a dataset consisting of data points si withi = 0,1,2, . . . ,N where the number of data points is N = 2p with pan integer. The Haar wavelet coefficients {ci} with i = 0,1,2, . . . ,Ncan be seen as the results of functional inner products of the datasequence with the Haar functions {Hi}, the first few of which arerepresented in Fig. 1. The calculation of the Haar wavelet coeffi-cients is fairly straightforward. The first one, c0 is the average ofall the data points (see H0 on Fig. 1). The next one, c1 is the differ-ence between the averages over the first and second halves of thedata (see H1). Then, c2 is the difference between the averages overthe first and second quarters of the dataset (see H2) while c3 is thedifference between the averages over the third and fourth quarters(see H3). Then we go to a smaller scale with c4, which is the differ-ence between the averages of the first and second eighth of thedataset (see H4). This goes on until differences between individualdata points are all recorded. Clearly, there are N/2 coefficients mea-suring the differences between consecutive data points, there areN/4 coefficients measuring the differences between consecutiveaverages of groups of two data points, and there are N/2m coeffi-cients measuring the differences between consecutive averagesover groups of 2m�1 data points.

The coefficients {ci} are linear combinations of the data points{si} to which measurement errors {dsi} may be associated. Underthe assumption that the errors are gaussian and independent, itis straight forward to propagate the errors through the coefficientscalculation and obtain errors {dci} for each coefficient. The ratio jci/dcij can be seen as the statistical significance with which variabilityis observed at the corresponding scale and time in the dataset.Since the Haar coefficients within a given timescale do not dependon any common data, they are statistically independent so they canbe used in a v2 analysis with v2 ¼

Pðci=dciÞ2. In the absence of var-

iability, the expected coefficient values are zero. With the numberof coefficients characterizing one specific timescale as the numberof degrees of freedom, the complement of the v2 probability pro-vides the CL with which variability over that timescale may bereported.

The statistical independence of the Haar wavelet coefficientswithin each timescale is one of their advantages. Another advan-tage is their computational and conceptual simplicity. The Haarwavelets provide coefficients, which can be directly understoodand described in simple terms of the up and down variations fromone region to the next. However, the sensitivity to a variability fea-ture with a given timescale may depend on the precise timing ofthat feature. For example, if two consecutive data points have ahigher signal than the rest of the data, the corresponding variabil-

1

−1

0 1

1

−1

0 1

1

−1

0 1

1

−1

0 1

H0 H1

H4 H5

Fig. 1. The first eight Haa

ity should ideally appear in the two data point scale. If the first ofthe two data points has an odd index (with the first data point in-dexed as 1) the variability will indeed appear in one coefficient ofthe two data point scale. However, if the first of the two data pointshas an even index, the variability will show up in two coefficientsof the one data point scale. This corresponds to an effective blur-ring of the specific timescale to be associated to the different wave-let scales. Section 3 shows simulated data and the correspondingHaar wavelet transforms.

3. Application to simulated data

In order to illustrate the Haar wavelet analysis method, dimen-sionless datasets of 64 points in arbitrary units (a.u.) were ran-domly generated. It is assumed the data points are recorded atequal time intervals, which we will use as a time unit. The time be-tween two consecutive data points is used as the time unit. Twoexamples are presented here. In both cases a mean baseline at a le-vel of 0.3 a.u. is superimposed with a Gauss function shaped mod-ulation. The data points are drawn randomly according to a Gaussdeviate of standard deviation equal to unity and the errors are alltaken equal to 1.0 a.u.

3.1. Slow variability example

The data presented on the left panel of Fig. 2 corresponds to aGauss modulation of amplitude 1.0 a.u. centered on data point 48with a standard deviation of 8 time units, which is indicated bythe solid line. The visual inspection of the data suggests the signalincreases in the second half of the dataset. However, the data devi-ates from the average (represented by the horizontal dashed line)with a very acceptable reduced v2 of 1.109 for 63 degrees of free-dom corresponding to a v2 probability of more than 0.25 or a CL forvariability of less than 75%, which is insufficient to report anyvariability.

The Haar wavelet coefficients statistical significances are shownin the right panel of Fig. 2. It is the coefficient value divided by itserror that is represented and therefore, all the error bars on thegraph have a unit amplitude. The first coefficient differs from zerowith a statistical significance of 6.3, indicating the average signal isdifferent from zero. The second coefficient is different from zerowith a statistical significance of 3.04, indicating an increase ofthe signal from the first half to the second half of the data points(since the coefficient is negative, see H1 on Fig. 1) with a v2 prob-ability or CL of more than 99.7%. This is to be compared to the CL of75% obtained from the simple v2 test applied on the original data,and indicates a greater sensitivity of the Haar wavelet analysis. The

H7

1

−1

0 1

1

−1

0 1

1

−1

0 1

1

−1

0 1

H2 H3

H6

r wavelet functions.

Fig. 2. Left: randomly generated data with a Gauss modulation of amplitude 1.0 centered on data point 48 with a standard deviation of 8 time units. The simulatedmodulation is indicated by the solid curve and the data average by the dashed line. Right: the significance of the Haar wavelet coefficients (coefficients divided by theirstatistical error) arranged in groups (separated by vertical dashed lines) the longest scale on the left to the shortest on the right. The second coefficient being different fromzero and negative indicates a signal increase from the first half to the second half of the dataset.

R. Price et al. / Astroparticle Physics 34 (2011) 871–877 873

second half of the groups of coefficients in the 16 and 1

32 scales (fourand two time units) display some deviation from zero but not at asufficiently significant level to reveal actual variability at thesetimescales. The global v2 probabilities under the null hypothesisare 69% and 46% respectively for these two timescales. Choosinga timescale, for example the one with the largest v2 probabilityfor variability, must be associated with a trial factor penalty. How-ever, the wavelet coefficients in different timescales are not statis-tically independent. Further more, a same variability feature islikely to affect the coefficients in more than one timescale. As aconsequence, the CL obtained for each and all timescales shouldbe reported with their timescale of relevance.

3.2. Fast variability example

Fig. 3 presents the result of a similar simulation with a Gaussmodulation of amplitude 6.0 a.u., centered on data point 37 witha standard deviation corresponding to one time unit. Visual inspec-tion draws attention to the simulated variability with one point atalmost five standard deviations from the baseline and its neighborsat more than one standard deviation. As long as external informa-tion (such as simultaneous observations in other energy bands) is

Fig. 3. Same as Fig. 2 for a modulation of amplitude 6.0, centered on point 37 and a standas well as in the Haar wavelet coefficients statistical significance (see the vertical dotted lestablished.

not available, the CL in the identification of this feature must fullyaccount for the implicit trial factor associated with the selection ofa specific position and scale in the data among all the possibilities.The v2 test for variability gives a reduced v2 of 1.13 for 63 degreesof freedom, corresponding to a v2 probability of more than 0.22 ora CL of less than 78%. The right panel of Fig. 3 presents the statis-tical significances of the Haar wavelet coefficients. The simulatedmodulation appears in one coefficient standing out at more thanthree standard deviations in the 1

32 scale (two time units). It mayalso appear in the 1

64 scale (one time unit) with a couple of coeffi-cients at a little more than two standard deviation from zero. Thesecould be interesting as they start to resolve the structure of thepulse. However, CL for these scales are of 88% and 75%, insufficientfor variability to be reported. The Haar wavelet analysis does notprovide better result than a v2 test for the shortest variabilitytimescales.

3.3. Statistical comparison with the direct v2 test for variability

In order to quantitatively characterize the differences in sensi-tivity noticed in the two above examples, we consider a large num-ber (1000) of simulations of the same variability patterns and

ard half width corresponding to 1 time unit. Although the pulse appears in the dataines), the CL associated with the modulation is not sufficient for any variability to be

874 R. Price et al. / Astroparticle Physics 34 (2011) 871–877

compare the distributions of the v2 probabilities in each timescaleof the Haar wavelet analysis to that of a direct v2 test on the origi-nal data.

The left panel of Fig. 4 shows the distribution of the v2 proba-bility of the 1

2 scale compared to that of a direct v2 test in the caseof a slow variability as in Fig. 2. The v2 probability is the comple-ment of the CL for variability. While the direct v2 test yields a 99%(99.9%) CL in 5.7% (1.1%) of the cases, the Haar wavelet coefficientanalysis results in a 99% (99.9%) CL in 41.2% (17.1%) of the cases,demonstrating a greater sensitivity. It should be noted that whenthe variability is centered in the dataset instead of concentratingin one half, the variability appears in the 1

4 scale with a reducedsensitivity (99% (99.9%) CL in 25.9% (9.9%) of the cases) while thev2 test gives the same result.

A similar comparison is shown on the right panel of Fig. 4 in thecase of the faster variability of Fig. 3. Since the variability extendsover �3 data points, it primarily affects the coefficients in the 1

16scale and it is the corresponding v2 probability that is comparedto the v2 probability of the direct v2 test for variability. The directv2 test yields a 99% (99.9%) CL in 45.3% (19.7%) of the cases and theHaar wavelet coefficients analysis results in a 99% (99.9%) CL in34.3% (15.1%) of the cases. The somewhat lower performance ofthe Haar wavelet coefficient analysis is another example of thesensitivity of the Haar coefficients to the position of the variabilityfeature. When the same variability is centered on data point 38 in-stead of 37, both the Haar analysis and direct v2 test result in thesame v2 and CL distributions. This suggests the CL for variability inthe different timescales should be all given.

The advantage of the Haar analysis over the direct v2 test forlonger variability timescales comes from taking into account theorder of the data points. When the variability is so fast it affectsonly one or a very few data points, the order of the data points be-comes irrelevant and both methods should have similar sensitivi-ties as we just observed.

4. Application to real data

4.1. Complications generally associated with the analysis of real data

4.1.1. If the number of data points is not a power of twoThe Haar wavelet analysis assumes the number of data points to

be a power of two while this may not be the case with real data. A

Fig. 4. The distribution of v2 probabilities are compared between the direct v2 test for vawavelet analysis. The v2 probability distributions are shown only for the most significantfrom the simulations of a variability such as in Fig. 2 while histograms shown on the ri

strict approach for circumventing this problem consists of truncat-ing the dataset to a subset with a power of two as the number ofdata points included. This may however be frustrating when thenumber of data points is close but just inferior to an integer powerof two. Another approach is to pad the end of the dataset withzeros until a power of two is obtained in such a way that the samealgorithm as described above may be used. The last non-zero aver-age used in each timescale is them biased toward zero. In order toavoid this that average can be calculated counting only the datapoints originally in the data, excluding any padding zeros. Thiseliminates any data pollution caused by the zero padding. How-ever, this also blurs the timescales as the last non zero coefficientof each timescale is calculated from a number of data points thatmay be different from what it is for the other coefficients in thesame timescale section.

4.1.2. The data points have different errorsThe data points’ errors were considered to be all the same in the

above simulations while this is generally not the case. In thecalculation of averages, data points with the largest errors shouldbe given a lesser weight. One can choose to use weighted averageswith 1

ds2i

as the weight of data point si. The error on the average is

then given by 1ffiffiffiffiffiffiffiffiffiP1

ds2i

q [14]. The wavelet coefficient errors are then

obtained, just as before, as the half square root of the sum of thevariances of the two averages involved. This is valid provided theprobability distributions are gaussian as we assumed from thebeginning. This was nevertheless tested by simulating 10,000 datasets of 64 samples without any variability and with the standarddeviation of each data point taken randomly and uniformly be-tween 0.1 and 10. It was then verified that the wavelet coefficientswithin each variability scale are normally distributed around zerowith a standard deviation corresponding to the calculated coeffi-cient error.

In some cases, for example in counting experiments, when thecounts are very finely binned in time so the individual data pointsderive from a Poisson statistics, attention may have to be given tothe non-Gaussian nature of the measurements constituting thedata. This is not investigated here and the errors are assumed tobe all Gaussian.

riability and the v2 test with the null hypothesis for selected times scales in the Haartimescales to avoid cluttering the figure. Histograms in the left panel were obtained

ght panel correspond to variability such as in Fig. 3.

Table 1For each timescale of the Haar coefficient analysis of the M87 data, the reduced v2,the number of degrees of freedom (NDOF) and the corresponding CL for variability aregiven in the table.

Scale Reduced v2 NDOF CL (%)

11

37.74 1 100.012

14.59 1 99.98714

0.51 2 40.018

1.98 3 88.51

161.20 7 70.2

132

1.73 13 95.1

R. Price et al. / Astroparticle Physics 34 (2011) 871–877 875

4.1.3. Uneven data samplingAnother difficulty arrises from the fact that data points are gen-

erally not recorded with a strict regularity. In order to apply a Haarwavelet analysis in these situations, one could subdivide the datain powers of two according to the time data points are recorded.However, one is then likely to rapidly encounter time intervalscontaining no data spreading sporadically throughout the entiretime interval. Alternatively, our approach consists of applying theHaar wavelet analysis as described above, ignoring irregularitiesin the data points’ times. The resulting Haar wavelet coefficientsmay then not correspond to comparison between data subsets ofexactly equal durations anymore. They however still provide a use-ful systematic comparison between different domains of the data-set. The calculation of CL for variability as described above is stillvalid but the variability timescales can only be identified on the ba-sis of a careful inspection of the data points’ time distribution asillustrated in the following application example.

4.2. Example application

Fig. 5 shows the light curve of M87 above 250 GeV recordedwith the VERITAS gamma ray observatory with 27 nightly averagemeasurements obtained between 12/14/2007 and 05/02/2008 [5].At first, we consider the data points in sequence, ignoring the dif-ferences between the time intervals separating them. A direct v2

test for variability gives a 99.840% CL indication for variability (re-duced v2 of 2.0 for 26 degrees of freedom). Without a more refinedanalysis, only a visual inspection may permit to develop some ideaabout the variability pattern and timescales. Here, it can be seenthat the data points are on average higher in the first half of thedataset and a flare may have been recorded with measurements11, 12 and 13 which seem to be higher than their neighbors. TheCL that may be reported about the different features of the variabil-ity pattern remain undefined.

This can in fact be clarified with the Haar coefficients analysispresented above. Since the data contains only 27 points, 5 paddingzeros were added at the end. The statistical significance of the Haarcoefficients is shown on the right panel of Fig. 5. Because of thepadding with zeros, the last three coefficients of the 1

32 timescaleare null as well as the last of the 1

16 and 18 timescales. Table 1 gives

Fig. 5. Left: light curve of M87 recorded with VERITAS at E > 250 GeV with 27 measuremwas higher in the first half of the dataset and that a two or three day flair may have beenof the Haar wavelet coefficients (wavelet coefficients divided by their statistical error so

the CL for variability over each scale of the Haar coefficients anal-ysis. The first coefficient, which corresponds to the average of theentire data-set, is different from zero with a statistical significanceof 6.14 which corresponds to a high CL for the detection of gammaray emission from M87. The second coefficient, which approxi-mately corresponds to the difference between the first and secondhalves of the data-set, is different from zero with a statistical sig-nificance of 3.82 corresponding to a 99.987% statistical CL the fluxchanged. The potential flare appears in the 1

16 (difference betweenconsecutive pairs of measurements) and 1

32 (difference betweenconsecutive measurements) scales. In the 1

16 scale, the 3rd and4th coefficients present deviations from zero of just under 2 stan-dard deviations. They correspond to differences between measure-ments 9 through 12 and 13 through 16 respectively. In the 1

32 scale,the 6th and 7th coefficients present deviations from zero of the or-der of 2 standard deviations. They correspond to differences be-tween consecutive measurements 11 and 12 and measurements13 and 14 respectively. The CLs for variability in the 1

16 and 132 time-

scales are found to be 70% and 95% respectively so the dataremains inconclusive regarding the shortest timescale variability.

The data points were not recorded evenly in time. The left panelof 6 presents the flux measurements as a function of time and theright panel of Fig. 6 shows the modified Julian date of eachobservation as a function of its index. Observations are regularlyinterrupted by periods of full moon and, because of weather condi-tions, observations were also impossible on occasional nights dur-ing moonless periods. The solid line represents the boundary

ents over the period between 12/14/2007 and 05/02/2008. It seems the average fluxrecorded around the 12th measurement (02/12/2008). Right: Statistical significance

all the error bars have unit amplitude) for the M87 data. See text for details.

Fig. 6. Left: M87 flux recorded with VERITAS at E > 250 GeV as a function of time with the measurement index numbers indicated next to each point. Right: measurementstimes presented as a function of their index. The solid, dashed and dotted line indicate the data subdivision in halves, quarters and eighths after zero padding as done in theHaar wavelet coefficients calculation. See text for details.

876 R. Price et al. / Astroparticle Physics 34 (2011) 871–877

between the two halves of the dataset as used in the calculation ofthe second Haar wavelet coefficient (the 1

2 scale). The data halveshave respective durations of 81 and 58 days indicating the 1

2 scalecorresponds to durations of �70 days. The dashed lines subdividethe data in quarters of durations 51, 26, 36 and 20 days respec-tively, and the 1

4 data scale is averaging to a �33 days timescale.The dotted lines subdivide the data in seven eighths with durationsof 29, 20, 5, 20, 26, 9 and 20 days respectively, and the 1

8 data scaleis averaging to a �18 days timescale. It should be noted that thetimescale of each wavelet domain is not strictly defined. In ourexample the 1

4 scale spans durations from 20 to 51 days while the18 scale spans durations from 5 to 29 days, showing an overlap be-tween them. Going to smaller data scales, this overlap becomesmore severe and the timescales less defined. The 1

16 scale spansdurations from 3 to 28 days while the 1

32 scale spans durations from1 to 20 days. This makes it impossible to associate them to actualspecific timescales. However, CL calculated from these data scalesare still good indicators for variability over the shortest timesscales available in the data.

In our example, the short data scale CLs are too small for the fewday flare to actually be claimed. However the change in flux be-tween the first and second half of the data can be reported witha 99.987% CL, greater than the indication provided by the non-timespecific v2 test which gave a 99.840% CL. Expressed in term ofgaussian standard deviations, this corresponds to changing theindication for variability from less than 3.2r to more than 3.8rwhile providing information on the variability timescale at thesame time.

5. Conclusions

It is often difficult to establish a CL with which variability canbe reported from the analysis of a dataset such as a light curve. Asimple v2 test may not be optimally sensitive to the long vari-ability timescales and does not provide any indication regardingthe timescales involved. The decomposition of simulated dataon the Haar wavelet basis with error propagation was shownto be a simple and useful tool for investigating variability overseveral times scales simultaneously. A v2 test under null hypoth-esis within each timescale provides the confidence level with

which variability occurs over that timescale. This simple treat-ment is made possible by the fact that the Haar wavelet coeffi-cients within one timescale are statistically independent, aproperty that is not shared with other wavelet families morecommonly used.

The method as described can be applied to data with irregular-ities, the most important of which pertain to the data points timedistribution. The interpretation of the results is then complicatedby the degradation of the actual definition of the timescales butthe method remains usable. This was illustrated by an exampleapplication to data from the VERITAS gamma ray observatory.The sensitivity gain appears on the largest scales while at theshortest scales, both the standard v2 test and the wavelet approachhave the same sensitivity. The Haar wavelet analysis provides asimple and systematic approach to extract variability scale infor-mation from the data. However, it was noted that the precise iden-tification of the variability timescales can be affected byirregularities in the time sequence of measurements.

We have considered the case of time variability studies. Thesame approach could be used in the analysis of energy spectrafor which the common practice consists in considering the v2

probability of a power law fit. When the v2 probability is large en-ough, there is typically no further investigation for a possible devi-ation from a straight power law [4,2]. The Haar wavelet analysiscould be applied to the residual of this power law fit to identifyspectral curvature or cutoff with improved sensitivity.

Several developments could be considered to improve this anal-ysis. It was for example observed that the sensitivity of this analy-sis maybe affected by the precise location of the variability patternin the data. This weakness may be alleviated by scanning the dataseveral times, each time shifting the starting data sample. In suchan analysis, additional trial factors should be taken into account inthe calculation of the CLs for variability. Also, in this paper, wemade use of v2 tests which are only justified under the assumptionof the Gaussian nature of the data points errors. A possible furtherdevelopment of the method could consist of replacing the v2 witha likelihood characterization for different statistics such as Poisson.Finally, the method could easily be transposed to multi-dimensional data such as sky-maps as a tool to search for featuresover different scales.

R. Price et al. / Astroparticle Physics 34 (2011) 871–877 877

Acknowledgements

The authors are grateful to Michelle Hui and the VERITAS collab-oration for providing the M87 light curve data as well as helpful sug-gestions. The authors also gratefully acknowledge support for thisresearch under National Science Foundation Grant PHY-0947088.

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