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How Sample Completeness Affects Gamma-Ray Burst Classification

a r X i v :a s t r o -p h /0209073v 1 4 S e p 2002How Sample Completeness A?ects Gamma-Ray Burst

Classi?cation

Jon Hakkila and Timothy W.Giblin

Department of Physics and Astronomy,College of Charleston,Charleston,SC,29424

Richard J.Roiger and David J.Haglin Department of Computer and Information Sciences,Minnesota State University,Mankato,MN 56001William S.Paciesas Department of Physics,University of Alabama in Huntsville,Huntsville,AL 35899and Charles A.Meegan NASA,Marshall Space Flight Center,Huntsville,AL 35899ABSTRACT Unsupervised pattern recognition algorithms support the existence of three gamma-ray burst classes;Class I (long,large ?uence bursts of intermediate spec-tral hardness),Class II (short,small ?uence,hard bursts),and Class III (soft bursts of intermediate durations and ?uences).The algorithms surprisingly as-

sign larger membership to Class III than to either of the other two classes.A known systematic bias has been previously used to explain the existence of Class III in terms of Class I;this bias allows the ?uences and durations of some bursts to be underestimated (Hakkila et al.,ApJ 538,165,2000).We show that this bias primarily a?ects only the longest bursts and cannot explain the bulk of the Class III properties.We resolve the question of Class III existence by demon-strating how samples obtained using standard trigger mechanisms fail to preserve the duration characteristics of small peak ?ux bursts.Sample incompleteness is thus primarily responsible for the existence of Class III.In order to avoid this incompleteness,we show how a new dual timescale peak ?ux can be de?ned in terms of peak ?ux and ?uence.The dual timescale peak ?ux preserves the du-ration distribution of faint bursts and correlates better with spectral hardness

(and presumably redshift)than either peak?ux or?uence.The techniques pre-

sented here are generic and have applicability to the studies of other transient

events.The results also indicate that pattern recognition algorithms are sensi-

tive to sample completeness;this can in?uence the study of large astronomical

databases such as those found in a Virtual Observatory.

Subject headings:gamma-rays:bursts—methods:data analysis,statistical—

instrumentation:miscellaneous

1.Introduction

In recent years,data mining algorithms have been used to aid the process of scienti?c classi?cation.Data mining is the extraction of potentially useful information from data using machine learning,statistical,and visualization techniques.Pattern recognition algorithms (or classi?ers)are data mining tools that search for clusters in complex,multi-dimensional spaces of attributes(observed and/or measured properties).These algorithms typically oper-ate in one of two modes:supervised(in which the classi?er is trained with known classi?ca-tion instances)and unsupervised(in which classi?cation occurs without training examples). Algorithms are designed to identify data patterns such as clustering and/or correlations, but their limitations must also be understood:it is up to the scientist to interpret physical mechanisms responsible for producing identi?ed clusters.Clusters found by classi?ers can represent source populations;this happens when the class properties are produced by phys-ical mechanisms pertaining to the sources.Clusters can also result from the way in which source properties are measured;sampling biases,systemic instrumentation errors,and corre-lated properties can all force data to cluster and thus give the appearance of class structure when there is none.

Data mining algorithms can be applied to gamma-ray burst classi?cation.Two gamma-ray burst classes have been recognized for years(Mazets et al.1981;Norris et al.1984; Klebesadel1992;Hurley1992;Kouveliotou et al.1993)on the basis of duration and spec-tral hardness.Class I(Long)bursts are longer,spectrally softer,and have larger?uences than Class II(Short)bursts.Recent classi?cation schemes have used data collected by BATSE(the Burst And Transient Source Experiment on NASA’s Compton Gamma-Ray Observatory;CGRO)(Meegan et al.1992)because this large database(2704bursts ob-served between1991and2000)was collected by a single instrument with known instrumen-tal characteristics.Three attributes of BATSE gamma-ray burst data have been identi?ed as being signi?cant(using techniques such as principal component analysis)in delineating gamma-ray burst classes(Mukherjee et al.1998;Bagoly et al.1998;Hakkila et al.2000a):

duration T90(the time interval during which90%of a burst’s emission is detected),?uence S(time integrated?ux in the50to300keV spectral range),and spectral hardness HR321 (the50to300keV?uence divided by the25to50keV?uence).Logarithmic measures of these values are typically used because classes are more clearly delineated when attributes are de?ned logarithmically.Historically,bursts with durations T90<2seconds have been typically considered to belong to Class II.

Data mining techniques allow a third gamma-ray burst class to be identi?ed in BATSE data.Three classes are preferably recovered instead of two by both statistical clustering techniques(Mukherjee et al.1998;Horv′a th1998)and unsupervised pattern recognition al-gorithms(Roiger et al.2000;Balastegui et al.2001;Rajaniemi and M¨a h¨o nen2002).The third class forms at the boundary between Class I and Class II,and primarily contains the softest and smallest?uence bursts from Class I.Since Class II appears to be relatively un-changed by the detection of the third class,the three classes are called Class II(short,small ?uence,hard bursts),Class I(long,large?uence bursts of intermediate hardness),and Class III(intermediate duration,intermediate?uence,soft bursts;also referred to as Intermediate bursts).

The boundaries between classes are fuzzy,as some bursts are not easily assigned to a speci?c class.Di?erent data mining algorithms do not necessarily assign individual bursts to the same classes because each classi?er operates under di?erent assumptions concerning correlations between data attributes and how these relate to clustering criteria.Some clas-si?ers are designed to work with nominal data while others are not;some employ Bayesian while others employ frequentist statistics;some assume a priori distributions of attribute values while others do not.The results of any classi?er can change if the size and makeup of the data set is altered.Data errors can in?uence the results since few classi?ers cur-rently include measurement error information in their analyses.However,irreproducibility is not necessarily a fault of machine learning methodology.Each classi?er provides di?erent insights into the way the data are structured.For any given data set,there is a good pos-sibility that some critical experiment or observation has not been performed,or that some key measurements have yet to be made,or that the relative importance of some attribute has been underestimated or overestimated.There is no correct way of classifying a dataset because the usefulness of the classi?cation depends on the insights gained from it by the user.

In a previous application of supervised classi?cation(Hakkila et al.2000a)to gamma-ray burst data we hypothesized that Class III does not necessarily represent a separate source population.Instead,instrumental and sampling biases have been proposed as a way in which some Class I bursts can take on Class III characteristics.Due to a known correlation between hardness and intensity(Paciesas et al.1992;Mitrofanov et al.1992;

Nemiro?et al.1994;Atteia et al.1994;Dezalay et al.1997;Qin et al.2001),small?uence Class I bursts are typically softer than bright Class I bursts;this is supported by prin-cipal component analysis(Bagoly et al.1998).Since the correlation results from a shift to smaller average peak spectral energy E p at lower peak?ux but not from changes in the average low-energy spectral index α or the average high-energy spectral index β (Mallozzi et al.1995;Hakkila et al.2000a;Paciesas et al.2002),this correlation has been attributed to the softer bursts being generally at larger cosmological redshift.(This con-clusion may not necessarily be correct because a broad range of gamma-ray burst luminosi-ties is suggested from redshifts of gamma-ray burst afterglows(van Paradijs et al.2000); however,it should be noted that only a small afterglow sample is available.)Addition-ally,?uences and durations of some Class I bursts can systematically be underestimated (Koshut et al.1996;Bonnell et al.1997;Hakkila et al.2000a);we refer to this as the?u-ence duration bias(Hakkila et al.2000a).Simply put,?uences and durations of some Class I bursts(particularly those with the smallest peak?uxes)can be underestimated due to the unrecognizability of low signal-to-noise emission;combined with their spectral softness,this gives them characteristics consistent with Class III.

Unfortunately,the?uence duration bias has been di?cult to quantify.The amounts by which the?uence and duration of an individual burst are a?ected depend on the?tted background rates and estimated burst durations at all energies;to remove the background properly assumes a priori knowledge of the burst’s intrinsic temporal and spectral structure. Such a priori knowledge can only be acquired in the absence of background,and gamma-ray burst observations are inherently noisy.Very high signal-to-noise estimates of a burst’s temporal and spectral structure can only be obtained for a small number of the bursts with the largest?uences.These well-measured quantities are not entirely intrinsic;it appears that even the brightest bursts require systematic correction because they are at large redshift (z≈1).

Our objective is to determine whether or not the?uence duration bias can account for the number of bursts with Class III characteristics.In order to do this,we determine the total number of bursts that exhibit Class III characteristics using several di?erent unsupervised classi?ers.Then,we statistically model the suspected bias and determine whether it is strong enough to produce the Class III bursts.

A number of pertinent questions will have to be addressed in pursuing this objective: Do theoreticians need to develop models for one,two,or three gamma-ray burst classes? How can data mining techniques be used to aid scienti?c classi?cation?Are systematic e?ects present in data collected by BATSE or other gamma-ray burst experiments that alter classi?cation structures?Can these e?ects be understood?Can information on intrinsic

properties of the source population be extracted if these e?ects are present?Can future instruments be designed to minimize or eliminate these e?ects?

2.Class III and the Fluence Duration Bias

2.1.The Signi?cance and Size of Class III

We systematically compare the output of various unsupervised algorithms in conjunction with a homogeneous gamma-ray burst data set obtained with one set of instrumental settings. We use the online gamma-ray burst ToolSHED(Haglin et al.2000)(SHell for Expeditions in Data mining)that we are developing to aid our analysis.This ToolSHED(currently ready for pre-beta testers at https://www.doczj.com/doc/265195001.html,/grbts/)provides a suite of supervised and unsupervised data mining tools and a large database of preprocessed gamma-ray burst attributes.It allows users to classify data using more than one algorithm in order to identify consistencies in the di?erent classi?cation techniques and thereby gain better insight into the heterogeneous nature of the data.

In order to further minimize the e?ects of instrumental biases,we have limited our database to bursts detected with a homogeneous set of BATSE trigger criteria.The database consists of bursts from the BATSE Current Burst Catalog(https://www.doczj.com/doc/265195001.html,/batse/ grb/catalog/current/).Bursts included are non-overwriting and non-overwritten bursts(e.g. those whose BATSE readout periods did not overlap detectable bursts immediately preceding or following their detection)triggering at least two BATSE detectors in the50to300keV energy range with the trigger threshold set5.5σabove background on any of the three trigger timescales.We require all classi?ers to use only the three attributes of log(T90),log(HR321), and log(S).

We apply four unsupervised ToolSHED algorithms with di?erent approaches to cluster-ing.These algorithms are ESX,a Kohonen neural network,the unsupervised EM algorithm, and the unsupervised Kmeans algorithm.

ESX(Roiger et al.1999)is a classi?er that forms a three-level tree structure.The root level of the tree contains summary information for all bursts.The second(concept)level of the tree sub-divides the root level into clusters formed as a result of applying a similarity-based evaluation function.The third tree level holds the individual bursts.

A Kohonen neural network(Kohonen1982)architecture is represented as a collection of input and output units.During training,the input units iteratively feed the burst instances to the output units.The output units compete for the burst instances.The output units

collecting the most bursts are saved.The saved units represent the clusters found within the data.

The unsupervised EM algorithm(Dempster et al.1977)assumes that the attribute space can be subdivided into a predetermined number of normally distributed clusters.An initial guess is made as to the properties of each random distribution,and this guess is used to calculate probabilities that bursts belong to each cluster.The cluster characteristics are iteratively adjusted until all clusters are optimally-de?ned.

The Kmeans(Lloyd1982)algorithm randomly selects K data points as initial cluster centers.Each instance is then placed in the cluster to which it is most similar.Once all instances have been placed in their appropriate cluster,the cluster centers are updated by computing the mean of each new cluster.The process continues until an iteration of the algorithm shows no change in the cluster centers.

Predetermined classi?cation signi?cance helps de?ne the number of classes that can be recovered.When allowed to?nd an optimum number of classes based on a default signi?cance,the aforementioned classi?ers typically recover three to four burst classes as opposed to the two traditionally accepted classes.This indicates that the two traditional classes are not considered to be the optimal solution.

We force all four classi?ers to recover two,three,and four classes because we hope that by studying the properties of these force-recovered classes we can determine why the three-class solution is preferred over the two-class solution.The properties of three force-recovered classes are indicated in table1.The properties of these classes are similar to those obtained using other clustering techniques(Mukherjee et al.1998;Horv′a th1998;Roiger et al.2000; Balastegui et al.2001;Rajaniemi and M¨a h¨o nen2002),so we again refer to these as Class I(Long),Class II(Short),and Class III(Intermediate).However,these previous results typically place fewer bursts in Class III than Class I,whereas three of our four classi?ers place the largest number in Class III.Therefore,our analysis?nds Class III to be the dominant class.

In order to explain why the percentage of Class III members is so large,we examine the placement of Class III bursts when classi?ers are forced to recover only two classes(Short and Long).The results are remarkably consistent:all four classi?ers fail to clearly delineate the traditionally-accepted Short and Long classes,and each places a large number of soft Class III bursts in with the hard Short class(see Figure1).This is surprising,since Class III clustering is not obvious to the naked eye in the hardness vs.duration parameter space whereas Short and Long burst clustering is.The hardness vs.duration boundary is not chosen by the classi?ers because a sharper one exists in the?uence vs.duration parameter

space(Figure2);the boundary separating short faint bursts from long bright bursts is more signi?cant than that separating short hard bursts from long soft bursts.

It is surprising that?uence plays such an important role in the classi?cation.First,?uence is an extrinsic attribute(since it represents a convolution of a burst’s luminosity and distance)as opposed to hardness and duration,so there is no reason why?uence clustering should relate to any physical di?erences between burst classes.Second,one would intuitively expect a burst with a longer duration to have a larger?uence,indicating that?uence and duration should be highly-correlated attributes.Thus,clustering in the duration attribute can also cause clustering in the?uence attribute,and the use of?uence as a classi?cation attribute magni?es the clustering importance of duration relative to hardness.The break between short faint and long bright bursts therefore appears due in part to the use of?uence, an attribute which is of questionable value.

To determine if the?uence bias can be removed,we eliminate this attribute and perform the classi?cation using only log(T90)duration and log(HR321)hardness ratio.Even without ?uence,the classi?ers again prefer to recover three classes instead of two,and Class III is not diminished in size.Examination of the three class properties indicates that log(T90) has been used almost exclusively to delineate the classes;hardness is almost ignored by the classi?ers.This is surprising,since the eye tends to delineate two burst classes.We check this result by supplying only the T90attribute to the classi?ers.Indeed,the classi?ers again return three classes rather than two(Class I bursts have T90>6sec.,Class II bursts have T90<1.4sec.,and Class III bursts have1.4sec.≤T90≤6sec.).However,the size of Class III has been diminished in this reclassi?cation and it is no longer the largest class;this result is consistent with that obtained earlier using only the duration attribute(Horv′a th1998). We conclude that strong evidence exists for the three-class structure.

Before accepting the new class as a separate source population,we must try to dis-count alternative explanations concerning its existence.It is possible that the classi?ers have detected a data cluster resulting from the way that the data have been collected,rather than from a separate and distinct source population.We consider it unlikely that Class III represents a statistical anomaly since it has been found by four classi?ers using di?erent algorithms,and since stringent requirements have been imposed for each classi?er to?nd additional classes.Thus,Class III could result from a systematic e?ect such as an instru-mental or sampling bias.The suspected bias appears to primarily a?ect duration and the coupled yet extrinsic attribute of?uence.

This conclusion leads us again to examine the hypothesized?uence duration bias.This bias could provide a mechanism for underestimating both?uence and duration of some Class I bursts(particularly faint soft ones),and could cause these bursts to take on Class III

characteristics.However,with the increased size of Class III,it is reasonable to think that the bias might be strong.

2.2.Inadequacy of the Fluence Duration Bias Model to Explain Class III

Properties

In an attempt to quantify the?uence duration bias,we have developed a simple model of the bias that can be applied statistically.The model only in?uences Class I and Class III bursts(as de?ned by the EM algorithm),since the bias has not been hypothesized to alter Class II properties.In a previous work(Hakkila et al.2000a)we estimated the maximum amount by which the?uences and durations of?ve bright bursts might need to be corrected if their signal-to-noise ratios were reduced;our simple model averages these values to obtain maximum corrections of?uence and duration as functions of p1024(peak?ux measured on the1024ms timescale).We do not know how much the?uence of an individual burst might need to be corrected,therefore we assume that the?uence of each burst should be corrected between0ergs cm?2sec?1and the maximum?uence correction S max,and that the duration of each burst should be corrected between0seconds and the maximum duration correction T90max.The amount of the maximum correction is dependent upon the signal-to-noise ratio and thus on the peak?ux;the suspected bias is more pronounced for bursts with peak?uxes near the detection threshold.We naively assume a probabilityρthat each burst’s measured ?uence and duration will be altered with equal probability in the intervals[0,log(S)max]and [0,log(T90)max].Thus,the modeled amount by which an individual burst’s?uence would be a?ected by the bias isρlog(S)max and the amount by which its duration would be a?ected is ρlog(T90)max.The problem can be inverted to estimate how much observed burst?uences and durations have been underestimated as a function of p1024.

If the?uence duration bias produces Class III properties,then(1)the faint Class I and Class III bursts(as measured by p1024)should show evidence of having their?uences(and durations)systematically underestimated,and(2)no evidence of this bias should be present if this combined distribution has been properly corrected for the e?ect.We would thus like to compare both the observed distribution and the corrected distribution with the“true”distribution.Unfortunately,we do not know the“true”distribution.

If we assume that the bias has not a?ected the?uence and duration distributions of bright bursts(as measured by p1024),then we can compare the corrected and uncorrected distributions of faint bursts to the observed distributions of bright bursts.The comparison can be made once we identify how the?uence and duration distributions scale with peak ?ux.

If a given burst’s intensity were decreased(either by decreasing the burst’s luminosity or if the burst were observed at a larger distance),then its?uence would decrease proportionally to its peak?ux.This generic statement is false only in the presence of sampling and/or instrumental biases.The e?ect of time dilation due to cosmological expansion is an example of a sampling bias that can systematically a?ect?uence count rates di?erently than peak?ux count rates.Since we measure the peak?ux and?uence in the same energy channels,the primary source of bias is that the observed peak?ux can be as little as(1+z)?1of its actual value due to time dilation,whereas the?uence would not be expected to be lessened.This bias would cause the peak?ux of distant bursts to be small relative to the?uence;note that this bias cannot explain Class III characteristics,since Class III bursts have?uences that are small relative to their peak?uxes.In going from bright bursts to faint bursts,a decreasing signal-to-noise ratio can cause?uences of faint bursts to decrease non-proportionally to peak ?uxes;this is an example of a statistical(rather than systematic)instrumental bias.

Sampling biases can cause the faint burst distribution(as measured by either peak?ux or?uence)to be di?erent than the bright burst distribution.Trigger biases can cause bursts with certain characteristics to trigger disproportionately relative to other bursts.However, trigger biases that have been proposed prior to this manuscript do not appear to alter the makeup of the BATSE dataset by large amounts(Meegan et al.2000).We therefore assume in testing the?uence duration bias that it is primarily responsible for causing a burst’s?uence to change not in proportion to the change in its peak?ux,and that the faint burst distribution of Class I+III bursts would be the same as the bright distribution in the absence of this e?ect.

The distribution of burst?uences at a given peak?ux indicates bursts with di?erent time histories;greater?uence typically belongs to longer bursts with more pulses and smaller ?uence typically belongs to shorter bursts with fewer pulses.If these burst peak?uxes were all decreased by the same amount,then their?uences would decrease proportionally along a line de?ned by log(S)line=log(p1024)+R(where R is an arbitrary constant).The di?erence ?log(S)=log(S)line?log(S)obs obtained for each burst indicates the?uence o?set of each burst from the line given its peak?ux.The distribution of?log(S)can be examined for bright bursts(e.g.those presumably una?ected by the bias)and for faint bursts(e.g.those a?ected by the bias).In the absence of any biases,the faint distribution will be similar to the bright distribution.If the?uence duration bias is present,then the faint distribution will di?er from the bright distribution.The aforementioned statistical correction should make the“corrected”?uence distribution?log(S)corr=ρlog(S)max?log(S)obs more compatible with the bright distribution than is the uncorrected faint distribution.

Figure3is a plot of log(S)vs.log(p1024)for the burst sample used in this study.

Class I,II,and III bursts have been identi?ed using the unsupervised EM algorithm.The proportional decrease of?uence and peak?ux is shown for a hypothetical Class I burst (diagonal line);the curving path indicates how the bias might a?ect the measured?uence of this burst as a function of p1024(curving line)in the case whereρ=1.The amount by which the?uence would need to be corrected?log(S)corr is also shown(vertical line).

We construct eight?log(S)bins for the set of bright bursts and eight bins for the faint bursts(the zero point for the?log(S)scale is arbitrary,so we use?log(S)=log(S)?log p1024+6).The dividing line between“bright”and“faint”bursts is set at log(p1024)≥1 photon cm?2sec?1since bursts brighter than this value should be essentially una?ected by the proposed?uence duration bias.The faint uncorrected?log(S)distribution is moder-ately di?erent than the bright distribution,with aχ2=13.8for7degrees of freedom and a corresponding probability of q=0.055.The?uence distribution(as determined from ?log(S))has been shifted to lower values consistent with the?uence duration bias.

In order to test the correction by the proposed model,we correct the?uence of each of the i bursts by di?ering amountsρi log(S max).Theχ2of the corrected faint burst distribution is again compared to the“control”sample of bright bursts.Since we might have overcorrected some bursts while undercorrecting others,we run the analysis a total of100times and average the results.The corrected?log(S)corr distribution is signi?cantly di?erent than the bright distribution(we obtain χ2 =34.0for7degrees of freedom and a corresponding probability of q=2×10?5that the two distributions are identical)indicating that our model has signi?cantly overcorrected for the suspected bias.Similar results are obtained using the ?log(T90)distributions.

Since we have apparently overestimated the amplitude of the?uence duration bias ρi log(S)max for typical bursts,we can decrease our estimate of the bias by introducing a free parameter D in the relationshipsρi D log(S)max andρi D log(T90)max,where D=0represents the uncorrected sample while D=1indicates the originally hypothesized bias.In table2, we demonstrate the e?ectiveness of the?uence duration bias for di?erent values of D chosen arbitrarily.The model?t is only improved when we reduce our estimates of log(S)max and log(T90)max signi?cantly.

We have shown previously(Hakkila et al.2000a)that the maximum time interval used to calculate burst?uences decreases dramatically near the BATSE detection threshold(es-sentially no bursts in the3B catalog with p1024<0.4photons cm?2sec?1have?uence durations≥100seconds).This indicates that the?uence duration bias causes?uences and durations of very long bursts with small peak?uxes to be underestimated.Our current analysis supports this hypothesis:the?log(S)and?log(T90)distributions suggest that shorter bursts with small peak?uxes have probably not been a?ected by the bias,whereas

some longer bursts have.Our experience with BATSE data analysis procedures is also in agreement:?uence duration intervals are rarely chosen to be shorter than many tens of seconds,and the time histories of only a few bursts are particularly susceptible to this bias (Koshut et al.1996).This should prevent a systematic bias from being introduced for short bursts with small peak?ux but not necessarily for long bursts with small peak?ux.

These results indicate that the?uence duration bias does not in?uence faint bursts to the extent hypothesized previously.The shorter Class I bursts,which were originally thought most likely to take on Class III characteristics via the bias,are apparently a?ected the least.The?uence duration bias appears to primarily in?uence the properties of some longer BATSE bursts.We conclude that the?uence duration bias is not responsible for the large number of shorter softer bursts comprising Class III.

3.Sample Incompleteness and the Duration Distribution

Although the?uence duration bias does not appear to be responsible for the creation of Class III,our analysis of the proportional decrease of?uence and peak?ux has unexpect-edly provided new insight into measured burst properties.The faint?uence and duration distributions used in classi?cation are truncated because the samples triggered using short-timescale peak?uxes.This truncation has the potential of biasing the sample via sample incompleteness.In order to study the potential e?ects of sample truncation,we consider the advantages of a?uence-limited sample relative to a peak?ux-limited sample.

An experiment triggering with a short integration window is more likely to detect a short burst than an experiment triggering with a long integration window,because in the latter case the entire burst?ux can be recorded in a single temporal bin.A peak?ux-limited sample is thus biased towards shorter bursts relative to longer bursts because it excludes longer bursts having large?uences but with peak?uxes too faint to trigger.However,a long timescale trigger(such as one that could trigger on?uence)would prove to be equally-biased.

A hypothetical experiment triggering on?uence(e.g.one with a10,000second integration window)would be more likely to detect a faint long burst(having little of its?uence in one temporal window)than would an experiment triggering on peak?ux.A?uence-limited sample would be biased towards longer bursts because it would include faint longer bursts but exclude faint shorter bursts with the same peak?ux.Figure3demonstrates that an excessive number of short Class III bursts are found near BATSE’s peak?ux trigger;the ?uence distribution of these bursts is acutely truncated by the peak?ux trigger.Thus,Class III occupies a?uence vs.peak?ux region where the instrumental(peak?ux)trigger favors detection of shorter bursts over longer ones.

We would like to identify a peak?ux measure that does not su?er from truncation of the duration distribution.The proportional relationship between?uence S and peak?ux p1024as a burst’s luminosity is decreased or as its distance is increased provides a method for identifying such a peak?ux measure.We can re-de?ne?uence to be a peak?ux by de?ning an extremely long temporal windowτ(τis a constant)that contains the entire ?ux of the sample’s longest burst.The?uence divided by this temporal window(S/τ)is a peak?ux(having units of photons cm?2sec?1or ergs cm?2sec?1,using an approximate transformation of A≈2.24×10?7ergs photon?1)(Hakkila et al.2000b).The equation governing this proportional decrease in peak?ux and?uence is

2log(F0)=log(S/(Aτ))+log(p1024)(1)

or F20=S/(Aτ)(p1024)where F0has units of?ux and is thus a measure of burst intensity. We de?ne this quantity as the dual timescale peak?ux(Hakkila et al.2002)since it uses two di?erent timescale measurements.The minimum value of the dual timescale peak?ux F0can be called the dual timescale threshold.The dual timescale peak?ux is merely a multiple of this threshold value,log(F)=log(F0)+K(or F=K F0),where K is a constant.A dual timescale threshold can be de?ned as an instrumental setting for a gamma-ray burst experiment(e.g.by requiring S/(τ)(p1024)to exceed a trigger value),as a selection process on previously-detected events in a standard experiment,or with archival data from an experiment triggering independently on one temporal trigger at a time(such as BATSE).This latter concept is not new;several studies have developed their own post facto BATSE triggers using archival time series data(Schmidt1999;Kommers et al.2001;Stern et al.2001).

The dual timescale peak?ux treats longer bursts having larger?uences and smaller peak ?uxes on an equal basis with shorter bursts having smaller?uences and larger peak?uxes: these bursts with di?erent temporal structures have something in common,which is that they have equal probability of detection using the dual timescale peak?ux.Their di?erences must therefore be de?ned by a line orthogonal to the dual timescale peak?ux,e.g.one that satis?es the relationship

log(Γ)=log(S/A)?log(p1024).(2) orΓ=S/(Ap1024)whereΓhas units of time and represents a duration.We callΓthe?ux durationΓ;it measures the total time that a burst could emit at its peak?ux in order to produce its?uence.Longer bursts typically have large S/p1024values and shorter bursts should typically have small S/p1024values.In fact,the correlation for Class I+III bursts demonstrated in Figure4has a Spearman Rank-Order correlation signi?cance0f10?118that Γand T90are uncorrelated.

The dual timescale peak?ux does not favor bursts of any duration(longer than the smallest1024ms integration window),whereas peak?ux or?uence do by truncating the

distribution and thereby favoring“faint”bursts(e.g.those near the threshold)of longer or shorter durations.Since the dual timescale peak?ux does not truncate the duration distribution,we can say that the dual timescale peak?ux-limited sample retains the duration characteristics of the sample by preserving the duration S/p1024relative to the intensity (S/τ)(p1024).

It was recognized soon after BATSE’s launch(Petrosian et al.1994)that long-timescale triggers underestimated the intensities of shorter bursts and biased the sample.However,our analysis demonstrates(perhaps surprisingly)that short temporal timescale triggers would also bias the sample against longer bursts.

Figure5demonstrates how BATSE’s peak?ux trigger in?uences the number of events placed in Class III relative to those placed in Class I.Shorter bursts(small S/p1024;denoted by region‘C’)have been detected while faint longer bursts(large S/p1024;denoted by‘A’) have been excluded by the trigger.A hypothetical?uence trigger allowing the faintest shorter bursts currently detected by BATSE to trigger would not resolve this problem:shorter bursts (large S/p1024;denoted by‘B’)would go undetected by the?uence trigger relative to longer bursts(small S/p1024;denoted by‘D’).A dual timescale threshold is shown(diagonal dashed line)that favors neither longer nor shorter BATSE bursts.The threshold excludes most of the bursts previously identi?ed as Class III because these have been favored by the one-second trigger window relative to longer bursts.It also excludes many Class II bursts which are both typically faint and shorter than the one-second trigger window.

We make a cut on our BATSE sample that is equally complete for both longer(large S/p1024)and shorter(small S/p1024)bursts and use this as our dual timescale threshold.This threshold follows the relation log(S)+log(p1024)=?6.5,and has been chosen so that even the longest bursts with the largest S/p1024values are detected by BATSE’s actual peak?ux trigger.This is demonstrated by the diagonal dotted line in Figure4,and corresponds to a dual timescale peak?ux(via equation1)of F0=0.048photons cm?2sec?1forτ=617 seconds(the T90of the longest burst in the sample).

We wish to determine how the sample properties vary with dual timescale peak?ux.We thus divide our sample of Class I+III into four subsamples containing similar numbers of bursts but with di?erent dual timescale peak?uxes:bright bursts(log S+log p1024≥?5.1, or K =7.49),moderately bright bursts(?5.8≤log S+log p1024

enough bursts to constitute a reasonable statistical sample.

We identify three?ux duration intervals from the sample:longer bursts(log S≥log p1024?5.6),middle bursts(log p1024?6.1≤log S

The attributeΓis closely related to GRB duty cycle(Hakkila et al.2000b).The duty cycleΨmeasures the persistence of burst emission via the relationship

Ψ=

This evidence supports our hypothesis that the bulk of the Class III bursts are short Class I bursts that have preferentially been detected by BATSE’s short timescale trigger.

If a corresponding sample of longer Class I bursts is detected(by having a lower peak ?ux trigger threshold and/or by having some bursts trigger on a longer timescale),then these bursts most likely would be as soft as the Class III bursts.We suggest that the FXTs (Fast X-ray Transients)found by BeppoSAX(Heise et al.2001)using an x-ray trigger and subsequently identi?ed in BATSE data(Kippen et al.2001)might be long soft bursts that previously escaped detection.

The slope of the log(HR321)vs.log(peak?ux)relation is largest when log(F)is used as the peak?ux measure as opposed to either log(S)or log(p1024);this is true regardless of whether the sample is peak?ux-limited,?uence-limited,or duration-limited.This result is demonstrated in table3,where hardness vs.intensity correlations are examined using a Spearman Rank-Order Correlation test for the three di?erent intensity measures:the1024 ms peak?ux p1024,the?uence S,and the dual timescale peak?ux F.Small probabilities indicate strong correlations between spectral hardness and the peak?ux measure.Spectral hardness(and presumably redshift)correlates better with the dual timescale peak?ux than with any other peak?ux measure,regardless of which measure is used to select the sample. Furthermore,the log(HR321)vs.log(peak?ux)slope is essentially identical for burst samples of di?erent T90durations when log(F)is used;it does not appear that the same can be said when either log(p1024)or log(S)are used as a peak?ux measure.Thus,F appears to more easily deconvolve the attributes of hardness,duration,and peak?ux than do either S or p1024. We take this to indicate that F is a preferred peak?ux indicator to S and p1024.

There are potentially far-reaching consequences to having F as a less-biased temporal ?ux measure.To date,essentially all statistical studies have used either log(p1024)or S as intensity measures(e.g.log(N>S)vs.log(S),log(N>p1024)vs.log(p1024),E peak vs. log(p1024)).These studies are potentially biased because S and p1024do not deconvolve the hardness intensity correlation as cleanly as does F.Presumably,studies made using S and p1024combine longer bursts measured at one value of HR321with short bursts measured at another value of HR321.The use of F in future modeling endeavors might improve our understanding of gamma-ray burst properties better than do either?uence or peak?ux.

We test our hypothesis that Class I bursts and Class III bursts belong to the same population by submitting all bursts in the original sample brighter than F0to the EM algorithm for unsupervised classi?cation,and using the attributes of S,T90,and HR321. The classi?er preferably recovers six classes as opposed to three;the original three-class structure is lost as a result of the new trigger.Despite this,Class II is still easily identi?able even though it contains only40members(Class II bursts in the BATSE Catalogs thus appear

to have been preferentially detected as a result of BATSE’s short timescale trigger).The remaining bursts are placed in?ve classes with properties not recognizable as belonging to the original Class I or Class III.These classes may provide interesting additional insights into burst properties,but they warrant no further discussion here because they are not identi?able as the original burst classes.

Thus,strong reasons exist that the Class III cluster arises primarily from the shape of the attribute space de?ned by BATSE’s peak?ux trigger,and not from a separate source population.Our results support the hypothesis that Class III does not represent a separate source population.We have demonstrated that both?uence and duration are truncated by BATSE’s peak?ux trigger.The truncation e?ectively oversamples short bursts relative to long bursts.As a result of this truncation,the database contains an excess of faint,short (soft)bursts.The use of the dual timescale trigger supports the hypothesis that Classes I and III are really one continuous duration distribution with faint bursts being softer than bright bursts.The properties of this continuous distribution become somewhat ambiguous at low signal-to-noise,where the?uence duration bias alters burst properties.

On the other hand,Class II appears to represent a separate source population from Class I(Hakkila et al.2000a).Neither sampling biases nor instrumental biases appear to be responsible for creating Class II characteristics from Class I bursts.However,it should be noted that BATSE’s short trigger timescales have aided in the large detection rate of these short events.

4.Conclusions

We have demonstrated that

1.Gamma-ray burst Class III does not have to represent a separate source population;

it can be produced by the integration time of the instrumental trigger,

2.the?uence duration bias by itself,as modeled from a sample of high signal-to-noise

bursts,is unlikely to be responsible for the existence of Class III.

3.Class III is likely produced by an excess of short,low?uence bursts detected by

BATSE’s short trigger temporal window.

4.The excess bursts can be eliminated via a selection process that is dual timescale peak

?ux-limited,rather than peak?ux-limited or?uence-limited.

5.The dual timescale peak?ux measure resulting from this selection process appears

to correlate better with hardness(and therefore with E peak and redshift)than either peak?ux or?uence.This adds support to the argument that dual timescale peak ?uxes correct the temporal limitations introduced by using single timescale peak?uxes.

Dual timescale peak?uxes can be established for many combinations of temporal measurements.

The results found here are important to gamma-ray burst astrophysics as well as to the general problem of scienti?c classi?cation.Data mining tools can help identify complex clusters in multi-dimensional attribute spaces.The tools are sensitive to clusters and data patterns,as evidenced here because they have allowed us to discover clusters produced arti?cially as a result of sample incompleteness.This sensitivity is advantageous,because a better understanding of instrumental response and sampling biases can be used to improve the design of future instruments.

We note that sample incompleteness is generic and applies to the detection of any transient sources identi?ed as the result of a temporal trigger.Examples of transient event statistics that might be biased by a temporal trigger include?are stars,soft gamma repeaters, x-ray bursts,and earthquakes.

However,the sensitivity of data mining tools can also cause problems.Data mining is central to the operation of planned Virtual Observatories,which will electronically combine data collected from a variety of instruments with a range of temporal,spectral,and inten-sity responses.Since sample incompleteness can cause a single instrument with one set of characteristics to?nd phantom classes,classes identi?ed using multiple instruments should be interpreted cautiously.The instrumental responses of Virtual Observatory components will have to be accurately known in order for newly-identi?ed classes to be recognized as separate source populations.

It is important to recognize that data mining techniques have their limitations.Principal component analysis has identi?ed?uence,duration,and hardness as being critical gamma-ray burst classi?cation attributes,while the trigger attribute of peak?ux was not chosen. Data mining classi?ers failed to recognize that attribute selection had removed the attribute that could have provided the most insight into the gamma-ray burst clustering structure.

We gratefully acknowledge NASA support under grant NRA-98-OSS-03(the Applied Information Systems Research Program)and NSF support under grant AST-0098499(Re-search in Undergraduate Institutions).We also thank James Ne?and Robert Dukes for valuable discussions.

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掌握基本的数学知识与技能、数学思想和方法,获得广泛的数学活动经验。学生是数学学习的主人，教师是数学学习的组织者、引导者与合作者。 5．评价的主要目的是为了全面了解学生的数学学习历程，激励学生的学习和改进教师的教学；应建立评价目标多元、评价方法多样的评价体系。对数学学习的评价要关注学生学习的结果,更要关注他们学习的过程;要关注学生数学学习的水平,更要关注他们在数学活动中所表现出来的情感与态度，帮助学生认识自我，建立信心。 6．现代信息技术的发展对数学教育的价值、目标、内容以及学与教的方式产生了重大的影响、数学课程的设计与实施应重视运用现代信息技术、特别要充分考虑计算器、计算机对数学学习内容和方式的影响,大力开发并向学生提供更为丰富的学习资源,把现代信息技术作为学生学习数学和解决问题的强有力工具,致力于改变学生的学习方式,使学生乐意并有更多的精力投入到现实的、探索性的数学活动中去。 1、数学教育是中小学的一门基础的学科教育，如同其他的学科一样，其教育意义并不局限于本学科的只是掌握，更反映在它有理地促进人的素质的发展，是人的文化修养的最深刻、最有效的部分之一。 2、经济发达国家的数学教育改革方向：学校数学的焦点从双重任务---对大多数人教最少的数学，而把高等数学教给少数人----- 过渡到单一中心，把数学的最重要的公共核心教给所有的学生。从基于传递只是的权威性的模式过渡到以启发学习为特征的，以学生为中

心理学分析研究方法纯手打整理重点

变量：属性地逻辑组合，通常用定义来解释或限定.自变量—实验条件（操控）; 调节变量：所要解释地是自变量在何种条件下会影响因变量，也就是说，当自变量与因变量间地相关大小或正负方向受到其它因素地影响时，这个其它因素就是该自变量与因变量之间地调节变量.中介变量：不可观察地，而在理论上又是影响所观察现象地因素因变量—所需测定地特征或方面（可测量）额外变量—对因变量有一定影响，但与该次实验研究目地无关地变量（控制：屏蔽、中和））随机误差：可见误差.偶然地、随机地无关变量引起，较难控制，无规律可循；影响信效度.系统误差：常定误差.常定地、有规律地无关变量引起，其方向和大小地变化是恒定而有规律地.影响效度. 无关变量地控制：、消除法排除或隔离无关变量对实验效果地影响.标准化指导语、双盲程序、内隐测量、恒定法实验期间，尽量使得所有地实验条件、实验处理、实验者及被试都保持恒定.研究在同一时间、地点举行，程序、拉丁方设计、平衡法设置使得无关变量对所有地实验组和对照组地影响都均等、统计控制：一种事后补救，统计隔离无关变量地影响,协方差分析，偏相关.操作定义：描述所界定地变量或事项如何测量，包括：工具，方法，程序.将变量或指标地抽象称述转化为具体地操作称述地过程. 、简单随机取样（不作任何预处理，适用范围：对总体中各类比例不了解或来不及了解地情况）) 抽彩法（充分搅匀））随机表法（随

机进入））随机函数法（种子问题）、分层随机取样：对总体进行预处理，分成若干层次后然后独立地从每一层次中选取样本. (）比例分层取样：按每一层次个体数量占总体中地比例决定该层次样本地数量. ()非比例分层取样：每层次中样本量不按该层次在总体中地比例抽取，而是根据研究者对不同层次个体地研究兴趣和侧重程度确定比例地大小. 、内部效度（逻辑度与额外因素影响度）：研究中自变量与因变量因果关系地明确程度. 影响因素：成熟因素、历史因素、被试选择上地差异、研究被试缺失产生地效应、前测地影响、实验程序不一致或处理扩散产生地效应、统计回归效应、研究条件与因素间地交互作用.、外部效度：可细分为总体效度和生态效度.研究结果能一般化或普遍化到样本来自地总体（总体效度）和其它变量条件、时间和背景（生态效度）中去地程度，即研究结果地普遍代表性和适用性.影响因素：.取样代表性；.变量地操作方式不准，致使研究地可重复性差；.研究对被试地反作用；.事前测量与实验处理地相互影响；.多种处理地干扰；.实验者效应.研究地人为性；.被试选择与实验处理地交互作用.提高方法：严格控制；做好取样工作，包括被试取样、实验情境、研究工具、

2第二章设计心理学的研究方法

第二章设计心理学的研究方法 2.1 设计心理学的研究方法人的消费活动是一种复杂的社会行为, 是人类心理活动的一部分。研究消费者心理活动规律的方法与整个心理学的一般研究方法是一致的, 心理学本身的发展, 为心理学应用分支的发展提供了科学的基础。但人类的消费活动是一种特殊领域。在运用心理学的某些研究方法了解消费行为规律时, 必然有一些新的内容和新的问题。因此探索设计心理学研究方法, 不仅有利于自身的发展, 也丰富了心理学主干研究方法的积累。设计心理学一般常用的研究方法有观察法、访谈法、问卷法、投射法、实验法、态度总加量表法、语义分析量表法、案例研究法、心理描述法、抽样调查法、创新思维法1 1 种方法。 2.1.1 观察法观察法是心理学的基本方法之一。观察是科学研究的最一般的实践方法, 同时也是最简便、易行的研究方法。所谓观察法是在自然条件下, 有目的、有计划地直接观察研究对象( 消费者) 的言行表现, 从而分析其心理活动和行为规律的方法。设计心理学借助观察法, 用以研究广告、商标、包装、橱窗以及柜台设计等方面的效果。例如, 为了评估商店橱窗设计的效果, 可以在重新布置橱窗的前后, 观察行人注意橱窗或停下来观看橱窗的人数, 以及观看橱窗的人数在过路行人中所占的比例。通过重新布置前后观看橱窗的人数变化来说明橱窗设计的效果。观察法的核心, 是按观察的目的, 确定观察的对象、方式和时机。观察时应随时记录消费者面对广告宣传、产品造型、包装设计以及柜台设计等方面所表现的行为举止, 包括语言的评价、目光注视度、面部表情、走路姿态, 等等。观察记录的内容应包括: 观察的目的、对象, 观察时间, 被观察对象的有关言行、表情、动作等的数量与质量等。另外, 还有观察者对观察结果的综合评价。观察法的优点是自然、真实、可靠、简便易行、花费低廉。在确定观察的时间和地点时, 要注意防止可能发生的取样误差。例如, 在了解商店消费者的构成时, 要区分休息日和非休息日, 也要区别上班时间和下班时间。有时商店消费者的构成也受周围居民成分的影响, 要观察少数民族消费者的特点, 就应该选择少数民族特需品的供应商店。在分析观察结果时, 要注意区分偶然的事件和有规律性的事实, 使结论具有科学性。观察法的缺点也是明显的。在进行观察时, 观察者要被动地等待所要观察的事件出现。而且, 当事件出现时, 也只能观察到消费者是怎样从事活动的, 并不能得到消费者为什么会这样活动, 他的内心是怎样想的资料。现代科技水平的发展, 使观察法能借用先进的观察设备诸如录像、录音、闭路电视的方式进行观察, 使观察效果更准确更及时, 并节省观察人员。但观察法只能记录消费者流露出来的言行、表情, 而对流露出这种言行、表情的原因, 是无法通过观察法直接获取, 因而必须结合其他的有关方法, 才能进一步了解消费行为规律。当研究的心理现象不能直接观察时, 可通过搜集有关资料, 间接了解消费者的心理活动, 这种研究方法叫调查法。调查法分为两种: 一种是口头调查法, 亦称谈话法、访谈法; 另一种是书面调查法, 亦称问卷法、调查表法。 2.1.2 访谈法访谈法是通过访谈者与受访者之间的交谈, 了解受访者的动机、态度、个性和价值

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