Processing of X-ray Microcalorimeter Data with Pulse Shape Variation using Principal Component Analysis

2.2 MB·

by  Daikang Yan

JournalofLowTemperaturePhysicsmanuscriptNo.
(willbeinsertedbytheeditor)
D.Yan · T.Cecil · L.Gades · C.Jacobsen ·
T.Madden · A.Miceli
Processing of X-ray Microcalorimeter Data
with Pulse Shape Variation using Principal
Component Analysis
6
1
0 thedateofreceiptandacceptanceshouldbeinsertedlater
2

n
a Abstract Wepresentamethodusingprincipalcomponentanalysis(PCA)topro-
J cessx-raypulseswithsevereshapevariationwheretraditionaloptimalfiltermeth-

7 odsfail.WedemonstratethatPCAisabletonoise-filterandextractenergyinfor-
mationfromx-raypulsesdespitetheirdifferentshapes.Weapplythismethodtoa

] datasetfromanx-raythermalkineticinductancedetectorwhichhasseverepulse
t
e shapevariationarisingfromposition-dependentabsorption.
d
- Keywords Principal Component Analysis (PCA), Pulse Processing, Shape
s
Variance,Microcalorimeter
n
i
.
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1 Introduction
i
s
y
Acommonmethodofpulseprocessingforlowtemperaturemicrocalorimetersis
h
the optimal filter [1], where one cross-correlates a pulse with a pulse model (or
p
convolves a pulse with the time reverse of a known model). This method maxi-
[
mizes the signal to noise under the conditions that the pulse shape and noise are

1
stationary.Howeverinrealdetectors,theseconditionsarenotalwayssatisfiedfor
v
avarietyofreasons.
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Wedescribeheretheuseofprincipalcomponentanalysis(PCA)[2]asanon-
5
6 parametric analysis approach that requires no prior knowledge of the dataset for
1 thepulseprocessingoflowtemperaturedetectors[3].Inthiswork,wedrawupona
0 PCA-basedapproachusedinx-rayspectromicroscopyanalysis[4,5]toexaminea
. simplesimulateddatasetconsistingofpulsesofdifferentdecaytimesanddifferent
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0 pulseheights.Wethenapplyourapproachtoarealdatasetwithseverepulseshape
6 variation.
1
: DepartmentofAppliedPhysics,NorthwesternUniversity,
v EvanstonIL60208,USA
i AdvancedPhotonSource,ArgonneNationalLaboratory,
X
ArgonneIL60439,USA
r E-mail:[email protected]
a
2
2 PrincipalComponentAnalysis
Whenx-raysareabsorbedinsuperconductingmicrocalorimeterdetectors,apulse
is generated over some finite time before equilibrium is restored. Consider a set
ofindividuallytriggereddetectorpulses(n=1,...N)whichareeachsampledin
time (t =1,...T), yielding a data matrix D . Our goal is to represent these
T×N
data using a basis setC with S characteristic pulse shape factors, with each
T×S
individualpulsebeingrepresentedbyaweightingR ofmembersofthisbasis
S×N
set,or
D =C ·R . (1)
T×N T×S S×N
IfwecanfindareducedsubsetwithS(cid:48) <T pulseshapefactorsthatthedatatell
usmustbepresent,wecanrepresenteachpulsenotwithallT timepointsbutin
terms of its S(cid:48) weighting factors. This gives a more compact representation of a
pulseoverfewervariables,andoncethematrixCT×S(cid:48) hasbeendeterminedandits
matrixinverted,wecanfindeachpulse’sweightingfactorsRS(cid:48)×N byasimpleand
rapidly-calculatedmatrixmultiplication
RS(cid:48)×N =CT−×1S(cid:48)·DT×N. (2)
ThisanalysisismadesimplerifthematrixC isconstructedtohaveorthogo-
T×S
nalvectors(toenablematrixinversionusingsimpletransposition)sortedinorder
ofdecreasingstatisticalsignificance(thusallowingthereducedbasissetCT×S(cid:48) to
be easily separated from the full basis set C ). This is precisely what is ac-
T×S
complishedbyPCA[2].TocalculateC ,wefirstcalculatethetimecovariance
T×S
abouttheoriginof
Z =D ·DT (3)
T×T T×N N×T
(therelationshipbetweenPCA,SVD,andcovariancematricesisdiscussedintext-
booksonthetopic[2]aswellasinAppendixBof[4]).Becausethistimecovari-
anceissymmetric,wecanrepresentitintermsofasetofeigenvectorsC and
T×S
eigenvalueweightingsΛ ,or
S×S
Z ·C =C ·Λ , (4)
T×T T×S T×S S×S
whereS=T attheoutsetofouranalysis.Mostnumericaleigenvalue-solvingrou-
tines sort their output in terms of decreasing eigenvalue weightings. As a result,
thefirsteigenvector(ortheeigenpulse)isessentiallyanaverageofthepulses.The
second eigenvector gives the first correction to that average, the third eigenvec-
torgivesthenextcorrectiontothefirsttwo,andsoon.Poorlycorrelatednoiseis
exiled to higher order eigenvectors [6]. In this way, one can arrive at a reduced
setCT×S(cid:48) ofeigenvectorswhichdescribeallofthesignificantcharacteristicpulse
shapecomponents,andbecausethisisanorthonormalmatrixitsinverseisgiven
bythetransposesothatEq.2canbecalculatedfromthereducedsetofeigenvec-
torsas
RS(cid:48)×N =CT−×1S(cid:48)·DT×N =CST(cid:48)×T·DT×N (5)
WiththereducedsetofS(cid:48) eigenvectors,onecanalsogenerateacompressedand
noise-filteredversionoftheoriginaldataas
D(cid:48)T×N =CT×S(cid:48)·RS(cid:48)×N. (6)
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Fig. 1 Illustration of eigenvector representation of some simulated pulses. At Left is shown
severalindividualpulsesfromthesimulateddataset.Theyhaveacombinationoftwodifferent
heightsandshapes,sotherearefourgroupsofthem.TheRightfigureshowstheeigenvalues
(whicharefromΛS×S),andtheinsertshowsthefirstthreeeigenvectors.(Colorfigureonline)
Fig.2 TheLeftfigureshowsfourPCAreconstructedpulses(Eq.6)forS(cid:48)=2.TheRightfigure
representsthedistributionoftheelementsfromtheweightingmatrixRS(cid:48)×N forS(cid:48)=2fromthe
PCAanalysisofthesimulateddata.(Colorfigureonline)
InordertogainintuitiononhowPCAtreatspulsedata,wehavesimulateda
datasetwhichcontainsexponentialpulseswithtwodecaytimes,twopulseheights
and white noise as shown in the left subfigure of Fig. 1. As shown in the right
subfigure,whendecomposedthisdatasetcontainstwoprimaryeigenvectors.The
third (and higher) eigenvector contains no shape information and corresponds to
noiseinthedataset.Thus,wecanrebuildthedatasetasD(cid:48)T×N=CT×S(cid:48)=2·RS(cid:48)=2×N.
AsshownintheleftsubfigureinFig.2,noiseisgreatlyfilteredyetthepulseshape
andheightfeaturesremain.TherightsubfigureofFig.2showsthedistributionof
elementsfromtheweightingmatrixRS(cid:48)=2×N,wherecomponents1and2respec-
tivelyaretheweightingfactorsofthe1st and2nd eigenvectors.
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Fig.3 FigureontheTopLeftshowsseveralindividualpulsesfromtheTKIDdevice.Aclear
shapedifferencecouldbeseenatthebeginning.Aftersomeequilibriumtimethepulsesgoto
twobranches;theloweroneisMnKα,andtheupperoneisMnKβ.TheFigureontheTop
RightisthefirstfifteeneigenvaluesandtheBottomLeftthefirstsixeigenvectors.TheBottom
Rightfigureshowstworawpulses(black)incomparisonwithPCAreconstructedpulses(Eq.6)
forS(cid:48)=1,2,3(red)(Colorfigureonline).
For pulses with the same shape and height, their weighting factors are the
same,soaplotofindividualpulsesasdotsattheirparticulareigenvectorweight-
ings shows four clusters in the right subfigure of Fig. 2. For pulses with same
shapeanddifferentheight,theyhavethesameratioofcomponent1tocomponent
2.Forpulseswithdifferentshapebutthesameheight,thelinearcombinationof
theirweightingsistheheight,sotheirdatapointsareonthesamelinewithlines
thatcorrespondtodifferentheightsparalleltoeachother.
3 AnalysisofTKIDDataUsingPCA
We now apply the PCA method to a real dataset from an x-ray thermal kinetic
inductancedetector(TKID).WhileothergroupshavereportedTKIDswith75eV
resolution at 6 keV [7], we worked here with a TKID [8, 9] from which pulse
shapes were strongly dependent on the location on the sensor at which an x-ray
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Fig.4 TheLeftfigureshowsthedistributionofelementsintheweightingmatrixRS(cid:48)=2×N from
thePCAanalysisoftheTKIDdata.TheUpperinsertshowsapileupeventandtheLowerinsert
showsalowenergyevent,bothwithapositionseparatefromthemaincluster.TheRightfigure
isthehistogramofthepulseset.
wasabsorbed(seeFig.3TopLeft).About30µsafterthestartofapulse,thepulse
shape does not vary and the amplitude is proportional to the energy, the Mn Kα
andMnKβ linesoftheFe-55sourcebecomeapparent.Insuchadataset,atradi-
tionalmatchedoroptimalfiltergivesnoenergyinformation,sincethepulseshapes
aresodifferentthatenergycouldnotbesimplyextractedfrompulseheightorarea.
ThishasmotivatedustoconsideraPCAanalysiswhichmakesnoassumptionsof
thedataset.
Following the PCA analysis presented in Sec. 2, the eigenvalues and eigen-
vectors are calculated and shown in Fig. 3. The first two eigenvalues are most
significant,buteigenvalues3–9encodesomesubtlevariationsinthedata.Inpar-
ticular, the fluctuations near a time of 1500 µs are related to the jitter in the rise
time;thesecomponentsarelikelyhighlycorrelatedwitharrivaltime.Avariantof
PCA analysis (using singular variant decomposition, or SVD) recently has been
studiedforthedetectionofnearly-coincidentpulses[10].Whilethecomponents
beyond the first two may show some correlation with photon energy, we restrict
the analysis in this paper to the first two components for simplicity. We can see
fromthebottomrightsubfigureinFig.3thatthereisqualitativelynolargediffer-
encebetweenS(cid:48)=2andS(cid:48)=3,thoughrigorousandrobustselectionmetricsfor
S(cid:48) needtobedevelopedinthefuture.
In order to extract energy information, we examined the weighting matrix
RS(cid:48)×N withS(cid:48)=2whichisa2DscatterplotshownintheleftsubfigureofFig4.
We see two clusters which we associate with the Mn Kα (black) and Mn Kβ
(blue) lines; black points are pulses in the lower Kα branch as in the Top Left
figure of Fig. 3, and blue ones (those who are not outliers) are in the higher Kβ
branch. These clusters can be automatically detected and separated [2], and we
havealreadyusedtheseautomatedapproachesinothercontexts[4,5].Byfittinga
line(red)totheMnKα cluster,wecangenerateanaxiswhichwasusedtorotate
the 2D scatter plot of the weighting matrix so that the clusters are vertical [11].
Theprojectionontothex-axisisusedtogeneratetheenergyhistogramintheright
subfigureinFig.4.Thus,theenergycanbecorrelatedtoalinearcombinationof
thefirsttwoPCAcomponents.
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Fig.5 TheLeftfigurerepresentstheentiredataset’sweightingmatrixdatadistribution,which
is calculated with a training eigenvector set from 200 pulses. The Right figure is the energy
histogramgeneratedafterrotatingthisweightingmatrix.
Weshouldnotethatthisdatasetincludespileup(i.e.,morethanonepulseina
singletimerecordT)andlowenergyevents.Theseevents,shownintheinsertion
in the left subfigure of Fig 4, result in PCA weights that are vastly different, or
points isolated from the main clusters. By using S(cid:48) >2 components, pileups can
be further distinguished from low energy events. This suggests that PCA can be
effectiveforpileuprejection.
One disadvantage of PCA is its time-consuming eigenvector calculation. A
solutionistouseasmallersetofpulsesasatrainingset.Asanexample,weused
thefirst200pulsestoperformthePCAdecompositionandobtainaneigenvector
setCtraining.SelectingS(cid:48)=2,andusingEq.2,weobtainedtheweightingmatrix
T×S(cid:48)
fortheremainingN=3088pulses,whichisshownintheleftsubfigureofFig.5.
ComparedtotheleftsubfigureofFig4,despiteaninverseofthefirstcomponent
the training data agrees well with what we obtain from direct PCA composition
of the entire dataset. The energy histogram also shows very little change. With
thetrainedsetofeigenvectors,thePCAreconstructionofthedatasimplifiedtoa
matrixmultiplication.Thismethodcouldenablefast,real-timepulseprocessing.
More work is needed to determine a sufficient number of pulses for the training
set.
4 Conclusions
Wehaveintroducedanon-parametricmethodforTKIDpulseprocessingbasedon
PCA,andhaveshownthatitisbeneficialfordatasetswithpulseshapevariation.
We have shown that PCA reduces data noise by the selection of a few number
of components, and provides energy information by converting the data into a
lowerdimensionbasissystem.Moreover,italsoprovidesanewmethodtoidentify
pileupeventsandforfast,real-timepulseprocessing.
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Acknowledgements UseoftheCenterforNanoscaleMaterialswassupportedbytheU.S.De-
partment of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No.
DE-AC02-06CH11357.WorkatArgonneNationalLaboratorywassupportedbytheU.S.De-
partment of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No.
DE-AC02-06CH11357.DevicesinthispaperwerefabricatedatCNM;wegratefullyacknowl-
edgeassistancefromRaluDivan,LeoOcola,DaveCzaplewskiandSuzanneMilleratCNM.
WethankMirnaLeroticandRachelMakforusefuldiscussiononPCA.Finally,wethankthe
reviewersfortheirusefulinsights.
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