ABabcdfghiejkl · 2020. 4. 24. · Homogeneous data are all alike; all heterogeneous data are heterogeneous in their own way

Usingdistancestoaddressthechallengesofheterogeneousdata

SusanHolmeshttp://www-stat.stanford.edu/˜susan/

Bio-X andStatistics, StanfordUniversity

July29, 2015

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Themesseswedealwith

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Homogeneous data are all alike;all heterogeneous data are

heterogeneous in their own way.

.

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GoalsinModernBiology: SystemsApproachLookatthedata/allthedata: dataintegration

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GoalsinModernBiology: SystemsApproachLookatthedata/allthedata: dataintegration

Tumor Cells

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Whatdostatisticiansdo?

▶ Designnewexperimentstotestscientifichypotheses.▶ Visualizeandsummarizedatainwaysthataccountfor

uncertainties.▶ Lookformeaningfuldifferencesorstructureinhigh

dimensionalnoisydata.▶ Predicttheclassofnewobservationsgivenpreviously

observedones.▶ Predictthevalueofaresponsevariablegivenawhole

setofotherexplanatoryvariables.▶ Combinedifferentsourcesofdatatounderstandcomplex

interactions.

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Today'schallenge

▶ Dataarenotuniformlydistributedfromsomemanifold.

▶ Dataarenotanidenticallydistributedrandomsample.

▶ Dataarenotindependent.

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Datacanoftenbeseenaspointsinastatespace

Rp

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DistancesinStatistics

▶ EuclideanDistances, spatialdistances.▶ WeightedEuclideandistances: Mahalanobisdistancefor

discriminantanalysis.▶ Chisquaredistancesforcontingencytablesanddiscrete

data.▶ Jaccarddistancesforpresenceabsenceisoneof50

distancesusedinEcology.▶ EarthMover'sdistanceontreesorgraphs.▶ Biologicallymeaningfuldistances(DNA,haplotype,

Proteins).

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Whatdostatisticiansusedistancesfor?

▶ SummariesthroughFréchetMeansandMediansandpseudovariances.

▶ CenterofCloudofObjects Tk (equalweights): Find T0

thatminimizeseither∑K

k=1 d2(T0,Tk) thisisthe (L2)definitionoftheFréchetmeanobject,

▶ or∑K

k=1 d(T0,Tk) (L1 orGeometricMedian).▶ Pseudovariance= 1

K−1

∑Kk=1 d2(T0,Tk) = s2. Dimension

reductionandvisualization.NearestNeighborMethods.Clustering.Makenetworkedgesfromclosepoints. Predictionbyminimizingweightedresidualdistances.Cross-products: correlations, autocorrelations.Generalizationsofanalysisofvariance.

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Whatdostatisticiansusedistancesfor?

▶ SummariesthroughFréchetMeansandMediansandpseudovariances.

▶ Dimensionreductionandvisualization.▶ NearestNeighborMethods.▶ Clustering.▶ Makenetworkedgesfromclosepoints.▶ Predictionbyminimizingweightedresidualdistances.▶ Cross-products: correlations, autocorrelations.▶ Generalizationsofanalysisofvariance.

Findingtherightdistanceusuallysolvesthestatisticalproblem.

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Part I

The Geometries of Data

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Firstexample: cellsegmentationJointworkwithAdamKapelnerandPP Lee.Stainedbiopsyslides. Multispectralimaging(8levels/wavelengths).StainedLymphNode Aimtoidentifycell.

Pointssimilarinfeaturespaceareofthesametype.

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Problem: Stainingisheterogeneous

Bothimagesarefromthesameimageset. ThestainedcellsarecancercellsstainedwithFastRedred.Someregionsofthetissuestainliketheimageontheleftandotherregionsstainastheleft.ThisshowsthelevelofheterogeneityThesearetwo``subclasses''ofthesamephenotype(theleftisnamedsubclass`À,''theright, subclass``B'').

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Problem: StainingisheterogeneousExtremevariabilityintheimagecolors/intensity/contrast.Pixelsfromasamecellnotindependentandidenticallydistributedacrossthedifferentslidesoracrossdifferentcelltypes.

Simplenearestneighborapproach:-Take8dimensionalpixelspoints.-Assigningthepointtotheclosestneighbor

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Problem: StainingisheterogeneousExtremevariabilityintheimagecolors/intensity/contrast.Pixelsfromasamecellnotindependentandidenticallydistributedacrossthedifferentslidesoracrossdifferentcelltypes. ?

Simplenearestneighborapproach:-Take8dimensionalpixelspoints.-Assigningthepointtotheclosestneighbor

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(3.2,2)

D12(p,m1)=19.7 D2

2(p,m2)=16

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MultivariateNormalData

MahalanobisTransformation.Severaldifferentclusterswithdifferentvariance-covariancematricesanddifferentmeans.(µ1,Σ1) (µ2,Σ2)

D21(x, µ1) = (x− µ1)

TΣ−11 (x− µ1)

D22(x, µ2) = (x− µ2)

TΣ−12 (x− µ2)

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CorrespondingDataTransformation

H = I− 1Dn1T, S = X′HDnHX

zi. = S− 12 (xi. − x)

Thisissometimescalled`datasphering'.

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Sphered O

Speh

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Red

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OutputDataTumor

Tumor Cells

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NumberofTumorcells: 27,822 . .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

WecanaddinformationthroughchoiceofdistancesSampledatacanoftenbeseen Variablesare`vectors'aspointsinastatespace. indatapointspaceRp Rn

x

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n

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x4 .

x 3.

xtQy =< x, y >Q xtDy =< x, y >DDuality: Transposabledata. . .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

DataAnalysis: GeometricalApproachi. Thedataare p variablesmeasuredon n observations.ii. X with n rows(theobservations)and p columns(the

variables).iii. D isan n× n matrixofweightsonthe`òbservations'',

whichismostoftendiagonalbutnotalways.iv Symmetricdefinitepositivematrix Q, weightson

. variables, often Q =

1σ21

0 0 0 ...

0 1σ22

0 0 ...

0 0. . . 0 ...

... ... ... 0 1σ2p

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x

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EuclideanSpaceanddimensionreduction

Thesethreematricesformtheessential``triplet" (X,Q,D)definingamultivariatedataanalysis.Q and D definegeometriesorinnerproductsin Rp and Rn,respectively, through

xtQy =< x, y >Q x, y ∈ Rp

xtDy =< x, y >D x, y ∈ Rn.

Thiscanbeextendedtomoreinnerproductsgivingwhatisknownas Kernel methods.

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PrincipalComponentAnalysis: DimensionReduction

PCA seekstoreplacetheoriginal(centered)matrix X byamatrixoflowerrank, thiscanbesolvedusingthesingularvaluedecompositionof X:

X = USV′, with U′DU = In and V′QV = Ip and S diagonal

XX′ = US2U′, with U′DU = In and S2 = Λ

PCA isalinearnonparametricmultivariatemethodfordimensionreduction. D and Q aretherelevantmetricsonthedualrowandcolumnspacesof n samplesand p variables.

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A CommutativeDiagramApproach

CaillezandPages, 1976. Escoufier, 1977.Statisticianssearchforapproximationswithcertainproperties, forthecaseofPCA forinstance, werephrasetheproblemasfollows:

▶ Q canbeseenasalinearfunctionfrom Rp toRp∗ = L(Rp), thespaceofscalarlinearfunctionson Rp.

▶ D canbeseenasalinearfunctionfrom Rn toRn∗ = L(Rn).

▶

V = XtDX

Rp∗ −−−−→X

Rn

Qx yV D

y xW

Rp ←−−−−Xt

Rn∗

W = XQXt

Thisdualitygives`transposable'data.

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PropertiesoftheDiagram

Rankofthediagram:X,Xt,VQ and WD allhavethesamerank.For Q and D symmetricmatrices, VQ and WD arediagonalisableandhavethesameeigenvalues.

λ1 ≥ λ2 ≥ λ3 ≥ . . . ≥ λr ≥ 0 ≥ · · · ≥ 0.

Eigendecompositionofthediagram: VQ is Q symmetric, thuswecanfind Z suchthat

VQZ = ZΛ,ZtQZ = Ip, where Λ = diag(λ1, λ2, . . . , λp). (1)

ModernextensionstothisapproachincludeKernelmethodsinMachineLearning.

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PredictingandSummarizingthroughdistances

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ComparingTwoDiagrams: theRV coefficientManyproblemscanberephrasedintermsofcomparisonoftwo``dualitydiagrams"orputmoresimply, twocharacterizingoperators, builtfromtwo``triplets", usuallywithoneofthetripletsbeingaresponseorhavingconstraintsimposedonit.Mostoftenwhatisdoneistocomparetwosuchdiagrams,andtrytogetonetomatchtheotherinsomeoptimalway.(O = WD)Tocomparetwosymmetricoperators, thereiseitheravectorcovarianceasinnerproductcovV(O1,O2) = Tr(Ot

1O2) =< O1,O2 > oravectorcorrelation(Escoufier, 1977)

RV(O1,O2) =Tr(Ot

1O2)√Tr(Ot

1O1)tr(Ot2O2)

.

Ifweweretocomparethetwotriplets(Xn×1, 1,

1nIn

)and(

Yn×1, 1,1nIn

)wewouldhave RV = ρ2.

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Part II

Dimension Reduction: theEuclidean embedding workhorse:

MDS

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MetricMultidimensionalScalingSchoenberg(1935)

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FromCoordinatestoDistancesandBack

Ifwestartedwithoriginaldatain Rp thatarenotcentered:Y, applythecenteringmatrix

X = HY, with H = (I− 1

n11′), and 1′ = (1, 1, 1 . . . , 1)

Call B = XX′, if D(2) isthematrixofsquareddistancesbetweenrowsofX intheeuclideancoordinates, wecanshowthat

−1

2HD(2)H = B

Schoenberg'sresult: exactEuclideandistance If B ispositivesemi-definitethen D canbeseenasadistancebetweenpointsinaEuclideanspace.

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ReverseengineeringanEuclideanembedding

Wecangobackwardsfromamatrix D to X bytakingtheeigendecompositionof B = −1

2HD(2)H inmuchthesameway

thatPCA providesthebestrank r approximationfordatabytakingthesingularvaluedecompositionof X, ortheeigendecompositionof XX′.

X(r) = US(r)V′ with S(r) =

s1 0 0 0 ...0 s2 0 0 ...0 0 ... ... ...0 0 ... sr ...... ... ... 0 0

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MultidimensionalScaling(MDS)

Simpleclassicalmultidimensionalscaling.▶ SquareD elementwise D(2) = D2.▶ Compute −1

2 HD2H = B.▶ Diagonalize B tofindtheprincipalcoordinates SV′.▶ Chooseanumberofdimensionsbyinspectingthe

eigenvalue'sscreeplot.Theadvantageisthattheoriginaldistancesdon'thavetobeEuclidean.

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TakingCategoricalDataandMakingitintoaContinuum

HorseshoeExample:JointwithPersiDiaconisandSharadGoel(AnnalsofAppliedStats, 2005). Datafrom2005U.S.HouseofRepresentativesrollcallvotes. Wefurtherrestrictedouranalysistothe401Representativesthatvotedonatleast90% oftherollcalls(220Republicans, 180Democratsand1Independent)leadingtoa 401× 669 matrixofvotingdata.

TheDataV1 V2 V3 V4 V5 V6 V7 V8 V9 V10

R1 -1 -1 1 -1 0 1 1 1 1 1 ...

R2 -1 -1 1 -1 0 1 1 1 1 1 ...

R3 1 1 -1 1 -1 1 1 -1 -1 -1 ...

R4 1 1 -1 1 -1 1 1 -1 -1 -1 ...

R5 1 1 -1 1 -1 1 1 -1 -1 -1 ...

R6 -1 -1 1 -1 0 1 1 1 1 1 ...

R7 -1 -1 1 -1 -1 1 1 1 1 1 ...

R8 -1 -1 1 -1 0 1 1 1 1 1 ...

R9 1 1 -1 1 -1 1 1 -1 -1 -1 ...

R10 -1 -1 1 -1 0 1 1 0 0 0 .... .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

L1 distance

Wedefineadistancebetweenlegislatorsas

d(li, lj) =1

669

669∑k=1

|vik − vjk|.

Roughly, d(li, lj) isthepercentageofrollcallsonwhichlegislators li and lj disagreed.

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!0.1!0.05

00.05

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0

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3-DimensionalMDS mappingoflegislatorsbasedonthe2005U.S.HouseofRepresentativesrollcallvotes. Weused

dissimilarityindices1-exp(−λd(R1,R2))

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!0.1!0.05

00.05

0.1!0.2

!0.1

0

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!0.15

!0.1

!0.05

0

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0.15

3-DimensionalMDS mappingoflegislatorsbasedonthe2005U.S.HouseofRepresentativesrollcallvotes. Colorhasbeenaddedtoindicatethepartyaffiliationofeachrepresentative.

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MDS Rank

Natio

nal J

ourn

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core

ComparisonoftheMDS derivedrankforRepresentativeswiththeNationalJournal'sliberalscore

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AnApplication: VisualizingGeodesicDistancesbetweenTrees

▶ NearestNeighborInterchange(NNI). RotationMoves

4

0

2 31

0

4321

0

41 32

▶ Fill-inofNNI moves: Billera, Holmes, Vogtmann(2001)(BHV).Theboundariesbetweenregionsrepresentanareaofuncertaintyabouttheexactbranchingorder. Inbiologicalterminologythisiscalledanùnresolved'tree.Moredetailshere

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4

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4

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A ConePath

A pathbetweentwotrees T and T′ alwaysexists. Sinceallorthantsconnectattheorigin, anytwotrees T and T′ canbeconnectedbyatwo-segmentpath, thisiscalledthecone-path.

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c

a

c

b ba

Theorem(Billera, Holmes, Vogtmann(BHV,2001)):TreespacewithBHV metricisaCAT(0)space, thatis, ithasnon-positivecurvature.Thisimpliestherearegeodesicbetweenanytwotrees(Gromov).Note: ThisspaceoftreesisnotanEuclideanspace.

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c

a

c

b baThesizeofthe``pseudo-variance''canbeestimatedfrom∑

pid(T0,Ti)2.PropertiesoftheFréchetmeanofasetoftreeshasbeen(Bhattacharyaetal.2010, Miller, Mattingley, Owen, Marron, al.2013).

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PhylogeneticTreesMalariaDataasseenusing ape

Pre1

Pme2

Plo6

Pga11

Pma3

Pbe5

Pfr7

Pkn8

Pcy9

Pvi10

Pfa4

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SamplingDistributionforTrees

Data 1

23

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Data 1

23

Treespace Tn

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Data1

23

4

True Sampling Distribution

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Data1

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4

Bootstrap Sampling Distribution (non parametric)

n

^

*

**

*

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BootstrapofMalariaData

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HierarchicalClusteringTrees

HEA2

5_EF

FE_3

MEL

39_E

FFE_

2HE

A31_

EFFE

_2M

EL67

_EFF

E_4

HEA5

5_EF

FE_4

HEA5

9_EF

FE_5

HEA2

6_EF

FE_1

MEL

51_E

FFE_

5M

EL36

_EFF

E_1

MEL

53_E

FFE_

3HE

A31_

NAI_

2HE

A55_

NAI_

4M

EL67

_NAI

_4M

EL53

_NAI

_3HE

A25_

NAI_

3M

EL51

_NAI

_5HE

A59_

NAI_

5HE

A26_

NAI_

1M

EL36

_NAI

_1M

EL39

_NAI

_2M

EL51

_MEM

_5HE

A26_

MEM

_1M

EL67

_MEM

_4HE

A31_

MEM

_2HE

A55_

MEM

_4HE

A25_

MEM

_3HE

A59_

MEM

_5M

EL53

_NAI

_3M

EL36

_MEM

_1M

EL39

_MEM

_2

Human AF5q31 protein (AF5q31) intracellular hyaluronan−bindiselectin L (lymphocyte adhesioHuman cDNA FLJ10470 fis, cloneHuman mRNA for KIAA0303 gene, KIAA0303 proteinSTAT induced STAT inhibitor 3KIAA0752 proteinGRB2−related adaptor proteindelta (Drosophila)−like 1stanninproteoglycan link proteinIncyte ESTamyloid beta (A4) precursor prHuman genomic DNA, chromosome follicular lymphoma variant trHuman, clone IMAGE:3875338, mRHuman, Similar to phosphodiestHuman sodium/myo−inositol cotrESTs, Weakly similar to MUC2_HHuman CpG island DNA genomic MPOU domain, class 2, transcripHuman zinc finger protein ZNF2Human 54 kDa progesterone receprotein tyrosine phosphatase, platelet/endothelial cell adheHuman cDNA FLJ20849 fis, cloneHuman mRNA for KIAA0972 proteiKIAA0290 proteinHuman clone 295, 5cM region sueukaryotic translation initiatferritin, heavy polypeptide 1Human cDNA: FLJ22008 fis, clonHuman insulin−like growth factgranzyme K (serine protease, gHuman Epstein−Barr virus inducPAS−serine/threonine kinaselymphotoxin beta (TNF superfamHuman mRNA for nel−related prochemokine (C−C motif) receptorPAS−serine/threonine kinaseHuman mRNA for alpha−actinin, Human mRNA encoding the c−myc Human RATS1 mRNA, complete cdshyaluronoglucosaminidase 2Human DNA for muscle nicotinicHuman epithelial V−like antigeinterferon gamma receptor 2 (iHomo sapiens clone 24775 mRNA syntaphilinHuman mRNA for endosialin protHuman zinc finger protein PLAGshort−chain dehydrogenase/reduHuman, short−chain dehydrogena

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0 20000 40000 60000 80000 120000

Eigenvalues of MDS for bootstrapped trees

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−40 −20 0 20 40

−60

−40

−20

020

40

Bootstrapped trees

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9697

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100o

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Part III

Combine and Compare Trees,Graphs and Contingent Count

Data

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LayersofDatainthe MicrobiomeJoshuaLederberg:`theecologicalcommunityofcommensal,symbiotic, andpathogenicmicroorganismsthatliterallyshareourbodyspaceandhavebeenallbutignoredasdeterminantsofhealthanddisease'Microbiome Completecollectionofgenescontainedinthe

genomesofmicrobeslivinginagivenenvironment.

Numbers Humansshelter100trillionmicrobes(1014), (wearemadeof10 ×1012 cells).

Metagenome Compositionofallgenespresentinanenvironment(soil, gut, seawater), regardlessofspecies.

Transciptome ThesearethemRNA transcriptsinthecell, itreflectsthegenesthatarebeingactivelyexpressedatanygiventime.

Metabolome Themetabolites(smallmolecules)nucleicorfattyacids, sugars,... presentinthesampleeitherendogenousorexogenous(medication, pollution).

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.

Source: YK LeeandSK MazmanianScience, 2010.. .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Bacteriaetc... andUs

Thehumanmicrobiomeorhumanmicrobiotaistheassemblageofmicroorganismsthatresideonthesurfaceandindeeplayersofskin, inthesalivaandoralmucosa, intheconjunctiva, andinthegastrointestinaltracts.

▶ Theyincludebacteria, fungi, andarchaea.▶ Someoftheseorganismsperformtasksthatareuseful

forthehumanhost. (liveinsymbiosis)▶ Majorityhavenoknownbeneficialorharmfuleffect.

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HumanMicrobiome: Whatarethedata?

DNA TheGenomicmaterialpresent(16sRNA-geneespecially, butalsoshotgun).

RNA Whatgenesarebeingturnedon(geneexpression), transcriptomics.

MassSpec Specificsignaturesofchemicalcompoundspresent(LC/MS,GC/MS).

Clinical Multivariateinformationaboutpatients'clinicalstatus, medication, weight.

Environmental Location, nutrition, drugs, chemicals,temperature, time.

DomainKnowledge Metabolicnetworks, phylogenetictrees,geneontologies.

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

HeterogeneousDataObjects

Objectorientedinputanddatamanipulationwith phyloseq

(McMurdieandHolmes, 2013, PlosONE)ObjectorienteddatainR:

Taxonomy Table taxonomyTableslots: .Data

OTU Abundanceclass: otuTableslots: .Data, speciesAreRows

Sample VariablessampleDataslots: .Data,names,row.names,.S3Class

Phylogenetic Treeclass: phyloslots: see ape package

matrix matrixdata.frame

phyloseqslots:otuTablesampleDatataxTabtre

Experiment-level data object:

Component data objects:

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Part IV

Heteroscedasticity: Mixturesand to Normalize them

Source: xkcd.. .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Pointsaremeasuredwithunequalvariance

x

x1

2x

x

x

x

x

2

.

.

.

.

.

.

p

i

1

3

.

xn ..

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Some real data (Caporoso et al, 2011)

> GlobalPatterns

phyloseq-class experiment-level object

otu_table() OTU Table: [ 19216 taxa and 26 samples ]

sample_data()Sample Data: [ 26 samples by 7 sample variables ]

tax_table()Taxonomy Table: [ 19216 taxa by 7 taxonomic ranks ]

phy_tree() Phylogenetic Tree:[ 19216 tips and 19215 internal nodes ]

> sample_sums(GlobalPatterns)

CL3 CC1 SV1 M31Fcsw M11Fcsw M31Plmr M11Plmr F21Plmr

864077 1135457 697509 1543451 2076476 718943 433894 186297

.....

NP3 NP5 TRRsed1 TRRsed2 TRRsed3 TS28 TS29 Even1

1478965 1652754 58688 493126 279704 937466 1211071 1216137

> summary(sample_sums(GlobalPatterns))

Min. 1st Qu. Median Mean 3rd Qu. Max.

58690 567100 1107000 1085000 1527000 2357000

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Pointsaremeasuredwithunequalvariance

x

x1

2x

x

x

x

x

2

.

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.

.

.

.

p

i

1

3

.

xn ..

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Equalizationofvariances

Inthisbinomialexamplethevarianceoftheproportionestimateis Var(Xn ) =

pqn =

qnE(

Xn ), afunctionofthemean.

Thisisacommonoccurrenceandonethatistraditionallydealtwithinstatisticsbyapplyingvariance-stabilizingtransformations.However, inordertofindtherighttransformation, weneedagoodmodelfortheerror.

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

VarianceStabilization

Prefertodealwitherrorsacrosssampleswhichareindependentandidenticallydistributed.Inparticularhomoscedasticity(equalvariances)acrossallthenoiselevels.Thisisnotthecasewhenwehaveunequalsamplesizesandvariationsintheaccuracyacrossinstruments.A standardwayofdealingwithheteroscedasticnoiseistotrytodecomposethesourcesofheterogeneityandapplytransformationsthatmakethenoisevariancealmostconstant.Thesearecalled variancestabilizingtransformations.

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MixtureModelingworksMiracles

▶ Beta-Binomial(deepSNV).▶ ZeroinflatedPoissonorGaussian.▶ Gamma-Poisson.

MixturesareubiquitousbecauseofamathematicaltheoremDeFinnetti'sTheorem

. .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

http://en.wikipedia.org/wiki/De_Finetti's_theorem

WolfgangHuber. .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Correcttransformationsareavailable

McMurdieandHolmes(2014)``WasteNot, WantNot: Whyrarefyingmicrobiomedataisinadmissible'', PLOSComputationalBiology, Methods.WeproposetomodelthereadcountsIftechnicalreplicateshavesamenumberofreads: sj,Poissonvariationwithmean µ = sjui.Taxa i incidenceproportion ui.Numberofreadsforthesample j andtaxa i wouldbe

Kij ∼ Poisson (sjui)

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A distanceontheknowntreeMonge-Kantorovichearthmover'sdistanceonthetree.Usedtocomparetwosamplesorbodysitesforinstance.Incorporatetaxaabundancesandphylogenetictree

Epulopiscium

Clostridium

Adlercreutzia

Lachnospira

Alistipes

Roseburia

Coprococcus

Clostridium

Blautia

Coprococcus

Dehalobacterium

Clostridium

Clostridium

Clostridium

Coprobacillus

Coprococcus

Clostridium

Clostridium

Moryella

Abundance

1

25

625

Class

Actinobacteria (class)

Bacilli

Bacteroidia

Clostridia

Erysipelotrichi

Gammaproteobacteria

Mollicutes

Verrucomicrobiae

YS2

Dualitydiagrammethodsthatcanuseanydependencystructure.

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UnifracDistance(LozuponeandKnight, 2005)

isadistancebetweengroupsoforganismsthatarerelatedtoeachotherbyatree.SupposewehavetheOTUspresentinsample1(blue)andinsample2(red).Question: Dothetwosamplesdifferphylogenetically?ItisdefinedastheratioofthesumofthelengthsofthebranchesleadingtomembersofgroupA ormembersofgroupB butnotbothtothetotalbranchlengthofthetree.

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WeightedUnifracdistance A modificationofUniFrac,weightedUniFracisdefinedin(Lozuponeetal., 2007)as

n∑i=1

bi × |AiAT− Bi

BT|

▶ n = numberofbranchesinthetree

▶ bi =lengthoftheithbranch

▶ Ai =numberofdescendantsofithbranchingroupA

▶ AT =totalnumberofsequencesingroupA

[7].[6]. . .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Costelloetal. 2010

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Rao'sDistance

Westartwithadistancebetweenindividuals.Theheterogeneityofapopulation(Hi )istheaveragedistancebetweenmembersofthatpopulation.Theheterogeneitybetweentwopopulations(Hij)istheaveragedistancebetweenamemberofpopulation i andamemberofpopulation j.Thedistancebetweentwopopulationsis

Dij = Hij −1

2(Hi + Hj)

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

DecompositionofDiversity

Ifwehavepopulations 1, . . . , k withfrequencies π1, . . . , πk,thenthediversityofallthepopulationstogetheris

H0 =

k∑i=1

πiHi +∑i

∑j

πiπjDij = H(w) + D(b)

. .. .. . . .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

DoublePrincipalCoordinateAnalysisPavoine, DufourandChessel(2004), Purdom(2010)andFukuyamaetal. (2011). .Supposewehavenspeciesinplocationsanda(euclidean)matrix ∆ givingthesquaresofthepairwisedistancesbetweenthespecies. Thenwecan

▶ Usethedistancesbetweenspeciestofindanembeddingin n− 1 -dimensionalspacesuchthattheeuclideandistancesbetweenthespeciesisthesameasthedistancesbetweenthespeciesdefinedin ∆.

▶ Placeeachoftheplocationsatthebarycenterofitsspeciesprofile. TheeuclideandistancesbetweenthelocationswillbethesameasthesquarerootoftheRaodissimilaritybetweenthem.

▶ UsePCA tofindalower-dimensionalrepresentationofthelocations.

Givethespeciesandcommunitiescoordinatessuchthattheinertiadecomposesthesamewaythediversitydoes.

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FukuyamaandHolmes, 2012.Method Originaldescription Newformula PropertiesDPCoA square root of Rao's distance

basedonthesquarerootofthepatristicdistances

[∑

i bi(Ai/AT − Bi/BT)2]1/2 Mostsensitivetooutliers, leastsensitive to noise, upweightsdeep differences, gives OTUlocations

wUniFrac∑

i bi |Ai/AT − Bi/BT|∑

i bi |Ai/AT − Bi/BT| Less sensitive to outliers/moresensitivetonoisethanDPCoA

UniFrac fractionofbranchesleadingtoexactlyonegroup

∑i bi1{

Ai/AT−Bi/BTAi/AT+Bi/BT

≥ 1} Sensitive to noise, upweightsshallowdifferencesonthetree

Summaryofthemethodsunderconsideration. `Òutliers"referstohighlyabundantOTUs, andnoisereferstonoiseindetectinglow-abundanceOTUs(seethetextformoredetail).

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AntibioticTimeCourseData

Measurementsofabout2500differentbacterialOTUsfromstoolsamplesofthreepatients(D,E,F)Eachpatientsampled ∼ 50timesduringthecourseoftreatmentwithciprofloxacin(anantibiotic).TimescategorizedasPreCp, 1stCp, 1stWPC (weekpostcipro), Interim, 2ndCp, 2ndWPC,andPostCp.

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UniFrac

Axis 1: 14.7%

Axi

s 2:

10.

3%

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weighted UniFrac

Axis 1: 47.6%A

xis

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2.3%

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weighted UF on presence/absence

Axis 1: 32.7%

Axi

s 2:

15.

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−0.10−0.050.000.050.100.150.20

subject

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E

F

ComparingtheUniFracvariants. Fromlefttoright:PCoA/MDS withunweightedUniFrac, withweightedUniFrac,andwithweightedUniFracperformedonpresence/absencedataextractedfromtheabundancedatausedintheothertwoplots

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(a) MDS of OTUs

Axis 1: 6.2%

Axi

s 2:

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%

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(c) DPCoA OTU plot

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−1.5 −1.0 −0.5 0.0 0.5 1.0

phylum

● 4C0d−2

● Actinobacteria

● Bacteroidetes

● Candidate division TM7

● Cyanobacteria

● Firmicutes

● Fusobacteria

● Lentisphaerae

● Proteobacteria

● Synergistetes

● Verrucomicrobia

(b) DPCoA community plot

Axis 1: 40.9%

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(a)PCoA/MDS oftheOTUsbasedonthepatristicdistance, (b)communityand(c)speciespointsforDPCoA afterremovingtwooutlyingspecies.

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AntibioticStress

Wenextwanttovisualizetheeffectoftheantibiotic.OrdinationsofthecommunitiesduetoDPCoA andUniFracwithinformationaboutthewhetherthecommunitywasstressedornotstressed(precipro, interim, andpostciprowereconsidered``notstressed'', whilefirstcipro, firstweekpostcipro, secondcipro, andsecondweekpostciprowereconsidered``stressed'').WeseethatforUniFrac, thefirstaxisseemstoseparatethestressedcommunitiesfromthenotstressedcommunities.DPCoA alsoseemstoseparatetheoutthestressedcommunitiesalongthefirstaxis(inthedirectionassociatedwithBacteroidetes), althoughonlyforsubjectsD andE.

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Axis1

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Antibiotic stress

● 1: not stressed

2: stressed

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PCoA/MDS withunweightedUniFrac. Thelabelsrepresentsubjectplusantibioticcondition.

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CommunitypointsasrepresentedbyDPCoA.Thelabelsrepresentsubjectplusantibioticcondition.

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ConclusionsforAntibioticStress

SinceUniFracemphasizesshallowdifferencesonthetreeandsincePCoA/MDS withUniFracseemstoseparatethesubjectsfromeachotherbetterthantheothertwomethods, wecanconcludethatthedifferencesbetweensubjectsaremainlyshallowones. However, DPCoA alsoseparatesthesubjectsandthestressedversusnon-stressedcommunities, andexaminingthecommunityandOTU ordinationscantellusaboutthedifferencesinthecompositionsofthesecommunities.

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Distancesenablestatisticiansto....

▶ Summarizedatawithmedians, meansandprincipaldirections.

▶ Encodesomevariationsinuncertainty.▶ Makecomparisonsofheterogeneoussourcesof

information.▶ Integratenetworkandtreeinformation.▶ Measurediversity, inertiaandgeneralizethenotionof

variance.

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Questionsformathematicians?▶ Howtomakeamethoddesignedforuniformlydistributed

pointsworkforpointsgeneratedbymixturesofheterogeneousdistributions?ExamplesfromworkbyEdelsbrunner, Carlsson,Zoromodianandco-authors.

Source:Zoromodian.. .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Questionsformathematicians▶ Howtobuilddistancesbetweenimagesthataccountfor

unequalmeasurementerrors, evenlocally?

x

x1

2x

x

x

x

x

2

.

.

.

.

.

.

p

i

1

3

.

xn ..

WorkbyAdler, TaylorandWorsley(2003,2005,2007)usingRandomFields.

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Questionsformathematicians

▶ HowwellcantheEuclideanembeddingapproximationsdo?

▶ Aretherebetterwaysofapproximatingthecommutativediagrams?ThisisalsoanimportantpointofcontactwiththeuseofStein'smethodinprobabilitytheory.

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▶ HowwellcantheEuclideanembeddingapproximationsdo?

▶ Aretherebetterwaysofapproximatingthecommutativediagrams?ThisisalsoanimportantpointofcontactwiththeuseofStein'smethodinprobabilitytheory.

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▶ Howtodistinguishbetweentheeffectofthecurvatureofastatespaceandtheeffectoftheunequalsampling?

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AnswerscomefromDifferentialGeometry.

XavierPennec, YannOllivier, TomFletcher, RabiBhattacharya.Inparticularenableustoincorporatetherelevantdatadependenttransformationsintolocalizedmetrics.

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Outputshowingposterioruncertaintymeasures

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BenefittingfromthetoolsandschoolsofStatisticians.......

Thankstothe R community:▶ RStudiofortoolsforreproducibleresearchandHadley

Wickhamforggplot2.▶ Ecologistsandbiologists: Chessel, Jombart, Dray,

Thioulouse ade4 andEmmanuelParadis ape.

Collaborators: DavidRelman, AlfredSpormann, YvesEscoufier,LesDethfelsen, JustinSonnenburg, PersiDiaconis, SergioBaccallado, ElisabethPurdom.

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LabGroup

PostdoctoralFellowsPaul(Joey)McMurdie, BenCallahan, SimonRubinstein-Salzado, ChristofSeiler.Students: JohnChakerian, JuliaFukuyama, KrisSankaran.Fundingfrom NIH/NIGMS R01, NSF-VIGRE andNSF-DMS.

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ReferencesL. Billera, S. Holmes, andK. Vogtmann.Thegeometryoftreespace.Adv.Appl.Maths, 771--801, 2001.

J. ChakerianandS. Holmes.distory:Distancesbetweentrees, 2010.

DanielChessel, AnneDufour, andJeanThioulouse.Theade4package-i: One-tablemethods.R News, 4(1):5--10, 2004.

P. Diaconis, S. Goel, andS. Holmes.Horseshoesinmultidimensionalscalingandkernelmethods.AnnalsofAppliedStatistics, 2007.

Y. Escoufier.Operatorsrelatedtoadatamatrix.InJ.R.et al.Barra, editor, RecentdevelopmentsinStatistics., pages125--131.NorthHolland,, 1977.

. .. .. .. .. .. .. . . .. .. .. . . .. .. .. . . .. . . .. .. .

Steven N EvansandFrederick A Matsen.ThephylogeneticKantorovich-Rubinsteinmetricforenvironmentalsequencesamples.arXiv, q-bio.PE,Jan2010.

M Hamady, C Lozupone, andR Knight.Fastunifrac: facilitatinghigh-throughputphylogeneticanalysesofmicrobialcommunitiesincludinganalysisofpyrosequencingandphylochipdata.TheISME Journal, Jan2009.

SusanHolmes.Multivariateanalysis: TheFrenchway.InD. NolanandT. P.Speed, editors, ProbabilityandStatistics: EssaysinHonorofDavidA.Freedman,volume 56ofIMS LectureNotes--MonographSeries.IMS,Beachwood, OH,2006.

RossIhakaandRobertGentleman.R:A languagefordataanalysisandgraphics.

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JournalofComputationalandGraphicalStatistics,5(3):299--314, 1996.

K. Mardia, J. Kent, andJ. Bibby.MultiariateAnalysis.AcademicPress, NY., 1979.

P. J.McMurdieandS. Holmes.Phyloseq: Reproduibleresearchplatformforbacterialcensusdata.PlosONE,2013.April22,.

SerbanNacu, RebeccaCritchley-Thorne, PeterLee, andSusanHolmes.Geneexpressionnetworkanalysisandapplicationstoimmunology.Bioinformatics, 23(7):850--8, Apr2007.

SandrinePavoine, Anne-BéatriceDufour, andDanielChessel.

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Fromdissimilaritiesamongspeciestodissimilaritiesamongcommunities: adoubleprincipalcoordinateanalysis.JournalofTheoreticalBiology, 228(4):523--537, 2004.

ElizabethPurdom.Analysisofadatamatrixandagraph: Metagenomicdataandthephylogenetictree.AnnalsofAppliedStatistics, Jul2010.

C. R.Rao.Theuseandinterpretationofprincipalcomponentanalysisinappliedresearch.SankhyaA,26:329--359., 1964.

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Documents

ABabcdfghiejkl · 2020. 4. 24. · Homogeneous data are all alike; all heterogeneous data are heterogeneous in their own way