Cluster ( Vehicle Example. Cluster analysis ( Terminology. Vehicle Clusters. Why cluster?

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1 Why luster? referene funton R R Although R and R both somewhat orrelated wth the referene funton, they are unorrelated wth eah other Cluster ( A number of smlar ndvduals that our together as a: two or more onseutve onsonants or vowels n a segment of speeh b: a group of houses (...) : an aggregaton of stars or galaxes that appear lose together n the sky and are gravtatonally assoated. Cluster analyss ( A statstal lassfaton tehnque for dsoverng whether the ndvduals of a populaton fall nto dfferent groups by makng quanttatve omparsons of multple haratersts. Vehle Example Vehle Top speed km/h Colour Ar resstane Weght Kg V red.3 3 V 3 blak.3 4 V3 6 red.9 5 V4 4 gray.35 8 V5 55 blue V6 3 whte.4 6 V7 blak.5 3 V8 5 red.6 5 V9 gray Vehle Clusters Termnology Objet or data pont feature spae label 3 Lorres 3 Lorres 5 5 luster Weght [kg] 5 Medum market ars Sports ars Weght [kg] 5 Medum market ars Sports ars feature Top speed [km/h] Top speed [km/h] feature

2 475Hz 557Hz Ok? Yes.43. Yes.97.3 Yes.78. Yes.579. Yes.3.5 No..748 No No.7. No.4.4 No Example: Classfy raked tles Table : frequeny ntenstes for ten tles. Tles are made from lay moulded nto the rght shape, brushed, glazed, and baked. Unfortunately, the bakng may produe nvsble raks. Operators an detet the raks by httng the tles wth a hammer, and n an automated system the response s reorded wth a mrophone, fltered, Fourer transformed, and normalsed. A small set of data s gven n TABLE (adapted from MIT, 997). - - Algorthm: hard -means (HCM) (also known as k means) - Plot of tles by frequenes (logarthms). The whole tles (o) seem well separated from the raked tles (*). The objetve s to fnd the two lusters Plae two luster entres (x) at random.. Assgn eah data pont (* and o) to the nearest luster entre (x). Compute the new entre of eah lass. Move the rosses (x)

3 Iteraton Iteraton 3 M = Iteraton 4 (then stop, beause no vsble hange) Eah data pont belongs to the luster defned by the nearest entre The membershp matrx M:. The last fve data ponts (rows) belong to the frst luster (olumn). The frst fve data ponts (rows) belong to the seond luster (olumn) Membershp matrx M -partton data pont k luster entre luster entre j All lusters C together flls the whole unverse U Clusters do not overlap f u m k k otherwse u k dstane j A luster C s never empty and t s smaller than the whole unverse U C U C C Ø Ø C U K j for all j for all There must be at least lusters n a -partton and at most as many as the number of data ponts K

4 Objetve funton Mnmse the total sum of all dstanes J J k, u k C u k Algorthm: fuzzy -means (FCM) Eah data pont belongs to two lusters to dfferent degrees. Plae two luster entres. Assgn a fuzzy membershp to eah data pont dependng on dstane Compute the new entre of eah lass. Move the rosses (x) Iteraton

5 Iteraton 5 Iteraton M = Iteraton 3 (then stop, beause no vsble hange) Eah data pont belongs to the two lusters to a degree The membershp matrx M:. The last fve data ponts (rows) belong mostly to the frst luster (olumn). The frst fve data ponts (rows) belong mostly to the seond luster (olumn) Fuzzy membershp matrx M Fuzzy membershp matrx M Pont k s membershp of luster mk / q dk j d jk dk u k Fuzzness exponent Dstane from pont k to urrent luster entre Dstane from pont k to other luster entres j mk / q dk j d jk / q / q / q d k dk d k dk d k d k / q dk / / q q / q dk d k d k Gravtaton to luster relatve to total gravtaton

6 Eletral Analogy Fuzzy -partton I U R R U RI R R R R R R R R R R U R R I U I Same form as m k All lusters C together fll the whole unverse U. Remark: The sum of membershps for a data pont s, and the total for all ponts s K A luster C s never empty and t s smaller than the whole unverse U C U C C Ø Ø C U K j for all Not vald: Clusters do overlap for all j There must be at least lusters n a -partton and at most as many as the number of data ponts K Example: Classfy aner ells Possble Features Normal smear Usng a small brush, otton stk, or wooden stk, a spemen s taken from the utern ervx and smeared onto a thn, retangular glass plate, a slde. The purpose of the smear sreenng s to dagnose pre-malgnant ell hanges before they progress to aner. The smear s staned usng the Papanolau method, hene the name Pap smear. Dfferent haratersts have dfferent olours, easy to dstngush n a mrosope. A yto-tehnan performs the sreenng n a mrosope. It s tme onsumng and prone to error, as eah slde may ontan up to 3. ells. Severely dysplast smear Dysplast ells have undergone preanerous hanges. They generally have longer and darker nule, and they have a tendeny to lng together n large lusters. Mldly dysplast els have enlarged and brght nule. Moderately dysplast ells have larger and darker nule. Severely dysplast ells have large, dark, and often oddly shaped nule. The ytoplasm s dark, and t s relatvely small. Nuleus and ytoplasm area Nuleus and yto brghtness Nuleus shortest and longest dameter Cyto shortest and longest dameter Nuleus and yto permeter Nuleus and yto no of maxma (...) Classes are nonseparable Hard Classfer (HCM) moderate A ell s ether one or the other lass defned by a olour. Ok lght Ok severe

7 Fuzzy Classfer (FCM) moderate Ok lght Ok severe A ell an belong to several lasses to a Degree,.e., one olumn may have several olours.

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9 Proessng Steps ) SPM Data was preproessed (Sle Tmng, INRIAlgn, Normalzaton, Smoothng, Fltered) + lnear detrendng and normalzaton. ) Two subjets were loaded at a tme. Both sessons wthn a subjet were regressed wth the other subjet s two sessons to aount for varane between runs. Prevously, we had onatenated the two runs of data and orrelated the two as a sngle sesson. 3) One the orrelaton maps were reated, they were ombned for eah subjet to reate a map that represented the sgnfant areas of the bran that were hghly orrelated for a partular subjet. 4) These maps were also saved as SPM mages for seond level analyss to fnd nterestng effets wthn the entre group and between groups. preproessed fmri Data Subjet Sesson Subjet Sesson Subjet Sesson Subjet Sesson Correlaton for eah Voxel of Interest (~75) Correlaton Maps Generated Subjvs subjvs3 Subjvs4 Subjvs3 Subjvs4 Subj3vs4

10 Healthy Control: Group Comparsons IPC Results for 35 Healthy Subjets durng AOD. Maps are dvded by ther standard devaton and the threholded at z = 3. SPM Results for 35 Healthy Subjets durng AOD n an ANOVA desgn. Maps are dvded by ther standard devaton and thresholded at z=3. Patents: Group Comparsons IPC Results for 35 Patents durng AOD. Maps are dvded by ther standard devaton and the thresholded at z = 3. SPM Results for 35 Patents durng AOD n an ANOVA desgn. Maps are dvded by ther standard devaton and the threholded at z = 3.