Multi-dimensional data scaling – dynamical cascade

7/31/2019 Multi-dimensional data scaling dynamical cascade

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I|cto)doil`zoe`nc dntn zjnco`k dy`niojnc jnzjndl

nrrxenjf

Iocn` Mewewoj# Kleggxly Gev

O`don`n \òwlxzoty

NHZTXNJT

O` tfoz xlrext n i|cto)doil`zoe`nc dntn zjnco`k nrrxenjf oz

rxerezld o` dntn ioò`k n`d aè{cldkl dozjewlxy nrrcojntoe`z.

[l dlxowl tfl iltfed hnzld e` n` n`nceky te tfl rfyzojnc

jeir|tntoe` eg zok`nc doztextoe`. N dy`niojnc jnzjndl

jeir|tntoe` donkxniz xlz|ct gxei tfl ztntoztojnc rfyzojz iedlc

jeir|tntoe` o` tfl gxll l`lxky dljeirezotoe`. [l nzzlzz tfl

zjncl o`wnxon`jl eg wnxoe|z dntn zltz# z|jf nz {otf tfl oinkl

ietoe` zl~|l`jlz# n`d {otf tfl fokf doil`zoe`nc jfliojnc dntnzltz. Tflexltojnc iedlc eg lxxex rxernkntoe` oz kowl` hy tfl

`|ilxojnc jeir|tntoe`nc zjflilz. Ztntoztojnc inrro`k eg tfl

dntn oz n`ncy}ld tfxe|kf dy`niojnc jnzjndlz# nz n {ny eg

nrrxenjfo`k otz jedo`k n`d je`txec dntn ztx|jt|xl. [l zfe{ fe{

ot jexxlcntlz hy zlkil`to`k zlt eg jfliojnc jeire|`dz

ehzlxwntoe`z o` n fokf doil`zoe`nc rxerlxty zrnjl. Tfl

rxerezld nckexotfi# ncze# oz z|otnhcl gex tfl oirclil`tntoe` o`

rnxncclc jeir|tlx nxjfotljt|xlz. N` lvnircl oirclil`tntoe` e`

tfl i|ctojexl rxejlzzexz oz kowl` o` tfl l`d eg tfoz xlrext.

>. O@TXED\JTOE@N i|ctogxnjtnc iedlc gexincozi oz dlxowld o` tfl Tfnc{lk

NXJ. rxemljt xlrext Q>8Y# te lvrcno` tfl dljeirezotoe` eg oinkl

zl~|l`jlz o`te tfl zo`k|cnx dntn zltz. Tfl rnxtotoe` g|`jtoe`

dlzjxohlz tfl rxehnhocoztoj iedlc eg dntn jc|ztlxz n`d oz n`ncy}ld

nz n i|ctogxnjtnc ilnz|xl o` tfl iltfed. Zo`k|cnxoty n`ncyzoz eg

jeir|tntoe`nc inrz eg jc|ztlxo`k wljtexz oz dlxowld te dlzjxohl

tfl jeir|tntoe`nc iln`z eg dljeirezo`k tfl oinkl o`gexintoe`

o`te dogglxl`t zo`k|cnx zltz. [l zfe{ ncze tfnt tfl rxernkntoe` eg

o`gexintoe` o` oinkl zl~|l`jlz oz kewlx`ld hy tfl zjncl)zrnjl

{nwl l~|ntoe`# tflxlgexl l`nhco`k |z te txlnt zo`k|cnx gxl~|l`jolz

eg dntn jc|ztlxz o` n` |ògold {ny# hetf o` zrnjl n`d o` toil.

Je`tlvt|nc o`gexintoe` eg tfl zrntonc jeflxl`jy eg dntn oz |zld o`

tfl zlkil`tntoe` rxejlzz o` tfl folxnxjfojnc zjncl jeir|tntoe` eg

glnt|xl wljtexz. Tfl zrntonc zlkil`tntoe` eg oinklz oz rlxgexild

{focl |zo`k tfl Kxll`+z g|`jtoe`# rnxniltlxo}ld {otf tfl zjncl

rnxniltlx# nz tfl o`tlkxntoe` g|`jtoe` o` tfl zlkil`tntoe`

rxejlzz. Tfl zjncl o`gexintoe` oz lwnc|ntld hy je meoò`k tfl t{e

rnxniltlxz= tfl zjncl rnxniltlxeg tfl zok`nc doztextoe`# n`d tfl

zrntonc zjncl rnxniltlx x. N cnxklx lvtl`t eg zrntonc o`tlkxntoe` eg

tfl ietoe` o`gexintoe` oz |zld e` n cnxklx zjncl# {focl ot

hljeilz lggljtowlcy iexl cejnc o` zrnjl nz {l dljxlnzl tfl zjncl

eg zlkil`tntoe`.

Dozto`jt zo`k|cnx glnt|xlz nxl zlkil`tld e` n jlxtno` zjncl n`d

tfl clnzt zo`k|cnx glnt|xl hljeil zlkil`tld o` t{e zrntonc

{o`de{z {otf tfl Cnrcnjon` zyztli xlk|cnxoty je`ztxno`tz# o` tfl

folxnxjfojnc zjncl jeir|tntoe`. Njjexdo`kcy# tfl xlje`ztx|jtoe`

gexi|cn oz dlxowld hnzld e` tfl Cnrcnjon` zyztli eg tfl dogg|zoe`

eg tfl xlzod|nc o`gexintoe` gxei tfl iezt zo`k|cnx zltz. Tfoz

kowlz |z n` lggljtowl {ny eg jeirxlzzo`k n`d rxekxlzzowl jedo`k

eg tfl o`gexintoe` o` oinkl zl~|l`jlz. Tfl ho`nxy txll dntn

ztx|jt|xl eg tfl jc|ztlxo`k rnxniltlxz oz z|otnhcl o` tfl jedo`k

zjflilz tfnt |zl tfl folxnxjfojnc ztx|jt|xl eg tfl ho`nxy oinklz

eg tfl zrntonc doztxoh|toe` eg jc|ztlx {o`de{z# nce`k {otf tfl

glnt|xl wljtexz n`d xlzod|nc oinkl o`gexintoe` tfnt inal |r gex

tfl reo`t glnt|xl wljtex lztointoe`.

[l kowl flxl n dlxowntoe` eg tfl jeir|tntoe`nc zjflil gex n 8)

doil`zoe`nc jnzl# coal o` oinkl zl~|l`jlz. [l tfl` je`zodlx n

dy`niojnc je|rco`k n`d tfl l`lxky lvjfn`kl hlt{ll` : jc|ztlxz

jeir|tld. Jexxlzre`do`k ztntoztojnc inrz nxl n`ncy}ld {.x.t. tfl

doil`zoe`ncoty eg tfl lokl`wnc|l dljeirezotoe` eg tfl jc|ztlxz

jewnxon`jlz.

Tfl xlz|ctz nxl zfe{` gex tfl jfliojnc jeire|`dz o` tfl > n`d 8 doil`zoe`nc

ztntoztojnc inrz.

8. ILTFEDWnxontoe`nc dntn doztxoh|toe`z

[l dlgo`l n jc|ztlx eg dntn reo`tz flxl {otf otz jeir|tld jc|ztlx

wljtex xlrxlzl`tntowl y# n`d tfl zlcljtld kxe|r {o`de{ egjeir|tntoe`# [. Clt d&v# y/ dlètlz n doztextoe` ilnz|xl

o`txed|jld te n dntn reo`t v hy tfl xlrxlzl`tntoe` y. Tfl

doztextoe` l`lxky# ex wnxon`jl Weg n jc|ztlx oz dlgo`ld hy=

.# [ vRyvdW Ot jn` hl zfe{` Q6Y tfnt tfl rxehnhocoty dl`zoty g|`jtoe` tfnt

invoio}l%ioòio}l tfl l`txery=


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jexxlzre`do`k Cnkxn`klz i|ctorcolxz.

Jewnxon`jl dogglxl`tonhocoty n`d n zjncl)zrnjl

jeir|to`k nrrxenjf

Tfl è`co`lnx dy`niojz eg jc|ztlxo`k# o` tfoz {exa oz dlxowld

gxei tfl iedlc eg gxll l`lxky# exoko`nccy |zld o` ztntoztojnc

rfyzojz te iedlc dogglxl`t jeirclv zyztliz. Tfl gxll l`lxky

dlzjxohlz tfl ztntl eg n jc|ztlx gex n kowl` rnxniltlx #

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Tfl rnxniltlx oz o`wlxzlcy rxerextoe`nc te tlirlxnt|xl

&9>%T/# o` rfyzojnc n`nceky. Tfl l~|ocohxo|i ztntlz nxl

jeir|tld te ioòio}l tfl l`lxky lvjfn`kl nie`k jc|ztlxz gex n

kowl` zrntonc doztxoh|toe` eg tfl jc|ztlxz.

Tfl doztextoe` ilnz|xl# nrrcold o` tfl nckexotfi# oz jfezl` te hl

tfl co`lnx je`ztxno`t l~|ntoe` e` tfl ietoe` wljtex w

# ncze

aè{` nz tfl lvtl`dld ertojnc gce{ je`ztxno`t l~|ntoe`=

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{fojf rxewodlz tfl inzz je`zlxwntoe` rxo`jorcl Q>Y. O` tfoz {exa

tfl jeflxl`jy eg dntn oz lztointld {otf otz Kxll`+z g|`jtoe`# te

je`txec tfl zieetf`lzz eg tfl ertojnc gce{ ndnrtowlcy o` tfl zjncl.

Tfl je`ztxno`ld l~|ntoe` eg ietoe` gex n je|rcld rnox eg jc|ztlxz

oz kowl` hy=

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{flxl tfl |rrlx zok` jexxlzre dz te tfl jeeco`k rnxt n`d ce{lx te

tfl ilcto`k rnxt eg tfl nckexotfi.

Tfoz zyztli eg l~|ntoe`z jn` hl n`ncy}ld hy tfl zlxolz lvrn`zoe`

eg tfl zyztli+z gxll l`lxkolz=

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Tfl nhewl zyztli eg l~|ntoe`z xlz|ctz o` tfl dltlxio`n`t eg tfl

inr=


3/6

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Gex >2 tfl lokl`wnc|lz eg tfl dltlxio`n`t eg tfl inrfnwl `lkntowl wnc|lz og tfl Flzzon`z eg tfl gxll l`lxkolz# G> n`d

G8# nxl rezotowl dlgoòtl# {fnt kowlz n je`dotoe` eg `|ilxojnc

ztnhocoty eg tfl je|rcld zyztliz l~|ntoe`z.

Zjncl)zrnjl rntf{nyz= xlze`n`jl jeir|to`k

Zo`k|cnxoty eg dntn jc|ztlxz oz lwnc|ntld hy tfl iln`z eg tfl

zjncnhocoty eg tfl inrz eg glnt|xl wljtex xlrxlzl`tntowlz o` zjncl

zrnjl. Gex n kowl` invoinc wnc|l eg tfl doztextoe` l`lxky tfl

ioòinc `|ihlx eg zo`k|cnxoty inògecdz oz ehtno`ld o` tfl

folxnxjfojnc zjncl jeir|tntoe`.

Gex n kowl` jc|ztlx# tfl rnxtotoe` g|`jtoe` x] # dlzjxohlz tfldoztxoh|toe` eg tfl dntn reo`tz {otf xlzrljt te tfl jc|ztlx wljtexxlrxlzl`tntowl. Tfl zjncl o`gexintoe` oz lwnc|ntld hy je`meoò`k

tfl t{e rnxniltlxz= tfl zok`nc l`lxky doztextoe` zjncl rnxniltlx

# n`d tfl zrntonc zjncl rnxniltlx x# {fojf l~|ncz te tfl `|ihlx

eg dntn reo`tz o`zodl tfl zrntonc {o`de{ eg jeir|tntoe` [. Gex n

kowl` reo`t doztextoe` ilnz|xl eg tfl zok`nc# yvd}

#8 # tflrnxtotoe` g|`jtoe` oz {xottl` hy=

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Tfoz g|`jtoe` oz n i|ctogxnjtnc ilnz|xl# kowo`k n {ny eg

dljeirezo`k tfl zok`nc o`te glnt|xl wljtex jc|ztlxz exdlxld hy

tfl zo`k|cnxoty lvre`l`tz# F98W 3 ># n`d zo`k|cnx gxl~|l`jy

gnjtexz# G%W 3 ># nz {xottl` o`=

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Tfl è`co`lnx dy`niojz eg jc|ztlxo`k oz kewlx`ld hy tfl t{e

l`lxky g|`jtoe`z. Gex n kowl` jc|ztlx otz gxll l`lxky# n`d tfl

doztextoe` l`lxky oz dlgo`ld hy=

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[l xlcntl tfl iljfnòzi eg tfl i|ctogxnjtnc dljeirezotoe` eg tfl

zok`nc te tfl ztnhocoty n`ncyzoz eg tfl inr.

Tfl ztnhocoty je`dotoe` eg tfoz inr oz kowl` hy tfl xlcntoe`= 8W 3

>. [l coiot tfl zo`k|cnxoty lvre`l`t eg tfl jc|ztlxz {otf tflinvoinc wnc|l F# hy zrcotto`k tfnt jc|ztlx o` t{e gex {fojf tfl

je`dotoe` oz xlnjfld= 8jW 9 F# nt tfl jxotojnc wnc|l eg tfl zjncl

rnxniltlxj.

O`tlxlzto`k reo`tz eg ehzlxwntoe` hljeil tfl zok`nc l`lxky

clwlcz=

GW 9 je`zt.# &N.>/

n`d otz rxernkntoe` o` toil. Tfl toil dlxowntowl eg tfoz l~|ntoe`

hljeilz=

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nxl |zld o` tfl jeir|tntoe nc zjflil gex tfl lztointoe` eg tfl

jc|ztlxo`k rnxniltlxz# n`d nxl ztocc rnxt eg tfl e`keo`k xlzlnxjf.

Nt tfl zok`nc zjncl l~|ocohxo|i 2

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jn` hl iedlcld hy tfl l~|ntoe`z=

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Tfl Kxll`z g|`jtoe` kowlz n iedlc eg zrntonc jeflxl`jy eg

o`gexintoe` gex n dntn jc|ztlx=

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n`d oz nrrcold cejnccy o` tfl zlkil`tntoe` rxejlzz.


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&n/ &h/ &j/

Gok|xl >= Jc|ztlxo`k eg jfliojnc dntn= >88< ehzlxwntoe`z o` >)doil`zoe`nc# n`d &h/ 8)doil`zoe`nc. Tfl ehzlxwntoe`z nxl cnhlcld {otf > n`d 8 te o`dojntl n ilihlxzfor

te tfl jexxlzre`do`k jc|ztlxz. Tfl dogglxl`jl o` tfl ilihlxzfor cnhlcz o` &n/ n`d &h/ oz zfe{` o` gok|xl >.&j/.

&n/ &h/ &j/

Gok|xl >= Jc|ztlxo`k eg tfl lvrn`do`k hncc oinkl zl~|l`jl. : jc|ztlxz ztntoztojnc inrz rxemljtoe`z nce`k tfl zo`k|cnx jeire l`tz {otf=

&n/ dowlxkl`jl# &h/ xetex# n`d# &j/ dowlxkl`jl n`d xetex reo`t zlkil`tntoe`.

:. XLZ\CTZN dntnzlt eg >8:7 jeire|`dz {otf >)

doil`zoe`nc jnzl# tfl >8:7 jeire|`dz nxl zlkil`tld o` t{e

kxe|rz {otf= @> 9 887# @8 9 >2>2 kxe|rz ilihlxz.

Jexxlzre`do`k jc|ztlxz wnxon`jlz nxl= W> 9 826# n`d W8 9:2v>26# xlzrljtowlcy. Tfl xlz|cto`k doztxoh|toe` eg tfl dntn oz

zfe{` o` Gok|xl >&n/.

O` tfl 8)doil`zoe`nc jnzl {l jeir|tld zrntonc inr e|t eg tfl 8

iezt zo`k|cnx jeire`l`tz eg tfl dntn. >88< jeire|`dz {lxl

zlkil`tld o` t{e kxe|rz {otf= @> 9 >27; ilihlxz.

N`d tfl xlz|cto`k wnxon`jlz eg tfl kxe|rz dljxlnzld nz= W> 9

>?v>26# n`d W8 9 86v>26# xlzrljtowlcy. Tfl xlz|cto`k jc|ztlxz

ilihlxzfor doztxoh|toe` eg tfl dntn oz zfe{` o` Gok|xl >&h/. Tfl

dogglxl`jl o` jc|ztlxo`k nz o` &n/ n`d &h/ oz zfe{` o` Gok|xl >.&j/.

N tlzt oinkl zl~|l`jl eg n` lvrn`do`k hncc oz |zld nz n`

lvnircl rnttlx` gex 8)doil`zoe`nc zok`nc dljeirezotoe`. Tfl

ztntoztojnc inrz eg tfl jc|ztlxo}ld : oinkl zlkil`tz nxl zfe{` o`

Gok|xl 8. Dogglxl`tonc erlxntexz= dowlxkl`jl# xetex# n`d n

jeiho`ntoe eg tfl t{e nxl |zld gex jc|ztlxo`k dntn. Tfl xlz|cto`k

ilihlxzfor doztxoh|toe`z eg dntn reo`tz nxl zfe{` e` Gok|xlz

8.&n)j/ gex tflzl erlxntexz# xlzrljtowlcy.

4. JE@JC\DO@K XLINXAZ[l fnwl dlzjxohld n `l{ iltfed gex i|cto)doil`zoe`nc dntn

zjnco`k# o` dntn io`o`k n`d a`e{cldkl dozjewlxy nrrcojntoe`z.

Tfl folxnxjfojnc zjncl dljeirezotoe` o`te dogglxl`t zo`k|cnx zltz

oz ehtno`ld hy lwnc|nto`k zjncl)zrnjl zjncnhocoty eg zo`k|cnx

glnt|xlz gex tfl jc|ztlxz eg dntn reo`tz. Tfl o`gexintoe` je tl`t oz

lwnc|ntld hy tfl zjncl)zrnjl gxl~|l`jolz eg jexxlzre`do`k dntn.


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Tfl Kxll`+z g|`jtoe` lvrxlzzlz zrntonc jeflxl`jy eg dntn jc|ztlxz

n`d oz |zld nz n` o`tlkxntoe` g|`jtoe` gex tfl zlkil`to`k dntn hy

nrrcyo`k cejnc erlxntexz. O` tfl iltfed rxerezld# n

dljeirezotoe` o` fnxieòj dntn zltz oz njfolwld hy folxnxjfojnc

zjncl jeir|tntoe`. [l |zl dy`niojnc jnzjndl donkxniz te gexi n

aè{cldkl hnzl dozjewlxld o` tfl dntn# hy tfoz iltfed.

[l fnwl x|` tfl nckexotfi e`cy hy nrrcyo`k tfl zec|hocoty nz n`

nddotoe`nc rxerlxty wnc|l o` tfl dntn zlt. Tfl oirxewlil`tz jn`

hl njfolwld hy x|`ò`k tfl dogglxl`tonc wlxzoe` zlkil`tntoe` e`

tfl zec|hocoty rxerlxty. Tfl nzzejontld ztx|jt|xl donkxniz jn` hl

|zld te xlcntl dogglxl`t z|hzltz eg rxerlxty wnc|lz# n`d tflxlgexl

inao`k n aè{cldkl hnzl z|otnhcl gex jedo`k n`d je`txec

r|xrezlz.

[l o`tl`d te o`wlztokntl tfoz teec o` dntn ioò`k nrrcojntoe`z

otz jedo`k n`d je`txec ztx|jt|xl oirclil`tntoe`. Tfl

ertoio}ntoe` eg tfl zrnjl)l`lxky lvjfn`kl ztlr nce`k {otf tfl

xlgo`lil`t eg e|x `|ilxojnc zjflilz oz n rnxt eg e|x e`keo`k

xlzlnxjf {exa# nz {lcc.

NJA@E[CLDKLIL@T

Tfoz {exa fnz hll` rlxgexild o` tfl Rlxwnzowl

Tljfèceky Cnhz+ ) Jeii|òty Kxodz Cnh# nt tfl O`don`n

\òwlxzoty.

. Dowodl dntn lwl`cy=@%@r te lnjf jexl# o&r/ n`d j 9 ># o`dlvld

Ztnxt {otf= Je|rcl 9 2.

8. Gex A Ainv %% Xlj|xzowl rxejld|xl

:. Dekxnd|nc dlzjl`t|`tocxlze`ntl

JewnxUrxemljt &/1 %%Jeir|tl rnxniltlxz= :D ! 8-djew zo`k|cnx

jeire`l`tz

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Multi-dimensional data scaling – dynamical cascade

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