recent advances in sta.s.cs: theory of rough paths · rough paths i when a path is rough, the...

RecentAdvancesinSta.s.cs:

TheoryofRoughPathsTessyPapavasiliou

DepartmentofSta.s.cs,UniversityofWarwick

Some Examples of Real Data

SeismicdataInternettrafficdata

Financialdata Albatrossflighttrajectory

I A path is ‘rough’ if change is space is not proportional tochange in time.

Why does roughness matter?

200 400 600 800 1000

-2 -1 1 2

Ahome-madeearthquake Twoofthesignalsplo8edasapathin2d

I To understand the e↵ects of such data to systems, we need tomake sense of di↵erential equations

dYt = f (Yt) · dXt ,

where X is rough.I In order to make sense of such equations, we need to make

sense of integrals Z t

sf (Xu) · dXu.

Integrals and Areas

What is the meaning of

sZudXu?

-1.0 -0.5 0.5 1.0

Integrals and Areas

sZudXu?

-1.0 -0.5 0.5 1.0

Integrals and Areas

sZudXu?

-1.0 -0.5 0.5 1.0

Integrals and Areas

sZudXu?

-0.5 0.5 1.0

Integrals and Areas

sZudXu?

-0.5 0.5 1.0

Integrals and Areas

sZudXu?

-0.5 0.5 1.0

Rough Paths

I When a path is rough, the integrals are not uniquely defined.

I The first result of the theory of rough paths was to identifythe minimum number of integrals needed to identify the rest:for a path of ‘roughness p’, any integral

0,Tf (Xu) · dXu

is uniquely defined, once we fix Z t

s<u1<u2<tdXu1dXu2 , . . . ,

s<u1<···<up<tdXu1 . . . dXup

(Terry Lyons, 1998).

I Extension to systems that involve changes in time and space:theory of regularity structures (Martin Hairer, Fields Medal2014).

Rough Paths

0,Tf (Xu) · dXu

s<u1<u2<tdXu1dXu2 , . . . ,

Rough Paths

0,Tf (Xu) · dXu

s<u1<u2<tdXu1dXu2 , . . . ,

Signature of a Path

I The ‘iterated integrals’ provide the basic blocks forunderstanding any system driven by any path.

I All these integrals together (the ‘signature’ of the path)provide an alternative way of describing the path.(Boedihardjo et al, 2016)

I In many cases, by replacing a data stream (path) by thecorresponding signature, it is possible to capture informationmore e�ciently.

I First application: Chinese Handwriting Recognition (BenGraham, winner of 2013 competition).

Signature of a Path

Capturing Sound (Daniel Wilson-Nunn, MMathStats’16)

4HE 3IGNATURE OF 3OUND

)NTRODUCTION

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3IGNATURE AND !REA

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S(N)[s,t](X) =

⇣1, X1

[s,t], X2[s,t], . . . , XN

r?2`2Xn

[s,t] =Z

u1Æ···Æunu1,...,unœ[s,t]

dXu1 ¢ · · · ¢ dXun.

h?2 }M�H 2H2K2Mi Q7 S(2)[s,t](X) +�M #2 2tT`2bb2/

2H2K2Mi rBb2 �b � bmKX (i,j)

[s,t] = A(i,j)[s,t] + B(i,j)

[s,t] ,

r?2`2A(i,j)

[s,t] = 12

u1Æu2u1,u2œ[s1,t1]

dX (i)u1 dX (j)

u2 ≠ dX (j)u1 dX (i)

�M/B(i,j)

[s,t] = 12

⇣X (i)

t ≠ X (i)s

⌘ ⇣X (j)

t ≠ X (j)s

7Q` i, j = 1, . . . , d- �M/ X (i)t Bb i?2 p�Hm2 Q7 i?2

i@i? +QQ`/BM�i2 Q7 i?2 T�i? �i iBK2 tXA(i,j)

[s,t] Bb i?2 �`2� 2M+HQb2/ #v i?2 i?2 TH�M�` +m`p2⇣X (i)

u , X (j)u

⌘7Q` u œ [s, t]- �M/ i?2 +?Q`/ 7`QK

⇣X (i)

s , X (j)s

⇣X (i)

t , X (j)t

⌘- �b b?QrM #2HQrX

⇣X(i)

s , X(j)s

⇣X(i)

t , X(j)t

A(i,j)[s,t]

%XTENDING A 0ATH

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!DAPTIVE 3AMPLING

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0 1 2 3 4 5 6 7 8 9 10≠0.6≠0.4≠0.2

00.20.40.60.8

hBK2 UbV

�KTH

q?2M � bQmM/ Bb ?QKQ;2M2QmbHv b�KTH2/- /�i�Bb `2+Q`/2/ r?2M MQ bQmM/ Bb T`Q/m+2/X "v MQi`2+Q`/BM; /�i� r?2M MQ bQmM/ Bb T`Q/m+2/- /�i� Bbb�p2/X AMbi2�/ Q7 b�KTHBM; ?QKQ;2M2QmbHv- b�K@TH2 �i iBK2b {·i}- r?2`2

·i+1 = minn

· :��A(1,2)

[·i,· ]

�� Ø Ko

�M/ ·0 = 0- 7Q` bQK2 i?`2b?QH/ K > 0- r?2`2A(1,2)

[·i,· ] Bb i?2 �`2� Q7 i?2 2ti2M/2/ T�i?- 7Q` }t2/ÁX

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0 1 2 3 4 5 6 7 8 9 100

hBK2 UbV

7b�K

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%RROR 1UANTIlCATION

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1 2 3 4 5 6 7 8

·10≠5

01,0002,0003,0004,0005,0006,0007,000

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0 10 20 30 40 500

hBK2 UbV

≠140≠120≠100≠80≠60

*QHQ` F2v

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&UTURE 7ORK

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2EFERENCES

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$ANIEL '� 7ILSON .UNN � $R !NASTASIA 0APAVASILIOU$EPARTMENT OF 3TATISTICS�5NIVERSITY OF 7ARWICK

Arabic Handwriting Recognition

Two types of handwriting recognition

OfflineImage (matrix) data

– Scanned documents

– Historical and modern documents

��

x11 x12 . . . x1n2

x21 x22 . . . x2n2

.... . .

...xn11 xn12 . . . xn1n2

��

OnlineTimeseries data

– Data from touch devices

– Handwritten with finger or stylus

X = {(x1, y1) , (x2, y2) , . . . , (xN , yN )}

October 10, 2017 The Alan Turing InstituteA Rough Path Signature Approach to Time Series Classification

I Di↵erent challenge than Chinese: sequence of strokes matters.

I By including the signature as a feature of the data, Danielachieved 92.5% recognition, which is an improvement to thestate of the art (D Wilson-Nunn et al, in IEEE 2ndInternational Workshop on Arabic and Derived Script Analysisand Recognition (ASAR), 2018.)

Arabic Handwriting Recognition

Two types of handwriting recognition

OfflineImage (matrix) data

– Scanned documents

– Historical and modern documents

��

x11 x12 . . . x1n2

x21 x22 . . . x2n2

.... . .

...xn11 xn12 . . . xn1n2

��

OnlineTimeseries data

– Data from touch devices

– Handwritten with finger or stylus

X = {(x1, y1) , (x2, y2) , . . . , (xN , yN )}

October 10, 2017 The Alan Turing InstituteA Rough Path Signature Approach to Time Series Classification

I Di↵erent challenge than Chinese: sequence of strokes matters.

I By including the signature as a feature of the data, Danielachieved 92.5% recognition, which is an improvement to thestate of the art (D Wilson-Nunn et al, in IEEE 2ndInternational Workshop on Arabic and Derived Script Analysisand Recognition (ASAR), 2018.)

RNA editing - an adapting mechanism

I Main Dogma: DNA to RNA to protein expression.

I Hypothesis: Random changes in the RNA (editing) createvariability, making cells behave di↵erently, thus allowing anindividual to adapt to change in the same way that DNAmutations help species adapt to change.

I Experiment: We are given RNA molecules from a largenumber of genetically identical cells (same DNA) of anindividual organism, subject to change. We do not knowwhich cell each RNA molecule comes from except for a smallnumber of cells (single cell experiments).

I Question: Are cells di↵erent based on the percentage ofediting of their RNA molecules?

I Statistical Challenge: We need to use single cell data toinfer variability in RNA editing among cells.

I Conclusion: RNA editing on specific sites varies sagnificantlyfrom cell to cell (Nature Communications, Harjanto et al - incollaboration with Immune Diversity group in DKFZ,Heidelberg).

I Main Dogma: DNA to RNA to protein expression.I Hypothesis: Random changes in the RNA (editing) create

variability, making cells behave di↵erently, thus allowing anindividual to adapt to change in the same way that DNAmutations help species adapt to change.

The Signature of an RNA molecule

UndergraduateResearchSupportSchemeprojectUndergraduateResearchSupportSchemeproject

The signature of a path An efficient way of capturing information

Author Nikolaos Constantinou / Supervisor Dr Anastasia Papavasileiou / Statistics Department

Motivation

Statistical Context

Results

Classification Rule

Conclusions

Data Simulation

References I.  Chevyrev and A. Kormilitzin, “A primer on the signature method in machine learning”, arXiv preprint arXiv:1603.03788, 2016.

II.  Chevyrev and H. Oberhauser, “Signature moments to characterise laws of stochastic processes”, arXiv preprint arXiv:1810.10971, 2018.

Nikolas Constantinou, 3rd year MORSE student.

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