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Rough Paths Theory.
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Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Learning from the past, predicting the statisticsfor the future, and learning an evolving system
Ni HaoJoint work with Prof. Terry Lyons
1Department of MathematicsUniversity of Oxford
2Oxford-Man Institute of Quantitative Finance
Stochastic analysis seminar, June 10, 2013
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Outline
1 Introduction and Motivation
2 Preliminary of Rough Paths Theory
3 Our researchEncode a time series to its signaturesCombine Time Series Model into Expected SignatureFramework
4 Numerical ExamplesAR modelsModel Misspecification: The mixture of two AR models
5 Conclusion
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Motivation
The setting of the problem
Under the probability space (Ω,F ,P), Xtt∈[0,T ] is a E-valuedstochastic process and Y is a real valued random variable,such that there exists a function f ,
Y = f (Xtt∈[0,T ]) + ε
where E[ε|Xtt∈[0,T ]] = 0.Now we observe a sequence of data pairs Xi ,YiNi=1 and weare interested in recovering the functional relationship f fromthe data set.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Useful tools1 Rough Paths Theory2 Regression analysis
The plan of my talk1 Encode a time series into the signature of a two
dimensional path which includes the time axis.2 Use the property of the signature to turn the nonlinear
problem to the linear problem.3 Numerical examples
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
A Brief Introduction to Rough Paths Theory
A brief introduction of Rough Path Theory1 A non-linear extension of the classical theory of controlled
differential equation.2 The essential object of Rough Path Theory is the signature
of the path.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
The signature of the path
Definition (The signature of the path)Let J denote a compact interval. Let X : J → E be a continuouspath with finite p−variation for some p < 2. The signature of Xof T ((E)) defined as follows
XJ = (1,X 1J ,X
2J , ...),
where, for each n ≥ 1,X n
J =∫
u1<...<un,u1,...,un∈J dXu1 ⊗ . . .⊗ dXun . The signature of X isalso denoted by S(X ).
Notation: ρn is defined as follows:
ρn : S(X )→ (1,X 1J , . . . ,X
nJ ).
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Remarks:1 Assume that E has finite dimension d and choose a basis
(e1, ...,ed ) of E . Then it implies that for every n,ei1 ⊗ . . .⊗ eini1,...in∈1,...d is the basis of E⊗n.
2 The signature of the path X can be regarded as anon-commuting polynomials of the variable e1, ...,ed andthe coefficient of the monomial ei1 ⊗ . . .⊗ ein is∫
u1<...<un,u1,...,un∈J dX (i1)u1
. . . dX (in)un . Thus we can also write
XJ =∞∑
n=0
∑i1,...in∈1,...d
∫u1<...<un,u1,...,un∈J
dX (i1)u1
. . . dX (in)un ei1⊗. . .⊗ein
or use πI defined byπI(XJ) =
∫u1<...<un,u1,...,un∈J dX (i1)
u1. . . dX (in)
un .
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Shuffle Product Property
DefinitionWe define the set Sm,n of (m,n) shuffles to be the subset ofpermutation in the symmetric group Sm+n defined by
Sm,n = σ ∈ Sm+n : σ(1) < · · · < σ(m), σ(m + 1) < · · · < σ(m + n)
DefinitionLet I = (i1, . . . , ik1), J = (j1, . . . , jk2) and (k1, . . . , km+n) = I ∗ J.Define the shuffle product of πI and πJ , denoted by πI
πJ ,which maps from VE to T ((E)), and for every X : [0,1]→ Ebeing a continuous path of bounded variation,
(πI πJ)(S(X )) =
∑σ∈Sm,n
πkσ−1(1)
,...,kσ−1(m+n)(S(X )).
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Theorem (Shuffle product property)
For every X : [0,1]→ E being a continuous path of boundedvariation,
(πI πJ)(S(X )) = πI(S(X ))πJ(S(X ))
RemarkAny polynomial in S(X ) can be rewritten as a linear function inS(X ).
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Outline
1 Introduction and Motivation
2 Preliminary of Rough Paths Theory
3 Our researchEncode a time series to its signaturesCombine Time Series Model into Expected SignatureFramework
4 Numerical ExamplesAR modelsModel Misspecification: The mixture of two AR models
5 Conclusion
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Embed the time series into its signature
Embed a time series into its signature
Step1 :Embed the time series (ti , ri)ni=m into the twodimensional axis path. Let us define the two dimensional axispath, which maps [2m,2n + 1] to R+ × R, and is given asfollows:
R(s) =
tme1 + rm(s − 2m)e2, if s ∈ [2m,2m + 1)
[ti + (ti+1 − ti )(s − 2i − 1)]e1 + rie2, if s ∈ [2i + 1,2i + 2)
tie1 + [ri + (ri+1 − ri )(s − 2i − 2)]e2, if s ∈ [2i + 2,2i + 3)
where i = m,m + 1, ...,n − 1 and eii=1,2 are orthonormalbasis of R2.Step2 :Compute the signature of this transformed continuouspath and denote it by S((ti , ri)ni=m).
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
ExampleLet us consider the time series(2,2), (3,5), (4,3), (5,4), (6,6), (7,3), (8,2) and we embed itto the lattice path as shown in the following figure.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
LemmaLet X denote the signature of the time series (ti , ri)ni=1.Assume that tini=1 are known, for every i = 1, . . . ,n,
∆R = T−1A
where
A =
0!π2(X)1!π12(X)
...(n − 1)!π1...12(X)
,T =
1 1 . . . 1t1 t2 . . . tn...
.... . .
...tn−11 tn−1
2 . . . tn−1n
,
∆R :=
r1
r2 − r1...
rn − rn−1
.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Combine linear time series models into the expectedsignature framework
Classical linear time series models:It includes (a) simple autoregressive(AR) models, (b)simple moving-average(MA) models, (c) mixedautoregressive moving-average (ARMA) models,(d)seasonal models, etc.Let rtt be a time series. The linear time series modelsattempts to capture the linear relationship between rt andthe information available prior to time t , e.g. the past pvalues rt−i , i = 1, ...,p.The foundation of time series analysis is stationarity.Roughly speaking, the stationarity of a time series saysthat the joint distribution of rt1 , ..., rtk is invariant under timechanges, for every integer k .Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Outline
1 Introduction and Motivation
2 Preliminary of Rough Paths Theory
3 Our researchEncode a time series to its signaturesCombine Time Series Model into Expected SignatureFramework
4 Numerical ExamplesAR modelsModel Misspecification: The mixture of two AR models
5 Conclusion
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Definition (Simple autoregressive models)
Let rt be a time series. The notation AR(p) indicates anautoregressive model of order p. The AR(p) model is definedas follows:
rt = Φ0 + Φ1rt−1 + · · ·+ Φprt−p + at
where p is a non-negative integer and at is assumed to be awhite noise with mean zero and variance σ2
a.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Standard Estimation Method to AR models1 It is in the same form as a multiple linear regression model
with lagged values serving as the explanatory variables.2 Given the sequence of rt−p, ..., rt−1, rtt∈Z, by the
standard linear regression (e.g. least square method), wewill obtain Φipi=0 and get the estimation of the conditionalexpectation of rt given the past p values rt−i(i = 0, ...,p).
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
How does classical time series models fit in our setting?
Suppose that rtt is a time series generated by the AR(p)model or other classical time series models. The meanequation is a polynomial of the p-lagged values, and then thereexists a linear functional F such that the mean equation can bewritten as the linear functional on S(rt−ipi=1) by the shuffleproduct property of signatures.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Recall our original problem
Y = f (Xss∈[0,T ]) + Noise = f (S(X )0,T ) + Noise
assuming that E [Noise|(S(X )0,T ] = 0. Suppose that f can bewell approximated by a polynomial. Then we assume that
Y ≈ A(S(X )0,T ) + Noise
where A is linear function.Given a sequence of samples
S(Xi)0,T ,Yi
, it is natural to use
the linear regression to estimate the function A, and obtain
E [Y |Xss∈[t−∆t ,t]] = A(Xss∈[0,T ])
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Encode a time series to its signaturesCombine Time Series Model into Expected Signature Framework
Models AR(p) Model Our ModelDependent Variable rt rtExplanatory Variable rt−1, . . . , rt−p S(rt−ipi=1)
Output E [rt |rt−1, ...rt−p] E [rt |S(rt−ipi=1)]
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Outline
1 Introduction and Motivation
2 Preliminary of Rough Paths Theory
3 Our researchEncode a time series to its signaturesCombine Time Series Model into Expected SignatureFramework
4 Numerical ExamplesAR modelsModel Misspecification: The mixture of two AR models
5 Conclusion
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Example
We generated a time series based on AR(3) model, withparameters [φ0;φ1;φ2;φ3] = [0; 0.5; 0.3; 0.2]. We use thestandard calibration methods of AR models to get theestimated model parameters[φ0; φ1; φ2; φ3] = [0.0331; 0.5052; 0.2747; 0.2222]. On the otherhand, we apply our methods to regress rt to S(rt−i3i=1).
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by AR calibration for the learning set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by AR calibration v.s. The trueconditional mean for the learning set.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by AR calibration for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by AR calibration v.s. The trueconditional mean for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by our approach for the learning set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by our approach v.s. The trueconditional mean for the learning set.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by our approach for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by our approach v.s. The trueconditional mean for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Outline
1 Introduction and Motivation
2 Preliminary of Rough Paths Theory
3 Our researchEncode a time series to its signaturesCombine Time Series Model into Expected SignatureFramework
4 Numerical ExamplesAR modelsModel Misspecification: The mixture of two AR models
5 Conclusion
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by AR calibration for the learning set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by AR calibration v.s. The trueconditional mean for the learning set.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by AR calibration for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by AR calibration v.s. The trueconditional mean for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by our approach for the learning set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by our approach v.s. The trueconditional mean for the learning set.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The Difference of the true conditional mean and estimatedmean by our approach for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
AR modelsModel Misspecification: The mixture of two AR models
Figure: The estimated mean by our approach v.s. The trueconditional mean for the backtesting set
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Reference
Conclusion1 Our method is non-parametric and systematic.2 There is still space to improve the performance of
regression, e.g. weighted least square methods.
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system
Introduction and MotivationPreliminary of Rough Paths Theory
Our researchNumerical Examples
Conclusion
Reference
Reference I
Terry.J. Lyons, Michael J. Caruana, Thierry LevyDifferential Eequations Driven by Rough Paths.Springer, 2006.
Terry Lyons, Hao Ni, Daniel LevinLearning from the past, predicting the statistics for thefuture, learning an evolving system
Ni Hao Learning from the past, predicting the statistics for the future, and learning an evolving system