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High Dimensionsal Time Series High-dimensional Time Series Models George Michailidis University of Florida Transdisciplinary Foundations of Data Science IMA, September 2016 Transdisciplinary Foundations of Data Science IMA, Septemb George

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High Dimensionsal Time Series

High-dimensional Time Series Models

George Michailidis

University of Florida

Transdisciplinary Foundations of Data Science IMA, September 2016

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Learning Tasks with Temporally Dependent Data Predictive inference, forecasting, segmentation,covariance estimation/graphical modeling Regression models: yt = Xtβ + ǫt, where the p-dimensional predictors X and error term ǫ is generated by a stationary process Autoregressive models: Xt = AXt−1 + Et, where the p-dimensional error process Et is white noise Related control problem: Xt = AXt−1 + BUt + Et, together with a cost/performance function

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Learning Tasks with Temporally Dependent Data Predictive inference, forecasting, segmentation,covariance estimation/graphical modeling (ctd) Factor models: Xt = ΛFt + Et, where Xt is a p-dimensional process, Ft a k-dimensional latent/factor process and Et a noise process A popular model in the economics/finance literature is for the factors to be changing dynamically over time; e.g. Ft = ΦFt−1 + Ut Given a multivariate time series Xt and identify structural breaks; i.e. identify points in time that the structure of the model changes e.g Xt = A1Xt−1I(t ≤ τ) + A2Xt−1I(t ≥ τ) + Et, for some τ ∈ [0, T] There is an online version of the problem for streaming data Estimate covariance matrix of temporally dependent data

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Application areas Macroeconomics/Finance Functional Genomics Neuroscience Control of large networks

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Application areas: Economics testing relationship between money and income (Sims, 1972, 1980) understanding stock price-volume relation (Hiemstra et al., 1994) dynamic effect of government spending and taxes on output (Blanchard and Jones, 2002) identify and measure the effects of monetary policy innovations on macroeconomic variables (Bernanke et al., 2005)

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Forecasting models in Economics

2 4 6 Feb-60 Aug-60 Feb-61 Aug-61 Feb-62 Aug-62 Feb-63 Aug-63 Feb-64 Aug-64 Feb-65 Aug-65 Feb-66 Aug-66 Feb-67 Aug-67 Feb-68 Aug-68 Feb-69 Aug-69 Feb-70 Aug-70 Feb-71 Aug-71 Feb-72 Aug-72 Feb-73 Aug-73 Feb-74 Aug-74

Employment Federal Funds Rate Consumer Price Index

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Application areas: Functional Genomics Identify regulatory mechanisms from time course data (panel data structure) HeLa gene expression regulatory network [From: Fujita et al., 2007]

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Application Areas: Neuroscience Identify brain connectivity regions

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Need for high-dimensional models Economics: forecasting with many predictors (De Mol et al., 2008) or understand strcutural relationships (Christiano et al., 1999) Finance: build large scale systemic risk models Functional Genomics: reconstruct gene regulatory networks based on limited experimental data Neuroscience: build detailed connectivity maps on temporal data exhibiting multiple structural changes Network control: for large sparse systems (Liu, Slotine, Barabasi, 2011)

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Key issues: Nature of the data measurements (numerical, count, binary) (see Raskutti et al., 2016, for models for count data) Capture the correct dynamics (see Chen and Shojaie, 2016 for models for self-exciting processes) How does the temporal dependence impact estimation and prediction accuracy?

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Introduction

Illustration of estimation accuracy

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Modeling Framework

Vector Autoregression Canonical model for understanding lead-lag cross-dependencies Successful for forecasting purposes and for intervention analysis (impulse response) Exhibits a number of technical challenges in high-dimensions

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Modeling Framework

The VAR Model p-dimensional, discrete time, stationary process X t = {X t

1, . . . , X t p}

X t = A1X t−1 + . . . + AdX t−d + ǫt, ǫt i.i.d ∼ N(0, Σǫ) (1) A1, . . . , Ad : p × p transition matrices (solid, directed edges) Σ−1

ǫ : contemporaneous dependence (dotted, undirected edges)

stability: Eigenvalues of A(z) := Ip − d

t=1 Atzt outside {z ∈ C, |z| ≤ 1}

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Modeling Framework

Detour: VARs and Granger Causality Concept introduced by Granger (1969) A time series X is said to Granger-cause Y if it can be shown, usually through a series of F-tests on lagged values of X (and with lagged values

f Y also known), that those X values provide statistically significant

information about future values of Y . In the context of a high-dimensional VAR model we have that X T−t

j

is Granger-causal for X T

i

if At

i,j = 0.

Granger-causality does not imply true causality; it is built on correlations Also, related to estimating a Directed Acyclic Graph (DAG) with (d + 1) × p variables, with a known ordering of the variables

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Modeling Framework

Estimating high-dimensional VARs through regression data: {X 0, X 1, . . . , X T} - one replicate, observed at T + 1 time points construct autoregression

     (X T )′ (X T−1)′ . . . (X d)′     

=      (X T−1)′ (X T−2)′ · · · (X T−d)′ (X T−2)′ (X T−3)′ · · · (X T−1−d)′ . . . ... . . . . . . (X d−1)′ (X d−2)′ · · · (X 0)′     

   A′

1

. . . A′

d

  

B∗

+      (ǫT)′ (ǫT−1)′ . . . (ǫd)′     

vec(Y) = vec(X B∗) + vec(E) = (I ⊗ X) vec(B∗) + vec(E) Y

Np×1

= Z

Np×q

β∗

q×1

+ vec(E)

Np×1

vec(E) ∼ N (0, Σǫ ⊗ I) N = (T − d + 1), q = dp2 Key Assumption : At are sparse, d

t=1 At0 ≤ k

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Modeling Framework

Estimation Methods ℓ1-penalized least squares (ℓ1-LS) argmin

β∈Rq

1 N Y − Zβ2 + λN β1 ℓ1-penalized log-likelihood (ℓ1-LL) (Davis et al., 2012) argmin

β∈Rq

1 N (Y − Zβ)′ Σ−1

ǫ

⊗ I

(Y − Zβ) + λN β1

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Modeling Framework

ℓ1-LL Algorithm Objective function jointly non-convex, but convex w.r.t. B’s and Σ−1

ǫ

Algorithm converges to stationary point near truth with high probability under high-dimensional scaling, provided it is initialized at a good point (details in Lin et al., 2016)

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Detour: Probabilistic Consistency of Lasso For regression models, the quality of the estimates of the regression parameters depends on relies crucially on two regularity conditions: 1. Restricted Eigenvalue (RE): The null space of the normalized design matrix X/ √ N avoids a cone set C(S, 3) := {v ∈ Rp : vSc 1 ≤ 3vS1} αRE := min

v∈Rp,v≤1,v∈C(S,3)

1 N Xv2 > 0 2. Deviation Condition: X ′E/Nmax ≤ Q(X, σ)

log p/N

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Detour: Probabilistic Consistency of Lasso For regression models, the quality of the estimates of the regression parameters depends on relies crucially on two regularity conditions: 1. Restricted Eigenvalue (RE): The null space of the normalized design matrix X/ √ N avoids a cone set C(S, 3) := {v ∈ Rp : vSc 1 ≤ 3vS1} αRE := min

v∈Rp,v≤1,v∈C(S,3)

1 N Xv2 > 0 2. Deviation Condition: X ′E/Nmax ≤ Q(X, σ)

log p/N

Under the above conditions Estimation error: ˆ β − β∗2 ≤ Q(X, σ) αRE

k log p

N with high probability

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Lasso Regression for Time Series Data It is unknown if the above conditions are satisfied in high-dimensional time series data

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Lasso Regression for Time Series Data It is unknown if the above conditions are satisfied in high-dimensional time series data Verifying RE type assumptions for a fixed design is NP-hard

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Lasso Regression for Time Series Data It is unknown if the above conditions are satisfied in high-dimensional time series data Verifying RE type assumptions for a fixed design is NP-hard For random design matrix X, existing results only provide guarantees when the samples are independent

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Lasso Regression for Time Series Data It is unknown if the above conditions are satisfied in high-dimensional time series data Verifying RE type assumptions for a fixed design is NP-hard For random design matrix X, existing results only provide guarantees when the samples are independent Even for a stationary process, the data share complicated dependence patterns -

◮

Rows are dependent

◮

Columns are dependent

◮

error term E and design matrix X dependent

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Vector Autoregression

Random design matrix X, correlated with error matrix E      (X T )′ (X T−1)′ . . . (X d)′     

=      (X T−1)′ (X T−2)′ · · · (X T−d)′ (X T−2)′ (X T−3)′ · · · (X T−1−d)′ . . . ... . . . . . . (X d−1)′ (X d−2)′ · · · (X 0)′     

   A′

1

. . . A′

d

  

B∗

+      (ǫT )′ (ǫT−1)′ . . . (ǫd)′     

vec(Y) = vec(X B∗) + vec(E) = (I ⊗ X) vec(B∗) + vec(E) Y

Np×1

= Z

Np×q

β∗

q×1

+ vec(E)

Np×1

vec(E) ∼ N (0, Σǫ ⊗ I) N = (T − d + 1), q = dp2 Question: How often does RE hold? How small is αRE ? How does the cross-correlation affect convergence rates?

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence in high-dimensional VAR: Existing approaches One can try to procced analogously to regression for iid data e.g. Negahban and Wainwright, 2011: for VAR(1) models, assume A1 < 1, where A :=

Λmax(A′A)

For univariate autoregression X t = ρX t−1 + ǫt, reduces to |ρ| < 1 - equivalent to stability assumption It turns out, this is a very restrictive assumption for most realistic VAR models

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence via the Spectral Density

Spectral density function of a covariance stationary process {X t}, fX (θ) = 1 2π

∞

l=−∞

ΓX (l)e−ilθ, θ ∈ [−π, π] ΓX (l) = E

X t(X t+l)′

, autocovariance matrix of order l

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence via the Spectral Density

Spectral density function of a covariance stationary process {X t}, fX (θ) = 1 2π

∞

l=−∞

ΓX (l)e−ilθ, θ ∈ [−π, π] ΓX (l) = E

X t(X t+l)′

, autocovariance matrix of order l If the VAR process is stable, it has a closed form (cf. equation (9.4.23), Priestley (1981)) fX (θ) = 1 2π

A(e−iθ)

−1 Σǫ

A∗(e−iθ)

−1 The two sources of dependence factorize in the frequency domain

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence via the Spectral Density

For univariate processes, “peak” of the spectral density measures stability of the process - (sharper peak = less stable)

−10 −5 5 10 0.0 0.2 0.4 0.6 0.8 1.0 lag (h) Autocovariance Γ(h)

ρ=0.1

ρ=0.5 ρ=0.7

Autocovariance of AR(1)

−3 −2 −1 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 θ f(θ) ρ=0.1 ρ=0.5 ρ=0.7

Spectral Density of AR(1)

For multivariate processes, a similar role is played by the maximum eigenvalue of the (matrix-valued) spectral density

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence via the Spectral Density

For a covariance stationary process {X t} with continuous spectral density fX (θ), the maximum eigenvalue of its spectral density captures its stability M(fX ) = max

θ∈[−π,π] Λmax (fX (θ))

The minimum eigenvalue of the spectral density captures dependence among its components m(fX ) = min

θ∈[−π,π] Λmin (fX (θ))

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence via the Spectral Density

For a covariance stationary process {X t} with continuous spectral density fX (θ), the maximum eigenvalue of its spectral density captures its stability M(fX ) = max

θ∈[−π,π] Λmax (fX (θ))

The minimum eigenvalue of the spectral density captures dependence among its components m(fX ) = min

θ∈[−π,π] Λmin (fX (θ))

For stable VAR(1) processes, M(fX ) scales with (1 − ρ(A1))−2, where ρ(A1) is the spectral radius of A1 m(fX ) scales with the capacity (maximum incoming + outgoing effect at a node)

f the underlying graph

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Quantifying Dependence via the Spectral Density

For a covariance stationary process {X t} with continuous spectral density fX (θ), the maximum eigenvalue of its spectral density captures its stability M(fX ) = max

θ∈[−π,π] Λmax (fX (θ))

The minimum eigenvalue of the spectral density captures dependence among its components m(fX ) = min

θ∈[−π,π] Λmin (fX (θ))

For stable VAR(1) processes, M(fX ) scales with (1 − ρ(A1))−2, where ρ(A1) is the spectral radius of A1 m(fX ) scales with the capacity (maximum incoming + outgoing effect at a node)

f the underlying graph

We can similarly measure stability of subprocesses M(fX , k) := max

J⊂{1,...,p},|J|=k M

fX(J)
M(fX , 1) ≤ M(fX , 2) ≤ · · · ≤ M(fX , p) = M(fX )

Allows us to derive concentration inequalities with dependent random variables

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Consistency of sparse VAR estimates It is established (see Basu and Michailidis, 2015)

d

Ah − Ah

≤ φ(At, Σǫ)
k (log dp2)/N
Consistency in high-dimension:

Even if d, p = O(N2), k log dp2/N → 0 as long as k = o(N) Error has two components:

1. φ(At, Σǫ) large ⇔ M(fX ) large, m(fX ) small 2.

k log dp2/N: Estimation error for independent data

Estimation error same as i.i.d. data, modulo a price for temporal dependence

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Recap of the Main Theoretical Results Assuming RE and deviation conditions we can establish the consistency of sparse estimates of high-dimensional VAR models For stable VAR models, the RE and deviation conditions hold with high probability The convergence rate has two components: (i) the component governed by the structural parameters of the problems and is identical to the iid case and (ii) the component governed by the temporal depencence

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Theoretical Considerations

Beyond VAR and Sparsity: High Dimensional Models for Time Series Data

The concentration bounds obtained can be used to prove estimation consistency for

ther regularized methods with high-dimensional, Gaussian time series

Regression with Lasso; non-convex penalty (SCAD, MCP) Generalized linear models (regression with non-continuous outcome variables [Raskutti et al., 2016] Sparse covariance estimation with time series data Regression / VAR with Group Lasso [Basu et al., 2015] Low rank and Low rank+ Sparse VAR [Basu, 2014] Tensor Regression with dependent data [Raskutti and Yuan, 2015] Time series with local dependence [Schweinberger et al., 2015] VAR models with grouped structure on the transition matrices [Mattesson et al., 2015] The results have a common theme estimation error for dependent data

Measure of narrowness
f spectrum

× estimation error for i.i.d. data

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

Models with Structural Breaks Increasing interest in using time series models and/or graphical models as network models derived from high-dimensional data Numerous applications both for the offline and online versions

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

A Canonical Statistical Problem: Change Point Detection Simplest Setup: Random vector observed in a time interval {1, · · · , T} offline version Xt = (X1t, X2t, · · · , Xpt) ∼ N(0, Σ1), t ≤ τ, Xt = (X1t, X2t, · · · , Xpt) ∼ N(0, Σ2), t > τ. Objectives:

1. Estimate the change point τ 2. Estimate the Gaussian graphical models Ω1 ≡ Σ−1

1 , Ω2 ≡ Σ−1 2

The iid assumption is simplifying, but can easily be mitigated through neighborhood selection techniques leveraging lasso regressions with temporally dependent errors (Basu and Michailidis, 2015)

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

Some Background on Low Dimensional Change Point Problems Assume a stump model: yi = αI(i ≤ τ) + βI(i > τ) + ǫi, i = 1, · · · , T, where ǫi ∼ N(0, σ2) and I(·) denotes the indicator function Then, under a condition on the signal-to-noise ratio |α−β|

σ

≥ C > 0, one can establish the following

1. ˆ α, ˆ β can be estimated at rate √ T (the usual parametric rate) 2.

|ˆ τ−τ| T

= O( 1

T )

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

Naive Algorithm 1. For each t = 1 + d, · · · , T − d, for some d > 0, calculate the joint Gaussian likelihood assuming that τcandidate = t which is given by L(Ω1; t = 1, · · · , τcandidate) + L(Ω2; t = τcandidate + 1, · · · , T) 2. Set ˆ τ = argmaxτcandidateL(Ω1, Ω2, τcandidate) Technical Challenges: Note that at any solution ˆ τ = τ, one term in the likelihood is misspecified Hence, a much more careful handling of the technical issues is needed to establish the results

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

Main Results Under a Restricted Eigenvalue condition it can be established that (Roy, Atchade and Michailidis, 2016) 1. ||ˆ Ωk − Ωk||F = O(

s log pT)

2. |ˆ τ − τ| T = O(log pT T ).

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

Extension to Multiple Change Points In a recent paper, Leonardi and Buhlmann, arXiv, 2016 look at the same problem, but allow multiple change points To identify the change points, they propose a dynamic programming algorithm, as well as a computationally faster binary search approximation Further, they look at estimation consistency properties of τk, k = 1, · · · , K and the corresponding Ωk’s in a slow regime where change points are sparse and far apart and in a fast regime, where change points grow as a function of T The rates for the Ωk’s are the usual ones, but even in the slow regime the

btained rate for ˆ

τk is worse than the one previously obtained. A related problem was studied in Kollar and Xing, Electronic J. of Statistics, 2012 where each node can experience multiple-change points and Soh and Chadrashekaran, arXiv 2014

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34

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High Dimensionsal Time Series Segmentation problems

Concluding Remarks Temporal data are present in a diverse set of applied areas Time series models pose a number of subtle technical challenges in high-dimensions A number of open questions:

1. Going beyond Gaussian data (heavy tailed distributions, mixed types of data) 2. Incorporation of prior information/Bayesian modeling 3. Inference framework for assessing both parameter and model significance 4. Better models for capturing intricate temporal dynamics 5. Intervention/control problems

George Michailidis High Dimensionsal Time Series Transdisciplinary Foundations of Data Science IMA, Septemb / 34