DYTEC - Dynamic Tail Event Curves - Hu

Transcription

DYTEC - Dynamic Tail Event Curves
Petra Burdejová
Wolfgang Karl Härdle
Ladislaus von Bortkiewicz Chair of Statistics
C.A.S.E. – Center for Applied Statistics
and Economics
Humboldt–Universität zu Berlin
http://lvb.wiwi.hu-berlin.de
http://case.hu-berlin.de
Motivation
Importance of tail event (curves)
Dynamic Tail Event Curves
1-1
Motivation
1-2
Intra-day trading volume
13
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cumulated 1-min volume
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volume
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250 trading days
10:00 - 16:00
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daily 01-12/2008
14
20080102
NASDAQ Market
- CISCO stocks
10:00
12:00
14:00
time
16:00
Motivation
1-3
Jain and Joh (1988)
the day of the week and the hour have effect on trading volume
Darrat et al. (2003) and Spierdijk et al. (2003)
lagged values of volatility and trading volume simultaneously
Bialkowski et al. (2008) and Brownlees et al. (2011)
dynamic volume approach for VWAP
Motivation
1-4
VWAP trading strategy
Buying/Selling fixed amount of shares at average price
pj I{pj >VWAP} that tracks the VWAP benchmark
PJ
VWAP =
j=1 vj
· pj
PJ
j=1 vj
with price pj and volume vj of the j-th transaction
50 % of trades are VWAP orders
Implementation requires model for intraday evolution
of volume
Motivation
1-5
20080102−20080103
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High frequency data
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High dimensions
Focus on dynamics
volume
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Model curves
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Make forecasts
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Comprising tail events
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12:00
14:00
time
16:00
Motivation
1-6
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20080102
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volume
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10:00
12:00
14:00
16:00
time
Figure: Expectiles for τ ∈ {0.01, 0.02, 0.05, 0.1, 0.2, 0.5, , . . .}
Motivation
1-7
0.00
0.01
cum. returns
0.02
0.03
0.04
CISCO 2/1/2008
10:00
12:00
14:00
16:00
time
Figure: Cumulative returns of VWAP strategy with weights based
on τ -expectiles of volume, τ = 0.05, 0.5, 0.95.
Motivation
1-8
Temperature data
Are we getting stronger extremes?
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Berlin 0.95 expectiles 1948−2013
4.0
Daily average
temperature
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Model
Usage: Pricing
weather derivatives
3.5
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month in year
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Figure: 0.95-expectile of Berlin temperature residuals
in 1948-1969 1970-1991 1992-2013 1948-2013
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3.0
Model residuals
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2.5
Berlin, 1948 - 2013
temperature residuals
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Motivation
Hurricane predictions
1-9
Motivation
1-10
Hurricane predictions
Are we getting stronger extremes?
observe different
trend pattern
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Years 1946 - 2011
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wind
30 40
West Atlantic
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Strength of wind in knots
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Strenght of Atlantic huricanes yearly
1946
1967
1987
Year
Figure: Yearly expectiles for τ = 0.25, 0.5, 0.75 and trend
2011
Motivation
Dynamic demand models
Electricity demand
I Quarter-hourly
I Jan.2010 - Dec.2012
I Amprion company
in west of Germany
Water demand
Gas demand
1-11
Motivation
Objectives & Challenges
High frequency data
Time-varying with intraday pattern
Dependent
Model curves
Focus on dynamics and dependence
Comprising tail events
Make forecasts
1-12
Outline
1. Motivation
X
2. Quantiles and Expectiles
3. Modeling Time-varying Curves
4. Outlook
5. Empirical Study
6. References
Quantiles and Expectiles
2-1
For r.v. Y obtain τ -quantile
qτ = arg min E {ρτ (Y − θ)}
θ
with asymmetric loss function
ρτ (u) = |u|α τ − I{u<0} where α = 1 for quantiles and α = 2 for expectiles
2-2
0.8
0.6
0.2
0.4
loss function
1.0
1.2
1.4
Loss function ρτ(u) = uκ τ − I(u<0)
0.0
LQRcheck
−3
−2
−1
0
1
2
Figure: Loss function of expectiles and quantiles
for τ = 0.5 (dashed) and τ = 0.9 (solid)
x
3
2-3
Expectile Curves
Y associated with covariates X
Define generalized regression τ -expectile
e τ (x) = arg min E {ρτ (Y − θ) | X = x}
θ
Use method of penalized splines to estimate vector of coefficients α
α
b = arg min
α
n
X
n
o
ρτ yi − α> b(xi ) + λα> Ωα
i=1
with b basis vector (e.g. B-splines)
penalization matrix Ω and shrinkage parameter λ
Definition of B-splines
Penalization matrix
2-4
Estimation of Expectile Curves
LAWS
Schnabel and Eilers (2009): iterative LAWS algorithm
Schnabel (2011): expectile sheets for joint estimation of curves
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20080102
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10:00
12:00
14:00
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time
Figure: CISCO trading volume expectiles (λ = 5)
Modelling Time-varying Curves
3-1
Fix τ and model
τ
e (t) =
K
X
αk φk (t)
k=1
with predefined basis Φ = (φ1 , . . . , φK )> and t = 1, . . . , T .
Add variation in time, s = 1, . . . , S, and model
esτ (t)
=
K
X
k=1
αsk φk (t)
3-2
Modeling Time-varying Curves
Guo et al. (2013)
es (t) independent realizations of stationary process
Functional princ. components
Following Karhunen-Loève expansion:
es (t) = µ(t) +
K
X
αsk φk (t) = µ(t) + αs> Φ(t)
k=1
Method of penalized splines for mean function and FPC with
empirical loss function
S X
T
X
n
o
ρτ Yst − θµ> b(t) − αs> ΘΦ b(t) + pen.mtx
s=1 t=1
Details
Temporal (weak) dependent curves
Sequence {Xn } is m-dependent if for any k the σ-algebras
+
Fk− =σ (. . . , Xk−1 , Xk ) and Fk+m
=σ (Xk+m , Xk+m+1 , . . .) are
independent.
Hörmann and Kokoszka (2010)
Karhunen-Loève expansion applicable for m-dependent
asymptotic properties of FPC estimates remain the same
most of time series ARE NOT m-dependent
fail if i.i.d. curves are too noisy
fail if curves are sufficiently regular but dependency is too
strong
How to model stronger dependency?
3-3
3-4
Cramér-Karhunen-Loève representation
Panaretos and Tavakoli (2013)
spectral decomposition of stationary functional time series
es (t) ≈
J
X
exp(iωj s)
j=1
−π =ω1 < . . . < ωJ+1 = π
φ
eigenfunctions
j,k k≥1
αj,k k≥1 corresponding coefficients
K
X
αjk φj,k (t)
k=1
Theoretical details
3-5
Simplification: Assume φj,k (t) = φk (t) for each j
Then:
es (t) ≈
J
X
exp(iωj s)
j=1
≈ U(s)> AΦ(t)
with
U(s) = (exp (iω1 s) , . . . , exp (iωJ s)) >
AJxK matrix of coefficients
Φ(t) = (φ1 (t), . . . , φK (t))>
K
X
k=1
αjk φk (t)
3-6
Empirical loss function
P
Method of penalized splines for φk (t) = Ll=1 βkl bl (t)
Then
Φ(t) = BKxL b(t)
Minimize loss function
S X
T
n
o
X
ρτ Ys,t − U(s)> Cb(t) + λkCJxL kGJ
s=1 t=1
with CJxL = AJxK BKxL matrix of coefficients
and Group lasso penalization
v
L
J
J u
uX
X
X
t
c2
kC k =
kc k =
JxL GJ
Gj 2
j=1
jl
j=1
l=1
3-7
Empirical loss function
S X
T
X
J
n
o
X
ρτ Ys,t − U(s)> Cb(t) +λ
kcGj k2
|s=1 t=1
j=1
{z
}
l(C)
l (C ) continuously differentiable
b to be a solution:
K-K-T conditions for C
b )G + λ CGj
∇l (C
j
kCG k2
=0
j
if CGj 6= 0
b )G k2 ≤ λ if CG = 0
k∇l (C
j
j
If τ = 0.5, closed form solution available
3-8
Block-Coordinate
Gradient Descent Algorithm (BCGD)
Tseng & Yun (2009)
Solve nonconvex nonsmooth optimization problem
min f (x) + λP(x)
x
where λ > 0
P : Rn → ( − ∞, ∞ ] block-separable convex function
f smooth on an open subset containing domP
combination of quadratic approximation
and coordinate descent algorithm
global convergence
Theoretical details
3-9
B-C-G-D Algorithm
Stepwise and blockwise minimize
b (t) )+ 1 d > H (t) d +λ
b (t) +d ) = l (C
b (t) )+d > ∇l (C
Sλ ( C
2
J
X
(t)
kCGj +d k2
j=1
Details
Notation:
b (t) + d )
min Sλ (C
dGj
minimization, where d = (dG1 , . . . , dGJ ) with dGk = 0 for k 6= j
3-10
B-C-G-D Algorithm
repeat
t=t+1
for j = 1 to J do
b (t) k2 ≤ λ then
b (t) )G − h(t) C
if k∇l (C
j
Gj
G
j
(t)
dGj
b (t)
= −C
Gj
else
(t)
b (t) + d )
dGj = min Sλ (C
dGj
end if
end for
until convergence criterion met
b (t+1) − C
b (t) − α(t) d (t) =0
update C
Details
3-11
Algorithm for computing DYTEC
Idea:
start with initial weights ws,t (obtained separately for each
s = 1, . . . , S) and iterate between following steps:
b using BCGD algorithm
compute C
update weights
ws,t =
τ
1−τ
b b(t),
if Ys,t > U(s)> C
otherwise.
stop if there is no change in weights ws,t .
3-12
Dynamic functional factor model
generalization (capture nonstationarity)
Hays et al.(2012) & Kokoszka et al.(2014)
extend with the idea of two spaces of basis function
Model
es (t) =
K
X
Zsk mk (t) = Zs> m(t)
k=1
with time-varying factor loadings Zk
and functional factors
3-13
Time basis:
Zsk =
J
X
αkj uj (s)
j=1
Zs = AKxJ · U(s)
Space basis:
mk (t) =
L
X
βkl bl (t)
l=1
m(t) = BKxL b(t)
es (t) = Zs> m(t) = U(s)> Cb(t)
with CJxL = A>
JxK BKxL matrix of coefficients,
space basis vector b(t) = {b1 (t), . . . , bL (t)}>
and time basis vector U(s) = {u1 (s), . . . , uJ (s)}>
Same loss function:
S X
T
X
n
o
ρτ Ys,t − U(s)> Cb(t) + λkCJxL kGJ
s=1 t=1
3-14
Time basis
capture periodic variation
capture trend
Proposal by Song et al. (2013):
I Legendre polynomial basis
I Fourier series
See Appendix
3-15
Space basis
capture daily patterns
capture specific structure
Proposal by Song et al (2013):
I Data driven
I Based on combination of smoothing techniques and FPCA
B-splines
Definition of B-splines
3-16
Empirical Study
4-1
Empirical Study
13
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9
cumulated 1-min volume
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volume
11
12
250 trading days
10:00 - 16:00
●
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●
10
daily 01-12/2008
14
20080102
NASDAQ Market
- CISCO stocks
10:00
12:00
14:00
time
16:00
Empirical Study
4-2
Trading volume - smoothing
GCV
110.05
log(volume)
11
12
13
14
110.15
15
B-splines with knots every 5 minutes (i.e. 76 splines)
Optimal λ = 164
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9
109.95
10
●
●
100
150
lambda
200
10:00
12:00
14:00
2008−01−23
16:00
Empirical Study
4-3
Trading volume - FPCA
1.0
Use 14 FPCs to explain 90% of variation
0.6
cumulative expl. variability
0.7
0.8
0.9
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0
14
40
Number of FPCA
60
Empirical Study
4-4
log(volume)
10
12
14
mean
10:00
12:00
14:00
time of day
16:00
log(volume)
0.00 0.06
1st FPCA
10:00
12:00
14:00
time of day
16:00
Empirical Study
4-5
1500
Periodogram
0
−30
500
1000
0
−20
−10
score
10
2000
20
2500
Fisher’s G-test: p-value=0.000
0
50
100
150
200
250
0.0
day
Figure: Scores of 1FPC and periodogram
0.1
0.2
0.3
Frequency
0.4
0.5
Outlook
Outlook
Algorithm - Code in R
Simulation and Empirical studies
5-1
DYTEC - Dynamic Tail Event Curves
Petra Burdejová
Wolfgang Karl Härdle
Ladislaus von Bortkiewicz Chair of Statistics
C.A.S.E. – Center for Applied Statistics
and Economics
Humboldt–Universität zu Berlin
http://lvb.wiwi.hu-berlin.de
http://case.hu-berlin.de
References
References
Bialkowski, J., Darolles, S., Le Fol, G.
Improving VWAP strategies: A dynamic volume approach
Journal of Banking & Finance 32(9): 1709-1722, 2008,
DOI: 10.1016/j.jbankfin.2007.09.023
Brownlees, Ch. T., Cipolliniy, F., Gallo, G.M.
Intra-daily Volume Modeling and Prediction for Algorithmic
Trading
Journal of Financial Econometrics 9(3): 489-518, 2011,
DOI: 10.2139/ssrn.1393993
6-1
References
Darrat, A. F., Rahman, S., Zhong, M.
Intraday trading volume and return volatility of the DJIA
stocks: A note
Journal of Banking & Finance 27: 2035-2043, 2003,
DOI: 10.2139/ssrn.348820
Engle, R. F., Granger, C.W.J., Rice, J., Weiss, A.,
Semiparametric Estimates of the Relation between Weather
and Electricity Sales
Journal of the American Statistical Association 81(394):
310-320, 1986,
DOI: 10.2307/2289218
6-2
References
6-3
Guo, M., Härdle, W. K.
Simultaneous confidence bands for expectile functions
Advances in Statistical Analysis, 96: 517-541 published online,
2012,
DOI: 10.1007/s10182-011-0182-1
Guo, M., Zhou, L., Huang, J. Z., Härdle, W. K.
Functional data analysis of generalized regression quantiles
Statistics and Computing, published online, 2013,
DOI: 10.1007/s11222-013-9425-1
References
6-4
Härdle, W. K., Hautsch, N. and Mihoci, A.
Local Adaptive Multiplicative Error Models for High-Frequency
Forecasts
Journal of Applied Econometrics, accepted for publication,
2014, DOI: 10.1002/jae.2376
Hays, S., Shen ,H., Huang, J.Z,
Functional dynamic factor models with application to yield
curve forecasting
The Anals of Applied Statistics 6(3): 870–894, 2012,
DOI: 10.1214/12-AOAS551
References
Hörmann, S., Kokoszka ,P.
Weakly dependent functional data
The Anals of Statistics 38(3): 1845-1884, 2010,
DOI: 10.1214/09-AOS768
Kokoszka ,P., Miao, H., Zhang, X.
Functional Dynamic Factor Model for Intraday Price Curves
Journal of Financial Econometrics 0(0): 1-22, 2014,
DOI: 10.1093/jjfinec/nbu004
6-5
References
6-6
Panaretos, V. M., Tavakoli, S.
Cramér-Karhunen-Loève representation and harmonic principal
component analysis of functional time series
Stochastic Processes and their Applications 123(7):
2779-2807, 2013, DOI: 10.1016/j.spa.2013.03.015
Jain, P. C., Joh, G. H.
The Dependence between Hourly Prices and Trading Volume
The Journal of Financial and Quantitative Analysis 23(3):
269-283, 1988, DOI: 10.2307/2331067
References
6-7
Schnabel, S.K., Eilers, P. H. C.
Optimal expectile smoothing
Computational Statistics and Data Analysis 53(12):
4168-4177, 2009, DOI: 10.1016/j.csda.2009.05.002
Song, S., Härdle, W. K., Ritov, Y.
Generalized Dynamic Semiparametric Factor Models for High
Dimensional Nonstationary Time Series
The Econometrics Journal, accepted for publication, 2013,
DOI: 10.1111/ectj.12024
References
6-8
Spierdijk, L., Nijman, T.E., van Soest, A.H.O.
Price Dynamics and Trading Volume: A Semiparametric
Approach
Department of Applied Mathematics, University of Twente,
2004, DOI: 10.2139/ssrn.512342
Tseng, P., Yun, S.
A coordinate gradient descent method for nonsmooth separable
minimization
Mathematical Programming 117(1-2): 387-423, 2009,
DOI: 10.1007/s10107-007-0170-0
Appendix
7-1
Model of Temperature data
−15 −10
−5
Temperature
0
5
10
15
20
Berlin temperature 1945/1978/2013
Jan
Apr
Jul
Day
Oct
Dec
Figure: Daily average temperature in Berlin in 1948, 1980, 2013
Back to Motivation - Temperature Data
Appendix
7-2
For days t = 1, . . . , 24090 (i.e. 66 years)
Xt = Tt − Λt
2 n
m · π · t m · π · t o
X
Λt = a + bt +
cm cos
+ dm sin
365
365
m=1
Xt =
L
X
βl Xt−l + εt
l=1
Appendix
7-3
Series X_adjust
â = 5.617
0.6
β̂1 = 0.786
−5
b̂ = 2.3 ∗ 10
β̂4 = 0.015
d̂1 = −7.932
β̂5 = 0.011
d̂2 = −3.109
β̂6 = 0.007
0.4
β̂3 = 0.024
0.2
ĉ2 = −7.14 ∗ 10
−2
0.0
ĉ1 = −4.15 ∗ 10
Partial ACF
β̂2 = −0.078
−2
0
β̂7 = 0.001
β̂8 = 0.019
10
20
30
Lag
40
Appendix
7-4
B-splines
Knot vector t = (t1 , . . . , tM ) as nondecreasing sequence in [0, 1]
Control points P0 , . . . , PN
Define i-th B-spline basis function Ni,j of order j as
(
1 if ti < t < ti+1
Ni,0 (t) =
0 otherwise
Ni,j (t) =
ti+j+1 − t
t − ti
Ni,j−1 (t) +
Ni+1,j−1 (t)
ti+j − ti
ti+j+1 − ti+1
j = 1, . . . , N − M − 1
Back to Estimation of Expectile Curves
Back to Space Basis
Appendix
7-5
Estimation of Quantile Curves
Definition of Penalty matrix
α
b = arg min
α
N
X
n
o
ρτ yi − α> b(xi ) + λα> Ωα
i=1
where b(x) = (b1 , . . . , bq (x))> is vector of B-spline basis functions
Denote b̃(x) = (b̃1 (x), . . . , b˜q (x))> the vector of second derivatives
of basis functions
and set
Z
Ω=
Back to Estimation of Expectile Curves
b̃(x)b̃(x)> dx
Appendix
7-6
LAWS estimation
Schnabel and Eilers (2009):
min
n
X
wi (τ )(yi − µi )2
i=1
where
wi (τ ) =
τ
1−τ
if yi > µi
if yi ≤ µi ,
µi expected value according to some model.
Iterations:
fixed weights, closed form solution of weighted regression
recalculate weights
until convergence criterion met.
Back to Expectiles Curves
Appendix
7-7
LAWS estimation
Example:
Classical linear regression model
Y = Xβ + ε
where E (ε |X ) = 0 and µ = E (Y |X ) = X β.
βb = arg min
β
n
X
wi (yi − µi )2
i=1
Then:
βb = (X > WX )−1 XWY
with W diagonal matrix of fixed weights wi .
Back to Expectile Curves
Appendix
7-8
Functional principal components
X (t) stochastic process on compound interval T
with mean function µ(t) = E {X (t)}
and covariance function K (s, t) = cov(X (s), X (t))
There exist orthogonal sequence of P
eigenfunctions φj and
eigenvalues λi such that K (s, t) = ∞
j=1 λj φj (s)φj (t)
We can rewrite process as
∞
X
p
X (t) = µ(t) +
λj κj φj (t)
j=1
where κj = √1
λj
R
X (t)φj (s)ds , E(κj ) = 0 and E(κj κK ) = δjk .
Back to Modelling of Time-varying Curves
Appendix
7-9
Guo(2013) - Empirical loss function
S ∗ = S + Mµ + MΦ
where
S=
D X
T
X
ρτ Ydt − b(t)> θµ − b(t)> ΘΦ αd
d=1 t=1
Mµ =
θµ>
MΦ =
Z
K
X
b̃(x)b̃(x)> dxθµ = θµ> Ωθµ
Z
θφ,k
b̃(x)b̃(x)> dx θφ,k
k=1
and b̃(x) vector of second derivatives
Back to Guo(2013) - Estimation of Quantile Curves
Appendix
7-10
Conditions (Panaretos and Tavakoli (2013))
Xt second order stationary time series in L2 ([0, 1], R)
with zero mean, E kX0 k22 < ∞ and autocovariance kernel at lag t :
rt (u, v ) = E (Xt (u)X0 (v ))
u, v ∈ [0, 1], t ∈ Z, inducing operator :
Rt : L2 ([0, 1], R) → L2 ([0, 1], R)
Assume:
P
i)
t∈Z kRt k1 < ∞
ii) (u, v ) → rt (u, v ) continuous t ∈ Z, and
Back to C-K-L representation
P
t∈Z krt k∞
<∞
Appendix
7-11
Theorem (Panaretos and Tavakoli (2013))
Xt admits representation
Z
π
exp(iωj t)dZω a.s.
Xt =
−π
where for fixed ω, Zω is random element of L2 ([0, 1], C)
and process ω → Zω has orthogonal increments.
Integral can be understood as a Riemann-Stieltjes limit in sense
E kXt −
J
X
exp(iωj t)(Zωj+1 − Zωj )k22 → 0 as J → 0
j=1
Appendix
7-12
Remark (Panaretos and Tavakoli (2013))
With spectral density operator Fω =
having eigenfunctions {φωn }t≥1
1
2π
P
t∈Z exp(−iωt)Rt
C-K-L representation can be interpreted as
Z
π
Xt =
exp(i)
−π
∞
X
n=1
hφωn , dZω i φωn
Appendix
7-13
Block-Coordinate
Gradient Descent Algorithm
Tseng & Yun (2009)
min f (x) + λP(x)
x
Solve
1 >
>
min d ∇f (x) + d Hd + λP(x + d )
2
d
P is block-separable then H is block-diagonal
solve subproblems
(for every j take d = (dG1 , . . . , dGJ ) with dGk = 0 for k 6= j)
Back to BCGD algorithm
Appendix
7-14
Block-Coordinate
Gradient Descent Algorithm
b (t)
b (t) )G − h(t) C
∇l (C
j
Gj
G

(t)
dGj = −

1 
j
b (t) )G − λ

∇l (C
j
(t)
(t)
(t)
b (t) k2
b )G − hG C
hGj
k∇l (C
j
Gj
j
(t)
(t)
H (t) has submatrices HGj = hGj IGj for scalars hG(t)
j
α(t) set by Amijo rule (See Details: Tseng & Yun (2009))
Back to BCGD algorithm
Appendix
Song et. al.(2013) - Time basis
Orthogonal Legendre polynomial basis
to capture the global trend in time
u1 (d ) = 1/C1 , u2 (d ) = d /C2 , u3 (d ) = (3t 2 − 1)/C3 , . . .
P
2
with generic constant Ci such that D
d=1 ui (d )/Ci = 1
Fourier series
to capture periodic variations
u4 = sin(2πd /p)/C4 , u5 = cos(2πd /p)/C5 ,
u6 = sin(2πd /(p/2))/C6 , . . .
with given period p
Back to Song(2013) -Time Basis
7-15

DYTEC - Dynamic Tail Event Curves - Hu

Transcription

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