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Models for spatial
and spatio-temporal volatility
Michele Nguyen, joint work with Almut E. D. Veraart
Department of Mathem...
Spatial and spatio-temporal volatility
• wind velocities;
• air pollution (Fuentes, 2002; Fuentes and Smith, 2001);
• vege...
Outline
Background: Ambit fields
Spatio-temporal OU processes
Definition and example
Simulation and estimation
Properties
Vo...
The “Ambit Fields” framework
• “ambire”, “ambitus” (Latin): boundary of influence
Spatial and spatio-temporal volatility Mi...
The “Ambit Fields” framework
• “ambire”, “ambitus” (Latin): boundary of influence
Yt (x) =µ +
Dt (x)
q(x, t; ξ, s)as(ξ)dξds...
The “Ambit Fields” framework
• “ambire”, “ambitus” (Latin): boundary of influence
Yt (x) =µ +
Dt (x)
q(x, t; ξ, s)as(ξ)dξds...
The “Ambit Fields” framework
• “ambire”, “ambitus” (Latin): boundary of influence
Yt (x) =µ +
Dt (x)
q(x, t; ξ, s)as(ξ)dξds...
• Motivated by turbulence
modelling.
• Ambit set ≈ causality cone.
• Other applications include tumour
growth and electric...
Outline
Background: Ambit fields
Spatio-temporal OU processes
Definition and example
Simulation and estimation
Properties
Vo...
Motivation
Stationary Ornstein-Uhlenbeck (OU) process {Zt }t∈R:
dZt = −λZt dt + dWt , λ > 0; (1)
Zt =
t
−∞
exp (−λ (t − s)...
Our model
Introduced in Barndorff-Nielsen and Schmiegel, 2004:
Yt (x) =
At (x)
exp − λ(t − s) L(dξ, ds), (3)
Spatial and s...
Our model
Introduced in Barndorff-Nielsen and Schmiegel, 2004:
Yt (x) =
At (x)
exp − λ(t − s) L(dξ, ds), (3)
where At (x) ...
Our model
Introduced in Barndorff-Nielsen and Schmiegel, 2004:
Yt (x) =
At (x)
exp − λ(t − s) L(dξ, ds), (3)
where At (x) ...
The canonical example
Yt (x) =
t
−∞
x+c|s−t|
x−c|s−t|
exp − λ(t − s) L(dξ, ds), where c > 0,
(4)
Spatial and spatio-tempor...
The canonical example
Yt (x) =
t
−∞
x+c|s−t|
x−c|s−t|
exp − λ(t − s) L(dξ, ds), where c > 0,
(4)
=
∞
0
∞
−∞
h(u, w)L(du, d...
Rectangular grid simulation
Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 10
Rectangular grid simulation
• {(tJ, xI) : I, J = 1, ..., 4}:
simulation grid;
• W
d
= L([0, x ] × [0, t ]):
L´evy noise ov...
Rectangular grid simulation
Types of simulation errors:
1. Kernel truncation;
2. Kernel discretisation;
3. Integration set...
Diamond grid simulation
Types of simulation errors:
1. Kernel truncation;
2. Kernel discretisation;
3. Integration set
app...
Simulate...
• Canonical process:
Yt (x) =
t
−∞
x+c|s−t|
x−c|s−t| exp − λ(t − s) L(dξ, ds);
• 500 data sets from each algor...
Simulate...
• Canonical process:
Yt (x) =
t
−∞
x+c|s−t|
x−c|s−t| exp − λ(t − s) L(dξ, ds);
• 500 data sets from each algor...
And moment-match
An extension of the estimation method in J´onsd´ottir et al., 2013.
Spatial and spatio-temporal volatilit...
And moment-match
An extension of the estimation method in J´onsd´ottir et al., 2013.
We match the theoretical and empirica...
And moment-match
An extension of the estimation method in J´onsd´ottir et al., 2013.
We match the theoretical and empirica...
Results
Rectangular grid data:
q
q
q
q
q
0.60.81.01.21.41.6
(a) λ
^
q
q
q
q
q
q
q
q
q
qq
qq
q
0.960.970.980.991.001.011.02...
Interesting theoretical properties
• Spatio-temporal stationarity;
• Markovianity;
• Exponential temporal autocorrelation ...
Interesting theoretical properties
• Spatio-temporal stationarity;
• Markovianity;
• Exponential temporal autocorrelation ...
Space-time clusters
Comparison to a random simulation
Canonical model: Linear approximation to regions of influence.
Spatia...
Outline
Background: Ambit fields
Spatio-temporal OU processes
Definition and example
Simulation and estimation
Properties
Vo...
Definition
Y(x) =
Rd
g(x − ξ)σ(ξ)W(dξ),
where:
• {Y(x) : x ∈ Rd } for some d ∈ N;
• g is a deterministic (kernel) function;...
Extension of a Gaussian moving average
GMA:
Rd
g(x − ξ)W(dξ).
Type-G LMA:
Rd
g(x − ξ)L(dξ),
where:
• L(dξ) = σ(ξ)W(dξ) is ...
A 2D example
Y(x) =
R2
λ
π
exp −λ (x − ξ)T
(x − ξ) σ(ξ)W(dξ),
where σ2
(ξ) =
R2
η
π
exp −η (ξ − u)T
(ξ − u) L(du),
and L i...
A 2D example
Y(x) =
R2
λ
π
exp −λ (x − ξ)T
(x − ξ) σ(ξ)W(dξ),
where σ2
(ξ) =
R2
η
π
exp −η (ξ − u)T
(ξ − u) L(du),
and L i...
Sample paths of a GMA and our VMMA with the same Gaussian kernel and volatility
mean, together with the simulated stochast...
Useful properties of a VMMA
Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 23
Useful properties of a VMMA
σ2 stationary and independent of W
⇒ Y(x) =
Rd
g(x − ξ)σ(ξ)W(dξ) is stationary.
However, Y(x)|...
• Var(Y(t)) =
Rd
g2(t − s)E σ2(s) ds.
2D example:
Y(t) =
R2
λ
π
exp −λ (t − s)T
(t − s) σ(s)W(ds),
where σ2
(s) =
R2
η
π
e...
For x, x∗ ∈ Rd ,
Corr(Y(x), Y(x∗)) does not depend on the parameters of σ2
but Cov(Y2(x), Y2(x∗)) and Corr(Y2(x), Y2(x∗)) ...
Inference approach
Direct moments-matching with higher order moments:
Parameter sign error due to fourth cumulant and Var ...
First step: Estimating a and λ
Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart ...
First step: Estimating a and λ
Analytical formula Estimator
Variance aλ
2π
1
D−1
D
i=1
Yvi
− ¯Y
2
N. variogram 2 1 − exp −...
Second step: Estimating σ2
I (x) =
Rd
g2
(x − ξ)σ2
(ξ)dξ
Spatial and spatio-temporal volatility Michele Nguyen, joint work...
Second step: Estimating σ2
I (x) =
Rd
g2
(x − ξ)σ2
(ξ)dξ
(a) Moving window on Y
qq
q
M
M
ξ1
(b) σ^
I
2
locations
M−q+1
M−q...
Practical considerations
Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 29
Practical considerations
1. How to choose q?
• Too small ⇒ too little data in subregion for reliable local
variance estima...
For fixed q, the maximum regional variance = max
i
ˆκI
2(ξi)
where:
ˆκI
2(ξi) =
1
|YAi
| − 1
x∈Ai
Y(x) − ¯YAi
2
,
and YAi
d...
Choosing q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
10 20 30 40 50
0.51.01.52.02.53.03.5
(a) VMMA (Seed 1)
q
Maximumregion...
Practical considerations
2. How to compute the local variance?
ˆκI
2(ξi) involves the sample mean of the subregion
⇒ Under...
One data set: ˆκI
2 vs ˆσ2
I
The red diamonds in Plots (b) and (e) denote the point corresponding to the maximum
regional ...
One data set: VMMA vs GMA
Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 34
Using ˆσ2
I to estimate b and η
• E σ2
I (x) = aλ(2π)−1 ⇒ ˆa2.
• Var σ2
I (x) = A := bλ3η(4π3(2λ + η))−1 ⇒ A.
•
E (σ2
I (x...
Summary of inference procedure
• Variance and normalised variogram of Y ⇒ ˆλ and ˆa.
• Moving window strategy with tuning ...
Results from 100 data sets
q
3.54.04.55.0
(a) λ
^
63 q
q
0.60.81.01.21.41.6
(b) a^
1
33
q
q
1520253035404550
(c) q
22
38
q...
What if we fixed q?
q
0.51.01.5
(a) B
^
38
q
0.51.01.5
(b) B
^
with q = 21
57 q
1234567
(c) η^
84
qq
1234567
(d) η^ with q ...
Application to sea surface temperature anomalies
(SSTA)
160 180 200 220
−60−40−200204060
(a) SSTA and test points
−4
−2
0
...
q
q
q
q
q
0.1 0.2 0.3 0.4 0.5
0.10.20.30.40.50.60.7
(a) γ^(dx) for Y
dx
qq
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
10 20 ...
Outline
Background: Ambit fields
Spatio-temporal OU processes
Definition and example
Simulation and estimation
Properties
Vo...
Current work
Extend the spatio-temporal OU process by mixing:
Yt (x) =
∞
0 At (x)
exp − λ(t − s) L(dξ, ds, dλ),
Allow for ...
More information
• Slides and links to papers at
www.michelenguyentd.com/projects.
• 06-125 until Friday, 13th January 201...
More information
• Slides and links to papers at
www.michelenguyentd.com/projects.
• 06-125 until Friday, 13th January 201...
References I
Barndorff-Nielsen, O. E., F. E. Benth, and A. E. D. Veraart
(2015). “Recent advances in ambit stochastics wit...
References II
Huang, W. et al. (2011). “A class of stochastic volatility models
for environmental applications”. Journal o...
References III
Wallin, J. and D. Bolin (2015). “Geostatistical Modelling Using
Non-Gaussian Mat´ern Fields”. Scandinavian ...
STOU: Comparing simulation grids
W81 W82 W83 W84 W85 W86
W71 W72 W73 W74 W75 W76
W61 W62 W63 W64 W65 W66
W51 W52 W53 W54 W...
STOU: Outgoing longwave radiation anomaly data
• {Yt (x)}x∈X,t∈T : OLR anomalies averaged over 5 days and
the latitudes 5◦...
VMMA: Simulation settings and timings
• Grid size = 0.05;
• Data region: [−1.5, 11.5] × [−1.5, 11.5] for σ2,
[0, 10] × [0,...
VMMA: Sea surface temperature anomaly data
• {Y(x) : x ∈ X}: SSTA anomalies in ◦C for the week 29th
May to 4th June 2016 (...
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Models for spatial and spatio-temporal volatility

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Stochastic volatility is commonly used in financial modelling to describe alternating periods of high and low fluctuations in logarithmic returns. The concept of volatility clusters can also be extended to space and space-time, and are associated with for example, variable yields across fields and volatile wind speeds. In this talk, we look at two classes of stochastic processes: a spatio-temporal Ornstein-Uhlenbeck (OU) process and a volatility modulated moving average (VMMA). Both can be seen as subclasses of a more general framework: ambit fields. While the spatio-temporal OU process is potentially a model for spatio-temporal volatility, the VMMA is a spatial model with volatility. The latter is constructed by introducing a stochastic volatility process into the Gaussian moving average (or process convolution) which is commonly used in Geostatistics. For each of these processes, we touch on theory, simulation as well as estimation methods.

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Models for spatial and spatio-temporal volatility

  1. 1. Models for spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart Department of Mathematics www.michelenguyentd.com/projects Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 1
  2. 2. Spatial and spatio-temporal volatility • wind velocities; • air pollution (Fuentes, 2002; Fuentes and Smith, 2001); • vegetation and desertification (Huang et al., 2011; Seekell and Dakos, 2015; Yan, 2007)... Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 2
  3. 3. Outline Background: Ambit fields Spatio-temporal OU processes Definition and example Simulation and estimation Properties Volatility modulated moving averages Definition and example Properties and estimation Application Summary and current work Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 3
  4. 4. The “Ambit Fields” framework • “ambire”, “ambitus” (Latin): boundary of influence Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 4
  5. 5. The “Ambit Fields” framework • “ambire”, “ambitus” (Latin): boundary of influence Yt (x) =µ + Dt (x) q(x, t; ξ, s)as(ξ)dξds where {Yt (x) : (x, t) ∈ S = Rd × R}, µ ∈ R and: • L: (homogeneous) L´evy basis; • Dt (x), At (x): ambit sets; • q, h: deterministic kernel functions; • σ ≥ 0, a: random fields. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 4
  6. 6. The “Ambit Fields” framework • “ambire”, “ambitus” (Latin): boundary of influence Yt (x) =µ + Dt (x) q(x, t; ξ, s)as(ξ)dξds + At (x) h(x, t; ξ, s)σs(ξ)L(dξ, ds), where {Yt (x) : (x, t) ∈ S = Rd × R}, µ ∈ R and: • L: (homogeneous) L´evy basis; • Dt (x), At (x): ambit sets; • q, h: deterministic kernel functions; • σ ≥ 0, a: random fields. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 4
  7. 7. The “Ambit Fields” framework • “ambire”, “ambitus” (Latin): boundary of influence Yt (x) =µ + Dt (x) q(x, t; ξ, s)as(ξ)dξds + At (x) h(x, t; ξ, s)σs(ξ)L(dξ, ds), where {Yt (x) : (x, t) ∈ S = Rd × R}, µ ∈ R and: • L: (homogeneous) L´evy basis; • Dt (x), At (x): ambit sets; • q, h: deterministic kernel functions; • σ ≥ 0, a: random fields. • L0 or L2 stochastic integration theory for existence. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 4
  8. 8. • Motivated by turbulence modelling. • Ambit set ≈ causality cone. • Other applications include tumour growth and electricity prices. • See Barndorff-Nielsen, Benth, and Veraart, 2015 for an overview. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 5
  9. 9. Outline Background: Ambit fields Spatio-temporal OU processes Definition and example Simulation and estimation Properties Volatility modulated moving averages Definition and example Properties and estimation Application Summary and current work Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 6
  10. 10. Motivation Stationary Ornstein-Uhlenbeck (OU) process {Zt }t∈R: dZt = −λZt dt + dWt , λ > 0; (1) Zt = t −∞ exp (−λ (t − s)) W(ds). (2) Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 7
  11. 11. Our model Introduced in Barndorff-Nielsen and Schmiegel, 2004: Yt (x) = At (x) exp − λ(t − s) L(dξ, ds), (3) Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 8
  12. 12. Our model Introduced in Barndorff-Nielsen and Schmiegel, 2004: Yt (x) = At (x) exp − λ(t − s) L(dξ, ds), (3) where At (x) ⊂ S satisfies certain conditions: 1. Translation invariant: At (x) = A0(0) + (x, t) ⊂ S; 2. As(x) ⊂ At (x), ∀ s < t; 3. Non-anticipative: At (x) ∩ X × (t, ∞) = ∅. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 8
  13. 13. Our model Introduced in Barndorff-Nielsen and Schmiegel, 2004: Yt (x) = At (x) exp − λ(t − s) L(dξ, ds), (3) where At (x) ⊂ S satisfies certain conditions: 1. Translation invariant: At (x) = A0(0) + (x, t) ⊂ S; 2. As(x) ⊂ At (x), ∀ s < t; 3. Non-anticipative: At (x) ∩ X × (t, ∞) = ∅. This means that the set has a temporal component equal to (−∞, t], that of a classical OU process. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 8
  14. 14. The canonical example Yt (x) = t −∞ x+c|s−t| x−c|s−t| exp − λ(t − s) L(dξ, ds), where c > 0, (4) Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 9
  15. 15. The canonical example Yt (x) = t −∞ x+c|s−t| x−c|s−t| exp − λ(t − s) L(dξ, ds), where c > 0, (4) = ∞ 0 ∞ −∞ h(u, w)L(du, dw), where w = t − s and u = x − ξ, and h(u, w) = 1|u|≤cw exp (−λw) , ≈ p j=0 q i=−q h(ui, wj)W (x,t) ij , where wj = j w and ui = i u for some w , u > 0. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 9
  16. 16. Rectangular grid simulation Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 10
  17. 17. Rectangular grid simulation • {(tJ, xI) : I, J = 1, ..., 4}: simulation grid; • W d = L([0, x ] × [0, t ]): L´evy noise over grid squares where t = w and x = u; • Shaded area: approximated integration set (true set outlined in bold red). W81 W82 W83 W84 W85 W86 W71 W72 W73 W74 W75 W76 W61 W62 W63 W64 W65 W66 W51 W52 W53 W54 W55 W56 W41 W42 W43 W44 W45 W46 W31 W32 W33 W34 W35 W36 W21 W22 W23 W24 W25 W26 W11 W12 W13 W14 W15 W16 W81 W82 W83 W84 W85 W86 q q q q q q q q q q q q q q q q xn x3 x2 x1 tm t3 t2 t1 p q q Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 10
  18. 18. Rectangular grid simulation Types of simulation errors: 1. Kernel truncation; 2. Kernel discretisation; 3. Integration set approximation. Mean-squared errors derived in Nguyen and Veraart, 2016b. W81 W82 W83 W84 W85 W86 W71 W72 W73 W74 W75 W76 W61 W62 W63 W64 W65 W66 W51 W52 W53 W54 W55 W56 W41 W42 W43 W44 W45 W46 W31 W32 W33 W34 W35 W36 W21 W22 W23 W24 W25 W26 W11 W12 W13 W14 W15 W16 W81 W82 W83 W84 W85 W86 q q q q q q q q q q q q q q q q xn x3 x2 x1 tm t3 t2 t1 p q q Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 11
  19. 19. Diamond grid simulation Types of simulation errors: 1. Kernel truncation; 2. Kernel discretisation; 3. Integration set approximation. Mean-squared errors derived in Nguyen and Veraart, 2016b. W71 W72 W73 W74 W75 W61 W62 W63 W64 W65 W51 W52 W53 W54 W55 W41 W42 W43 W44 W45 W31 W32 W33 W34 W35 xn x4 x3 x2 x1 tm t4 t3 t2 t1 Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 12
  20. 20. Simulate... • Canonical process: Yt (x) = t −∞ x+c|s−t| x−c|s−t| exp − λ(t − s) L(dξ, ds); • 500 data sets from each algorithm; Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 13
  21. 21. Simulate... • Canonical process: Yt (x) = t −∞ x+c|s−t| x−c|s−t| exp − λ(t − s) L(dξ, ds); • 500 data sets from each algorithm; • Grid spacing: t = x = w = u = 0.05; • Kernel truncation parameters: p = q = 300; • [0, 10] × [0, 10] space-time region; • λ = c = 1; • L: Gaussian basis with seed mean and standard deviation µ = 0.2 and τ = 0.1, e.g. for E ⊂ S, L(E) ∼ N(0.2Leb(E), 0.12Leb(E)). Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 13
  22. 22. And moment-match An extension of the estimation method in J´onsd´ottir et al., 2013. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 14
  23. 23. And moment-match An extension of the estimation method in J´onsd´ottir et al., 2013. We match the theoretical and empirical values of: • the normalised spatial and temporal variograms: E[(Yt (x)−Yt (x−dx ))2 ] Var[Yt (x)] = 2 1 − exp −λ|dx | c , E (Yt (x)−Yt−dt (x)) 2 Var[Yt (x)] = 2 1 − exp(−λ|dt |) ; • and the cumulants, κl (Yt (x)), such that log (E [exp (iθYt (x))]) = ∞ l=1 κl(Yt (x))(iθ)l l! : κ1 Yt (x) = 2cµ λ2 , κ2 Yt (x) = cτ2 2λ2 . Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 14
  24. 24. And moment-match An extension of the estimation method in J´onsd´ottir et al., 2013. We match the theoretical and empirical values of: • the normalised spatial and temporal variograms: E[(Yt (x)−Yt (x−dx ))2 ] Var[Yt (x)] = 2 1 − exp −λ|dx | c , E (Yt (x)−Yt−dt (x)) 2 Var[Yt (x)] = 2 1 − exp(−λ|dt |) ; • and the cumulants, κl (Yt (x)), such that log (E [exp (iθYt (x))]) = ∞ l=1 κl(Yt (x))(iθ)l l! : κ1 Yt (x) = 2cµ λ2 , κ2 Yt (x) = cτ2 2λ2 . Estimators consistent under increasing domain asymptotics. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 14
  25. 25. Results Rectangular grid data: q q q q q 0.60.81.01.21.41.6 (a) λ ^ q q q q q q q q q qq qq q 0.960.970.980.991.001.011.02 (b) c^ q q q q q q q q q q 0.10.20.30.40.50.6 (c) µ^ q q q q qq 0.080.090.100.110.120.130.14 (d) τ^ Diamond grid data: q qq q q q 0.60.81.01.21.41.61.8 (a) λ ^ q q q 0.970.991.011.03 (b) c^ q q q q q q q q q 0.10.20.30.40.50.6 (c) µ^ q qq q q q 0.080.090.100.110.120.130.14 (d) τ^ Similar results observed for c = 0.5, 2 and L Inverse Gaussian, Normal Inverse Gaussian and Gamma. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 15
  26. 26. Interesting theoretical properties • Spatio-temporal stationarity; • Markovianity; • Exponential temporal autocorrelation exp(−λdt ); • Equality in law to classical OU process; Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 16
  27. 27. Interesting theoretical properties • Spatio-temporal stationarity; • Markovianity; • Exponential temporal autocorrelation exp(−λdt ); • Equality in law to classical OU process; • Probability distribution through cumulant generating functions, C{θ ‡ Yt (x)} := log (E [exp (iθYt (x))]); • Flexible spatial autocorrelations; • Non-separable spatio-temporal autocorrelations: e.g. for canonical process, min exp (−λdt ) , exp −λ|dx | c . Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 16
  28. 28. Space-time clusters Comparison to a random simulation Canonical model: Linear approximation to regions of influence. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 17
  29. 29. Outline Background: Ambit fields Spatio-temporal OU processes Definition and example Simulation and estimation Properties Volatility modulated moving averages Definition and example Properties and estimation Application Summary and current work Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 18
  30. 30. Definition Y(x) = Rd g(x − ξ)σ(ξ)W(dξ), where: • {Y(x) : x ∈ Rd } for some d ∈ N; • g is a deterministic (kernel) function; • {W(E) : E ∈ Bb(Rd )}: white noise, e.g. W(E) ∼ N(0, Leb(E)); • and {σ2(ξ) : ξ ∈ Rd } is a stationary stochastic volatility field, independent of W. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 19
  31. 31. Extension of a Gaussian moving average GMA: Rd g(x − ξ)W(dξ). Type-G LMA: Rd g(x − ξ)L(dξ), where: • L(dξ) = σ(ξ)W(dξ) is a Type-G L´evy basis; • and {σ2(ξ) : ξ ∈ Rd }: a set of positive, independent and infinitely divisible random variables which are independent of W. (See for example, Bolin, 2014 and Wallin and Bolin, 2015.) Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 20
  32. 32. A 2D example Y(x) = R2 λ π exp −λ (x − ξ)T (x − ξ) σ(ξ)W(dξ), where σ2 (ξ) = R2 η π exp −η (ξ − u)T (ξ − u) L(du), and L is a homogeneous Inverse Gaussian (IG) basis such that the random variable L = L(E) where Leb(E) = 1 has an IG distribution with mean a and variance b. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 21
  33. 33. A 2D example Y(x) = R2 λ π exp −λ (x − ξ)T (x − ξ) σ(ξ)W(dξ), where σ2 (ξ) = R2 η π exp −η (ξ − u)T (ξ − u) L(du), and L is a homogeneous Inverse Gaussian (IG) basis such that the random variable L = L(E) where Leb(E) = 1 has an IG distribution with mean a and variance b. Simulation via two-step discrete convolution; mean-squared errors derived in Nguyen and Veraart, 2016a. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 21
  34. 34. Sample paths of a GMA and our VMMA with the same Gaussian kernel and volatility mean, together with the simulated stochastic volatility. The GMA and VMMA share the same realisation of the white noise. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 22
  35. 35. Useful properties of a VMMA Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 23
  36. 36. Useful properties of a VMMA σ2 stationary and independent of W ⇒ Y(x) = Rd g(x − ξ)σ(ξ)W(dξ) is stationary. However, Y(x)|σ is not: Y(x)|σ ∼ N(0, σ2 I (x)) where σ2 I (x) = Rd g2(x − ξ)σ2(ξ)dξ. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 23
  37. 37. • Var(Y(t)) = Rd g2(t − s)E σ2(s) ds. 2D example: Y(t) = R2 λ π exp −λ (t − s)T (t − s) σ(s)W(ds), where σ2 (s) = R2 η π exp −η (s − u)T (s − u) L(du). Since E σ2(s) = a = E [L ], Var(Y(t)) = aλ 2π . Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 24
  38. 38. For x, x∗ ∈ Rd , Corr(Y(x), Y(x∗)) does not depend on the parameters of σ2 but Cov(Y2(x), Y2(x∗)) and Corr(Y2(x), Y2(x∗)) do. 2D example: Corr(Y(x), Y(x∗ )) = e− λ(x−x∗ )T (x−x∗ ) 2 and Cov(Y2 (x), Y2 (x∗ )) = Ae−λ(x−x∗)T (x−x∗) + Be −λη 2λ+η (x−x∗)T (x−x∗) , where A := (bλη+a2(2λ+η)π)λ2 2π3(2λ+η) and B := bλ3η 4π3(2λ+η) . Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 25
  39. 39. Inference approach Direct moments-matching with higher order moments: Parameter sign error due to fourth cumulant and Var Y2(t) estimates. ⇒ Two-step moments-matching method. For a note on the other approaches we tried, see Nguyen and Veraart, 2016a. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 26
  40. 40. First step: Estimating a and λ Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 27
  41. 41. First step: Estimating a and λ Analytical formula Estimator Variance aλ 2π 1 D−1 D i=1 Yvi − ¯Y 2 N. variogram 2 1 − exp −λ|x−x∗ |2 2 1 #N(|x−x∗|) (i,j)∈N(|x−x∗|) (Yvi −Yvj )2 Var(Yv) Here, we have that: • {Yv : v ∈ V} is our data set and V is the set of observation sites; • D denotes our sample size; • and N(|x − x∗ |) is the set containing all the pairs of indices of sites with spatial distance |x − x∗ | > 0. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 27
  42. 42. Second step: Estimating σ2 I (x) = Rd g2 (x − ξ)σ2 (ξ)dξ Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 28
  43. 43. Second step: Estimating σ2 I (x) = Rd g2 (x − ξ)σ2 (ξ)dξ (a) Moving window on Y qq q M M ξ1 (b) σ^ I 2 locations M−q+1 M−q+1 The q × q moving window for the M × M data region and the (M − q + 1) × (M − q + 1) field of local variance estimates (ˆσ2 I ) created. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 28
  44. 44. Practical considerations Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 29
  45. 45. Practical considerations 1. How to choose q? • Too small ⇒ too little data in subregion for reliable local variance estimates. • Too large ⇒ underestimating non-stationarity and too little local variance estimates for inference. • A physical understanding: b and η govern the variance and correlation of σ2 ⇒ Choose q to reflect size of the volatility clusters. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 29
  46. 46. For fixed q, the maximum regional variance = max i ˆκI 2(ξi) where: ˆκI 2(ξi) = 1 |YAi | − 1 x∈Ai Y(x) − ¯YAi 2 , and YAi denotes the data values within the subregion Ai (demarcated by the moving window) whose centre is ξi. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 30
  47. 47. Choosing q qq q q q q q q q q q q q q qqq q q q q q 10 20 30 40 50 0.51.01.52.02.53.03.5 (a) VMMA (Seed 1) q Maximumregionalvariance q q q qqqqqqqqqq q qq q q q q q q 10 20 30 40 50 0.51.01.52.02.53.03.5 (b) GMA (Seed 1) q Maximumregionalvariance Maximum regional variance as a function of q for one VMMA data set and its corresponding GMA data set. The red vertical lines mark the peaks. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 31
  48. 48. Practical considerations 2. How to compute the local variance? ˆκI 2(ξi) involves the sample mean of the subregion ⇒ Underestimate local variance at clusters. A better estimator would be: ˆσ2 I (ξi) = 1 |YAi | x∈YAi Y2 (x). Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 32
  49. 49. One data set: ˆκI 2 vs ˆσ2 I The red diamonds in Plots (b) and (e) denote the point corresponding to the maximum regional variance for q = 21. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 33
  50. 50. One data set: VMMA vs GMA Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 34
  51. 51. Using ˆσ2 I to estimate b and η • E σ2 I (x) = aλ(2π)−1 ⇒ ˆa2. • Var σ2 I (x) = A := bλ3η(4π3(2λ + η))−1 ⇒ A. • E (σ2 I (x)−σ2 I (x∗)) 2 Var[σ2 I (x)] = 2 1 − exp(−B(ξi − ξj)T (ξi − ξj)) ⇒ B, where B := λη(2λ + η)−1. A and B ⇒ ˆb and ˆη. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 35
  52. 52. Summary of inference procedure • Variance and normalised variogram of Y ⇒ ˆλ and ˆa. • Moving window strategy with tuning parameter q ⇒ ˆσ2 I (x). • Expectation of ˆσ2 I (x) ⇒ ˆa2. • Variance of ˆσ2 I (x) ⇒ ˆA. • Normalised variogram of ˆσ2 I (x) ⇒ ˆB. • ˆA and ˆB ⇒ ˆb and ˆη. Estimators consistent under increasing domain and infill asymptotics. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 36
  53. 53. Results from 100 data sets q 3.54.04.55.0 (a) λ ^ 63 q q 0.60.81.01.21.41.6 (b) a^ 1 33 q q 1520253035404550 (c) q 22 38 q q 0.60.81.01.21.41.61.8 (d) a^ 2 1 33 q q 0.00.51.01.52.02.53.0 (e) A ^ 33 73 q 0.51.01.5 (f) B ^ 38 q q q q q q q 051015202530 (g) b ^ 11 22 33 65 73 96 98 q 1234567 (h) η^ 84 Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 37
  54. 54. What if we fixed q? q 0.51.01.5 (a) B ^ 38 q 0.51.01.5 (b) B ^ with q = 21 57 q 1234567 (c) η^ 84 qq 1234567 (d) η^ with q = 21 1 52 Better precision of B and ˆη ⇒ Is there a better way to determine q? Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 38
  55. 55. Application to sea surface temperature anomalies (SSTA) 160 180 200 220 −60−40−200204060 (a) SSTA and test points −4 −2 0 2 4 q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 160 180 200 220 −60−40−200204060 (b) Median polish surface −4 −2 0 2 4 160 180 200 220 −60−40−200204060 (c) Residuals and test points −4 −2 0 2 4 q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Fit a VMMA to the mean polish residuals of SSTA data (reserve test points). Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 39
  56. 56. q q q q q 0.1 0.2 0.3 0.4 0.5 0.10.20.30.40.50.60.7 (a) γ^(dx) for Y dx qq q q q q q q q q q q q q qq q q q q q q q 10 20 30 40 50 0.81.01.21.4 (b) Maximum Regional Variance q 160 180 200 220 −60−40−2002040 (c) σ^ I 2 with q = 21 0.5 1.0 1.5 q q q q q 0.1 0.2 0.3 0.4 0.5 0.020.060.100.14 (d) γ^(dx) for σ^ I 2 Fitted variograms, identified volatility cluster and MRV plot. Despite narrower intervals where volatility is low, 93 test points lie within 95% confidence intervals constructed by VMMA compared to the 89 for the fitted GMA. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 40
  57. 57. Outline Background: Ambit fields Spatio-temporal OU processes Definition and example Simulation and estimation Properties Volatility modulated moving averages Definition and example Properties and estimation Application Summary and current work Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 41
  58. 58. Current work Extend the spatio-temporal OU process by mixing: Yt (x) = ∞ 0 At (x) exp − λ(t − s) L(dξ, ds, dλ), Allow for long-range dependence structures. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 42
  59. 59. More information • Slides and links to papers at www.michelenguyentd.com/projects. • 06-125 until Friday, 13th January 2017. • Feedback/suggestions; current/future projects. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 43
  60. 60. More information • Slides and links to papers at www.michelenguyentd.com/projects. • 06-125 until Friday, 13th January 2017. • Feedback/suggestions; current/future projects. Thank you for listening! Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 43
  61. 61. References I Barndorff-Nielsen, O. E., F. E. Benth, and A. E. D. Veraart (2015). “Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency”. Banach Center Publications 104, pp. 25–60. Barndorff-Nielsen, O. E. and J. Schmiegel (2004). “L´evy-based spatial-temporal modelling, with applications to turbulence”. Russian Mathematical Surveys 59.1, pp. 65–90. Bolin, D. (2014). “Spatial Mat´ern Fields Driven by Non-Gaussian Noise”. Scandinavian Journal of Statistics 41.3, pp. 557–579. Fuentes, M. (2002). “Spectral methods for nonstationary spatial processes”. Biometrika 89.1, pp. 197–210. Fuentes, M. and R. L. Smith (2001). A new class of nonstationary spatial models. Tech. rep. Technical report, North Carolina State University, Raleigh, NC. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 44
  62. 62. References II Huang, W. et al. (2011). “A class of stochastic volatility models for environmental applications”. Journal of Time Series Analysis 32.4, pp. 364–377. J´onsd´ottir, K. ´Y. et al. (2013). “L´evy-based Modelling in Brain Imaging”. Scandinavian Journal of Statistics 40.3, pp. 511–529. ISSN: 1467-9469. DOI: 10.1002/sjos.12000. Nguyen, M. and A.E.D. Veraart (2016a). “Modelling spatial heteroskedasticity by volatility modulated moving averages”. Under review. Preprint available at arXiv:1609.04682. — (2016b). “Spatio-temporal Ornstein-Uhlenbeck Processes: Theory, Simulation and Statistical Inference”. Scandinavian Journal of Statistics, pp. –. URL: http://dx.doi.org/10.1111/sjos.12241. Seekell, D. A. and V. Dakos (2015). “Heteroskedasticity as a leading indicator of desertification in spatially explicit data”. Ecology and Evolution. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 45
  63. 63. References III Wallin, J. and D. Bolin (2015). “Geostatistical Modelling Using Non-Gaussian Mat´ern Fields”. Scandinavian Journal of Statistics. Yan, Jun (2007). “Spatial stochastic volatility for lattice data”. Journal of agricultural, biological, and environmental statistics 12.1, pp. 25–40. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 46
  64. 64. STOU: Comparing simulation grids W81 W82 W83 W84 W85 W86 W71 W72 W73 W74 W75 W76 W61 W62 W63 W64 W65 W66 W51 W52 W53 W54 W55 W56 W41 W42 W43 W44 W45 W46 W31 W32 W33 W34 W35 W36 W21 W22 W23 W24 W25 W26 W11 W12 W13 W14 W15 W16 W81 W82 W83 W84 W85 W86 q q q q q q q q q q q q q q q q xn x3 x2 x1 tm t3 t2 t1 p q q W91 W92 W93 W94 W95 W96 W97 W81 W82 W83 W84 W85 W86 W87 W71 W72 W73 W74 W75 W76 W77 W61 W62 W63 W64 W65 W66 W67 W51 W52 W53 W54 W55 W56 W57 W41 W42 W43 W44 W45 W46 W47 W31 W32 W33 W34 W35 W36 W37 W21 W22 W23 W24 W25 W26 W27 W11 W12 W13 W14 W15 W16 W17 xn x4 x3 x2 x1 tm t4 t3 t2 t1 p q q Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 47
  65. 65. STOU: Outgoing longwave radiation anomaly data • {Yt (x)}x∈X,t∈T : OLR anomalies averaged over 5 days and the latitudes 5◦S - 5◦N in W/m2 (base period 1979-1995); • X = {1.25, 3.75, . . . , 356.25, 358.75}: longitude in ◦E; • T = {1, 2, . . . , 52, 53}: set of five days starting from 1-5 January 2014 to 18-22 September 2014; • Negative anomalies imply increased cloudiness and an enhanced likelihood of precipitation; positive anomalies imply decreased cloudiness and a lower chance of precipitation. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 48
  66. 66. VMMA: Simulation settings and timings • Grid size = 0.05; • Data region: [−1.5, 11.5] × [−1.5, 11.5] for σ2, [0, 10] × [0, 10] for Y; • Model parameters: λ = η = 4, a = 1 and b = 2 (L is Inverse Gaussian); • Time taken to generate one data set ≈ 2 seconds; • Time taken to select q ≈ 22 seconds. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 49
  67. 67. VMMA: Sea surface temperature anomaly data • {Y(x) : x ∈ X}: SSTA anomalies in ◦C for the week 29th May to 4th June 2016 (with respect to the 1971-2000 climatology); • 1◦ latitude/longitude grid in the Pacific Ocean between 150.5◦E and 234.5◦E, and −69.5◦N and 59.5◦N. Spatial and spatio-temporal volatility Michele Nguyen, joint work with Almut E. D. Veraart 50

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