the laplace driven moving average – a non-gaussian stationary
TRANSCRIPT
The Laplace driven moving average– a non-Gaussian stationary process
Sofia Aberg1, Krzysztof Podgorski2, Igor Rychlik1
1Mathematical Sciences, Mathematical Statistics, Chalmers2Centre for Mathematical Sciences, Mathematical Statistics, Lund
Smogen, August 2008
Sofia Aberg Laplace driven MA
Objectives
Background:
I Gaussian process very convenient in environmental sciencessince they allow for covariance/spectral modelling.
I Sometimes not sufficient, does not allow for skewed marginaldistributions and has often too light tails.
Goals:
I Construct non-Gaussian stationary process...
I ... possessing a spectrum,
I a skewed marginal distribution,
I and heavier tails than the Gaussian distribution.
Starting point:the Laplace distributions
Sofia Aberg Laplace driven MA
The Laplace distribution – from a historical point of view
First and second Laplace law of error.
I. The Laplace distribution. (Laplace, 1774).
II. The normal (Gauss) distribution. (Laplace, 1778).
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Sofia Aberg Laplace driven MA
Generalized Laplace distributions
I Laplace distribution:
φ(t) =
(1
1 + σ2t2
2
)
I Generalization:
φ(t) =
(1
1− iµt + σ2t2
2
)τ
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Standard Laplace: τ=1
StandardGaussian: τ=∞
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Generalized asymmetricLaplace: τ = 3
Sofia Aberg Laplace driven MA
Laplace motion - a counterpart to Brownian motion
A stochastic process Λ(t) is calledasymmetric LM if
1. it starts at the origin
2. it has independent andstationary increments
3. the increments have ageneralized asymmetric Laplacedistribution
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Brownian motion
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Laplace motion: τ = σ=1
Sofia Aberg Laplace driven MA
The Laplace driven moving average
Using the Laplace motion Λ(x) one can define a Laplace drivenmoving average by
X (t) =
∫ ∞
−∞f (t − x)dΛ(x).
The function f is called a kernel and should satisfy∫f 2(x) dx < ∞. A similar definition is possible in higher
dimensions.
Sofia Aberg Laplace driven MA
Spectral properties of Laplace moving averages
I The spectrum of X (t) is givenby
S(ω) =τ(σ2 + µ2)
2π|F f (ω)|2
I By requiring that f is asymmetric function one canestimate the kernel from thespectral density.
I By ”minimum phase”assumptions one can get causalkernels.
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angular frequency
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Sofia Aberg Laplace driven MA
Marginal distribution
I The marginal distribution is givenin terms of its characteristicfunction.
I Moments of the distribution can becomputed.
I The method of moments (with firstfour moments) can be used infitting the marginal distribution todata.
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StandardGaussian: τ = ∞
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Symmetric
Asymmetric
Sofia Aberg Laplace driven MA
Simulation
The LMA can be seen as a convolution of Laplace noise with akernel f . Discrete version:
∫f (t − x)dΛ(x) ≈
∑f (t − xi )∆Λ(xi )
Time domain simulation:
I simulate iid Laplace noise
I convolve the noise with the kernel
Frequency domain simulation:
I simulate iid Laplace noise
I Fourier transform the noise and the kernel f using FFT
I Take product of the Fourier transforms
I Take inverse Fourier transform of the product
Sofia Aberg Laplace driven MA
Example: Random fields with Matern covariance
The Matern family of covariances is commonly used to describespatial dependence in geostatistics. It has covariance
r(x) =φ
2β−1Γ(β)(α|x |)βKβ(α|x |),
and spectrum
S(ω) =Γ(β + d
2 )α2β
Γ(β)πd/2
φ
(α2 + |ω|2)β+ d2
.
d is the dimension φ is variance, α a range parameter, β asmoothness parameter and K is a modified Bessel function.
Sofia Aberg Laplace driven MA
Matern correlations
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distance
corr
elat
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Matérn correlations for different parameters
β=0.5,α=0.9986β=1.5,α=1.5813β=2.5,α=1.9729β=3.5,α=2.2922
Sofia Aberg Laplace driven MA
... and kernels
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distance
kern
el v
alueMatérn kernels for different parameters
β=0.5,α=0.9986β=1.5, α=1.5813β=2.5, α=1.9729β=3.5, α=2.2922
Sofia Aberg Laplace driven MA
Asymmetric fields
Laplace parameters: [τ, σ, µ, c] = [1, 1/√
2, 1/√
2,−1/√
2].
Sofia Aberg Laplace driven MA
Marginal densities
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β=0.5,α=0.9986β=1.5,α=1.5813β=2.5,α=1.9729β=3.5,α=2.2922
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β=0.5,α=0.9986β=1.5,α=1.5813β=2.5,α=1.9729β=3.5,α=2.2922
All distributions have mean zero and variance one
Sofia Aberg Laplace driven MA
Relation to Gaussian processes
A generalized Laplace distributed random variable Λ can berepresented using a Γ(τ, 1)-distributed random variable Γ andstandard Gaussian variable B:
ΛD= c + µΓ + σ
√ΓB.
Similarly, the Laplace motion can be represented as
Λ(t)D= c · t + µΓ(t) + σB(Γ(t)),
where Γ(t) is a Gamma-process with parameter τ and B(t) isstandard Brownian motion.
Sofia Aberg Laplace driven MA
Conditioning on the Gamma process – a smart trick
Conditional on a specific realisation γ of the gamma-process theLaplace moving average becomes a non-stationary Gaussianprocess! It will have mean
m1(t) = E [X (t) | Γ(x) = γ(x)] = c
∫f (t−x) dx+µ
∫f (t−x) dγ(x)
and variance
σ211(t) = Var [X (t) | Γ(x) = γ(x)] = σ2
∫f 2(t − x) dγ(x)
both depending on time t.
Sofia Aberg Laplace driven MA
Example: Rice’s formula
I Formula forcomputing theexpected number oflevel crossings
I Very important inreliability applications
I For Gaussianstationary processesthere is a closed formsolution 0 10 20 30 40 50 60 70 80 90 100
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Sofia Aberg Laplace driven MA
Rice’s formula - non-stationary case
For a non-stationary process
E [N+T (u)] =
∫ T
0
∫ ∞
0zfY (t),Y ′(t)(u, z) dz dt
I N+T (u) - number of upcrossings of level u during [0, T ].
I For a Gaussian process the innermost integral can beevaluated.
I The outer integral can be computed numerically
Sofia Aberg Laplace driven MA
Monte-Carlo approach to Rice’s formula
I N+T (u)- number of upcrossings of level u in time interval [0,T ]
I Condition on Γ(x) = γ(x):
µ+(u) = E [N+1 (u)] = E [E [N+
1 (u) | Γ(x) = γ(x)]]
I Approximate by forming a Monte-Carlo average:
µ+(u) ≈ 1
n
n∑
k=1
E [N+1 (u) | Γ(x) = γk(x)]
I The terms in the sum are level-crossing intensities fornon-stationary Gaussian processes.
Sofia Aberg Laplace driven MA
Upcrossing intensity
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upcr
ossi
ng in
tens
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GaussianLMA: skewness = 0, kurtosis = 3.01LMA: skewness = 0.3, kurtosis = 4LMA: skewness = 0.6, kurtosis = 5
Sofia Aberg Laplace driven MA
Summary
I The Laplace driven moving average can be used to modelsecond order stationary loads with skewed marginaldistribution.
I The model can be fitted to data using a moment matchingapproach.
I Simulation can either be done in time or in frequency domain.
I Conditional on a realisation of a gamma-process the LMAbecomes a non-stationary Gaussian process.
I Rice’s formula can be evaluated by a Monte-Carlo method.
Sofia Aberg Laplace driven MA