Download Nonparametric estimation of a shot

Nonparametric
estimation of a
shot-noise process
Institut
Mines-Telecom
Paul Ilhe, [email protected]
Joint work with François Roueff, Éric
Moulines & Antoine Souloumiac
Journée TSI, 2016.
Introduction
Statistical estimation
Numerical results
Outline
Introduction
Shot-noise processes on the real line
γ-spectroscopy
Statistical estimation
Assumptions
Estimator
RKHS
B-spline method
Numerical results
2/23
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
Outline
Introduction
Shot-noise processes on the real line
γ-spectroscopy
Statistical estimation
Numerical results
3/23
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
Shot-noise process: definition
Definition
A stationary shot-noise process is a stochastic process X = (Xt )t∈R
that can be written
X
(1)
Xt :=
Yk h(t − Tk )
k : Tk ≤t
4/23
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
Shot-noise process: definition
Definition
A stationary shot-noise process is a stochastic process X = (Xt )t∈R
that can be written
X
(1)
Xt :=
Yk h(t − Tk )
k : Tk ≤t
where N :=
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P
k∈Z δ(Tk ,Yk )
is a Poisson random measure such that
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the sequence of time events (Tk )k follows a Poisson point
process on R with intensity λ,
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(Yk )k are independent of (Tk )k and mutually i.i.d. with
density f ,
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
Shot-noise process: definition
Definition
A stationary shot-noise process is a stochastic process X = (Xt )t∈R
that can be written
X
(1)
Xt :=
Yk h(t − Tk )
k : Tk ≤t
where N :=
P
k∈Z δ(Tk ,Yk )
is a Poisson random measure such that
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the sequence of time events (Tk )k follows a Poisson point
process on R with intensity λ,
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(Yk )k are independent of (Tk )k and mutually i.i.d. with
density f ,
and h is a measurable real-valued function.
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07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
The elements of a shot-noise
Properties
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A shot-noise is fully characterized by its triplet (λ, f , h).
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The process is well defined as soon as
Z Z
λ
|yh(s)| f (y )dy ds < ∞ ,
R
I
R
The characteristic function ϕ of X0 is given for u ∈ R by
Z
h
i
(2) ϕX0 (u) = E eiuX0 = exp λ [ϕY (uh(s)) − 1] ds .
R
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07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
Illustration
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Signal m esuré
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Tem ps (en µs)
Figure: Sample path of a shot-noise process.
Vocabulary
Intensity: λ, Marks’ density: f , Impulse response: h.
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Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
γ-spectroscopy
Interpretation
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(Tk )k : interaction photon-matter events,
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(Yk )k : energies recorded by the detector,
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h: electrical current generated by the detector.
Inputs
A low frequency sample of the shot-noise Xδ , . . . , Xnδ
Objectives
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Estimate the mean number of photons in one second: λ,
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Estimate the density f of the marks (Yk )k .
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
γ-spectroscopy: example 1
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Source: Americium 241
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Intensity: 5.105 cps
1600
Signal mesuré (unité arbitraire)
1400
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1000
800
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0
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400
Temps (en 10 −7 s)
600
800
1000
Example of a temporal signal in γ-spectroscopy.
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07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Shot-noise processes on the real line
γ-spectroscopy
γ-spectroscopy: example 2
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Source: Americium 241
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Intensity: 4.106 cps
6000
Signal mesuré (unité arbitraire)
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1000
0
0
200
400
Temps (en 10 −7 s)
600
800
1000
Example of a temporal signal in γ-spectroscopy.
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07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Assumptions
Estimator
RKHS
B-spline method
Outline
Introduction
Statistical estimation
Assumptions
Estimator
RKHS
B-spline method
Numerical results
10/23
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Assumptions
Estimator
RKHS
B-spline method
Shot-noise process: statistical problem
Problem and assumptions
Based on
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A low frequency sample Xδ , . . . , Xnδ ,
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A complete knowledge of the impulse response h,
we aim to estimate the flow function
fλ := λf .
11/23
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Assumptions
Estimator
RKHS
B-spline method
Assumptions
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the mark’s density f belongs to
(H-1) W 2 := {f : [0, 1] → R+ : f , f 0 absolutely continuous,
Z 1
00 2
f (t) dt < ∞} .
f (1) = f 0 (1) = 0 and
0
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There exists two positive constants C0 and α s.t.
(H-2)
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|h(t)| ≤ C0 e−αt ,
Institut Mines-Telecom
t∈R.
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Assumptions
Estimator
RKHS
B-spline method
Estimation problem (1/3)
Main tool
If E[|Y0 |] < ∞, we have for every u ∈ R
0
Z
1Z
g (u) := ϕ (u)/ϕ(u) =
0
ixh(s)eiuxh(s) λf (x)dxds =: Kh [fλ ](u) .
R
Functional inverse problem
We transformed the estimation problem into a linear inverse
problem with
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Kh a BLE acting from W 2 to another Hilbert space,
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we can construct an empirical estimator ĝn of g .
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Assumptions
Estimator
RKHS
B-spline method
Introduction
Statistical estimation
Numerical results
Estimation problem (2/3)
Plug-in estimator
Pn
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iX eiuXk
k
k=1
ĝn (u) := ϕ̂0n (u)/ϕ̂n (u) = P
, u ∈ R.
n
iuXk
k=1 e
√
error terms: n (u) = n (ĝn (u) − g (u)) , u ∈ R.
Theorem
Under Assumption (H-2), we have for all positive K that
n ⇒ Z
in C([−K , K ], C) ,
where Z is a Gaussian process with covariance matrix Ω.
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07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Assumptions
Estimator
RKHS
B-spline method
Estimation problem (3/3)
Discrete sampling
ĝn and g are evaluated on a discrete grid u = {u1 , . . . , uN }.
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ĝn (u) = [ĝn (u1 ), · · · , ĝn (uN )]T ,
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Ku [ξ] = [Kh [ξ](u1 ), · · · , Kh [ξ](uN )]T .
Minimization problem
Given a positive definite matrix W of size N and µ a positive
number, we define
(3)
fˆn,λ := argminkW 1/2 · {ĝn (u) − Ku [ξ]}k22 + µkξk2W 2 .
2
ξ∈W+
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Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Assumptions
Estimator
RKHS
B-spline method
Reproducing Kernel
The functional space W 2 is a RKHS with r.k.
Z 1
Rx (y ) := R(x, y ) :=
(u − x)+ (u − y )+ du ,
x, y ∈ [0, 1] .
0
Corollary
The solution of (3) can be rewritten fˆn,λ =
PN
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ηi : x → Kh [Rx (·)](ui ) for i = 1, . . . , N.
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The coefficients α̂n are solution of
k=1 α̂n,i ηi
with
argminkW 1/2 · {ĝn (u) − G α}k22 + µαT G α .
α∈CN
with G the Gram matrix associated to η1 , . . . , ηN .
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07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Assumptions
Estimator
RKHS
B-spline method
Introduction
Statistical estimation
Numerical results
B-splines
We approximate W 2 by B-splines spaces Sk3
k
of order 3.
O(k −2 ),
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the approximation error is in
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the functions are locally supported,
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the penalty term kξk2W 2 is easily expressible.
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B-splines spaces: k = 10, q = 3
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B-splines spaces: k = 20, q = 3
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Figure: B-splines basis of order 3. Left: k = 10. Right: k = 20.
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Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Assumptions
Estimator
RKHS
B-spline method
Introduction
Statistical estimation
Numerical results
Practical procedure
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Choice of N = k = 2blog(n)c ,
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Choice of W = Ω̂−1
n with Ω̂n a positive definite estimator of
Ω,
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Selection of the regularizing parameter µ by generalized
cross-validation
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Outline
Introduction
Statistical estimation
Numerical results
19/23
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Simulated shot-noise
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λ=3, δ = 0.01,
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h : t → t e−10t 1{t≥0} ,
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f = 0,7N 0,3; 2,5.10−3 + 0,3N 0,6; 2,5.10−3
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shot noise
sample points
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Simulated shot-noise and associated sample points.
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Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
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Introduction
Statistical estimation
Numerical results
RKHS
Mitigated results:
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Retrieves the modes of the distribution
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Positivity constraints hard to handle
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Basis function not adapted
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fλ
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f̂ n, λ = 10 6
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f̂ n, λ = 5. 10 6
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f̂ n, λ = 10 7
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Figure: Density estimation using the representer’s theorem.
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Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
B-splines
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k = N = 2blog(n)c ,
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Computation of Ω̂n by bootstrap,
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Choice of the penalization µ by generalized cross-validation.
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fn, λ , n = 10
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f̂ n, λ − fλ 22
fn, λ , n = 10 5
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fn, λ , n = 10 7
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Integrated squared error
fλ
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n = 10 5
n = 10 6
n = 5. 10 6
Sample size
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Institut Mines-Telecom
Nonparametric estimation of a shot-noise process
Introduction
Statistical estimation
Numerical results
Nuclear dataset
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source: mixture of Am 241 and Pu 238,
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intensity: 5.105 cps.
07/2016
Institut Mines-Telecom
Nonparametric estimation of a shot-noise process