A kernel estimator and associated confidence bands in the - - PowerPoint PPT Presentation

A kernel estimator and associated confidence bands in the Spektor-Lord-Willis problem

Bogdan Ćmiel, Zbigniew Szkutnik and Jakub Wojdyła

Faculty of Applied Mathematics AGH University of Science & Technology, Kraków

Be ¸dlewo, 2016

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 1 / 16

Wicksell’s corpuscule problem and SLW problem

balls of random radii randomly placed in an opaque 3-d medium

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Wicksell’s corpuscule problem and SLW problem

balls of random radii randomly placed in an opaque 3-d medium random 2-d planar or 1-d linear section through the medium unfold the distribution of spheres’ radii (stereological inverse problem)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Wicksell’s corpuscule problem and SLW problem

balls of random radii randomly placed in an opaque 3-d medium random 2-d planar or 1-d linear section through the medium unfold the distribution of spheres’ radii (stereological inverse problem) ⇒ from measured planar sections radii: Wicksell’s problem (1925)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Wicksell’s corpuscule problem and SLW problem

balls of random radii randomly placed in an opaque 3-d medium random 2-d planar or 1-d linear section through the medium unfold the distribution of spheres’ radii (stereological inverse problem) ⇒ from measured planar sections radii: Wicksell’s problem (1925) ⇒ from measured linear sections radii: Spektor-Lord-Willis problem (1950/51)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 2 / 16

Applications

medicine (microtumors in a tissue — original Wicksell’s motivation)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

Applications

medicine (microtumors in a tissue — original Wicksell’s motivation) geology (mineral deposits in a rock)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

Applications

medicine (microtumors in a tissue — original Wicksell’s motivation) geology (mineral deposits in a rock) metallurgy (graphite grains in spheroidal graphite cast iron)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

Applications

medicine (microtumors in a tissue — original Wicksell’s motivation) geology (mineral deposits in a rock) metallurgy (graphite grains in spheroidal graphite cast iron)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 3 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c) balls’ radii density ρ on [0; 1] and f := cρ

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c) balls’ radii density ρ on [0; 1] and f := cρ n - known size of the experiment (related to line probe length)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c) balls’ radii density ρ on [0; 1] and f := cρ n - known size of the experiment (related to line probe length) Xi - observed ”radii” (half-lengths) of intersections

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c) balls’ radii density ρ on [0; 1] and f := cρ n - known size of the experiment (related to line probe length) Xi - observed ”radii” (half-lengths) of intersections then X 2

1 , . . . , X 2 N(n) form a Poisson process on [0; 1] with intensity ng

g(u) =

1

√u

f (x)dx =: (Gf )(u)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c) balls’ radii density ρ on [0; 1] and f := cρ n - known size of the experiment (related to line probe length) Xi - observed ”radii” (half-lengths) of intersections then X 2

1 , . . . , X 2 N(n) form a Poisson process on [0; 1] with intensity ng

g(u) =

1

√u

f (x)dx =: (Gf )(u) N(n) - Poissonian rv, EN(n) = n

1

0 g(u)du

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

SLW problem — Poisson process framework

homogeneous Poisson process of ball centers (”low” intensity c) balls’ radii density ρ on [0; 1] and f := cρ n - known size of the experiment (related to line probe length) Xi - observed ”radii” (half-lengths) of intersections then X 2

1 , . . . , X 2 N(n) form a Poisson process on [0; 1] with intensity ng

g(u) =

1

√u

f (x)dx =: (Gf )(u) N(n) - Poissonian rv, EN(n) = n

1

0 g(u)du

f (x) = (G −1g)(x) = −2xg′(x2) - ill-posed inverse problem

Goal: Confidence bands for f

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 4 / 16

Some existing constructions of asymptotic confidence bands

In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

Some existing constructions of asymptotic confidence bands

In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator In inverse problems: Bissantz et al. (JRSS, 2007), Bissantz & Holzmann (IP, 2008) — smooth density deconvolution

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

Some existing constructions of asymptotic confidence bands

In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator In inverse problems: Bissantz et al. (JRSS, 2007), Bissantz & Holzmann (IP, 2008) — smooth density deconvolution Birke et al. (IP, 2010) — univariate inverse regression with convolution operator Lounici & Nickl (AS, 2011) — deconvolution Delaigle et al. (JRSS, 2015) — nonparametric regression with errors in explanatory variables Proksch et al. (Bernoulli, 2015) — multivariate inverse regression with convolution operator

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

Some existing constructions of asymptotic confidence bands

In direct problems: many results starting from seminal Bickel & Rosenblatt (AS, 1973) — for standard kernel density estimator In inverse problems: Bissantz et al. (JRSS, 2007), Bissantz & Holzmann (IP, 2008) — smooth density deconvolution Birke et al. (IP, 2010) — univariate inverse regression with convolution operator Lounici & Nickl (AS, 2011) — deconvolution Delaigle et al. (JRSS, 2015) — nonparametric regression with errors in explanatory variables Proksch et al. (Bernoulli, 2015) — multivariate inverse regression with convolution operator In stereological inverse problems: Wojdyła, Szkutnik (SS, 2017) — Wicksell’s problem

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 5 / 16

SLW problem — the ”central” estimator

idea: apply G −1 to a kernel estimator of the intensity g

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16

SLW problem — the ”central” estimator

idea: apply G −1 to a kernel estimator of the intensity g problem: g(0) =

f > 0 =

⇒ g discontinuous at zero, boundary effects

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16

SLW problem — the ”central” estimator

idea: apply G −1 to a kernel estimator of the intensity g problem: g(0) =

f > 0 =

⇒ g discontinuous at zero, boundary effects remedy: reflection device ˆ fn(x) = − 2x nh2

N(n)

• i=1
• K ′
• x2 − X 2

i

h

• + K ′
• x2 + X 2

i

h

• 1[0;1](x)
• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16

SLW problem — the ”central” estimator

idea: apply G −1 to a kernel estimator of the intensity g problem: g(0) =

f > 0 =

⇒ g discontinuous at zero, boundary effects remedy: reflection device ˆ fn(x) = − 2x nh2

N(n)

• i=1
• K ′
• x2 − X 2

i

h

• + K ′
• x2 + X 2

i

h

• 1[0;1](x)

Theorem (B.Ć. , Z.Sz. & J.W, 2016)

ˆ fn is L2-rate minimax with the rate n−2m/(2m+3) over the class of f ∈ W (m, L) such that |f (m)(x)| axm+ξ for x ∈ [0; ǫ), if h ≍ n−1/(2m+3) and K is ”regular”.

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 6 / 16

SLW problem — Assumptions for confidence bands

f (hence, also g) is compactly supported in [0, 1]

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 7 / 16

SLW problem — Assumptions for confidence bands

f (hence, also g) is compactly supported in [0, 1] confidence bands to be constructed on [a, b] ⊂ [0, 1], with a > 0, b < 1

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 7 / 16

SLW problem — Assumptions for confidence bands

f (hence, also g) is compactly supported in [0, 1] confidence bands to be constructed on [a, b] ⊂ [0, 1], with a > 0, b < 1

• 1. Assumptions on the problem:

For some m 1 and ∆ > 0, f is (m − 1)-times continuously differentiable in (a − ∆; b + ∆) and there exists bounded f (m) in (a − ∆; b + ∆).

1

b f (x) dx > 0

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 7 / 16

Assumptions (continued)

• 2. Assumptions on the kernel and solution:

K is a kernel of order at least m, supported and twice continuously differentiable on [−1; 1]

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 8 / 16

Assumptions (continued)

• 2. Assumptions on the kernel and solution:

K is a kernel of order at least m, supported and twice continuously differentiable on [−1; 1] the bandwidth h satisfies h ≍ N(n)−γ , with γ ∈ (

1 2m+3; 1 2)

(undersmoothing)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 8 / 16

Assumptions (continued)

• 2. Assumptions on the kernel and solution:

K is a kernel of order at least m, supported and twice continuously differentiable on [−1; 1] the bandwidth h satisfies h ≍ N(n)−γ , with γ ∈ (

1 2m+3; 1 2)

(undersmoothing) ˜ qk is an estimator of the density q of observed X 2

i ’s such that

˜ qk − q∞ = op (1/ log k)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 8 / 16

Assumptions (continued)

• 2. Assumptions on the kernel and solution:

K is a kernel of order at least m, supported and twice continuously differentiable on [−1; 1] the bandwidth h satisfies h ≍ N(n)−γ , with γ ∈ (

1 2m+3; 1 2)

(undersmoothing) ˜ qk is an estimator of the density q of observed X 2

i ’s such that

˜ qk − q∞ = op (1/ log k)

Example (kernel that satisfies the ssumptions with m = 2)

K0(x) = (15/16)(1 − x2)2I[−1,1](x) (biweight kernel)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 8 / 16

Confidence bands

Theorem (B.Ć, Z.Sz. & J.W, 2016)

Under the assumptions, for each x ∈ R, P

ˆ

fn(t) − bn(t, x) f (t) ˆ fn(t) + bn(t, x) for all t ∈ [a, b]

• → exp{−2 exp(−x)},
• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 9 / 16

Confidence bands

Theorem (B.Ć, Z.Sz. & J.W, 2016)

Under the assumptions, for each x ∈ R, P

ˆ

fn(t) − bn(t, x) f (t) ˆ fn(t) + bn(t, x) for all t ∈ [a, b]

• → exp{−2 exp(−x)},

where bn(t, x) = 2tCK,1[˜ qN(n)(t2)N(n)]1/2 nh3/2[2 log(1/h)]1/2

• x + log CK,2

2πh2

• ,
• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 9 / 16

Confidence bands

Theorem (B.Ć, Z.Sz. & J.W, 2016)

Under the assumptions, for each x ∈ R, P

ˆ

fn(t) − bn(t, x) f (t) ˆ fn(t) + bn(t, x) for all t ∈ [a, b]

• → exp{−2 exp(−x)},

where bn(t, x) = 2tCK,1[˜ qN(n)(t2)N(n)]1/2 nh3/2[2 log(1/h)]1/2

• x + log CK,2

2πh2

• ,

C 2

K,1 =

• K ′(x)2 dx,

C 2

K,2 = b2 − a2

CK,1

• K ′′(x)2 dx

(h = h[N(n)] depends on the Poissonian number of observed squared radii)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 9 / 16

ERM and bandwidth selection

Minimize w.r.t. h ||ˆ fn − f ||2

2 − ||f ||2 2

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 10 / 16

ERM and bandwidth selection

Minimize w.r.t. h ||ˆ fn − f ||2

2 − ||f ||2 2 = ||ˆ

fn||2

2 − 2

1

ˆ fn(x)f (x) dx

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 10 / 16

ERM and bandwidth selection

Minimize w.r.t. h ||ˆ fn − f ||2

2 − ||f ||2 2 = ||ˆ

fn||2

2 − 2

1

ˆ fn(x)f (x) dx = ||ˆ fn||2

2 + 2

1

ˆ fn(√x) dg(x)

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 10 / 16

ERM and bandwidth selection

Minimize w.r.t. h ||ˆ fn − f ||2

2 − ||f ||2 2 = ||ˆ

fn||2

2 − 2

1

ˆ fn(x)f (x) dx = ||ˆ fn||2

2 + 2

1

ˆ fn(√x) dg(x) ≈ ||ˆ fn||2

2 + 2 M−1

• i=0

ˆ fn(√xi)[˜ g(xi+1) − ˜ g(xi)] where 0 = x0, . . . , xM = 1 is a uniform grid (M = 210 in simulations) ˜ g(x) = 1 nH

N(n)

• i=1
• Kep
• x − X 2

i

H

• + Kep
• x + X 2

i

H

• (in simulations: H = 5(N(n)/n)2n−1/5)
• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 10 / 16

Simulations: Beta(4,2) and ERM efficiency, n = 104

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(a) true,best,worst

0.00 0.05 0.10 0.15 0.20 0.00 0.04 0.08 0.12 0.16 0.20

(b) risk: ERM vs oracle

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 11 / 16

Simulations: SML-B and ERM efficiency, n = 104

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(c) true,best,worst

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

(d) risk: ERM vs oracle

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 12 / 16

Simulations: 80% and 95% conf. bands for Beta(4,2)

0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(e) n = 104

0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(f) n = 105

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 13 / 16

Simulations: 80% and 95% conf. bands for SML-B

0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(g) n = 104

0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(h) n = 105

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 14 / 16

Simulations: 80% and 95% conf. bands for NM

0.2 0.4 0.6 0.8 1 2 3 4

(i) n = 104

0.2 0.4 0.6 0.8 1 2 3 4

(j) n = 105

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 15 / 16

Summary

L2-minimax estimator of the intensity function in the SLW stereological problem

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 16 / 16

Summary

L2-minimax estimator of the intensity function in the SLW stereological problem ERM-based bandwidth selection

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 16 / 16

Summary

L2-minimax estimator of the intensity function in the SLW stereological problem ERM-based bandwidth selection asymptotic confidence bands for intensity functions

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 16 / 16

Summary

L2-minimax estimator of the intensity function in the SLW stereological problem ERM-based bandwidth selection asymptotic confidence bands for intensity functions shown to work reasonably well in simulation

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 16 / 16

Summary

L2-minimax estimator of the intensity function in the SLW stereological problem ERM-based bandwidth selection asymptotic confidence bands for intensity functions shown to work reasonably well in simulation together with recent results for Wicksell’s problem, filling a methodological gap in stereological inverse problems

• B. Ćmiel, Z. Szkutnik, J. Wojdyła (AGH)

Confidence bands in the SLW problem Be ¸dlewo, 2016 16 / 16