DIFFERENT KINDS OF ESTIMATORS OF THE MEAN DENSITY OF RANDOM CLOSED SETS: THEORETICAL RESULTS AND NUMERICAL EXPERIMENTS.

Transcription

1 DIFFERENT KINDS OF ESTIMATORS OF THE MEAN DENSITY OF RANDOM CLOSED SETS: THEORETICAL RESULTS AND NUMERICAL EXPERIMENTS Elena Villa Dept. of Mathematics Università degli Studi di Milano Toronto May 22, 2014 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

2 In this talk... A random set in R d of locally finite H n -measure induces a random measure μ Θn (A) := H n ( A), A B R d, and the corresponding expected measure E[μ Θn ](A) := E[H n ( A)], A B R d. Whenever E[μ Θn ] H d on R d, its density, say λ Θn is called mean density of A crucial problem is the pointwise estimation of λ Θn (x). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

3 In this talk... A random set in R d of locally finite H n -measure induces a random measure and the corresponding expected measure μ Θn (A) := H n ( A), A B R d, E[μ Θn ](A) := E[H n ( A)], A B R d. Whenever E[μ Θn ] H d on R d, its density, say λ Θn is called mean density of A crucial problem is the pointwise estimation of λ Θn (x). We present here 3 different kinds of estimators of λ Θn (x): ν,n Natural estimator λ (x) It will follow as a natural consequence of the Besicovitch derivation theorem Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

4 In this talk... A random set in R d of locally finite H n -measure induces a random measure and the corresponding expected measure μ Θn (A) := H n ( A), A B R d, E[μ Θn ](A) := E[H n ( A)], A B R d. Whenever E[μ Θn ] H d on R d, its density, say λ Θn is called mean density of A crucial problem is the pointwise estimation of λ Θn (x). We present here 3 different kinds of estimators of λ Θn (x): ν,n Natural estimator λ (x) It will follow as a natural consequence of the Besicovitch derivation theorem κ,n Kernel estimator λ (x) It will follow as a generalization to the n-dimensional case of the classical kernel density estimator of random vectors Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

5 In this talk... A random set in R d of locally finite H n -measure induces a random measure and the corresponding expected measure μ Θn (A) := H n ( A), A B R d, E[μ Θn ](A) := E[H n ( A)], A B R d. Whenever E[μ Θn ] H d on R d, its density, say λ Θn is called mean density of A crucial problem is the pointwise estimation of λ Θn (x). We present here 3 different kinds of estimators of λ Θn (x): ν,n Natural estimator λ (x) It will follow as a natural consequence of the Besicovitch derivation theorem κ,n Kernel estimator λ (x) It will follow as a generalization to the n-dimensional case of the classical kernel density estimator of random vectors μ,n Minkowski content -based estimator λ (x) It will follow by a local approximation of λ Θn based on a stochastic version of the n-dimensional Minkowski content of. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

6 We remind that... A compact set S R d is called n-rectifiable, if there exist a compact K R n and a Lipschitz function g : R n R d such that S = g(k); countably H n -rectifiable if there exist countably many Lipschitz maps g i : R n R d such that H n( ) S \ g i (R n ) = 0. A random closed set (r.c.s.)θ in R d is a measurable map i=1 Θ : (Ω, F, P) (F, σ F ), where F is the class of the closed subsets in R d, and σ F is the σ-algebra generated by the so called Fell topology, or hit-or-miss topology. IN WHAT FOLLOWS: is a countably H n -rectifiable r.c.s. of locally finite H n -measure Θ 1 n,..., Θ N n i.i.d. random sample for Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

7 We remind that... A compact set S R d is called n-rectifiable, if there exist a compact K R n and a Lipschitz function g : R n R d such that S = g(k); countably H n -rectifiable if there exist countably many Lipschitz maps g i : R n R d such that H n( ) S \ g i (R n ) = 0. A random closed set (r.c.s.)θ in R d is a measurable map i=1 Θ : (Ω, F, P) (F, σ F ), where F is the class of the closed subsets in R d, and σ F is the σ-algebra generated by the so called Fell topology, or hit-or-miss topology. IN WHAT FOLLOWS: is a countably H n -rectifiable r.c.s. of locally finite H n -measure Θ 1 n,..., Θ N n i.i.d. random sample for NOTE THAT: if n = 0 and Θ 0 = X random vector with pdf f X, then E[H 0 (X A)] = P(X A) = f X (x)dx A B R d and so λ X (x) = f X (x). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32 A

8 ν,n The natural estimator λ (x) The Besicovitch derivation theorem implies that if E[H n ( A)] = λ Θn (x)dx, A B R d, then A λ Θn (x) = lim r 0 E[H n ( B r (x))] b d r d H d -a.e. x R d. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

9 ν,n The natural estimator λ (x) The Besicovitch derivation theorem implies that if E[H n ( A)] = λ Θn (x)dx, A B R d, then A λ Θn (x) = lim r 0 E[H n ( B r (x))] b d r d H d -a.e. x R d. This suggests the following natural estimator for the mean density λ Θn (x) of, λ ν,n (x) := 1 Nb d r d N N H n (Θ i n B rn (x)). i=1 Here and in the following r N is called the bandwidth associated with the sample size N, as usual in literature. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

10 The kernel estimator We remind that κ,n λ (x) A measurable function k : R d R is said to be a multivariate kernel if it satisfies the following conditions: 0 k(z) M for all z R d, for some M > 0; k is radially symmetric; R d k(z)dz = 1. Given X 1,..., X N i.i.d. random sample for X random vector with p.d.f f X, the multivariate kernel density estimator of f X based on a chosen kernel k, and scaling parameter r N (0, + ), is defined by f N X (x) := 1 N N i=1 k rn H 0 Xi (x) = 1 Nr d N N ( ) x Xi k i=1 r N Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

11 The kernel estimator We remind that κ,n λ (x) A measurable function k : R d R is said to be a multivariate kernel if it satisfies the following conditions: 0 k(z) M for all z R d, for some M > 0; k is radially symmetric; R d k(z)dz = 1. Given X 1,..., X N i.i.d. random sample for X random vector with p.d.f f X, the multivariate kernel density estimator of f X based on a chosen kernel k, and scaling parameter r N (0, + ), is defined by f N X (x) := 1 N N i=1 k rn H 0 Xi (x) = 1 Nr d N N ( ) x Xi k As a natural extension to the n-dimensional r.c.s, we define the following kernel estimator for the mean density of : λ κ,n (x) := 1 N N i=1 k rn H n Θ i n (x) = 1 Nr d N N i=1 Θ i n i=1 r N ( x y ) k H n (dy) r N Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

12 The kernel estimator We remind that κ,n λ (x) A measurable function k : R d R is said to be a multivariate kernel if it satisfies the following conditions: 0 k(z) M for all z R d, for some M > 0; k is radially symmetric; R d k(z)dz = 1. Given X 1,..., X N i.i.d. random sample for X random vector with p.d.f f X, the multivariate kernel density estimator of f X based on a chosen kernel k, and scaling parameter r N (0, + ), is defined by f N X (x) := 1 N N i=1 k rn H 0 Xi (x) = 1 Nr d N N ( ) x Xi k As a natural extension to the n-dimensional r.c.s, we define the following kernel estimator for the mean density of : λ κ,n (x) := 1 N N i=1 k rn H n Θ i n (x) = 1 Nr d N N i=1 Θ i n i=1 r N ( x y ) k H n (dy) r N NOTE THAT: λν,n κ,n (x) = λ (x) by choosing as kernel k(z) = 1 b d 1 B1 (0)(z). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

13 The Minkowski content -based estimator We remind that μ,n λ (x) The n-dimensional Minkowski content of a closed set S R d is the quantity M n (S) so defined M n H d (S r ) (S) := lim r 0 b d n r d n provided the limit exists finite, where A r :={x R d : dist(x, A) r}. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

14 The Minkowski content -based estimator We remind that μ,n λ (x) The n-dimensional Minkowski content of a closed set S R d is the quantity M n (S) so defined M n H d (S r ) (S) := lim r 0 b d n r d n provided the limit exists finite, where A r :={x R d If S R d is countably H n -rectifiable and compact, and η(b r (x)) γr n x S, r (0, 1) : dist(x, A) r}. holds for some γ > 0 and some Radon measure η in R d, η H n, then M n (S) = H n (S). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

15 The Minkowski content -based estimator We remind that μ,n λ (x) The n-dimensional Minkowski content of a closed set S R d is the quantity M n (S) so defined M n H d (S r ) (S) := lim r 0 b d n r d n provided the limit exists finite, where A r :={x R d If S R d is countably H n -rectifiable and compact, and η(b r (x)) γr n x S, r (0, 1) : dist(x, A) r}. holds for some γ > 0 and some Radon measure η in R d, η H n, then M n (S) = H n (S). We say that a r.c.s. admits mean local n-dimensional Minkowski content if for all A B R d such that E[μ Θn ]( A) = 0 E[H d ( r A)] lim = E[H n (Θ r 0 b d n r d n n A)] Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

16 The Minkowski content -based estimator We remind that μ,n λ (x) The n-dimensional Minkowski content of a closed set S R d is the quantity M n (S) so defined M n H d (S r ) (S) := lim r 0 b d n r d n provided the limit exists finite, where A r :={x R d If S R d is countably H n -rectifiable and compact, and η(b r (x)) γr n x S, r (0, 1) : dist(x, A) r}. holds for some γ > 0 and some Radon measure η in R d, η H n, then M n (S) = H n (S). We say that a r.c.s. admits mean local n-dimensional Minkowski content if lim r 0 A P(x r ) E[H d ( r A)] dx = lim = E[H n (Θ b d n r d n r 0 b d n r d n n A)]= λ Θn (x)dx A for all A B R d such that E[μ Θn ]( A) = 0 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

17 It can be proved, under suitable regularity assumptions on, that [EV, Bernoulli, 2014] λ Θn (x) = lim r 0 P(x r ) b d n r d n, H d -a.e. x R d This suggests the following Minkowski content -based estimator of λ Θn (x) N λ μ,n i=1 1 Θ (x) := i n B r N (x) = #{i : x Θi n rn }. Nb d n r d n N Nb d n r d n N Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

18 It can be proved, under suitable regularity assumptions on, that [EV, Bernoulli, 2014] λ Θn (x) = lim r 0 P(x r ) b d n r d n, H d -a.e. x R d This suggests the following Minkowski content -based estimator of λ Θn (x) N λ μ,n i=1 1 Θ (x) := i n B r N (x) = #{i : x Θi n rn }. Nb d n r d n N Nb d n r d n N NOTE THAT: if Θ 0 = X random point in R d, then λ ν,n X (x) = λ μ,n X (x) = #{i : X i B rn (x)} Nr d N Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

19 It can be proved, under suitable regularity assumptions on, that [EV, Bernoulli, 2014] λ Θn (x) = lim r 0 P(x r ) b d n r d n, H d -a.e. x R d This suggests the following Minkowski content -based estimator of λ Θn (x) N λ μ,n i=1 1 Θ (x) := i n B r N (x) = #{i : x Θi n rn }. Nb d n r d n N Nb d n r d n N NOTE THAT: if Θ 0 = X random point in R d, then λ ν,n X (x) = λ μ,n X (x) = #{i : X i B rn (x)} Nr d N PROBLEM 1: regularity properties on such that the 3 proposed estimators are asymptotically unbiased and consistent. PROBLEM 2: optimal bandwidth r N. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

20 Assumptions and notation Every random closed set Θ in R d can be represented as particle process (or germ-grain process ), and so described by a marked point process Φ = {(X i, S i )} i N in R d with marks in a suitable mark space K so that Θ(ω) = x i + Z(s i ), ω Ω, (x i,s i ) Φ(ω) where Z i = Z(S i ), i N is a random set containing the origin. (If Φ is a marked Poisson point process, then Θ is called Boolean model) Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

21 Assumptions and notation Every random closed set Θ in R d can be represented as particle process (or germ-grain process ), and so described by a marked point process Φ = {(X i, S i )} i N in R d with marks in a suitable mark space K so that Θ(ω) = x i + Z(s i ), ω Ω, (x i,s i ) Φ(ω) where Z i = Z(S i ), i N is a random set containing the origin. (If Φ is a marked Poisson point process, then Θ is called Boolean model) We assume that Φ has intensity measure and second factorial moment measure Λ(d(x, s)) = f (x, s)dxq(ds) ν [2] (d(x, s, y, t)) = g(x, s, y, t)dxdyq [2] (d(s, t)) It follows that λ Θn (x) = f (y, s)h n (dy)q(ds) K x Z(s) Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

22 Assumptions and notation Every random closed set Θ in R d can be represented as particle process (or germ-grain process ), and so described by a marked point process Φ = {(X i, S i )} i N in R d with marks in a suitable mark space K so that Θ(ω) = x i + Z(s i ), ω Ω, (x i,s i ) Φ(ω) where Z i = Z(S i ), i N is a random set containing the origin. (If Φ is a marked Poisson point process, then Θ is called Boolean model) We assume that Φ has intensity measure and second factorial moment measure Λ(d(x, s)) = f (x, s)dxq(ds) ν [2] (d(x, s, y, t)) = g(x, s, y, t)dxdyq [2] (d(s, t)) It follows that λ Θn (x) = f (y, s)h n (dy)q(ds) K x Z(s) Note that: if Boolean model: g(x, s, y, t) = f (x, s)f (y, t), and Q [2] (d(s, t)) = Q(ds)Q(dt) if Θ 0 = X random point with pdf f X : Φ = (X, s), Z(s) := s R d, and Λ(d(y, s)) = f X (y)dyδ 0(s)ds, ν [2] (d(x, s, y, t)) 0 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

23 In what follows: α := (α 1,..., α d ) multi-index of N d 0 ; α := α α d ; α! := α 1! α d! y α := y α 1 1 y α d d ; Dy α f (y, s) := α f (y, s) y α 1 1 y α ; d D (α) (s) := disc(dy α f (y, s)). k will be a kernel with compact support d Moreover, under regularity assumptions will mean that some of the following assumptions are satisfied: (A1) for any (y, s) R d K, y + Z(s) is a countably H n -rectifiable and compact subset of R d, such that there exists a closed set Ξ(s) Z(s) such that K Hn (Ξ(s))Q(ds) < and for some γ > 0 independent of s; (A1) as (A1), replacing (1) with H n (Ξ(s) B r (x)) γr n x Z(s), r (0, 1) (1) γr n H n (Ξ(s) B r (x)) γr n x Z(s), r (0, 1) for some γ, γ > 0 independent of s; μ,n κ,n ν,n (A1) for λ (x); (A1) for λ (x) and λ (x) in proving asymptotical unbiasedness and consistency Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

24 (A2) for any s K, H n (disc(f (, s))) = 0 and f (, s) is locally bounded such that for any compact K R d sup x K diam(z(s)) f (x, s) ξ K (s) for some ξ K (s) with K H n (Ξ(s)) ξ K (s)q(ds) < (A2*) for α = 2, for any s K, H n (D (α) (s)) = 0 and D α y f (y, s) is locally bounded such that, for any compact C R d, for some ξ (α) C (s) with sup y C diamz(s) D α y f (y, s) K (α) ξ C (s) H n (Ξ(s)) ξ (α) C (s)q(ds) < (A2) for all the 3 estimators in proving asymptotical unbiasedness and consistency; κ,n ν,n (A2*) for λ (x) and λ (x) in finding the optimal bandwidth Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

25 (A3) for any (s, y, t) K R d K, H n (disc(g(, s, y, t))) = 0 and g(, s, y, t) is locally bounded such that for any compact K R d and a R d, for some ξ a,k (s, y, t) with 1 (a Z(t)) 1 (y) sup x K diam(z(s)) g(x, s, y, t) ξ a,k (s, y, t) R d K 2 H n (Ξ(s))ξ a,k (s, y, t)dyq [2] (ds, dt) <. (2) (A3) for any s, t K, g(, s,, t) is locally bounded such that, for any C, C R d compact sets: sup sup g(x, s, y, t) ξ C,C (s, t) x C diamz(s) y C diamz(t) for some ξ C,C (s, t) with K 2 H n (Ξ(s))H n (Ξ(t))ξ C,C (s, t)q [2] (ds, dt) <. (3) μ,n κ,n ν,n (A3) for λ (x); (A3) for λ (x) and λ (x) in proving asymptotical unbiasedness and consistency Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

26 Statistical properties and optimal bandwidths of the proposed estimators Under regularity assumptions on Θ, the 3 proposed estimators of λ Θn (x) are asymptotically unbiased and weakly consistent for H d a.e. x R d. In order to find the optimal bandwidth of the 3 proposed estimators, we proceed along the same lines as what is commonly done for the kernel density estimator f N X (x) of the pdf f X (x) of a random variable X ; that is where r o,amse N (x) := arg minamse ( λ N (x)), r N MSE( λ N (x)) :=E[( λ N (x) λ Θn (x)) 2 ] = Bias( λ N (x)) 2 + Var( λ N (x)) is the mean square error of λ N (x), and AMSE is the asymptotic MSE, obtained by Taylor series expansion of Bias and Variance. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

27 Under regularity assumptions on we obtain κ,n for λ [Camerlenghi F, Capasso V, EV, J. Multivariate Anal., 2014] r o,amse (d n)c N (x) = 4+d n Var (x) 4NCBias 2 (x), H d -a.e. x R d with C Bias (x) := 1 k(z)z α dz Dy α f (y, s)h n (dy)q(ds) α! α =2 R d K x Z(s) C Var (x) := k(z)k(z + w)f (y, s)h n (dw)h n (dy)dzq(ds) where π x,s y K R d x Z(s) π x,s y G n is the approximate tangent space to x Z(s) at y x Z(s). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

28 Under regularity assumptions on we obtain κ,n for λ [Camerlenghi F, Capasso V, EV, J. Multivariate Anal., 2014] r o,amse (d n)c N (x) = 4+d n Var (x) 4NCBias 2 (x), H d -a.e. x R d with C Bias (x) := 1 k(z)z α dz Dy α f (y, s)h n (dy)q(ds) α! α =2 R d K x Z(s) C Var (x) := k(z)k(z + w)f (y, s)h n (dw)h n (dy)dzq(ds) where π x,s y K R d x Z(s) π x,s y G n is the approximate tangent space to x Z(s) at y x Z(s). ν,n for λ the same as above with k(z) = 1 b d 1 B1 (0)(z) Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

29 μ,n for λ, if s K, reach(z(s)) > R for some R > 0, [Camerlenghi F, EV, work in progress, 2014] N (x) = r o,amse ( (d n)λ Θn (x) ) 1 d n+2 2Nb d n (A 1(x) A 2(x)) 2 if d n > 1 ( λ Θd 1 (x) ) 1 3 4N ( A 1(x) A ) 2 3(x) if d n = 1 with A 1(x) := b d n+1 f (x y, s)φ n 1(Z(s); dy)q(ds) b d n K Z(s) A 2(x) := d n Dx α f (x y, s)u α μ n(z(s); d(y, u))q(ds) d n + 1 α =1 K N(Z(s)) A 3(x) := g(y 1, s 1, y 2, s 2)H d 1 (dy 2)H d 1 (dy 1)Q [2] (d(s 1, s 2)) K 2 (x Z(s 1 )) (x Z(s 2 )) where Φ n(z(s), ) is the n-dimensional curvature measure of Z(s) and μ n(z(s), ) is the n-dimensional support measure of Z(s). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

30 Particular cases d = 1, n = 0, Θ 0 = X random variable with pdf f X C 2 We reobtain the well known results for kernel density estimates of f X r o,amse N (x) = 5 N f X (x) ( f X (x) R 9f 5 X (x) 2N(f, X (x))2 k(z) 2 dz ) 2, for z 2 k(z)dz λ κ,n X ν,n μ,n for λ (x) = λ (x) X X Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

31 Particular cases d = 1, n = 0, Θ 0 = X random variable with pdf f X C 2 We reobtain the well known results for kernel density estimates of f X r o,amse N (x) = 5 N f X (x) ( f X (x) R 9f 5 X (x) 2N(f, X (x))2 k(z) 2 dz ) 2, for z 2 k(z)dz λ κ,n X ν,n μ,n for λ (x) = λ (x) X X Θ 0 = Ψ point process in R d with intensity λ Ψ C 2 ν,n If N = 1, λ Ψ (x) coincides with the well-known classic and widely used Berman-Diggle estimator λ κ,n Ψ(Br (x)) Ψ (x) = b d r d Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

32 stationary Φ with intensity measure Λ(d(x, s)) = cdxq(ds). It follows that: λ Θn (x) ce[h n (Z)] H d -a.e. x R d, λ κ,n λ κ,n and and λ ν,n are unbiased for any bandwidth r > 0, and any sample size N; λ ν,n are strongly consistent for H d -a.e. x R d, as N. if Boolean model with E[(H n (Z)) 2 ] <, then + for ce[h n (Z)] r o,mse 3 = N ( πce[φ n 1(Z)] 2(cE[H n (Z)]) 2) 2 for λ κ,n and λ ν,n λ κ,n if d n = 1 d n+2 (d n)bd n ce[h n (Z)] 2N ( cb d n+1 E[Φ ) 2 n 1(Z)] for λ κ,n if d n > 1 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

33 Summarizing: λ κ,n (x) = 1 Nr d N kernel estimator N i=1 Θ i n Θ 0 =X k ( x y r N ) H n (dy) f N X (x) = 1 Nr d N kernel density estimator N i=1 k ( x X i r N ) k(z) = 1 b d 1 B1 (0)(z) k(z) = 1 1 b B1 d (0)(z) natural estimator λ ν,n (x) = 1 N H n (Θ i Nb d rn d n B rn (x)) i=1 naive estimator Θ 0 =X N f X (x) = #{i : X i B rn (x)} Minkowski content -based estimator NrN d λ μ,n (x) = #{i : x Θi n rn } Nb d n r d n N Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

34 Some numerical experiments Θ 1 = inhomogeneous Boolean model of segments of the type [0, l] {0} in R 2, with random length l U(0, 0.2) in the compact window W = [0, 1] 2, where the underlying Poisson point process has intensity f (x 1, x 2 ) = 700x 2 1. λ Θ1 (x 1, x 2) = (0.2) (0.2)2 x x 2 1 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

35 Natural estimator and Minkowski content -based estimator at point x = (0.5, 0.5) as function of the bandwidth expressed in pixels (1pixel = ), for N = 10 and N = 100 λ Θ1 (0.5, 0.5) = Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

36 Natural estimator and Minkowski content -based estimator at point x = (0.5, 0.5) as function of the bandwidth expressed in pixels (1pixel = ), for N = 10 and N = 100 λ Θ1 (0.5, 0.5) = Natural estimator λ Θ1 r N o, AMSE 50 λ Θ1 r N o, AMSE λν,n Θ 1 30 λν,n Θ r (pixel) r (pixel) N = 10 N = 100 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

37 Minkowski content -based estimator λ Θ1 r N o, AMSE 50 λ Θ1 r N o, AMSE λμ,n Θ 1 30 λμ,n Θ r (pixel) r (pixel) N = 10 N = 100 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

38 λ Θ1 (0.5, 0.5) = theoretical value λ ν,n Θ 1 (0.5, 0.5) λ Θ1 (0.5, 0.5) = for N = 10, r o,amse 10 77pixel(0.2973) λ ν,n Θ 1 (0.5, 0.5) λ Θ1 (0.5, 0.5) = for N = 100, r o,amse pixel(0.1425) λ μ,n Θ 1 (0.5, 0.5) λ Θ1 (0.5, 0.5) = for N = 10, r o,amse 10 9pixel(0.0271) λ μ,n Θ 1 (0.5, 0.5) λ Θ1 (0.5, 0.5) = 0.70 for N = 100, r o,amse 100 4pixel(0.0126) Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

39 Θ 1 = homogeneous Boolean model of segments of the type [0, l] {0} in R 2, with random length l U(0, 0.2) in the compact window W = [0, 1] 2, where the underlying Poisson point process has intensity f (x 1, x 2 ) = 300. λ Θ1 (x 1, x 2) = 30 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

40 Natural estimator and Minkowski content -based estimator at point x = (0.5, 0.5) as function of the bandwidth expressed in pixel (1pixel = ), for N = 10 and N = 100 λ Θ1 (x) 30; we choose x = (0.5, 0.5). Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

41 Natural estimator and Minkowski content -based estimator at point x = (0.5, 0.5) as function of the bandwidth expressed in pixel (1pixel = ), for N = 10 and N = 100 λ Θ1 (x) 30; we choose x = (0.5, 0.5). Natural estimator r o,amse N = + N = 10 N = 100 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

42 Minkowski content -based estimator N = 10 N = 100 Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

43 λ Θ1 30 theoretical value λ ν,n Θ 1 λ Θ1 = with r = 105pixel for N = 10, r o,amse 10 = + λ ν,n Θ 1 λ Θ1 = with r = 105pixel for N = 100, r o,amse 100 = + λ μ,n Θ 1 λ Θ1 = 6.13 for N = 10, r o,amse 10 5pixel(0.016) λ μ,n Θ 1 λ Θ1 = 1.50 for N = 100, r o,amse 100 3pixel(0.0074) Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

44 Θ 0 = Ψ inhomogeneous Poisson point process with intensity f (x 1, x 2) = x x 2 2 in the compact window W = [ 2, 2] 2. Ψ 1,..., Ψ N i.i.d. random sample for Ψ kernel estimator λ κ,n Ψ (x) = 1 Nr 2 N N k i=1 y j Ψ i ( x yj r N ) with kernel of Epanechnikov : { 2 k(t) = π (1 x2 1 x2 2 ), if (x 1, x 2 ) B 1 (0) 0, otherwise natural estimator λ ν,n Ψ (x) = 1 Nπr 2 N N i=1 Minkowski content -based estimator H 0 (Ψ i B rn (x)) = 1 Nπr 2 N λ μ,n Ψ (x) = #{i : x Ψi r N } NπrN 2 = 1 NπrN 2 N i=1 N 1 BrN (x)(y j ) i=1 y j Ψ i 1 {Ψ i (B rn (x))>0} Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

45 Epanechnikov-kernel estimator, Natural estimator and Minkowski content -based estimator in W = [ 2, 2] 2 with grid step size=0.2, for N = 1000 and N = Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

46 Epanechnikov-kernel estimator, Natural estimator and Minkowski content -based estimator in W = [ 2, 2] 2 with grid step size=0.2, for N = 1000 and N = Epanechnikov-kernel estimator N = 1000 N = Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

47 Natural estimator N = 1000 N = Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

48 Minkowski content -based estimator N = 1000 N = Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

49 λ Ψ (1.8, 1.8) = 6.48 theoretical value λ κ,n Θ 1 (1.8, 1.8) λ Ψ (1.8, 1.8) = for N = 1000, r o,amse pixel(0.4809) λ κ,n Ψ (1.8, 1.8) λ Ψ(1.8, 1.8) = for N = 10000, r o,amse pixel(0.3277) λ ν,n Ψ (1.8, 1.8) λ Ψ(1.8, 1.8) = for N = 1000, r o,amse pixel(0.4005) λ ν,n Ψ (1.8, 1.8) λ Ψ(1.8, 1.8) = for N = 10000, r o,amse pixel(0.2728) λ μ,n Ψ (1.8, 1.8) λ Ψ(1.8, 1.8) = for N = 1000, r o,amse pixel(0.0789) λ μ,n Ψ (1.8, 1.8) λ Ψ(1.8, 1.8) = for N = 10000, r o,amse pixel(0.0537) κ,n max λ Ψ (x) λ Ψ(x) = for N = 1000 x W max x W λ κ,n Ψ (x) λ Ψ(x) = for N = ν,n max λ Ψ (x) λ Ψ(x) = for N = 1000 x W max x W λ ν,n Ψ (x) λ Ψ(x) = for N = μ,n max λ Ψ (x) λ Ψ(x) = for N = 1000 x W max x W λ μ,n Ψ (x) λ Ψ(x) = for N = Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

50 Concluding Remarks Kernel estimators: pro: they extend in a natural way the corresponding kernel estimators for random objects of dimension n = 0 (random variables - univariate and multivariate, point processes) to random closed sets of any integer Hausdorff dimension n < d, in R d ; cons: practical applicability; non-trivial computation of integrals is required Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

51 Concluding Remarks Kernel estimators: pro: they extend in a natural way the corresponding kernel estimators for random objects of dimension n = 0 (random variables - univariate and multivariate, point processes) to random closed sets of any integer Hausdorff dimension n < d, in R d ; cons: practical applicability; non-trivial computation of integrals is required Natural estimators: pro: direct derivation from the Besicovitch Theorem; generalization of the notion of histogram estimators for the case Θ 0 = X random variable; more stable with respect to the choice of the bandwidth, than the Minkowski content -based estimators cons: include the nontrivial evaluation of H n (Θ i n B rn (x)) for any element Θ i n of the sample; for segment processes (n = 1) it seems more feasible, but for other sets of dimension n 1 it results of higher computational complexity. Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

52 Concluding Remarks Kernel estimators: pro: they extend in a natural way the corresponding kernel estimators for random objects of dimension n = 0 (random variables - univariate and multivariate, point processes) to random closed sets of any integer Hausdorff dimension n < d, in R d ; cons: practical applicability; non-trivial computation of integrals is required Natural estimators: pro: direct derivation from the Besicovitch Theorem; generalization of the notion of histogram estimators for the case Θ 0 = X random variable; more stable with respect to the choice of the bandwidth, than the Minkowski content -based estimators cons: include the nontrivial evaluation of H n (Θ i n B rn (x)) for any element Θ i n of the sample; for segment processes (n = 1) it seems more feasible, but for other sets of dimension n 1 it results of higher computational complexity. Minkowski content -based estimators: pro: easy computational evaluation; cons: quite sensitive to the choice of the bandwidth; low rate of convergence Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32

53 Essential Bibliography: Ambrosio L., Capasso V., Villa E. (2009). On the approximation of mean densities of random closed sets, Bernoulli, 15, Camerlenghi F., Capasso V., Villa E. (2014). On the estimation of the mean density of random closed sets. J. Multivariate Anal., 125, Camerlenghi F., Capasso V., Villa E.: (2014). Numerical experiments for the estimation of mean densities of random sets. In: Proceedings of the ECS11. Image Anal. Stereol. (to appear) Villa E. (2010). Mean densities and spherical contact distribution function of inhomogeneous Boolean models, Stoch. An. Appl., 28, Villa E. On the local approximation of mean densities of random closed sets, Bernoulli, 20, 1 27 (2014)...and references therein Elena Villa (University of Milan) Estimation of the mean density of RACS Toronto, May 22, / 32