Roberto D'ambrosio
email: r.dambrosio@unicampus.it

Richard Nock
email: rnock@martinique.univ-ag.fr

Wafa Bel

Haj Ali

Frank Nielsen
email: frank.nielsen@acm.org

Michel Barlaud
email: barlaud@i3s.unice.fr

Boosting Nearest Neighbors for the Efficient Estimation of Posteriors

It is an admitted fact that mainstream boosting algorithms like AdaBoost do not perform well to estimate class conditional probabilities. In this paper, we analyze, in the light of this problem, a recent algorithm, unn, which leverages nearest neighbors while minimizing a convex loss. Our contribution is threefold. First, we show that there exists a subclass of surrogate losses, elsewhere called balanced, whose minimization brings simple and statistically efficient estimators for Bayes posteriors. Second, we show explicit convergence rates towards these estimators for unn, for any such surrogate loss, under a Weak Learning Assumption which parallels that of classical boosting results. Third and last, we provide experiments and comparisons on synthetic and real datasets, including the challenging SUN computer vision database. Results clearly display that boosting nearest neighbors may provide highly accurate estimators, sometimes more than a hundred times more accurate than those of other contenders like support vector machines.

Introduction

Boosting refers to the iterative combination of classifiers which produces a classifier with reduced true risk (with high probability), while the base classifiers may be weakly accurate [?]. The final, strong classifier h, satisfies im(h) ⊆ R. Such an output carries out two levels of information. The simplest one is the sign of the output. This discrete value is sufficient to classify an unknown observation x: h(x) predicts that x belongs to a class of interest iff it is positive. The most popular boosting results typically rely on this sole information [?,?,?] (and many others). The second level is the real value itself, which carries out as additional information a magnitude which can be interpreted as a "confidence" in the classification. This continuous information may be fit into a link function f : R → [0, 1] to estimate conditional class probabilities, thus lifting the scope of boosting to that of Bayes decision rule [?]:

Pr[y = 1|x] = f (h(x)) .
(1)

To date, estimating posteriors with boosting has not met the same success as predicting (discrete) labels. It is widely believed that boosting and conditional class probability estimation are, up to a large extent, in conflict with each other, as boosting iteratively improves classification at the price of progressively overfitting posteriors [?,?]. Experimentally, limiting overfitting is usually obtained by tuning the algorithms towards early stopping [?].

Very recently, a new algorithm was proposed to leverage the famed nearest neighbor (nn) rules [?]. This algorithm, unn, fits real-valued coefficients for examples in order to minimize a surrogate risk [?,?]. These leveraging coefficients are used to balance the votes in the final k-nn rule. It is proven that, as the number of iterations T → ∞, unn achieves the global optimum of the surrogate risk at hand for a wide class of surrogates called strictly convex surrogates [?,?]. An explicit convergence rate is obtained for the specific case of the exponential loss, under a so-called "weak index assumption" [?].

Our contribution is threefold. First, we show that there exists a subclass of surrogate losses, elsewhere called balanced, whose minimization brings simple and efficient estimators for Bayes posteriors (1). Second, we show explicit convergence rates for unn for any such surrogate loss under a Weak Learning Assumption which parallels that of classical boosting results [?]. Third and last, we provide experiments on simulated and real domains, displaying that boosting nearest neighbors brings very good results from the conditional class probabilities estimation standpoint, without the overfitting problem of classical boosting approaches. A serious challenger to the popular logistic estimator for posteriors estimation also emerges, beating it by orders of magnitude on simulated data. We end up with the conclusion that learning posteriors with boosting nearest neighbors benefits from two advantages. First, the weak classifiers being simple examples, they naturally limit the risk of overfitting compared to more complex weak learners. Second, we end up learning posteriors using a natural, fixed topology of data, and not an ad hoc topology relying on an induced classifier.

The remaining of the paper is structured as follows: the next Section presents definitions, followed by a Section on convex losses and the estimation of posteriors. Then, a Section presents algorithms and results on boosting nearest neighbors. The two last Sections present experiments with discussions, and conclude.

Definitions

Estimation

Our setting is that of multiclass multilabel classification (See e.g. [?]). We have access to an input set of m examples, also called prototypes, S . = {(x i , y i ), i = 1, 2, ..., m}. Vector y i ∈ {-1, 1} C encodes class memberships, assuming y ic = 1 means that observation x i belongs to class c. S is sampled i.i.d. according to an unknown distribution D. Given an observation x ∈ O, we wish to estimate the conditional class probabilities for each class c, also called (estimated) posteriors:

pc (x) . = Pr[y c = 1|x] .
(2)

We note p c (x) .

= Pr D [y c = 1|x] the corresponding Bayes (true) posteriors.

Surrogates, losses and risks

Perhaps the simplest road towards computing these estimators consists in first crafting C separate classification problems, each of which leads to estimators for one class (2). Normalizing estimators to 1 over the C classes yields the values in (2). Each of these C problems is a one-versus-all classification task, say for class c, with corresponding sample S (c) = {(x i , y ic ), i = 1, 2, ..., m}. For each of these problems, we learn from S a classifier h : O → R out of which we may accurately compute (2), typically with pc (x) = f (h(x)) for some relevant function f . More sophisticated approaches exist that reduce the number of classifiers by folding classes in observation variables [?,?]. Each of them equivalently learn on a sample of Ω(mC) examples, and it is an easy task to craft from their output a set of C classifiers that fit into the framework we consider.

There exists a convenient approach to carry out this path as a whole, for each class c = 1, 2, ..., C: learn h by minimizing a surrogate risk over S [?,?,?]. A surrogate risk has general expression:

ε ψ S (h, c) . = 1 m m i=1 ψ(y ic h(x)) , (3)
for some function ψ that we call a surrogate loss. Quantity y ic h(x) ∈ R is called the edge of classifier h on example (x i , y i ) for class c. The surrogate risk is an estimator of the true surrogate risk computed over D:

ε ψ D (h, c) . = E D [ψ(y ic h(x))] . (4)
Any surrogate loss relevant to classification [?] has to meet sign(

h opt (x * )) = sign(2Pr D [y c = 1|x = x * ] -1), where h opt minimizes E D [ψ(y c h(x))|x = x ⋆ ].
Hence, the sign of the optimal classifier h opt is as accurate to predict class membership as Bayes decision rule. This Fisher consistency requirement for ψ is called classification calibration [?]. We focus in this paper on the subclass of classification calibrated surrogates that are strictly convex and differentiable.

Definition 1.

[?] A strictly convex loss is a strictly convex function ψ differentiable on int(dom(ψ)) satisfying (i) im(ψ) ⊆ R + , (ii) dom(ψ) symmetric around 0, (iii) ∇ ψ (0) < 0.

Definition 1 is extremely general: should we have removed conditions (i) and (ii), Theorem 6 in [?] brings that it would have encompassed the intersection between strictly convex differentiable functions and classification calibrated functions. Conditions (i) and (ii) are mainly conveniences for classification: in particular, it is not hard to see that modulo scaling by a positive constant, the surrogate risk ( 3) is an upperbound of the empirical risk for any strictly convex loss.

Minimizing the surrogate risk amounts thus to minimize the empirical risk up to some extent. We define the Legendre conjugate of any strictly convex loss ψ as ψ ⋆ (x)

. = x∇ -1 ψ (x) -ψ(∇ -1 ψ (x)
). There exists a particular subset of strictly convex losses of independent interest [?]. A function φ : [0, 1] → R + is called permissible iff it is differentiable on (0, 1), strictly concave and symmetric around x = 1/2 [?,?]. We adopt the notation φ = -φ [?].

Definition 2. [?] Given some permissible φ, we let ψ φ denote the balanced convex loss with signature φ as:

ψ φ (x) . = φ ⋆ (-x) -φ(0) φ ( 1 /2) -φ(0) . ( 5
)
Balanced convex losses have an important rationale: up to differentiability constraints, they match the set of symmetric lower-bounded losses defining proper scoring rules [?], that is, basically, the set of losses that fit to classification problems without class-dependent misclassification costs. Table 1 provides examples of surrogate losses, most of which are strictly convex surrogates, some of which are balanced convex surrogates. We have derived Amari's α-loss from Amari's famed α divergences [?] (proof omitted). The linear Hinge loss is not a balanced convex loss, yet it figures the limit behavior of balanced convex losses [?]. Remark that all signatures φ are well-known in the domain of decision-tree induction : from the top-most to the bottom-most, one may recognize Gini criterion, the entropy (two expressions), Matsushita's criterion and the empirical risk [?,?].

One dimensional exponential families and posteriors estimation

A (regular) one dimensional exponential family [?] is a set of probability density functions whose elements admit the following canonical form:

p[x|θ] . = exp (xθ -ψ(θ)) p 0 (x) , (6)
where p 0 (x) normalizes the density, ψ is a strictly convex differentiable function that we call the signature of the family, and θ is the density's natural parameter. It was shown in [?] that the efficient minimization of any balanced convex surrogate risk -i.e. a surrogate risk with a balanced convex loss -amounts to a maximum likelihood estimation θ = H(x) at some x for an exponential family whose signature depends solely on the permissible function φ.

[?] suggest to use the corresponding expected parameter of the exponential family as the posterior:

Pr[y = 1|x] = Pr φ [y = 1|x; H] . = ∇ -1 φ (H(x)) ∈ [0, 1] . (7)
∇ -1 φ plays the role of the link function (1). The quality of such an estimator shall be addressed in the following Section. = ln(x)/ ln(z) denote the base-z logarithm). From top to bottom, the losses are known as: squared loss, (normalized) logistic loss, binary logistic loss, Matsushita loss [?,?], linear Hinge loss, exponential loss, Amari's α-loss, for α ∈ (-1, 1) [?]. Strictly convex losses are A, B, C, D, F, G. Balanced convex losses are A, B, C, D (E corresponds to a limit behavior of balanced convex losses [?]). For each ψ, we give the corresponding estimators pc (x) (Theorem 1 and Eqs (9, 11) below: replace x by h opt (x)), and if they are balanced convex losses, the corresponding concave signature φ (See text for details).

ψ pc(x) φ A (1 -x) 2 1 2 (1 + x) x(1 -x) B log 2 (1 + exp(-x)) [1 + exp(-x)] -1 -x ln x -(1 -x) ln(1 -x) C log 2 (1 + 2 -x ) 1 + 2 -x -1 -x log 2 x -(1 -x) log 2 (1 -x) D -x + √ 1 + x 2 1 2 1 + x √ 1+x 2 x(1 -x) E 1 2 x(sign(x) -1) 1 if x > 0 0 if x < 0 2 min{x, 1 -x} F exp(-x) [1 + exp(-2x)] -1 N/A G 1 + 1-α 2 4 x -1+α 1-α 1 + 4-(1-α 2 )x 4+(1-α 2 )x 2 1-α -1 N/A

Strictly convex losses and the efficient estimation of posteriors

There is a rationale to use (7) as the posterior: the duality between natural and expectation parameters of exponential families, via Legendre duality [?,?], and the fact that the domain of the expectation parameter of one dimensional exponential families whose signature is (minus) a permissible function is the interval [0, 1] [?]. We improve below this rationale, with the proof that Bayes posteriors satisfy (7) for the classifier which is the population minimizer of (7).

Theorem 1. Suppose ψ strictly convex differentiable. The true surrogate risk

E D [ψ(y ic h(x))
] is minimized at the unique h opt (x) satisfying:

∇ ψ (-h opt (x)) ∇ ψ (h opt (x)) = p c (x) 1 -p c (x) . ( 8
)
Furthermore, is ψ is a balanced convex loss, then the population minimizer h opt of E D [ψ φ (y ic h(x))] satisfies:

p c (x) = ∇ -1 φ (h opt (x)) , (9)
for which

E D [ψ φ (y ic h opt (x))] = φ(p c (x)) -φ(0) φ( 1 /2) -φ(0) . (10)
(Proof omitted) Table 1 provides examples of expressions for p c (x) as in (9). Eq. ( 8) in Theorem (1) brings that we may compute an estimator pc (x) as:

pc (x) = ∇ ψ (-h(x)) ∇ ψ (h(x)) + ∇ ψ (-h(x)) . ( 11
)
This simple expression is folklore, at least for the logistic and exponential losses [?,?]. The essential contribution of Theorem 1 relies on bringing a strong rationale to the use of ( 7), as the estimators converge to Bayes posteriors in the infinite sample case. Let us give some finite sample properties for the estimation (7). We show that the sample-wise estimators of ( 9) are efficient estimators of (9); this is not a surprise, but comes from properties of exponential families [?].

What is perhaps more surprising is that the corresponding aggregation of classifiers is not a linear combination of all estimating classifiers, but a generalized ∇ -1 φ -mean.

Theorem 2. Suppose we sample n datasets S (c)

j , j = 1, 2, ..., n. Denote ĥopt,j the population minimizer for E S (c) j [ψ φ (y ic h(x))]. Then each pc,j (x) . = ∇ -1 φ ( ĥopt,j (x)
) is the only efficient estimator for p c (x). The corresponding classifier ĥopt aggregating all ĥopt,j , is: ĥopt (x)

. = ∇ φ 1 nx j:(x,.)∈S (c) j
∇ -1 φ ( ĥopt,j (x)) , ∀x ∈ ∪ j S j , where 1 ≤ n x ≤ n is the number of subsets containing x. = p c (x * ) (remark that µ ∈ dom(φ) = [0, 1] because φ is permissible) the expectation parameter of the exponential family, and thus θ = ∇ φ (µ). Using the fact that ∇ φ ⋆ = ∇ -1 φ , we get the score:

s(x|θ) . = ∂ ln p[x|θ] ∂θ = x -∇ φ ⋆ (θ) ,
and so x is an efficient estimator for ∇ φ ⋆ (θ) = µ; in fact, it is the only efficient estimator [?]. Thus, pc (x * ) is an efficient estimator for p c (x * ). There remains to use (9) to complete the proof of Theorem 2.

Leveraging and boosting Nearest Neighbors

The nearest neighbor rule belongs to the oldest, simplest and most widely studied classification algorithms [?,?]. We denote by NN k (x) the set of the k-nearest neighbors (with integer constant k > 0) of an example (x, y) in set S with respect to a non-negative real-valued "distance" function. This function is defined on Algorithm 1 Algorithm Universal Nearest Neighbors, unn(S, ψ, k)

Input: S = {(x i , y i ), i = 1, 2, ..., m, x i ∈ O, y i ∈ {-1, 1} C }, ψ strictly convex loss (Definition 1), k ∈ N * ; Let α j ← 0, ∀j = 1, 2, ..., m; for c = 1, 2, ..., C do Let w ← -∇ ψ (0)1; for t = 1, 2, ..., T do [I.0] Let j ← Wic(S, w); [I.1] Let δ j ∈ R solution of: i:j∼ k i y ic y jc ∇ ψ δ j y ic y jc + ∇ -1 ψ (-w i ) = 0 ; ( 12
) [I.2] ∀i : j ∼ k i, let w i ← -∇ ψ δ j y ic y jc + ∇ -1 ψ (-w i ) , (13)
[I.3] Let α jc ← α jc + δ j ; Output: H(x) . = j∼ k x α j • y j
domain O and measures how much two observations differ from each other. This dissimilarity function thus may not necessarily satisfy the triangle inequality of metrics. For the sake of readability, we let j ∼ k x denote the assertion that example (x j , y j ) belongs to NN k (x). We shall abbreviate j ∼ k x i by j ∼ k i. To classify an observation x ∈ O, the k-nn rule H over S computes the sum of class vectors of its nearest neighbors, that is: H(x) = j∼ k x 1 • y j , where • is the Hadamard product. H predicts that x belongs to each class whose corresponding coordinate in the final vector is positive. A leveraged k-nn rule is a generalization of this to:

H(x) = j∼ k x α j • y j , (14)
where α j ∈ R C is a leveraging vector for the classes in y j . Leveraging approaches to nearest neighbors are not new [?,?], yet to the best of our knowledge no convergence results or rates were known, at least until the algorithm unn [?].

Algorithm 1 gives a simplified version of the unn algorithm of [?] which learns a leveraged k-nn. Oracle Wic(S, w) is the analogous for nn of the classical weak learners for boosting: it takes learning sample S and weights w over S, and returns the index of some example in S which is to be leveraged.

[?] prove that for any strictly convex loss ψ, unn converges to the global optimum of the surrogate risk at hand. However, they prove boosting-compliant convergence rates only for the exponential loss. For all other strictly convex losses, there is no insight on the rates with which unn may converge towards the optimum of the surrogate risk at hand. We now provide such explicit convergence rates under the following Weak Learning Assumption:

WLA: There exist some ϑ > 0, ̺ > 0 such that, given any k ∈ N * , c = 1, 2, ..., C and any distribution w over S, the weak index chooser oracle Wic returns an index j such that the following two statements hold:

(i) Pr w [j ∼ k i] ≥ ̺; (ii) Pr w [y jc = y ic |j ∼ k i] ≤ 1 /2 -ϑ or Pr w [y jc = y ic |j ∼ k i] ≥ 1 /2 + ϑ.
Requirement (i) is a weak coverage requirement, which "encourages" Wic to choose indexes in dense regions of S. Before studying the boosting abilities of unn, we focus again on surrogate risks. So far, the surrogate risk (3) has been evaluated with respect to a single class. In a multiclass multilabel setting, we may compute the total surrogate risk over all classes as:

ε ψ S (H) . = 1 C C c=1 ε ψ S (h c , c) , ( 15
)
where H is the set of all C classifiers h 1 , h 2 , ..., h C that have been trained to minimize each ε ψ S (., c), c = 1, 2, ..., C. We split classifiers just for convenience in the analysis: if one trains a single classifier H : O × {1, 2, ..., C} → R like for example [?], then we define h c to be H in which the second input coordinate is fixed to be c. Minimizing the total surrogate risk is not only efficient to estimate posteriors (Section 3): it is also useful to reduce the error in label prediction, as the total surrogate risk is an upperbound for the Hamming risk [?]:

ε H S (H) . = (1/(mC)) C c=1 m i=1 I[y ic h c (x i ) < 0]
, where I[.] denotes the indicator variable. It is indeed not hard to check that for any strictly convex surrogate loss ψ, we have ε H S (H) ≤ (1/ψ(0)) × ε ψ S (H). We are left with the following question about unn:

"are there sufficient conditions on the surrogate loss ψ that guarantee, under the sole WLA, a convergence rate towards the optimum of (15) with unn ?"

We give a positive answer to this question when the surrogate loss meets the following smoothness requirement.

Definition 3.

[?] ψ is said to be ω strongly smooth iff there exists some ω > 0 such that, for all x, x ′ ∈ int(dom(ψ)),

D ψ (x ′ x) ≤ ω 2 (x ′ -x) 2
, where

D ψ (x ′ x) . = ψ(x ′ ) -ψ(x) -(x ′ -x)∇ ψ (x) ( 16
)
denotes the Bregman divergence with generator ψ [?].

Denote n j .

= |{i : j ∼ k i}| the number of examples in S of which (x j , y j ) is a nearest neighbor, and n * . = max j n j . Denote also H opt the leveraged k-nn which minimizes ε ψ S (H); it corresponds to the set of classifiers ĥopt of Section 3 that would minimize (3) over each class. We are now ready to state our main result (remark that ε ψ S (H opt ) ≤ ψ(0)).

Theorem 3. Suppose (WLA) holds and choose as ψ is any ω strongly smooth, strictly convex loss. Then for any fixed τ ∈ [ε ψ S (H opt ), ψ(0)], unn has fit a leveraged k-nn classifier H satisfying ε ψ S (H) ≤ τ provided the number of boosting iterations T in the inner loop satisfies:

T ≥ (ψ(0) -τ )ωmn * 2ϑ 2 ̺ 2 . ( 17
)
Proof sketch: To fit unn to the notations of ( 15), we let h c represent the leveraged k-nn in which each α j is restricted to α jc . We first analyze ε ψ S (h c , c) for some fixed c in the outer loop of Algorithm 1, after all α jc have been computed in the inner loop. We adopt the following notations in this proof: we plug in the weight notation the iteration t and class c, so that w (c) ti denotes the weight of example x i at the beginning of the "for c" loop of Algorithm 1.

ψ is ω strongly smooth is equivalent to ψ being strongly convex with parameter ω -1 [?], that is,

ψ(w) - 1 2ω w 2 is convex, ( 18
)
where we use notation ψ(x) .

= ψ ⋆ (-x). Any convex function h satisfies h(w ′ ) ≥ h(w) + ∇ h (w)(w ′ -w). We apply this inequality taking as h the function in (18). We obtain, ∀t = 1, 2, ..., T, ∀i = 1, 2, ..., m, ∀c = 1, 2, ..., C:

D ψ w (c) (t+1)i ||w (c) ti ≥ 1 2ω w (c) (t+1)i -w (c) ti 2 . ( 19
)
On the other hand, Cauchy-Schwartz inequality and (12) yield:

∀j ∈ S, i:j∼ k i r (c) ij 2 i:j∼ k i (w (c) (t+1)i -w (c) ti ) 2 ≥   i:j∼ k i r (c) ij w (c) ti   2 . ( 20
)
Lemma 1. Under the WLA, index j returned by Wic at iteration t satisfies

i:j∼ k i w (c) ti r (c) ij ≥ 2ϑ̺. ( proof
omitted) Letting e(t) ∈ {1, 2, ..., m} denote the index of the example returned at iteration t by Wic in Algorithm 1, we obtain:

1 m m i=1 D ψ w (c) (t+1)i ||w (c) ti ≥ 1 2ωm i:e(t)∼ k i w (c) (t+1)i -w (c) ti 2 (21) ≥ 1 2ωm i:e(t)∼ k i r (c) ie(t) w (c) ti 2 i:e(t)∼ k i r (c) ie(t) 2 (22) ≥ 2ϑ 2 ̺ 2 ωm × 1 i:e(t)∼ k i r (c) ie(t) 2 (23) = 2ϑ 2 ̺ 2 ωmn e(t) ≥ 2ϑ 2 ̺ 2 ωmn * . (24)
Here, (21) follows from ( 19), ( 22) follows from ( 20), ( 23) follows from Lemma 1, and ( 24) follows from the fact that r (c) ie(t) = ±1 when e(t) ∼ k i. Summing these inequalities for t = 1, 2, ..., T yields:

T t=1 1 m m i=1 D ψ w (c) (t+1)i ||w (c) ti ≥ 2T ϑ 2 ̺ 2 ωmn * . (25)
Now, unn meets the following property ([?], A.2):

ε ψ S (h (t+1)c , c) -ε ψ S (h tc , c) = - 1 m m i=1 D ψ w (c) (t+1)i ||w (c) ti , (26)
where h (t+1)c denotes h c after the t th iteration in the inner loop of Algorithm 1. We unravel (26), using the fact that all α are initialized to the null vector, and obtain that at the end of the inner loop, h c satisfies:

ε ψ S (h c , c) = ψ(0) - T t=1 1 m m i=1 D ψ w (c) (t+1)i ||w (c) ti ≤ ψ(0) - 2T ϑ 2 ̺ 2 ωmn * , (27)
from ( 25). There remains to compute the minimal value of T for which the right hand side of ( 27) becomes no greater than some user-fixed τ ∈ [0, 1] to obtain that ε ψ S (h c , c) ≤ τ . The aggregation of the bounds for each c = 1, 2, ..., C in ε ψ S (H) is immediate as it is an average of ε ψ S (h c , c) over all classes. Hence, this minimal value of T , used for each c = 1, 2, ..., C, also yields ε ψ S (H) ≤ τ . This ends the proof of Theorem 3. Section 3 has underlined the importance of balanced convex losses in obtaining simple efficient estimators for conditional class probabilities. Coupled with Theorem 3, we now show that unn may be a fast approach to obtain such estimators.

Corollary 1. Consider any permissible φ that has been scaled without loss of generality so that φ( 1 /2) = 1, φ(0) = φ(1) = 0. Then for the corresponding balanced convex loss ψ = ψ φ and under the WLA, picking

T > mn * 2ϑ 2 ̺ 2 min x∈(0,1) ∂ 2 φ ∂x 2 (28)
in the inner loop of unn, for each c = 1, 2, ..., C, guarantees to yield an optimal leveraged k-nn H, satisfying ε ψ S (H) = ε ψ S (H opt ). This leveraged k-nn yields efficient estimators for conditional class probabilities, for each class, by computing:

pc (x) = ∇ -1 φ (h c (x)) . ( 29
)
(Proof omitted) For the most popular permissible functions (Table 1), quantity min x∈(0,1) ∂ 2 φ ∂x 2 does not take too small value: its values are respectively 8, 4/ ln 2, 4 for the permissible functions corresponding to the squared loss, logistic loss, Matsushita loss. Hence, in these cases, the bound for T in ( 28) is not significantly affected by this term.

δ jc , see ( 30)

g : w i ← g(w i ) A 2W jc -1 w i -2δ jc y ic y jc B ln W jc 1-W jc w i w i ln 2+(1-w i ln 2)×exp(δ jc y ic y jc ) C log 2 W jc 1-W jc w i w i +(1-w i )×2 δ jc y ic y jc D 2W jc -1 2 √ W jc (1-W jc ) 1 - 1-w i + √ w i (2-w i )δ jc y ic y jc 1+δ 2 jc w i (2-w i )+2(1-w i ) √ w i (2-w i )δ jc y ic y jc E N/A N/A F 1 2 ln W jc 1-W jc exp(-δ jc y ic y jc ) G 4 1-α 2 (Wjc) 2 1-α -(1-W jc ) 2 1-α (Wjc) 2 1-α +(1-W jc ) 2 1-α 4 1-α 2 × 1-α 2 4 δ jc y ic y jc + 1+α 2 √ w i 1-α -2 1-α
Table 2: Computation of δ jc and the weight update rule of our implementation of unn, for the strictly convex losses of Table 1. unn leverages example j for class c, and the weight update is that of example i (See text for details and notations).

Experiments

Computing leveraging coefficients and weights update

Fix for short S

∇ ψ δ + ∇ -1 ψ (-w i ) = i∈S (c) j- ∇ ψ -δ + ∇ -1 ψ (-w i ) .
There is no closed form solution to this equation in the general case. While it can be simply approximated with dichotomic search, it buys significant computation time, as this approximation has to be performed for each couple (c, t). We tested a much faster alternative which produces results that are in general experimentally quite competitive, consisting in solving instead: i∈S (c) δ). We get equivalently that δ satisfies:

j+ w i ∇ ψ (δ) = i∈S (c) j- w i ∇ ψ (-
∇ ψ (-δ) ∇ ψ (δ) = W jc 1 -W jc , (30) with W jc .
= ( i∈S (c)

j+ w i )/( i∈S (c) j+ w i + i∈S (c) j- w i ).
Remark the similarity with (8). Table 2 gives the corresponding expressions for δ and the weight updates.

General experimental settings

We have tested three flavors of unn: with the exponential loss (F in Table 1), the logistic loss (B in Table 1) and Matsushita's loss (D in Table 1). All three a respectively referred to as unn(exp), unn(log) and unn(Mat). It is the first time this last flavor is tested, even from the classification standpoint. We chose support vector machines (SVM) as the contender against which to compare unn:

-4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 -2 -1 0 1 2 -1 0 1 2 3 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 -2 -1 0 1 2 -0.5 0 0.5 1 1.5 S unn(exp) unn(Mat)
Fig. 1: From left to right: example of simulated dataset with σ = 1.1; the estimated posterior for class 1 obtained by unn(exp); the corresponding gridwise kl divergence for class 1; the estimated posterior for class 1 obtained by unn(Mat); the corresponding gridwise kl divergence for class 1 (see ( 32) and text for details). SVM are large margin classifiers with convenient methods to obtain estimators for the posteriors [?]. For all these algorithms, we compute the estimation of posteriors as follows: we use (11) for unn(exp), (29) for unn(log) and unn(Mat). For SVM, we use the method of [?], which, given a SVM f for class c, forms the posterior:

pc (x) . = 1 1 + exp(af (x) + b) , (31)
where a and b are estimated by maximizing the log-likelihood of the training sample with a five-fold cross validation. We use two metrics to evaluate the algorithms. On simulated data, we compute an estimate of the Kullback-Leibler (kl) divergence between the true and estimated posterior which is a class-wise average of the divergence:

D KL (p p) . = c Pr[c] Pr[x]p c (x) ln pc (x) p c (x) dµ . (32)
Our estimate, DKL (p p) relies on a simple fine-grained grid approximation of the integral over the subsets of O of sufficient mass according to µ. On real data, we compute a couple of metrics. First, we compute the F-measure of the classifiers (the harmonic average of precision and recall), based on thresholding the probabilistic output and deciding that x belong to class c iff pc (x) ≥ κ, for varying κ ∈ ( 1 /2, 1). Second, we compute the rejection rate, that is, the proportion of observations for which pc (x) < κ. Either we plot couples of curves for the F-measure and rejection rates, or we summarize both metrics by their average values as κ ranges through ( 1 /2, 1), which amounts to compute the area under the corresponding curves.

Results on simulated data

We evaluated the goodness-of-fit of the estimates on simulated datasets with the following experiments. We crafted a general domain consisting of C = 3 equiprobable classes, each of which follows a Gaussian N(µ, σI), for σ ∈ [0.1, 1.1] with steps of 0.005, and µ remains the same. For each value of σ, we compute the average over ten simulations, each of which consists of 1500 training examples and 4500 testing examples. We get overall several thousands datasets, on which all algorithms are tested. Figure 1 presents an example of such datasets, along with results obtained by unn(exp) and unn(Mat) from the standpoints of the posterior estimates and kl-divergence on the same class. The estimators are rather good, with the largest mismatches (kl-divergence) located near the frontiers of classes. Also, unn(Mat) tends to outperform unn(exp).

Figure 2 synthesizes the results from the kl-divergence standpoints. Two clear conclusions can be drawn from these results. First, unn is the clear winner over SVM for the posteriors estimation task. The results of each flavor of unn is indeed better than those of SVM, with linear or radial basis functions kernel, by orders of magnitude. This is all the more important as the kernels we used are the theoretical kernels of choice given the way we have simulated data. The second conclusion is that unn(Mat) is the best of all flavors of unn, a fact also confirmed by the synthetic results of Table 3. The kl divergences of unn(Mat) are in general of minute order with respect to the others. Its behavior (Figure 2) is also monotonous: it is predictable that it increases with the degree of overlap between classes, that is, with σ. From the classification standpoint, the average F-measure metrics display a very slight advantage to SVM, and in particular to linear SVM. The results of SVM with radial basis functions kernel are approximately the same as those of unn(Mat).

The most important conclusion that can be drawn from the simulated data is shown in Figure 3: as the number of boosting iterations T increase, unn does not overfit posteriors in general. The only hitch -not statistically significant -is the case σ > 0.7 for unn(Mat), but the differences are of very small order compared to the standard deviations of the kl-divergence. Table 4: Area under the (F)-measure (in percentage) and (R)ejection rate on the SUN databases. For each database, the best F and R are written in bold faces.

Results on the SUN database domains

We have crafted, out of the challenging SUN computer vision database [?], three datasets, consisting in taking all pictures from the first ten (SUN 10), twenty (SUN 20) or thirty (SUN 30) classes. We have compared unn(exp), unn(log), unn(Mat) and SVM on each dataset, by computing the average values, over the threshold κ, of the F-measure and the rejection rate. Table 4 summarizes the results obtained. This table somehow confirms that classification and posterior estimation may be conflicting goals when it comes to boosting [?,?], as unn(Mat) achieves very poor results compared to the other algorithms. Furthermore, unn(exp) appears to the clear winner over all algorithms for this classification task. These results have to be appreciated in the light of the rejection rates: in comparison with the other algorithms, unn(Mat) rejects a very 3 for notations).

small proportion of the examples, this indicating a high recall for the algorithm. Figure 4 completes the picture by detailing F-measure and rejection rates plots. The F-measure plots clearly display the better performances of unn(exp) compared to the other algorithms, and the fact that unn(Mat) displays very stable performances. The rejection rates plots show that unn(Mat) indeed rejects a very small proportion of examples, even for large values of κ.

Conclusion

Boosting algorithms are remarkably simple and efficient from the classification standpoint, and are being used in a rapidly increasing number of domains and problems [?]. In some sense, it would be too bad that such successes be impeded when it comes to posterior estimation [?]. Experimental results display that this estimation is possible, but it necessitates a very fine tuning of the algorithms [?]. The point of our paper is that estimating class conditional probabilities may be possible, without such tedious tunings, and sometimes even without overfitting, if we boost topological approaches to learning like nearest neighbors. There is a simple explanation to this fact. For any classifier, the conditional class probability estimation for some x in ( 7) is be the same as for any other observation in the vicinity of x, where the "vicinity" is to be understood from the classifier standpoint. When boosting decision trees, the vicinity of x corresponds to observations classified by the same leaf as x. As the number of leaves of the tree increases, the vicinity gets narrowed, which weakens the estimation in (7) and thus overfits the corresponding estimated density. Ultimately, linear combinations of such trees, such as those performed in AdaBoost, make such a fine-grained approximation of the local topology of data that the estimators get irreparably confined to the borders of the interval [0, 1] [?]. Nearest neighbors do not have such a drawback, as the set of k-nearest neighbors in S of some observation x spans a region of O which does not change throughout the iterations. Furthermore, nearest neighbor rules exploit a topology of data which, under regularity conditions about the true posteriors, also carries out information about these posteriors. For these reasons, nearest neighbors might be a key entry for a reliable estimation of posteriors with boosting. Because of the wealth of "good" surrogates, this opens avenues of research to learn the most accurate surrogate on a data-dependent way, such as when it is parameterized (Amari's α-loss, see Table 1).

Acknowledgments

R. Nock acknowledges a visiting grant from Institut Universitaire de France / Université de Nice.

  Proof. Let us pick ψ = φ ⋆ in (6) and condition p[x|θ] . = p[x|θ; x * ] for each x * ∈ O. We let µ .

=

  {i : j ∼ k i ∧ y ic = by jc } for b ∈ {+, -}. (12) may be simplified as i∈S (c) j+

Fig. 2 :

 2 Fig. 2: Average kl-divergence as a function of σ on simulated datasets, for unn(exp), unn(log), unn(Mat) (left, k = 10) and SVM (right). Notice the differences in y-scales.

Fig. 3 :

 3 Fig.3: Average kl-divergence as a function of σ on simulated datasets, for unn(exp) (left), unn(log) (center), unn(Mat) (right), when the number of boosting iterations T varies in {m, 2m, 5m, 10m}. The color code in the same on each plot. Notice the differences in the y-scale for unn(Mat) (see text for details).

Fig. 4 :

 4 Fig. 4: F-measure (top row) and rejection rates (bottom row) on the SUN domains, with C = 10 (left), C = 20 (center) and C = 30 (right, see Table3for notations).

Table 1 :

 1 Examples of surrogates ψ (Throughout the paper, we let ln denote the base-e logarithm, and log z (x)

	.

Table 3 :

 3 Average results over simulated data, for unn(exp), unn(log), unn(Mat) with four different values of k, and for support vector machines with linear (SVM l ) or radial basis functions (SVM r ) kernel.


est destinée au dépôt

  Béatrice David. Introduction: La conquête de l'urbain et les esquisses d'un nouveau rapport à la " nature ". L'éclatement de la ville sur les campagnes ? Les recompositions du rapport urbain-rural dans les mobilités de loisirs en Chine au début du XXIe siècle Journée d'études organisée dans le cadre du Pôle Ville de l'Université Paris-8, en partenariat avec le Centre d'études sur la Chine moderne et contemporaine (CECMC-UMR 8173 Chine Corée Japon/EHESS-CNRS) et le Pôle de Recherches pour l'Organisation et la Diffusion de l'Information Géographique (PRODIG-UMR 8686/ Paris1-Panthéon-CNRS) Université Pari, Dec 2010, Paris St-Denis, France. hal-00702803 L'éclatement de la ville sur les campagnes ? Journée d'étude sur les recompositions du rapport urbain-rural dans les mobilités de loisirs en Chine au début du XXIe siècle. Nouveaux lieux, nouvelles pratiques et représentations Introduction: La conquête de l'urbain et les esquisses d'un nouveau rapport à la « nature » Béatrice David, anthropologue, maître de conférences au département de sociologie, Université paris 8, UFR HLSG

2 .

 2 Le tourisme et la Chine. La rencontre de deux phénomènes massifs L'essor fulgurant des mobilités du loisir en Chine, depuis le début des réformes engagées au début des années 1980, n'a pas tardé à placer l'étude du tourisme au centre de la réflexion des études en sciences sociale. Le caractère massif du phénomène qui est essentiellement domestique, en fait aussi le lieu privilégier d'où examiner les mutations en cours. Je laisserai à nos jeunes collègues géographes qui ouvriront la première séance le soin de nous rappeler l'expression chiffrée de cette autre manifestation de l'immensité chinoise. Un des temps forts de la rencontre des études d'anthropologie avec la Chine fut l'organisation en 2001 d'une conférence internationale à Kunming qui réunit les théoriciens de la discipline tels que le sociologue israélien Erik Cohen et l'anthropologue américain Nelson Graburn aux côtés des chercheurs chinois et étrangers. Les actes en anglais et ceux plus fournis en Chine Tourism, Anthropolgy and China (Tan Chee-Beng, CH Cheung et Yang Hui) ont ouvert la voie à des recherches fécondes. Le renouveau de l'anthropologie en terre chinoise coïncida avec les premiers signes d'un phénomène que les sciences sociales non chinoises, particulièrement en anthropologie, ont surtout sondé dans les régions de la Chine intérieure où se concentrent les nationalités minoritaires de l'Etat nation plurinational chinois. J'y reviendrai dans un instant. La réflexion doit considérablement aux travaux du géographe Tim Oakes de l'Université de Colorado dont les travaux dans le Guizhou sur les Dong ont débattu la question qui, à mon avis, justifie et légitime l'intérêt croissant porté par les sciences sociales à un phénomène au coeur des mondes contemporains. Pal Nyri, que je remercie également de sa présence parmi nous, a avec bonheur élargi cette exploration dans son ouvrage sur les « scenic spots » qui met particulièrement l'accent sur le rôle de l'Etat dans la production du paysage où se façonne l'expérience touristique.3. Le postulat tourisme et modernité et son application à l'étude des expériences chinoises de la modernité Si l'on admet que la pratique du tourisme est « une des caractéristiques qui définissent l'homme (post)moderne », le tourisme, envisagé comme lieu d'une expérience concrète de la modernité, offre « la métaphore succincte » (Oakes, 1998) à partir de laquelle appréhender la modernité chinoise, ou plus exactement les différentes multiples de la modernité chinoise qui se réalise dans le signifiant que recouvre le mot Chine, saisi dans sa contemporanéité. Ce postulat de départ d'une relation entre tourisme et modernité, qui inspire et irrigue la réflexion et l'approche des sciences sociales du monde chinois contemporain, doit être reposé à la lumière des situations locales en mutation dans une « Chine en mouvement » profondément remodelée par l'essor des mobilités contemporaines du travail et d'agrément. Le tourisme, regard urbain sur quoi exactement ? par qui exactement ? Les mobilités de loisirs sont un important vecteur d'une modernité qui impose un regard que je qualifierai d'essentiellement urbain sur les lieux fréquentés temporairement. Vers quoi s'est porté le regard ? Durant les premières décennies qui ont marqué l'envol de l'activité touristique en Chine au début des années 1980, l'approche du tourisme dans les travaux de sciences sociales (principalement en anthropologie et géographie) a surtout porté sur les dynamiques identitaires à l'oeuvre dans les lieux du « tourisme ethnique », dans les régions des nationalités minoritaires, particulièrement celles du sud-ouest[1]. La saillance du rôle de l'Etat dans la production des « cultures » réifiées des «minorités ethniques » engageait en effet à mettre au centre de l'observation et de l'analyse les lieux par excellence de la mise en scène de ces « altérités internes ». Cette mise en tourisme des ethnicités minoritaires informe souvent davantage sur les représentations du groupe dominant dans une société chinoise en mutations[2], portée par une quête d'exotique par laquelle confirmer la centralité politique et économique des Han dans la définition de la nation chinoise moderne. Le tourisme est aussi une importante ressource pour la mise en scène de l'ethnicité, aussi bien l'ethnicité officielle fixée dans le cadre de la catégorie politique et culturelle des Han dans la définition de la nation chinoise moderne[3]. C'est dans cette perspective que la fréquentation des « régions à minorités ethniques » par les touristes han participait d'un projet culturel et politique d'affirmation d'une idée de la nation chinoise une et indivise[4]. Les contributions présentées aujourd'hui de Frédérique Guyader et de Pascale-Marie Milan s'inscrivent dans la ligne directe de ces travaux pionniers, et saisissent le phénomène à un moment d'intensification d'une politique culturelle d'Etat (chez les Naxi), et d'extension continue du tourisme chez les Mosuo du lac de Lugu. La conquête de l'urbain et les esquisses d'un nouveau rapport à la « nature » Cette journée d'étude invite donc à poursuivre la réflexion sur le rôle des mobilités du tourisme dans la construction d'une « expérience chinoise de la modernité » qui recompose les paysages, réels et imaginaires, de l'urbanité et de la ruralité. La « ville » et la totalité des expériences et des images qui lui sont associées, sont devenues le signifiant conquérant d'une condition moderne chinoise contemporaine (et universelle.) L'essor des mobilités touristiques produit « un éclatement de la ville sur les campagnes » qui, sans abolir les distances entre les deux, rogne l'opposition radicale et arbitraire issue des expériences du siècle dernier. Pour les urbains de la Chine en transition depuis le début des réformes de la fin des années 1970, le rural, sous ses formes localisées diverses, fut longtemps (et le demeure) synonyme de pauvreté, d'un style de vie arriéré. Le signifié est-il si négatif ? Mlle Ren Yi, doctorante en économie du développement, nous invite, avec raison, à considérer le signifié plus ancien du mot ancien nong dans l'ancienne civilisation chinoise agraire. Il est exact que sous le mot « campagne » n'a pas toujours eu la consonance négative que lui ont donné les expériences dramatiques de la Révolution culturelle. On peut se demander dans quelle mesure la campagne réinventée pour les touristes de la génération post-révolution culturelle n'a pas pour modèle les images bucoliques de la campagne auxquelles restent associé le poèteTao Yuanming. Le développement des flux touristiques encouragés par le développement et le réaménagement du temps des loisirs depuis la fin des années 1990 (re)conduit désormais son lot, chaque année plus nombreux, d' « urbains » vers des espaces ruraux réaménagés et offerts à la fréquentation touristique sous les étiquettes de « tourisme rural », de « tourisme vert », « bleu », « blanc », « d'écotourisme », shengtai lüyou, de « tourisme environnemental » et d'un « tourisme rouge » aux accents patriotiques qui, au prétexte de conduire vers les sites révolutionnaires, parcourt aussi les lieux et les paysages emblématiques de l'Etat-nation (David, 2007.) Ces mots nouveaux résonnent comme autant d'espoirs d'expériences inédites qui inscrivent les acteurs sociaux dans le « monde (ré)enchanté » du tourisme, lieu d'une expérience de la modernité partagée à l'échelle universelle. Comme toute destination touristique, la « campagne » fréquentée dans le cadre des mobilités de loisirs est un lieu inventé, re-dessiné par les imaginaires et les attentes de visiteurs pour l'essentiel en provenance des zones urbaines. Les formes de la rencontre entre le rural et l'urbain qui se construit dans le temps et l'espace de la fréquentation touristique des lieux (re)façonnés pour la consommation touristique doivent être examinées, particulièrement dans les nouveaux sites où se réalisent ces « désirs d'ailleurs. » Quels sites, quelles pratiques sont explorées aujourd'hui. Comment se transforme le regard sur les campagnes ? Les thèmes et les sites présentés aujourd'hui, et les démarches des intervenants Le tourisme chinois est entré dans une période de maturité. Ce qui se manifeste par une diversification des sites, l'émergence de nouvelles pratiques. Benjamin Taunay et Emmnanuel Véron nous montrerons qu'il n'est plus envisageable de considérer le tourisme intérieur de manière monolithique (Veron p. 2.) Benjamin Taunay, dont on a pu lire l'étude récente sur les pratiques émergentes balnéaire en Chine du sud, montera à partir de plusieurs exemples dans le Guangxi, le développement d'un « tourisme rural » dans les régions plus éloignées des métropoles des plaines centrales et surtout côtières. Une extension à maints égards symptomatique d'une transformation du regard sur les campagnes (p. 9) Emmanuel Véron développera ensuite les différentes temporalités d'un tourisme périurbain à la périphérie de la métropole émergente de Shanghai qui s'inscrit aujourd'hui dans le cadre de la politique de « construction des nouvelles campagnes socialistes » engagée par le XI plan quinquennal en cours. Ces travaux montrent le rôle décisif de l'Etat dans la recomposition des espaces qui offrent un paysage qui n'est ni strictement rural ni urbain. L'intervention de Caroline Bodelec sur le musée dans le Shaanxi en Chine centrale apportera dans l'après-midi l'exemple d'une initiative locale de patrimonialisation qui doit peu à l'action des autorités locales, et surtout à une chaîne de personnages au premier plan duquel le très charismatique Jin Zhilin, relayé localement par un enseignant. Mais aussi le rôle financier d'une fondation étranger Ford dans la promotion d'un « musée privé » dont le développement semble aujourd'hui menacé. L'exemple des Mosuo présenté par Pascale-Marie Milan à la suite des travaux de Walsh est également exemplaire des ressources identitaires du tourisme pour échapper, ou contourner les catégorisations de l'Etat. Intégré aux Naxi, les Mosuo en devant onbjet touristique ont aussi acquis une semi reconnaissance de leur ethnicité. Ren Yi ensuite s'arrêtera plus longuement, sur l'exemple d'un eco-agriculture, l'occasion de nouveau de confirmer le rôle saillant, non plus d'un rertraité, mais d'un secrétaire du parti local dans la création d'une nouvelle forme de loisir à propos de laquelle on retient également qu'elle attire un vaste public en raison de sa faible coût. Le tourisme une indicateur de statut aura interrogé le matin Nathalie Claude-Sollier. Le mot « tourisme » est-il soluble dans le vocable au demeurant flou « classe moyenne », personnes à revenus moyens. La diversité des pratiques que recouvrent les mobilités loisirs invite à se méfier des généralisations. Enfin, Laure Dumont, dont je me ferai aujourd'hui l'interprète, explore les sentiers encore peu battus d'un écotourisme dans les steppes enneigées des éleveurs de rennes evenke de la Mongolie intérieure. La production « d'un nouvel espace naturel et protégé pour penser vert et écologique ».

1. Le tourisme, objet des sciences sociales. La réhabilitation

L'intérêt croissant dans les études en sciences sociales pour le tourisme n'est plus à démontrer, après une longue période de défiance voire de mépris, particulièrement au sein des sciences sociales francophones. Le tourisme n'avait pas la cote, envisagé dans ses dimensions économiques ou à travers ses impacts négatifs dans des sociétés vues essentiellement comme des récepteurs passifs. De « cette maturation tardive du paradigme touristique sous la loupe de l'anthropologie », selon l'expression heureuse du québécois Jean Michaud (2000) qui en fut l'un des pionniers en Asie du sud-est avec Michel Picard sur Bali, notre collègue anthropologue Saskia Cousin, que je remercie d'avoir accepté notre invitation, a largement rendu compte dans son ouvrage (Sociologie du tourisme), et la liste qu'elle anime et son séminaire témoigne de ce que l'on a pu appeler ailleurs, cette récente réhabilitation (Doquet & Evrard, 2008, numéro thématique de Civilisations) du tourisme comme objet de recherche après une longue période d'hésitations de la discipline face à cet objet. Une réhabilitation qui, à mon avis, est due à deux facteurs principaux. 1. d'

abord l'explosion des mobilités touristiques dans

	un monde « globalisé », confronté de manière inégale aux effets multiples
	de la mise en relation généralisée qui caractériserait nos conditions humaines contemporaines.
	La réhabilitation du tourisme comme objet de recherche signe aussi le changement du regard
	porté par les sciences sociales sur le contemporain, particulièrement par une anthropologie qui
	ne se reconnaît guère dans les terrains et les situations de la discipline dont elle est issue,
	l'ethnologie. La rencontre, longtemps retardée, était inévitable. A défaut sans doute de
	contribuer à renouveler voire à dynamiser l'approche épistémologique du phénomène, l'on
	reconnaîtra ici à l'étude de ces « tourismes émergents » la vertu de nous conduire « en
	touristes des sciences sociales » vers les lieux de productions identitaires et de réinvention de
	soi d'où éclairer les processus à l'oeuvre dans la Chine d'aujourd'hui.


Thermo-hydro-mechanical modelling of unsaturated porous media coupling damage and plasticity Le Pense S., Gatmiri B., Pouya A.

Université Paris-Est, Laboratoire Navier (ENPC/IFSTTAR/CNRS), Ecole des Ponts ParisTech,6 & 8 av Blaise Pascal,France. e-mail : solenn.le-pense@cermes.enpc.fr Cracking Modes: a. Traction, "splitting effects". b. Compression, "crossing effects". (Ortiz, 1985) If we consider three orthogonal sets of parallel non-interacting microcracks :

Second order damage tensor: Kachanov, 1992) Objective: to determine relevant nucleation and propagation criteria of microcracks and kinetic laws in microscopic level (REV)

Ω = 3 j=1 d i • n i ⊗ n i (

Advantages

◮ Ability to account for physical mechanisms involved in nucleation and growth of microcracks

Weaknesses

◮ Homogenisation procedure =⇒ difficulties of use in practical applications Coupling of damage and plasticity in unsaturated geomaterials

Micromechanical approach ε σ E 0 E 0 E(Ω) ε irr ε ed ε e ε σ E 0 E 0 E(Ω)

◮◮

  Solid: soil skeleton ◮ Liquid: water, dissolved air ◮ Gas: air and water vapour Typical Desorption and absorption curves for a silty soil(Fredlund, Xing & Huang, 1994) Hydraulic behaviour◮ Suction ր =⇒ soil stiffening ◮ Collapse phenomenon Choice of the stress variables ◮ Elastic behaviour ◮ Isotropic compression virgin line ◮ Yielding surface + hardening laws ◮ Plastic flow rule (associated or non-associated)◮ Critical state Most of the current models are derived from the Barcelona Basic Model (BBM) (p, q, s) stress space(Alonso, Gens & Josa, 1990) Elasto-plastic models Classic models: use of stress and suction as variables Generalized effective stress: work-conjugate stress and strain variables.σ -(u a -S r (u a -u w )) ←→ ǫ φ(u a -u w )

◮

  Non-linearity of stress-strain relationship ◮ Deterioration of elastic properties ◮ Induced material anisotropy ◮ Irreversible damage strains due to residual crack opening ◮ Unilateral response due to crack closure effect Objective: to use internal variables to represent material damage state; formulated in the irreversible thermodynamics frameworkAdvantages To develop a thermodynamically consistent thermo-hydro-mechanical model for unsaturated geomaterials coupling elasto-plasticity and damageMain issues:◮ What is the relative importance of plasticity and damage phenomena? Which one appears first? ◮ How does plasticity influence damage apparition and evolution? ◮ How does damage influence plasticity yield surface and plastic flow rules?


Yoshimitsu Fukushima

Fabián Luis

Bonilla

Oona Scotti

John Douglas

Site classification using horizontal-to-vertical response spectral ratios and its impact when deriving empirical groundmotion prediction equations

Keywords: H, V, response spectral ratio, site classification, attenuation relation, predominant period

, for which stations were originally classified as simply rock or soil. The calculation of average H/V response spectral ratios permits the majority of sites in the database to be unambiguously classified. Soft soil conditions are clearly apparent using this technique.

Ground-motion prediction equations are then computed using this alternative classification scheme. The aleatoric variability of these equations (measured by their standard deviations) is slightly lower than those derived using only soil and rock classes. However, perhaps more importantly, predicted response spectra are radically different to those predicted using the soil/rock classification. In addition, since the H/V response spectral ratios were used to classify stations the predicted spectra for different sites show clear separation. Thus, site classification using the predominant period appears to be partially mapped into the site coefficients of the ground-motion model.

Introduction

It is well known that precise site classification is important in determining accurate empirical ground-motion prediction relations. However, possessing a good knowledge of site conditions is rather exceptional. Even well characterized sites do not always have complete geotechnical information down to the bedrock. For example in Japan, the surface array K-net has geotechnical characterization down to a maximum depth of 20 m. Thankfully, its complementary borehole array KiK-net, has information down to 100 or 200 m depth depending on the site [START_REF] Pousse | Eurocode 8 design response spectra evaluation using K-net Japanese database[END_REF]. This important, but expensive, information is almost non-existent for most strong-motion networks in the world, and Europe is not an exception. Some previous European empirical ground motion prediction relations (e.g. [START_REF] Berge-Thierry | New empirical response spectral attenuation laws for moderate European earthquakes[END_REF][START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF] used a general rock/soil classification scheme due to this lack of detailed site information. This site classification, based mainly on geological information, came with the strong-motion data provided by the data contributing agencies. A general soil class, however, may contain both stiff and soft soils, and a general rock class may include sites with weathered rock or thin soil layers at the surface and hard bedrock stations. In addition, the difference between predicted ground motions at soil and rock stations predicted by these relations is not remarkable, being less than 70% [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF]. Therefore, a better site classification is required for precise strong ground motion estimation.

Site classification is ultimately related to site amplification effects. These can vary from simple 1D to complex 3D effects depending on the geometry and impedance contrast of the ground structure. Such effects can only be calculated for a few specific locations, for example the Los Angeles basin and Ashigara valleys, where 3D ground structural information has been acquired (e.g. [START_REF] Olsen | Three-dimensional simulation of earthquakes on the Los Angeles fault system[END_REF][START_REF] Pitarka | Modeling of strong motion in Ashigara Valley for the 1990 Odawara, Japan, earthquake[END_REF]. Nonetheless, even at these locations, the maximum frequency of the computed basin response is about 1 Hz, being proportional to the degree of knowledge of the geological structure for wavelengths of a few hundred meters.

In particular situations, an approximation of the theoretical site amplification can be computed using 1D multiple reflection theory. Yet, the cost of deep boring, down to the bedrock, at all sites is unrealistic. In the USA, average shear-wave velocity in the first 30 m, here after called Vs30, is used as an alternative parameter to classify soil conditions (Table 1). This parameter provides more information than surface geological data since impedance contrast at least down to this depth is reflected in the site classification.

However, the transfer function should not be limited to the first 30m; on the contrary, it should be continued down to bedrock.

An alternative, but less well-known, site classification scheme originally proposed by [START_REF] Kanai | On Microtremors. VIII[END_REF] is based on the site's predominant period. This is used in Japan for the seismic design code of highway bridges (Japan Road Association 1980[START_REF]Specifications for Highway Bridges Part V, Seismic Design[END_REF], and it is also given in Table 1. This classification does not exactly correspond to the Vs30 scheme, but it represents a combination between Vs and the thickness of soil layers. One advantage of this classification is that deep geological profiles and high shear-wave velocities are mapped to the resonance frequency. A recent study using Japan data [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF] proposed four site categories using this method. These authors used this site classification rather than the traditional description based on Vs30 because thick stiff soil layers overlying hard rock amplify long-period motions.

Conversely, thin soft soil layers amplify short periods. This means that Vs30 does not always capture the predominant period of the site, since it represents only the shallowest portion of the geological profile. A similar technique was successfully applied by [START_REF] Lang | Instrumental subsoil classification of Californian strong motion sites based on single-station measurements[END_REF] for sites in California. Using a classification scheme based on predominant period presents two advantages.

The first one is that strong-motion prediction will account for the entire site frequency content and not only shallow geological effects. In addition, at many sites easily available direct field measurements can be performed to estimate the site predominant period using either ambient noise measurements or H/V spectral ratios on earthquake data (e.g. [START_REF] Bonnefoy-Claudet | H/V ratio: a tool for site effects evaluation. Results from 1-D noise simulations[END_REF]. Furthermore, when computing empirical ground-motion prediction relations, [START_REF] Lussou | Seismic design regulation codes: Contribution of K-net data to site effect evaluation[END_REF] showed that, for K-net data, site classification by a combination of Vs30 and H/V predominant period reduced the standard deviation of the computed equation. [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF] classified K-net sites using average H/V response spectral ratios and obtained a relation that well modeled local site effects and also reduced the model's uncertainty. This procedure potentially has application to data classified into general categories such as rock or soil, or that use no classification at all.

In the present study, we use the H/V response spectral ratio, which was proposed by [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF], to attempt a site classification based on predominant period. It is not the objective of this paper to derive new empirical ground-motion estimation equations. We rather focus on the site classification procedure, its shortcomings and its general impact on the shape of the predicted response spectra.

Data

In this paper, we use the data used by [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF] to derive empirical ground-motion prediction equations. This dataset mainly consists of recorded strong motion from west Eurasia (mainly Europe). However, to better constrain near-fault saturation parameters, earthquakes from California and the 1995 Hyogo-ken Nanbu (Kobe) (Japan) earthquakes were included. Only shallow crustal earthquakes are selected in order to derive models for a homogeneous tectonic environment. We also adopt the same magnitude scale and distance metric as [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF]. Namely, magnitude M is the moment magnitude Mw when available; otherwise Ms (the surface-wave magnitude) is used (this is used for a number of smaller European events). The distance is defined as the closest distance from the site to the rupture for the Kobe and Kocaeli earthquakes (taken from [START_REF] Fukushima | Characteristics of observed peak amplitude for strong ground motion from the 1995 Hyogoken Nanbu (Kobe) earthquake[END_REF][START_REF] Fukushima | Attenuation characteristics of peak ground acceleration from fault trace of the 1999 Kocaeli (Turkey) earthquake and comparison of spectral acceleration with seismic design code[END_REF], otherwise it corresponds to hypocentral distance. Two stations were eliminated from the original database because they were suspected of having soil-structure interaction (Yonago and Ohsaka of BRI from the Kobe data). The number of events considered in this study is 50 recorded at 341 strong-motion stations.

The data ranges are, for distance, 0.5 to 235 km and, for magnitude, 5.5 and 7.4 (Figure 1). [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF] classified their sites according to the predominant period computed using the H/V response spectral ratio. They used the 5% damping response spectra ratio between the horizontal and vertical components as an approximation for smoothed Fourier spectral ratios. Their reasoning was that the response spectra are closer to engineering needs than Fourier spectra. Moreover, individual site H/V amplitudes are not important, but the predominant period is. In this way, they proposed four site categories for Japan (Figure 2,top). Figure 2 (bottom) shows the standard deviation of their average H/V response spectral ratios. Note that all site classes have a standard deviation value close to 0.4, representing a variability of 50%.

Site classification

Following the same procedure in this article, only records having three components (two horizontal and one vertical) were used to compute the H/V response spectral ratio. In the present study, average H/V response spectral ratios were used to classify stations, and in this regard, the two horizontal components were considered individually. Mean residuals between observed and predicted response spectra from the equations of [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF] were subsequently inspected, since analysis of residuals proved to be useful when the observed response spectra showed amplification with respect to the predicted accelerations. A first attempt was made to classify the sites according to the Japanese soil classification. Figure 3 shows an example of H/V response spectral ratios (top) and residuals (bottom) (for the Messina station in Italy). The Japanese H/V response spectral ratios from [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF] are also plotted. This station is actually classified as a rock site in the original database of [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF]. However, a peak around 0.3s is observed on the H/V plot as well as a remarkable amplification in the residuals near 0.4s. This station could then be classified as SC-II according to the Japanese spectral shapes.

Classification is, however, not always straightforward. The observed H/V response spectral ratio may not present a peak, but rather a broadband behavior as shown in Figure 4 for the San Francisco Airport station. This site is located on soft ground [START_REF] Mcgarr | Ground motion at the San Francisco international airport from the Loma Prieta earthquake sequence[END_REF], therefore, complex soil amplification could be expected, such as through nonlinear behavior, and the current site classification does not account for such soil behavior making this station impossible to classify. For stations that recorded the Kobe earthquake in Japan, the first author had a priori geotechnical information. However, in order to have a homogeneous procedure, all stations were classified as if only the H/V information was known. Since few (roughly 30%) stations have multiple records the average H/V response spectral ratio per site is not statistically robust. Therefore, automatic classification as proposed by [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF], based on a probabilistic distribution of the spectral shape, cannot be used. Thus in this study, all sites were manually classified.

During the classification procedure, each co-author independently provided their opinion. It became apparent that the frequency band for soil class SC-III is too narrow, and for all purposes inapplicable. Therefore, we decided to combine SC-II and SC-III into a single class for our study. The final site categories are SC-1 (SC-I), SC-2 (SC-II plus SC-III), and SC-3 (SC-IV), respectively. Finally, those sites that could not be classified maintained their original denomination of rock (SC-4) and soil (SC-5), respectively (Table 2).

These last two classes, however, are only retained so as to have enough data to compute ground-motion prediction equations.

Once the definitions of the site classes were finalized, each co-author manually classified each station. For roughly 90% of the stations, the classifications agreed with each other. Different classifications were made for stations close to the limits of the site classes. When opinions differed, stations were re-classified after further discussion. Final statistics of the classification are summarized in Table 3, which shows the number of stations that switched from original rock and soil categories to the new classes based on predominant period. Note that sites originally classified as rock were reclassified into all three new categories, SC-2 being the more common. Similarly, the original soil category was composed of a combination of all three new types of sites; in this case, SC-2 and SC-3 classes most represented. Note that we were able to classify 47 sites out of 91 rock sites and 171 sites out of 250 soil sites using the procedure (64% in total). Following [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF], we computed the mean H/V response spectral ratio and its standard deviation for each site class (Figure 5). Note the remarkable similarity between the Japanese H/V averages and those derived here. It should be noted that less than 4% of Japanese stations belong to soil class SC-III [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF] suggesting that class SC-2 closely corresponds to class SC-II in Japan because, as noted above, SC-III was found to be too narrow, here, to use for classification of stations (i.e. very few stations fell within this category). With respect to the standard deviations, the ones computed in this study show less variation with respect to period than those of [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF]; however, their absolute levels are similar. This is rather surprising considering the fewer records used in this study compared to those used by [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF]. 4 for all 341 sites. Many SC-1 and SC-2 stations are located in west Eurasia, whereas SC-3 stations are equally represented in the three regions.

Statistics of individual classes and regions are indicated in Table

Regression analysis

In this study, the same functional form as [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF] is used, namely:

log 10 (Sa(f)) = a(f)M -log 10 (R + d(f)*10 e(f)M ) + b(f)R + Σc j (f)δ j (1)
where Sa(f) is the elastic response spectral acceleration for 5% damping. M is magnitude, R is hypocentral distance or shortest distance to the rupture (for the Kocaeli and Kobe earthquakes) and a, b, c j and d, are frequency-dependent regression coefficients. The suffix j corresponds either to the general rock/soil classes (reference computation) or to the proposed 5 site classes (predominant period classification). The variable δ j is a dummy variable, which is equal to 1 if data are observed at j-th site category and 0 otherwise. The coefficient e(f) is fixed as 0.42 to ensure the stability of the regression procedure. A magnitudedistance filter is applied in order to avoid distant records from small events biasing the results. For the calculation of the empirical ground-motion equations, we use both horizontal records, which are considered to be independent. The interested reader is referred to [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF] for details.

Results and Conclusions

Recall that we do not seek to derive a new ground-motion model for use in seismic hazard assessment. Therefore, we limit ourselves to commenting on the effect of the new site classification on: the derived site coefficients, the standard deviations of the model, and predicted response spectra. Data were band-pass filtered with cut-offs of 0.25 and 25Hz [START_REF] Fukushima | Attenuation relation for west Eurasia determined with recent nearfault records from California, Japan and Turkey[END_REF]. Thus, all results are shown up to 3 s in order to avoid possible bias due to filter effects (e.g. [START_REF] Boore | Processing of strong-motion accelerograms: needs, options and consequences[END_REF].

Site Coefficients

Figure 6 shows the relative amplification of site classes SC-1, SC-2, and SC-3 with respect to the general rock site classification SC-4. The SC-1/rock ratio is close to unity, except for a slight increase at long periods, which indicates that the general rock class captures the main features of rock/stiff soil behavior. The slight difference at long periods occurs since the general rock category also contains medium to soft soil sites, as mentioned above.

Nonetheless, the overall effect of mixing site conditions within a general rock category is rather small. Conversely, the SC-2/rock ratio shows deamplification of 50% at short periods and an amplification of two at long periods. The SC-3/rock ratio demonstrates a slightly greater deamplification than the SC-2/rock results and, in addition, it clearly indicates an amplification factor up to 3.3 around 1s. Note that the general soil/rock ratio does not display deamplification and the maximum amplification is only about 70% at 1s. Another interesting result is the order of PGA amplification using this new classification: PGA at SC-3 sites is less amplified than at SC-2 stations, which, in turn, is less amplified than at sites in the SC-1 class. Unfortunately, this method has limitations since there are sites that do not exhibit clear H/V peaks, on the contrary they show response, which is not captured by any of the classes. Site response at these stations remains difficult to quantify irrespective of the classification scheme.

Deviation

Figure 7 shows a slight reduction (on average 2%) in the standard deviation compared to the original rock/soil classification scheme. This result suggests that better knowledge of local site conditions can reduce aleatoric variability in empirical ground-motion models. Therefore, more effort should go into site characterization in order to derive more accurate strong-motion models with lower uncertainties.

Predicted response spectra

Figure 8 shows that the shape of the predicted response spectra follows the shape of their corresponding average H/V response spectral ratios since these ratios were used in the classification process. This is less evident for SC-3 sites, although the amplification of spectral accelerations at long periods is remarkable. Incidentally, note that the period intervals defining such classes (Table 3) are apparent in the predicted response spectra. This suggests that site classification using predominant periods could be partially mapped into the site coefficients. This provides a useful link between strong ground motion predictions and earthquake-engineering design against soil-structure interaction.

Predicted spectral accelerations at SC-1 sites are slightly higher than those at general rock sites. This is an interesting result considering that SC-1 and the general rock class are similar in terms of site amplification (Figure 6).

This shows that the choice of site classification scheme has considerable impact on predicted ground motions. Recall that most SC-1 sites are located in west Eurasia (Table 4), thus these results may be regionally biased. Conversely, response spectra on sediments show distinct behavior depending on whether the ground is composed of medium or soft material (SC-2 and SC-3 classes).

Note that sites of the SC-3 class are almost equally represented in the three geographical regions, thus predicted response spectra may represent the average behavior of such sites and tectonic environments.

the quality of these dataset and the lack of quantitative knowledge of the velocity profile at each site, we cannot say which empirical median predictions are more correct. Additional studies should be performed, in Japan for example, to help resolve this practical question. Nonetheless, we can see that this site classification clearly reproduces some seismological observations. For example, predicted PGA for the three site classes are well separated. This separation is also maintained at other periods, from high spectral accelerations at short periods (stiff sediments) to low spectral amplitudes at long periods (soft soils). These results already show an improvement with respect to general rock/soil classes whose predicted spectral accelerations show little difference at short periods, and the PGA at rock sites is lower than at soil sites (Figure 8).

Good knowledge of site conditions is important for deriving accurate empirical ground motion prediction equations. In this study we used a site classification procedure based on the predominant resonance period computed from average H/V response spectral ratios and applied it to the database of et al. (2003). The most remarkable result is the similarity of both spectral amplitudes and shapes of average H/V response spectral ratios to those found in Japan using the same technique [START_REF] Zhao | An Empirical Site-Classification Method for Strong-Motion Stations in Japan Using H/V Response Spectral Ratio[END_REF]. This suggests that a common site classification scheme may be possible regardless of the geology and geographical location of a given site. However, the physical meaning of such shapes and their correspondence with Vs30 classification still remains to be clarified. We also found that previous general rock/soil categories are, in fact, composed of rock/stiff, medium and soft soils. The relative amplification of these new classes (SC-2 and SC-3) with respect to the general rock category is about 2.0 and reaches a maximum of 3.3 at 1s. On the other hand, SC-1 and general rock classes share similar amplification effects. The standard deviations of the derived ground-motion model using these new site classes are slightly lower than that obtained using general rock/site classes.

Despite this small reduction, this result encourages the pursuit of a better site classification in order to improve the accuracy of empirical models. Predicted response spectra using the new classes have similar spectral shapes to the average H/V response spectral ratios that define each class. This is useful for predicting the period of the maximum spectral acceleration. This, however, represents a shortcoming of the method because sites having a broadband response cannot be accurately classified. In any case, this simple technique allows a fast evaluation of site effects when geotechnical characterization of the site does not exist. In addition, the method can allow the direct incorporation of effects involving deep soil structure and average shear-wave velocity of the whole soil deposit into empirical ground-motion prediction equations.

Our scheme may be useful for site classification even when information on the soil profile is lacking. However, strong-motion data are not always available in low seismicity areas. Alternative classification using the H/V spectral ratios from microtremors [START_REF] Nakamura | A method for dynamic characteristics estimation of subsurface using microtremor on the ground surface[END_REF]) may be useful in this regard. The practical use of microtremor was first attempted by [START_REF] Kanai | On Microtremors. VIII[END_REF] as well as by Lang and Schwartz (20006) recently. However, extending the application of the procedure followed here to microtremors will require more careful investigation of comparisons between strong motion and microtremor, H/V spectral ratios.

Figure 1 .

 1 Figure 1. Magnitude-distance distribution of the records used.

Figure 2 .

 2 Figure 2. Average H/V response spectral ratio for the Japanese site classes (top) and standard deviation for each class (bottom). The standard deviation is computed using natural logarithms of the ratios. The vertical dashed lines represent the limits of the period intervals for each site class. Results from Zhao et al. (2006).

Figure 3 .

 3 Figure 3. Average H/V response spectral ratios for the Messina station in Italy (top) showing, for comparison, the results for the four Japanese four site classes. Mean residuals between observed and predicted response spectra from the Fukushima et al. (2003) model (bottom). In both plots, estimates for each component are shown. Most stations had only one record contributing two horizontal components, which were assumed to be independent.

Figure 4 .

 4 Figure 4. Same as Figure4, for station at San Francisco Airport in the USA. Note the broadband site response. This kind of behavior is not taken into account in the Japanese site classification; therefore the site cannot be classified.

Figure 6 .

 6 Figure 6. Relative amplification of classes used in this study, SC-1, SC-2, and SC-3, with respect to general rock site classification. For comparison, general soil-to-rock amplification is also shown. Note that amplification for SC-1 and general rock classes are similar.

Figure 8 .

 8 Figure8. Predicted response spectra a magnitude 6.0 earthquake at 30 km using the site classes proposed in this study and general rock/soil categories.

Table 1 .

 1 Site class definition used in Japan for engineering design practice (Japan Road Association 1980 and 1990) and the approximate correspondence with NEHRP site classes(BSSC, 2000).

	Site classes	Site natural period (s)	Average shear-wave velocity	NEHRP class
	SC-I	T G < 0.2s	V s30 > 600 m/s	A+B
	SC-II	0.2s ≤ T G < 0.4s	300 m/s ≤ V s30 < 600 m/s	C
	SC-III	0.4s ≤ T G < 0.6s	200 m/s ≤ V s30 < 300 m/s	D
	SC-IV	0.6s ≤ T G	V s30 < 200 m/s	E
	T G of K-net station is simply estimated by T G =4H/Vs, where H is thickness of
	upper layer than rock or stiff soil in (m), and Vs is average S wave velocity of
	the layer in (m/s)

Table 2 .

 2 Site class definition used in this study and the approximate correspondence with NEHRP site classes(BSSC 2000).

	Site classes	Site natural period (s)	Average shear wave velocity	NEHRP class
	SC-1	T G < 0.2s		V s30 > 600 m/s	A+B
	SC-2	0.2s ≤ T G < 0.6s	200 m/s ≤ V s30 < 600 m/s	C+D
	SC-3	0.6s ≤ T G		V s30 ≤ 200 m/s	E
	SC-4	Unknown		V s30 > 800 m/s	A+B
	SC-5	Unknown		300 m/s ≤ V s30 < 800 m/s	C
	SC-4 and SC-5 are general rock and soil classes that were impossible to
	classify using the new procedure.
	3 Number of sites in each category using the proposed and original
	classification schemes.
	Site classes	SC-1	SC-2	SC-3	SC-4	SC-5	Total
	Rock	12	23	12	44		91
	Soil	11	77	83		79	250
	Total	23	100	95	44	79	341

Table 4 .

 4 Number of sites per region and site class.

	Region	SC-1	SC-2	SC-3	SC-4	SC-5	Total
	West Eurasia	20	63	29	34	51	197
	Japan	1	10	39	6	6	62
	USA	2	27	27	4	22	82
	Total	23	100	95	44	79	341

Acknowledgements

We would like to express our gratitude to Dr. J. Zhao from GNS, New Zealand, for his help with this study. We would also like to acknowledge all organizations that provided strong-motion data. This manuscript was improved by thoughtful comments of an anonymous reviewer and Dr. D. Boore.