René Blacher

Une Nouvelle Condition d'indépendance pour le théorème de la limite centrale

Keywords: Central Limit Theorem, strongly mixing sequence, martingale, moments

On prouve un théorème de la limite centrale avec des conditions d'indépendance asymptotique beaucoup plus faibles que les hypothèses classiques.

I Introduction

Notations 1-1 : Let {X n } be a sequence of random variables defined on a probability space ( , ,P) such that, for all n∈N , E{X n } = 0 and 0 < E{|X n | 2 } <∞ .

Let σ(n)

2 be the variance of X 1 +X 2 +...+X n . One sets S n = (X 1 +X 2 +...+X n )

/σ(n) .

______________ There is two classical ways to study the central limit theorem (CLT). One can use mixing conditions or martingales.

Strong Mixing condition 1-2 : The process {X n } is said to be strongly mixing with coefficient α if where for a<b, b a is the σ-field generated by (X a ,X a+1 ,...,X b ) .

________________

In 1962 Ibragimov has obtained a necessary and sufficient condition under which strongly mixing sequences satisfy the CLT (cf [START_REF] Ibragimov | Some limit theorems for stationary processes[END_REF] and Denker [START_REF] Eberlein | Dependence in Probability and Statistics[END_REF] p 269-274) ). Theorem 1-1 : Assume that {X n } is stricly stationary such that (I-1) holds. Assume that (n) ∞ as n ∞ .

Then, S n d → N(0,1) if and only if (S n )

2 is uniformly integrable.

In this case, σ 2 (n) = nh(n) where h is a slowly varying function ____________________

Ibragimov has deduced some results for functionals of mixing sequences. For example, Ibragimov has proved the theorem 18-6-2 of [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF].. These results have been developed by several authors , e.g. Bradley [START_REF] Bradley R | On a very weak Bernouilli condition[END_REF]- [START_REF] Bradley R | A Central limit theorem for ρ-mixing sequences with infinite variance[END_REF], Peligrad [START_REF] Peligrad M | On Ibragimov-Iosifescu conjecture for -mixing sequences[END_REF], Dehling-Denker-Philips [START_REF] Dehling H. Denker M | Versik Processes and very weak Bernouilli processes with summable rates are independent[END_REF], Davydov [START_REF] Davidov | The invariance principe for stationary processes[END_REF], Chan [START_REF] Chaan | A note on the geometric ergodicity of a Markov chain[END_REF], Tjøsteim [START_REF] Tjøstheim | Non linear times series and Martkov chains[END_REF], Liebscher [START_REF] Liebscher | Central Limit Theorems for sums of α-mixing random variables[END_REF], Utev [START_REF] Utev | On the central limit theorem for -mixing arrays of random variables[END_REF], and many others , Mervelede-Peligrad [START_REF] Merlevede | The functional central limit theorem under the strong mixing condition[END_REF], Neumeyer [START_REF] Neumeyer | A Central Limit theorem for two sample U-processes[END_REF], Dedecker-Rio [START_REF] Rio E | On the functional central limit theorem for stationary processes[END_REF], Johnson Barron [START_REF] Barron A | Fisher information inequalities and the Central Limit Theorem[END_REF] ((cf also Eberlein-Taqq [START_REF] Eberlein | Dependence in Probability and Statistics[END_REF], Hall-Heyde [START_REF] Heyde | Martingale Limit Theory and its applications[END_REF]], Ibragimov -Linnik [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF] and Doukhan [START_REF] Doukhan | Mixing. Properties and examples[END_REF] and [START_REF] Dudzinsky | A Note on the almost sure Central Limit Theorem for some dependent random variables[END_REF]- [START_REF] Szeidi | The Central Limit Theorem without the condition of Independence[END_REF]).

The CLT has been proved for martingales also cf [START_REF] Heyde | Martingale Limit Theory and its applications[END_REF]. Theses results have been developed by Chao [40], De Meyer [START_REF] De Meyer | The maximal variation of a bounded martingale and the central limit theorem[END_REF], Wang-Yang-Zhou [START_REF] Wang Y | A random functional central limit theorem for processes of product sums of liner processes generated by martingale differences[END_REF], Mervelede [START_REF] Mervelede | On the central limit theorem and its weak invariance principle for strongly mixing sequences with values in a Hilbert space via martingale approximation[END_REF], Ouchti [START_REF] Ouchti | On the rate of convergence in the central limit theorem for martingales difference sequences[END_REF]).

In the most part of these papers, the CLT is studied with the convergence in distribution. However, some authors have studied the moment's convergence (Bernstein [START_REF] Berstein | Quelques remarques sur le théorème limite Liapounoff[END_REF], Brown [START_REF] Brown | Characteristics functions, moments and the Central Limit Theorem[END_REF], Eissein-Janson [START_REF] Esseen | On moments conditions for normed sums of independent random variables and martingales differences[END_REF], Hernndoff [START_REF] Herrndorf | A functional Central Limit Theorem for ρ-mixing sequences[END_REF], Birkel [START_REF] Birkel | Moment bounds for associated sequences[END_REF], Krugov [START_REF] Krugov | The convergence of moments of random sums[END_REF] , Mairoboda [START_REF] Mairoboda | The Central limit Theorem for empirical moment generating functions[END_REF], Yokohama [START_REF] Yokohama R | Moment bounds for stationary mixing sequences[END_REF], [START_REF] Yokohama R | The convergence of moments in the Central limit Theorem for stationary -mixing processes[END_REF] , Ibragimov [START_REF] Ibragimov I | On the convergence of generalized moments in almost sure central limit theorem[END_REF], Soulier [START_REF] Soulier | Moment bounds and the central limit theorem for functions of Gaussian vectors[END_REF], Rozovsky [START_REF] Rozovsky L | An estimate of the remainder in the central limit theorem for a sum of independent random variables with infinite momernts of a higher ordrer[END_REF]). For example, Yokohama has obtained necessary conditions in order to the moments converges. Cox-Kim have studied the moments bounds [START_REF] Cox D | Moment bounds for mixing random variables useful in nonparametric function estimation[END_REF].

Then, under some assumptions (cf [START_REF] Yokohama R | The convergence of moments in the Central limit Theorem for stationary -mixing processes[END_REF] and [START_REF] Blacher R | Théorème de la limite centrale par les moments[END_REF], and [START_REF] Heyde | Martingale Limit Theory and its applications[END_REF] p 71) the MCLT holds (Moments's Central Limit theorem) : S n M → N(0,1) , that is, for all p∈N , {(S n ) p } µ p as n ∞ , where µ p is the p-th moment of N(0,1) . Now, mixing condition or martingale condition are the most used assumptions. But, theses conditions are not necessary conditions. As a mater of fact, those are strong assumptions : e.g. Strong mixing condition does not hold for some AR(1) processes (cf [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF] p 360-362, [START_REF] Eberlein | Dependence in Probability and Statistics[END_REF] p 180). However the CLT holds (cf th 18-6-5 of [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF])

Then, classical conditions are too strong. As a matter of fact, those are a tiny minority of asymptotic independence conditions which are sufficient on order that the CLT holds .

Because strong mixing condition is too strong and martingale condition is too specific, some authors have introduce weaker hypotheses : Versik Ornstein (cf [START_REF] Bradley R | On a very weak Bernouilli condition[END_REF] [5]) , Withers [START_REF] Withers | Central limit theorems for dependent random variables[END_REF], Cogburn [START_REF] Cogburn R | Asymptotic properties of statinaory sequences[END_REF] (cf also Rosenblatt [START_REF] Rosenblatt | Uniform ergodicity and strong mixing[END_REF]), Pinsker [START_REF] Pinsker | Information and information stability of random variables and processes[END_REF] Doukhan-Prieur [START_REF] Coulon-Prieur | A triangular central limit theorem under a new weak dependence condition[END_REF].

Theses conditions are more general than mixing conditions or martingale conditions but they are not necessary conditions. As matter of fact they are not founded on a measure of dependence which determines completely dependence. Then, in order to obtain a full solution, we have used a such measure : the higher order correlation coefficients ρ j 1 , j 2 ,...., j n . Theses coefficients are perfectly adapted to MCLT.

Because dependence is completely determined by the ρ j 1 , j 2 ,...., j n 's , one can ask what is their part in the MCLT. In [START_REF] Blacher R | Central limit theorem by polynomial dependence coefficients[END_REF] one has obtained necessary and sufficient conditions for MCLT . One has deduced almost minimal assumptions for the MCLT.

II Generalization of mixing conditions for MCLT

Now, this type of assumptions is different of 1.2. Then, one wants a condition whose the writing is near of that of mixing condition.

With this aim, one decomposes X 1 +X 2 +.....+X n in X 1 +X 2 +.....+X u , X u+1 +X u+2 +.....+X u+t and X u+t+1 +X u+t+2 +.....+X u+t+u , where u and t are functions of n : u = u(n), t=t(n), u+t+u = n.

Notations 2-1 :

We denote by κ(n) ∈N , an increasing sequence such that κ(1)= 0, κ(n)≤n and κ(n)/n 0 as n ∞ . We define the sequences u=u(n) and t=t(n) by : u

(1)=1, u(n) = max{m∈N * | 2m+κ(m) ≤n } and t(1)=0, t(n) = n-2u(n) if n≥2. ________________ We shall choose κ such that u(n) ∞, n/u(n) 2 and t(n)/u(n) 0 as n ∞ (cf 5- 18). Moreover, if κ(n) ∞, t(n) ∞ as n ∞.
Now, we need to normalize the associated sums. Then, one tolerates the following notations (cf 1-1). } 0 for all (p,q) as asymptotic independence condition. Indeed, if ξ u is negligible S n M → N(0,1) . Now in order to prove the MCLT for some distributions, one generalize this condition by the following way (cf [START_REF] Blacher R | Central Limi Theorem By Moment[END_REF]).

Theorem 2-1 : Assume E{X n }= 0 for all n∈N *. Let k∈N *. Assume that, for all p∈N , p<2k+1, E{(X n ) p }< ∞ for all n∈N * and that E{(ξ u ) p } 0 as n ∞ .

Assume that there exists two sequences of random variables {υ' u } and {υ' u }, E{(υ u ) p }+E{(υ' u ) p } 0 as n ∞ for all p∈N , p<2k+1, such that the following assumptions hold.

H mS (2k) : p∈N , p<2k+1, E{(S u +υ u ) p } -E{(S' u +υ' u ) p } 0 as n ∞ .

H mI (2k) : (p,q)∈N* 2 , p+q<2k+1, E{(S u +υ u ) p (S' u +υ' u ) q } -E{(S u +υ u ) p }E{(S' u +υ' u ) q } 0 as n ∞ .

Then, for all p∈N , p≤2k, E{(S n ) p } µ p .

_____________________________

If k=∞, we set H mS (∞) =H mS and H mI (∞)=H mI . Then, compare H mI and H mS with classical assumptions. Of course, if {X n } is strictly stationary, H mS holds.

Assume that {X n } is a strictly stationary φ-mixing process. By theorem 17-2-3 of [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF],

|E{(S u ) p (S' u ) q } -E{(S u ) p }E{(S' u ) q }| ≤ 2φ(t

) E{|S u | pa } 1/a E{|S u | qb } 1/b
, where a,b>1 a -1 +b -1 = 1.

Similarly, if {X n } is a strictly stationary strongly mixing process, by theorem 17-2-2 of [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF], there exists a function : R 3 R + such that, for all (p,q)∈N * 2 , for all n∈N *, (β,p,q) |E{(S u ) p (S' u ) q } -E{(S u ) p }E{(S' u ) q }| ≤ α(t) 1-β , where β≥0.

Remark that it is not needed that H mI holds if {X n } is strong mixing. But it is the case if S n M → N(0,1). In this case H mI is weaker than the strong mixing assumption : no conditions are dictated to the rate of convergence of E{(S u ) p (S' u ) q } -E{(S u ) p }E{(S' u ) q }.

Example 2-3 : For example, if Θ 1 has the standart normal distribution and g i (Θ 1 ) = i -3 sin(2πiΘ 1 ), S n M → N(0,1) (cf [START_REF] Blacher R | Central Limi Theorem By Moment[END_REF]).

Example 2-4 : Let {Θ t } be an IID sequence of random variables such that Θ 1 has the uniform distribution on [-1,1]. Let {Ψ t } be a strictly stationary process independent of {Θ t }. Let X t be the process

X t = ∑ i≥0 i -3/2 L 2 i (Θ t+i )f i (Ψ t+i )
, where {L i } is the family of Legendre polynomials such that L i (x) = x i +b i-1 x i-1 +......+b 0 , and where |f i (Ψ 1 )|≤ 1. One assumes that {X t } satisfies the first assumptions of theorem 1-7 of [START_REF] Blacher R | Central limit theorem by polynomial dependence coefficients[END_REF].

Then, E{L i (Θ 1 )π(Θ 1 )} = 0 for all polynomial π such that deg(π}<i. One deduces that X t satisfies the second assumptions of theorem 1-7 of [START_REF] Blacher R | Central limit theorem by polynomial dependence coefficients[END_REF]., but is not a martingale.

Then, by theorem 1-7 of [START_REF] Blacher R | Central limit theorem by polynomial dependence coefficients[END_REF], S n M → N(0,1) .

One deduce that H mI holds.

Remark that the dependence between {X t } and {X t+h } can be strong, e.g., if ∞ and σ(n)/σ(u) 2 1/2 as n ∞ .

f i (Ψ t )=Ψ t , it is enough that |E{(Ψ t ) 2 (Ψ t+h ) 2 } -E{(Ψ t ) 2 }E{(Ψ t+h ) 2 } |≤ e(h) 0 in order that S n M → N(0,1) .

III Application to CLT

____________________

Remark that n/u 2 (cf lemma 5-18). Then, "σ(n)/σ(u) 2 1/2 " is weaker than "h is a slowly varying function", that is σ(tn) 2 /σ(n) 2 t" for all t>0 (cf [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF], 18-2-3, p 325 and 394).

One can also compare H I with the other generalizations of the strong mixing condition : [START_REF] Bradley R | On a very weak Bernouilli condition[END_REF], [START_REF] Dehling H. Denker M | Versik Processes and very weak Bernouilli processes with summable rates are independent[END_REF], [START_REF] Withers | Central limit theorems for dependent random variables[END_REF], [START_REF] Cogburn R | Asymptotic properties of statinaory sequences[END_REF], [START_REF] Rosenblatt | Uniform ergodicity and strong mixing[END_REF] , [START_REF] Pinsker | Information and information stability of random variables and processes[END_REF] and [START_REF] Coulon-Prieur | A triangular central limit theorem under a new weak dependence condition[END_REF] .

Of course, in H I the uniformity of the strong mixing condition is suppressed.

Moreover, if {X n } is strongly mixing, H I holds : sup|P{A k,j B k,j }-P{A k,j }P{B k,j }| ≤ α{t(n)).
Now, in some cases, it may be simpler to use H mI than H I . k }converges to the k-th moment of N(0,a 2 +b 2 ). One deduces that (S u +υ u ,S' u +υ' u ) d → N 2 (0,I 2 ) = N(0,1)⊗N(0,1).

____________________

IV The central limit theorem for functionals.

Let {X t } be the process X t = ∑ i≥0 g i+1 (Θ t+i )f i+1 (Ψ t+i ) , where {Ψ t } is a strictly stationary process and where {Θ t } is an IID sequence independent of {Ψ t } . We assume that, for all i∈N , g i and f i are measurable, E{g i (Θ 1 )f i (Ψ 1 ) } = 0 and E{g i (Θ 1 )

2 f i (Ψ 1 ) 2 } <∞ .
Then, one can write S n =M n +r n with

M n = 1 σ(n) g s (Θ r )f s (Ψ r ) ∑ s=1 r ∑ r=1 n , r n = 1 σ(n) g s (Θ n+r )f s (Ψ n+r ) ∑ s=r+1 r+n ∑ r=1 ∞ : E{M n }= E{r n }=0 . Of course, S u(n) = M u(n) + r u(n) . Moreover, one can write S' u(n) = M' u(n) + r' u(n) with M' u = 1 σ(u) g s (Θ r+u+t )f s (Ψ r+u+t ) ∑ s=1 r ∑ r=1 n and r' u = 1 σ(u) g s (Θ n+r+u+t )f s (Ψ n+r+u+t ) ∑ s=r+1 r+n ∑ r=1 ∞ .
Under some assumptions, one can proves that H I holds. In particular that is the case if Ψ t = θ t where {θ t } is a strongly mixing process. Indeed, it is easy to prove the following proposition.

Proposition 4-1 : Assume that X t = ∑ i≥0 g i+1 (Θ t+i )f i+1 (θ t+i ) with E{g i (Θ 1 )}= 0 for all i∈N . Assume that ∑ 1≤i≤r g i (Θ 1 )f i (θ 1 ) converges in L 2 (Ω) to G such that ∫G 2 .dP > 0 with . Remark that no condition on α(n), the mixing coefficient of {θ t } is necessary.

4-3 Generalization :

For example, let X t = ∑ i≥0 f i+1 (θ t+i ) where F r (θ 1 ) = ∑ 1≤i≤r f i (θ 1 ) converges in L 2 (Ω) to F such that ∫F 2 .dP > 0 . Let e(s) = ∫( ∑ i=s ∞ f i (θ 1 )) 2 .dP .

If {θ t } is strong mixing, one can assume ∑ s=1 ∞ e(s) 1/2 < ∞ in order that H I holds.

V : Generalization of Theorem 3-1.

V-1 : Notations

In this section , one uses assumptions more general. Remark that we do not assume E{(X n )} = 0 and E{(X n ) 2 } <∞ . We also write Moreover, we write the uniform integrability of (S n +υ n ) 2 in the form :

S n = S 0,n = Ψ 0 (n) -1 (X n,1 +X n,1 +...+X n,N ) .
E |S n +υ n |≥k {(S n +υ n ) 2 } ≤ ∆ k , with ∆ k 0 as k ∞, where E |S n +υ n |≥k {(S n +υ n ) 2 } = ∫ ¶ [-∞,-k[ [k,∞[ (S n +υ n ). (S n +υ n ) 2 .dP .

_______________________

As a matter of fact, we shall decompose S 0,n in the form Then, we shall prove that theorem 3-1 holds for events A k,j = {σ*(u) -1 (S u +υ u )∈ kj } and B k,j = {σ*(u) -1 (S' u +υ' u )∈ kj } .

(Ψ 0 (n)/ Ψ 1 (n)) S 0,n = Ψ 1 (n) -1 (X n,1 +X n,1 +...+X n,N ) = S 1,n +ξ 1,n +S' 1,n = Ψ 1 (n) -1 [(X n
ε k,h I (n) = sup n'≥n { sup '∈ k , r≤h {|P{(S r,n' ∈ ) (S' r,n' ∈ ')}-P{S r,n' ∈ )}P{S' r,n' ∈ '}|}} , ε k,h S (n) = sup n'≥n { sup ∈ k , r≤h {|P{S r,n' ∈ } -P{S' r,n' ∈ '}|}} , ε τ,h ψ (n) = sup n'≥n, 0<r≤h { 1 -ψ r-1 {N(n')} 2 τ ψ r {N(n')} } .
Moreover, one denotes by ε h ξ (n) a non increasing sequence such that P{|ξ r,n |>ε h

ξ (n)} < ε h ξ (n)
for all r=1,2,...,h.

One sets

ε k τ (n) = b 2k ε τ,k ψ (n)+ε k ξ (n) +ε 2k,k I (n) +ε 2k,k S (n) .

___________________

In paragraph V-3, we shall suppose that P{(S r,n ∈ k,j ) (S' r,n ∈ k,j' )}-P{S r,n ∈ k,j )}P{S' r,n ∈ k,j' } 0 as n ∞ for all k,j,j',r. Then, because k is finite,

ε k,h I (n) 0 as n ∞.
Moreover, we shall suppose also that, for all r∈N , there exists a decreasing sequence ε r ξ (n) such that P{|ξ r,n |>ε r ξ (n)} < ε r ξ (n) for all r=1,2,..,h. Then, ε h ξ (u) is also decreasing. These assumptions more general as those of §III will can be used for a more complete study of the part of dependence coefficients in limit distributions (in particular, for the laws of large numbers and for the convergence to the Poisson distribution). We shall study these generalizations in full detail later.

Recall also that some CLT have been obtained for triangular array of random variables or if σ(n) 2 ≥ n r , r>2 (cf [START_REF] Tjøstheim | Non linear times series and Martkov chains[END_REF], [START_REF] Liebscher | Central Limit Theorems for sums of α-mixing random variables[END_REF], [START_REF] Utev | On the central limit theorem for -mixing arrays of random variables[END_REF]). Non stationarity is studied in [START_REF] Chaan | A note on the geometric ergodicity of a Markov chain[END_REF].

V-2 : Lemmas

In this paragraph, one proves some inequalities under the previous hypotheses. Moreover, we assume that the following assumption holds.

Hypothesis 5-3 : In this paragraph V-2, we assume that, for all r=1,2,...,h,

P{S r,n ∈ k )} ≤ 4 -k (V-1). ____________
Then the following result holds.

Lemma 5-4 : For all r=1,2,...,h,

P{S' r,n ∈ k )} ≤ 4 -k + ε k,h S (n) (V-2).

____________

One proves the following inequalities by using the same way as Volkonskii and Rosanov ( [START_REF] Volkonskii V | Some limit theorems for random functions I Theory of Probability and its applications[END_REF] : condition I'. cf also [START_REF] Blacher R | New form for Central Limit Theorem II[END_REF]).

|E{ζ(S r,n )} -E{ζ(S' r,n )}| ≤ 2Dε k,h S (n)) (V-3) , |E E{ζ(S r,n )η(S' r,n )} -E E{ζ(S r,n )}E {η(S' r,n )}| ≤ 4D 2 ε k,h I (n) (V-4) .

____________

Lemma 5-6 : Let t∈R and k and h∈N . Then, for all r=1,2,...,h, |E{e itS r,n } -E{e itS' r,n } | ≤ 6ε k,h S (n) + 4 .4 -k + 4|t|4 -k (V-5) , |E{e itS r,n +itS' r,n } -E{e itS r,n }E{e itS' r,n } | ≤ 16ε k,h I (n)+ 8ε k,h S (n) + 16 .4 -k + 16|t|4 -k (V-6).

} -E{cos(tS' r,n )} | ≤ |E S r,n ∈ k {cos(tS r,n )} -E S' r,n ∈ k {cos(tS' r,n )} | + 2.4 -k + ε k,h S (n) ≤ |E S r,n ∈ k {C k;t (S r,n )} -E S' r,n ∈ k {C k;t (S' r,n )} |+ 2.4 -k + 2.4 -k |t| + ε k,h S (n) ≤ 2.4 -k + 2.4 -k |t| + 3ε k,h S (n) .
One uses the same way with E{sin(tS r,n )}, E{cos(tS r,n )sin(tS r,n )}, etc (cf [START_REF] Blacher R | New form for Central Limit Theorem II[END_REF]).

_____________

Lemma 5-7 : Let t∈R and k and h∈N . Then, for all r=1,2,...,h,

|E{e itψ r-1 (N).S r-1,n /ψ r (N) } -E{e itS r,n } 2 | ≤ (2+|t|)ε h ξ (n) + 16ε k,h I (n) + 14ε k,h S (n) + 20.4 -k + 20|t|4 -k (V-7) .
______________

Proof : We know that |e ib -e ia | for all (a,b)∈R 2 . Then, |E{e itψ r-1 (N).S r-1,n /ψ r (N) } -E{e itS r,n +itS' r,n } | ≤ E{|e itS r,n +itS' r,n | |e itξ r,n -1| } ≤ E{ |e itξ r,n -e 0 | } ≤ E |ξ r,n |>εh ξ (n) {2} + E |ξ r,n |≤εh ξ (n) { |t.ξ r,n -0| }≤ (2+|t|)ε h ξ (n) .
Then, it is enough to use this inequality and (V-5) and (V-6) about |E{e itψ r-1 (N).S r-1,n /ψ r (N) } -E{e itS r,n +itS' r,n } | |E{e itS r,n +itS' r,n } -E{e itS r,n } E{

| E{e itS r-1,n } -(E{e itS r,n / 2 t } ) 2 | ≤ b k |t|ε τ,h ψ (n) + (2+2 -τ |t|)ε h ξ (n) + 16ε k,h I (n) + 14ε k,h S (n) +4 -k (22 + 20|t|2 -τ ) (V-8) . ______________ Proof : By (V-1), one can write | E{e itS r-1,n } -E{e it2 -τ ψ r-1 (N).S r-1,n /ψ r (N) } -| ≤ E |S r-1,n |≤b k { | e itS r-1,n } -e it2 -τ ψ r-1 (N).S r-1,n /ψ r (N) | } + 2.4 -k ≤ E |S r-1,n |≤b k { |tS r-1,n -t.2 -τ ψ r-1 (N).S r-1,n /ψ r (N)| } + 2.4 -k ≤ t.b k ε τ,h ψ (n) + 2.4 -k .
The result follows from (V-7). __________________ Lemma 5-9 : For all r∈R , we set δ r = 2 -rτ . For all r∈R and all k and h∈N ,

| E{exp(itS 0,n )} -E{exp(δ h itS h,n )} 2 h | ≤ 2 h (b k+h |t|ε τ,h ψ (n) + (2+|t|)ε h ξ (n) + 16ε k+h,h I (n) + 14ε k+h,h S (n) )+4 -k (44 + 40|t|) (V-9) . ___________________ Proof : Let k'≤k. Clearly, k k' . Then, ε k,h S (n) ≤ ε k',h S (n) and ε k,h I (n) ≤ ε k',h I (n).
Therefore, by using (V-8),

| E{exp(itS 0,n )} -E{exp(δ h itS h,n )} 2 h | ≤ | ∑ r=0 h-1 [E{exp(δ r itS r,n )} 2 r -E{exp(δ r+1 itS r+1,n )} 2 r+1 ] | ≤ ∑ r=0 h-1 2 r |E{exp(δ r itS r,n )} -E{exp(δ r+1 itS r+1,n )} 2 r | ≤ ∑ r=0 h-1 2 r b k+r 2 -rτ |t|ε τ,h ψ (n) + ∑ r=0 h-1 2 r (2+2 -(r+1)τ |t|)ε h ξ (n) + 16 ∑ r=0 h-1 2 r ε k+r,h I (n) + 14 ∑ r=0 h-1 2 r ε k+r,h S (n) + 22 ∑ r=0 h-1 2 -2k-r + 20 ∑ r=0 h-1 2 -2k-r 2 -(r+1)τ |t| ≤ |t| ∑ r=0 h-1 2 r b k+h ε τ,h ψ (n) + (2+|t|) ∑ r=0 h-1 2 r ε h ξ (n) + 16 ∑ r=0 h-1 2 r ε k+h,h I (n) + 14 ∑ r=0 h-1 2 r ε k+h,h S (n) + 4 -k (44+40|t|) .

__________________

Lemma 5-10 : Let h n ∈N * be a non-decreasing sequence. Then, for all t∈R ,

| E{exp(itS 0,n )} -E{exp(δ h n itS h n ,n )} 2 h n | ≤ 2 h n (b 2h n |t|ε τ,h n ψ (n) + (2+|t|)ε h n ξ (n) + 16ε 2h n ,h n I (n) + 14ε 2h n ,h n S (n) ) +4 -h n (44 + 40|t|) (V-10) .
__________________

V-3 : Some propositions

By using the previous lemmas, we obtain the following results.

Proposition 5-11: Assume that, for all k∈N * , ε k τ (n) 0 as n ∞ and that, for all k∈N *, all τ>0, all n∈N * , P{S r,n ∉ k )} ≤ 4 -k .

Then, S 0,n converges in distribution to a random variable L if and only if there exists a non decreasing sequence k n ∈R * , k n ∞ as n ∞ , such that the two following assertions hold. We deduce that E{exp(itS 0,n )} E{exp(itL)} as n ∞.

Assume that S 0,n converges to L. It is always possible to choose an increasing sequence {k n } which increases enough slowly in order that (V-11) holds. For a such sequence, by (V-10), Assumptions 5-13 : We assume that, for all n∈N , E{X n,t } = 0 , E{(X n,t ) 2 } <∞ and σ r (n) 2 = Ψ r n,1 +X n,2 +.....+X n,u r (n) ) 2 } <∞. We assume that, for all k∈N , k is the samallest subset of N such that [-2 k ,2 k ] k . We assume also that, for all k and ∈N , ε k,h I (n) 0, ε k,h S (n) 0 and ε h ξ (n) 0 as n ∞.

E{exp(δ k n itS k n ,n )}

______________

Then, one generalizes the theorem 1-3 of [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF]. In this case, τ=1/2.

: E |S k n ,n |>2 k n /4 {(S k n ,n ) 2 } ≤ ∆' k n .
Therefore, (V-13) holds for all sequences k n ≤ h n .

Moreover, one can choose k n which increase slowly in order that (V-11) holds. On the other hand, the proof of the necessary condition is classical (e.g. cf [START_REF] Ibragimov | Independent and stationary sequences of random variables[END_REF], p 339). _____________ V-4 : Proof of theorem 3-1.

In order to prove theorem 3-1, the following lemma is needed.

Notations 2 - 2 :

 22 Let σ(u) 2 be the variance of X 1 +X 2 +...+X u . One sets S u = (X 1 +X 2 +...+X u ) /σ(u), ξ u = (X u+1 +X u+2 +.....+X u+t )/σ(u) and S' u = (X u+t+1 +X u+t+2 +.....+X u+t+u )/σ(u). If t(n)=0, we set ξ u = 0. ________________ Then, one can assume E{(S u )

Theorem 2 -Definition 3 - 1 :∞

 231 1 suggests that one can replace the strong mixing condition by the following way. We define condition H S and H I by the following way. There exists two sequences of random variables υ u and υ' u , E{υ u }=E{υ' u }=0, E{(υ u ) 2 }+E{(υ' u ) , such that H S : k∈N , j∈N , P{A k,j } -P{B k,j } 0 as n ∞ , H I : k∈N , (j,j')∈N 2 , P{A k,j B k,j' } -P{A k,j }P{B k,j' } 0 as n ∞ , where A k,j and B k,j are the events A k,j = {σ*(u) -1 (S u +υ u )∈[4 -k j,4 -k (j+1)[} and B k,j = {σ*(u) -1 (S' u +υ' u )∈[4 -k j,4 -k (j+1)[} with σ*(u) 2 = E{(S u +υ u ) 2 }. ____________________ In particular, if υ u = υ' u = 0 , A k,j = {S u ∈ [4 -k j , 4 -k (j+1)[ } and B k,j = {S' u ∈[4 -k j , 4 -k (j+1)[} . Remark also that σ*(u) 2 1. Moreover, if the CLT holds and if H S and H I hold for υ u = υ' u = 0, then, H S and H I hold also for all sequences υ u and υ' u such that E{υ u 2 }+ E{υ' u 2 } 0. Now, generally, it is complicated to prove that H I holds for υ u = υ' u = 0. Then it is simpler to use the above assumptions. For example, if Ψ t = θ t , H I and H S hold if t(n) ∞ with υ u = -r u and υ' u = -r' u .Then the following theorem holds. Theorem 3-1 : We keep the previous notations. Assume that H S and H I hold. Assume that 0,1) if and only if (S n +υ n ) 2 is uniformly integrable .In this case, σ(n)

Corollary 3 - 2 : 3 :

 323 Assume that E{|X n |4 }< ∞ and that E{(ξ u ) 4 } 0 . Assume that H S , H I , H mS (4) and H mI (4) hold. Then , S n d → N(0,1). Moreover, E{(S n ) Assume that the assumptions of theorem 2-1 hold with k=∞. Then, H I and H S hold. ____________________ Proof : By our assumptions, E{(S u +υ u ) k } µ k . By H mS , E{(S' u +υ' u ) k } µ k . ByH mI , for all (a,b)∈R 2 , E{(a(S u +υ u )+b(S' u +υ' u ))

Proposition 4 - 2 :→

 42 Θ 1 )f i (θ 1 ))2 .dP< C <∞ .Then, σ(n)∞ and E{(r n ) 2 } 0 as n ∞ . Moreover, H I and H S hold with υ u = -r u and υ' u = -r' u if t(n) ∞ . ____________ Therefore, under the assumptions of 4-1, {X n } satisfies the CLT if and only if (M n ) 2 is uniformly integrable. Now,we know that (S n +υ n ) 2 is uniformly integrable if E{|S n +υ n | 2+δ } ≤ C <∞ where δ>0 ([11], p 270). Then, one can use the following property. Assume that the assumptions of 4-1 hold. We set G r (Θ 1 ,θ 1 θ 1 ) . Then, there exists K>0 such that E{(M n ) 4 }< K if and only there exists B>0 such that n -2 ∑ Θ 1 )f i (θ 1 )| = ε(s) with ∑ s=1 ∞ ε(s) < C< ∞ . Then, there exists M* >0 such that |G r (Θ r ,θ r ) | ≤M* . Therefore, there exists K>0 such that E{(M n )4 } ≤K.4-3 Example: Assume g i (Θ 1 )f i (θ 1 ) = (1/i) 3 exp{-i(Θ 1 ) 2 }cos(2πiθ 1 ). Then, S n d

Notations 5 - 1 :

 51 Let N(n) be an increasing sequence. Let {X n,t }, n,t ∈N *, 1≤t≤N(n), be a triangular array of random variables defined on (Ω, ,P). For any r∈N *, we denote by Ψ r (n) a real sequence such that Ψ r (n)>0.For all n∈N * and all r ∈N *, such that u r (n)>1, one sets S r,n = Ψ r (n) -1 (X n,1 +X n,2 +...+X n,u r (N) ) , ξ r,n = Ψ r (n) -1 (X n,u r (N)+1 +X n,u r (N)+2 +......+X n,u r (N)+t r (N) ) , S' r,n = Ψ r (n) -1 (X n,u r (N)+t r (N)+1 +X n,u r (N)+t r (N)+2 +......+X n,u r (N)+t r (N)+u r (N) ) , where N= N(n), t r (n) = t(u r-1 (N)) , and u r (N)= u[u[....[u(N)]...]] .

  Let k,j , k∈N , be a sequence of nested partitions of R : R = j∈ k kj where the kj 's are intervals and k N . One assumes that, for all k∈N , and all bounded interval , there exists a finite set k k such that j∈ k k,j . One assumes also that | kj | ≤ e k where e k 0 as k ∞, where | kj | is the length of kj .

  ,1 +...+X n,u(N) )+(X n,u(N)+1 +...+X n,u(N)+t(N) )+(X n,u(N)+t(N)+1 +...+X n,n )] .After, we shall decompose S 1,n in(Ψ 1 (n)/ Ψ 2 (n))S 1,n = S 2,n +S' 2,n +ξ 2,n = Ψ 2 (n) -1 [(X n,1 +......+X n,u(u(N)) )+(X n,u(u(N))+1 +....;+X n,u(u(N))+t(u(N)) ) +(X n,u(u(N)))+t(u())+1 +.....+X n,n )].Moreover we replace partitions [4 -k j,4 -k (j+1)[ by more general partition kj which have the same useful properties. For example, if kj =[4 -k j,4 -k (j+1)[ , | kj | = e k =4 -k 0 as k ∞. Moreover, for all bounded interval there exists P,Q ∈ Z such that [P,Q[ . Then, [P,Q[= [4 -k j,4 -k (j+1)[ where P4 k ≤j<Q4 k .

Assumptions 5 - 2 :

 52 We assume that e k = 4 -k . Let k be an increasing sequence of bounded intervals such that k = j∈ k k,i where k k is a finite set for all k∈N and R = k∈N k . We set b k = sup{|x| | x∈ k } . We denote by k the -algebra generated by { k,j } j∈ k . Let k,h∈N and τ>0. One sets

Lemma 5 - 5 :

 55 Let k and h∈N and D>0. Let ζ = ∑ D and |β j |≤ D. Then, for all r=1,2,....,h, the following inequalities hold :

______________Proof:

  There exists C k;t = ∑ j∈ k α j k,t ¶ k,j , |C k;t |≤1, such that |cos(ts) -C k;t | ≤ 4 -k |t| if s∈ k' . Then, by (V-1), (V-2) and (V-3), |E{cos(tS r,n )

  the ∑ j n 's , j=1,2,....,2 k n , are independent random variables which have the same distribution as S k n ,n . ___________ Proof : Assume that (V-11) and (V-12) hold.By (V-12), E{exp(itL)} -E{exp(δ k n itS k n ,n )} 2 k n 0 as n ∞.By (V-10) and (V-11), E{exp(itS 0,n )} -E{exp(δ k n itS k n ,n )

Corollary 5 - 12 : 2 r

 5122 )} as n ∞. ___________ Now, by (V-9) the following result holds. Assume that all the assumptions of proposition 5-11 hold. Then, for all r∈N , E{exp(2 -rτ itS r,n )} E{exp(itL)} as n ∞. ______________In particular, if E{exp(itL)} = exp(-t 2 /2) and τ=1/2, S r,n d → N(0,1) for all r∈N . Then, we can apply proposition 5-11 for the convergence to the normal distribution. In this cas, we choose the following assumptions.

Corollary 5 - 14 :

 514 Assume that the assumptions of 5-13 hold. Then, S r,n d → N(0,1) for all r∈N , if and only if there exists a non-decreasing sequence k n ∈N * , k n ∞ as n ∞ , such that (V-11) and (V-13) hold for τ=1/2, where (V-13) is the Lindeberg condition : d>0, E |S k n ,n |>d2 k n /2 {(S k n ,n ) 2 } 0 as n ∞ (V-13). ______________ Proof : By the Bienaymé-Tschébyscheff Inequality, P{S r,n ∉ k )} ≤ 4 -k . Moreover, by the theorem of page 103 of [39], (V-12) is equivalent to (V-13) if L N(0,1) and τ=1/2 . Assume that (V-11) and (V-13) hold. By proposition 5-11, S 0,n d → N(0,1). By (V-9), S r,n d → N(0,1) for all r∈N N . Assume that S r,n d → N(0,1) for all r∈N . By (V-7), E{eitσ r-1 (N).S r-1,n /σ r (N) } exp(-t 2 /2) as n ∞ . Therefore, σ r-1 (N)S r-1,n /σ r (N) d → L' N(0,2) . Therefore, for all r∈N *, σ r-1 (N)/σ r (N) 2 as n ∞ (V-14) .

Lemma 5 - 16 : 5 - 17 : 2 . 2 .

 51651722 Assume that the assumptions of 5-13 hold. Moreover, we assume that, for all r∈N , E{(S' r,n )2 } -E{(S r,n ) 2 } = ε 2 r (n) 0 and E{(ξ r,n ) 2 } 0 as n ∞ , and that E |S r,n |>k {(S r,n ) 2 } ≤ ∆ k r 0 as k ∞ (V-16) .Then, (V-14) holds. ______________Proof : One can assume that π k r =∆ 2k r is non-increasing. We know that b k ≤2 k+1 andP{S r,n ∉ k )} ≤ 4 k | ≤ 4 k+1 such that |s 2 -I k 2 (s)| ≤ (2.2 k+1 )4 -k if s∈ k .Then, by (V-3), for r>0,|E S r,n ∉ k {(S r,n ) 2 } -E S' r,n ∉ k {(S' r,n ) 2 }| ≤ | 1 -E S r,n ∈ k {(S r,n ) 2 } -(1 + ε 2 r (n) -E S' r,n ∈ k {(S' r,n ) 2 })| ≤ |E S r,n ∈ k {I k 2 (S r,n )} -E S' r,n ∈ k {I k 2 (S' r,n ) 2 }| + 8.2 -k + |ε 2 r (n)| ≤ 2.4 k+1 ε k,r S (n) + 8.2 -k + |ε 2 r (n)| .Then, for all k∈N , |ES' r,n ∉ k {(S' r,n ) 2 }| ≤ π k r + 2.4 k+1 ε k,r S (n) + 8.2 -k + |ε 2 r n)| .Let N k r such that, for all n ≥ N k r , ε 2 r (n) ≤ 2 -k and ε k,r S (n) ≤ 8 -k . We assume N k r increasing.We define µ k r >0 by (µ k r ) 2 = max{π k r + 17.2 -k , Max n<N k r (E S' r,n ∉ k {(S' r,n ) 2 }) } . Then, the following lemma is needed. Lemma Under the previous assumptions µ k r 0 as k ∞ (V-17) . ______________ Proof : let γ>0. There exists K such that π K r +17.2 -K ≤ γ. Then, there exists K'>0 such that Max n<N K r (E S' r,n ∉ K' {(S' r,n ) 2 }) ≤ γ. Let k ≥ max(K,K') . Then, ,n ∉ k {(S' r,n ) 2 } ≤ (µ k r ) 2 and E S r,n ∉ k {(S r,n ) 2 } ≤ (µ k r ) Moreover, there exists B r >0 such that E{|S' r,n |}≤ B r and E{|S' r,n | 2 } ≤ (B r ) Then, by Schwartz Inequality, |E S r,n ∉ k {S r,n S' r,n }| ≤ B r µ k r , for example. Then, |E{S r,n S' r,n } -E{S r,n }E{S' r,n }| ≤ |E S r,n ∈ k {S r,n S' r,n } -E S r,n ∈ k {S r,n }E{S' r,n }| + |E S r,n ∉ k {S r,n S' r,n }| + B r |E S r,n ∉ k {S r,n }| ≤ |E S r,n ∈ k {S r,n S' r,n } -E S r,n ∈ k {S r,n }E{S' r,n }| + 2 B r µ k r ≤ |E {S r,n ∈ k } {S' r,n ∈ k } {S r,n S' r,n }-E S r,n ∈ k {S r,n }E S' r,n ∈ k {S' r,n }| + 2B r µ k r + 2µ k r . Moreover, there exists I k = ∑ k | ≤ 2 k+1 , such that |s -I k (s)| ≤ 4 -k and |I k (s)| ≤s if s∈ k . Then, |E {S r,n ∈ k } {S' r,n ∈ k } {S r,n S' r,n }-E S r,n ∈ k {S r,n }E S' r,n ∈ k {S' r,n }| ≤ |E {S r,n ∈ k } {S' r,n ∈ k } {I k (S r,n )S' r,n }-E S r,n ∈ k {I k (S r,n )}E S' r,n ∈ k {S' r,n }| + 4 -k |E {S r,n ∈ k } {S' r,n ∈ k } {|S' r,n |}| + 4 -k |E S' r,n ∈ k {S' r,n }| ≤ |E {S r,n ∈ k } {S' r,n ∈ k } {I k (S r,n )S' r,n }-E S r,n ∈ k {I k (S r,n )}E S' r,n ∈ k {S' r,n }| + 2B r 4 -k

  For all t∈R all k and h∈N , and all r=1,2,...,h,

	itS' r,n } |
	and
	|E{e itS r,n } (E{e itS' r,n } -E{e itS r,n } )| .
	_____________
	Lemma 5-8 :

  Then, we can use theorem 5-11 with τ=1/2 . _____________ Now, one can always build a proper sequence k n if the Lindeberg Condition holds.

Corollary 5-15 : Assume that the assumptions of 5-13 hold. Then, S r,n d → N(0,1) for all r∈N , if and only if (V-14) holds and there exists increasing sequence N k such all n≥N k , d>0, E |S r,n |>2 k/4 {(S r,n ) 2 } ≤ ∆' k for all r≤k (V-15) where ∆' k 0 as k ∞ . ______________ Proof : Assume that N k is strictly increasing. We define h n by h n if n<N 2 and h n =s if N s ≤ n <N s+1 . Then, for all m≥ N h n , E |S r,m |>2 h n /4 {(S r,m ) 2 } ≤ ∆' h n for all r≤h n . Let k n be a sequence such that k n ≤ h n and k n ∞ . Because h n+1 = h n or h n+1 = h n +1 , there exists t such that k n = h t . Therefore, for all m≥N h t , E |S r,m |>2 k n /4 {(S r,m ) 2 } = E |S r,m |>2 h t /4 {(S r,m ) 2 } ≤ ∆' h t for any r ≤ h t = k n . Moreover, n≥ N hn t ≥ N h t . Then, one can apply the previous inequality with m=n and r ≤ k n

Assume that E{(S' u )

2 } -E{(S u ) 2 } 0 and E{(ξ u ) 2 } 0 . Then, 1 = 2 2 as n ∞ . Therefore, there exists N 0 such that, for n≥ N 0 , σ(n) 2 /σ(u) 2 ≥ 5/3 and 7/3 ≥ n/u(n) . ∞ where r n = r if M r <n≤M r+1 . We set

If n≤M 1 , we define X n,t by X n,t = X t if t<n and X n,n = X n + V n .

If M 1 <n≤M 2 , we set X n,t = X t if t≠u,u+1,n and X n,u = X u + V u , X n,u+1 = X u+1 +V n -V u -V' u , X n,n = X n +V' u .

If M h <n≤M h+1 , we set X n,t = X t if t≠u r (n),u r (n)+1, for r=0,1,...,h and

for r=1,2,...,h (cf also [START_REF] Blacher R | New form for Central Limit Theorem II[END_REF]).

Then, for n>M r ,

Therefore, σ r (N) 2 = E{(σ r (N)S r,n )

Therefore, S r,n = σ*(u r (n))

-1

(S u r (n) +υ u r (n) ) and S' r,n = σ*(u r (n))

-1

(S' u r (n) +υ' u r (n) ).

Then, ε k,h I (n) 0 and ε k,h S (n) 0 .

By our assumptions, σ*(u r (n)) Then, E{(S r,n )

2 }-E{(S' r,n ) 2 } 0 .

Moreover, by lemma 5-18, σ(u r (n))

2 /σ(u r-1 (n))


Andy Dong
email: andy.dong@sydney.edu.au

Bernard Yannou
email: bernard.yannou@centralesupelec.fr

A model of product-induced pain

Keywords: Creativity tools, innovation, user needs

published or not. The documents may come

A model of product-induced pain

Introduction

Solving customer "pain points" has almost become a mantra in industry. The Value Map by (Osterwalder, Pigneur, Bernarda, & Smith, 2014, p. 9) recommends business models that produce "pain relievers", that is, products and services that reduce "bad outcomes, risks, and obstacles related to customer jobs". When industry refers to pain points, what is meant are the pressing and urgent problems that need to be solved. This article takes as a general premise that pain reduction is a productive position from which to think about possible product and service innovations. Scholars have collaborated with disabled persons as lead users to identify needs that may not be evidence in a broad user base [START_REF] Conradie | Disabled persons as lead users in product innovation: a literature overview[END_REF]. Preventing pain and injury is both ethically and economically responsible. One study from the UK in the mid-1990s identified that products such as a bicycle, ladder, stepladder, and DIY knife caused the highest total medical cost [START_REF] Hayward | Risk of injury per hour of exposure to consumer products[END_REF]; in 1996, bicycle injuries cost £81270 per million persons per year, for example. In the US, power tools cause about 30% of all eye injuries each year [START_REF] Mcgwin | Consumer product-related eye injury in the United States[END_REF]. Where ethical and economic imperatives fail, legal requirements compel product designers and manufacturers to eliminate the risk of injury and pain to those who use their products [START_REF] Vernick | Role of Litigation in Preventing Product-related Injuries[END_REF]. Knowing that solving a customer pain is economically and ethically important is, of course, not sufficient to identify the pain. While there is deep literature about specific products and how to design them to minimize the potential of injury, we have been unable to locate a general framework to identify potential situations of pain induced by a product. Therefore, in this paper, we propose a generic model and ontology to describe situations of pain induced by a product for the purpose of designing them out. Anticipating all possible sources and situations that may result in pain is challenging, though. In recent research on improving medical products safety, it was identified that the lack of systems thinking about medical errors that could result in patient pain and injury led to the increased likelihood of errors [START_REF] Clarkson | Design for patient safety: A review of the effectiveness of design in the UK health service[END_REF]. The U.S Centers for Disease Control initiative of Prevention through Design (PtD) (National Institute for Occupational Safety and Health Education and Information Division, 2014) aims to "design out" possible injury. While the initiative regards the design of safety as a transdisciplinary problem, it does not offer a useful causal model through which to identify potential pains caused by products [START_REF] Ertas | Prevention through design: transdisciplinary process[END_REF].

A systematic method is needed to describe the mechanisms that can lead to accidental misuse, error and accidents, and possibly injuries. The challenge to designers is systematically finding predictable mechanisms that could induce pain as the basis for a design intervention that averts individuals from pain. While it is possible that some visionary individuals have some intrinsic and non-imitable skills to recognize highly valuable pain points to solve, this is an approach to innovation that depends upon chance rather than organizational capabilities.

Our perspective on pain reduction differs from the conceptual basis of design orientated around user needs, sometimes called user-centered design. While there is no single definition of usercentered design, user-centered design generally places a focus on the activities and goals of the end-user of a product [START_REF] Gulliksen | Key principles for user-centred systems design[END_REF] and generally defines the system boundary by the end-user(s), the product, and the tasks engaged between the end-user(s) and the product. Predominantly, the literature in user-centered design has placed attention toward improving usability and positive experiences. We were not able to identify any papers in which the minimization (or elimination) of predictable mechanisms that would create pain as the objective of the design process.

This paper develops a model to describe the sources and situations of pain associated with a product. The model contains semantic elements and relations between the elements to model the relations between elements of situations in which pain occurs. The aim of the model is to establish the basis for systems thinking about usage scenarios or segments for new or revised products and services that are likely to bring essential health, social, and economic values [START_REF] Bekhradi | Usefulness Simulation of Design Concepts[END_REF]. The philosophy underlying our design process is to try to design products and services that achieve their nominal performance requirements without creating unintended consequences and negative externalities. By removing potentially painful situations, we hope to promote more inclusive design [START_REF] Clarkson | Inclusive design[END_REF].

The first part of the paper reviews definitions of pain points and differentiates this concept from user needs. The paper then presents a model for pain induced by products and services. We present an evaluation of the model on a real-world database of consumer product injuries. The paper concludes with a discussion on the use of the model in a pain-minimizing design process that we call PxD.

Theoretical development

Defining pain

Physical and mental pain

This article takes a perspective on innovation as being about finding solutions that maximize the well-being of individuals by minimizing their pain. The philosophical basis of our position is based upon the legal philosophy of Jeremy [START_REF] Bentham | Introduction to the principles of morals and legislation[END_REF]. While most of Bentham's treatise is on the morality of meting pain as a form of punishment, he provides a characterization of pain (Bentham, 1823, p. 65):

A man's happiness, then, may be said to depend more or less upon the relation he bears to any sensible object, when such subject is in a way that stands a chance, greater or less, or producing to him, or averting from him, pain or pleasure.

Interpreting Bentham's position on pain from the perspective of product design, pain is induced when a situation causes actual ("pain of actual sufferance" (Bentham, 1823, p. 125) or perceived ("the pleasure or pain may result immediately from the perception which it accompanies" (Bentham, 1823, p. 65) mental or physical harm. Bentham (1823, pp. 125-126) develops a list of nine kinds of personal injuries. He points out that pain can be both physically (the first eight) and mentally injurious (the last type of pain).

Simple corporal injuries Irreparable corporal injuries Simple injurious restraint Simple injurious compulsion Wrongful confinement Wrongful banishment Wrongful homicide Wrongful menacement Simple mental injuries

The distinction between and acknowledgement of both physical and mental pain is reflected in current medical definitions of pain. According to the International Association for the Study of Pain, pain is (emphasis added), "An unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage." [START_REF] Coderre | The Biological Psychology of Pain[END_REF]; International Association for the Study of Pain, 2011). Individuals experience pain as a reaction to noxious stimulation [START_REF] Coderre | The Biological Psychology of Pain[END_REF].

Similar ideas are embodied in definitions of hazards. In systems engineering, a hazard is, "A system state or set of conditions that, together with a particular set of worst-case environment conditions, will lead to an accident (loss)." (Leveson, 2011, p. 184) In other words, a hazard is a set of conditions that could cause pain when the accident is inflicted upon the user.

While physical pains are generally readily observed, except in circumstances requiring specialized detection equipment, mental injuries are phenomenologically difficult to observe and describe. They are no less real than physical pains. One way to characterize negative mental pain is through the concept of emotions. Models of emotion [START_REF] Thamm | The Classification of Emotions[END_REF] generally follow either the classical approach of a set of mutually exclusive dimensions or the prototype [START_REF] Russell | In defense of a prototype approach to emotion concepts[END_REF] approach, which aims to identify emotions based upon exemplars of similar ways in which people label emotions. As our aim is to identify how products and situations may cause pain, or negative emotion, we follow the classical approach, and specifically, structural models of emotion that are based upon objects and situations [START_REF] Ortony | The Cognitive Structure of Emotions[END_REF] and individuals' appraisals of the objects and situations of significance to their well-being [START_REF] Ellsworth | Appraisal Processes in Emotion[END_REF]. The structural approach is appropriate in this context since the purpose of our model is to describe the mechanisms [START_REF] Edwards | On the nature and direction of relationships between constructs and measures[END_REF] linking antecedents of products, situations, and personal matters to pain. Emotional pain may be observed externally even if the person being observed does not 'feel' any pain; observing someone as being self-destructive, neglected, oppressed or persecuted reflects an evaluation by the observer that, according to [START_REF] Ortony | The Cognitive Structure of Emotions[END_REF], does not directly refer to the psychological state of the person observed, but may indicate sympathy or empathy by the observer. Pain can be an Internal -Nonmental -Physical & Bodily State such as being nauseous and weary. Given the preceding discussions, for the purposes of this paper and our model of pain, we will adopt the notion that pain can be nonmental, in which case it is affectmental, or physical.

Economic pain

Lastly, we consider pain that is neither physical nor mental. In most product liability laws, consumers are eligible for compensation due to 'loss' and 'damage' as a result of using a defective product. It is noteworthy that product liability laws permit compensation for economic loss including, e.g. in Australia, damage to or destruction of other property (Product Safety Australia, 2015), such as another product. Thus, while the previous concepts of pain dealt with physical or emotional injury, product liability laws also include economic pain. Recent scholarship has proposed that injury should be extended to social harms; companies should have a legal obligation to reduce the negative social costs of their products under performancebased regulation [START_REF] Sugarman | Performance-Based Regulation: Enterprise Responsibility for Reducing Death, Injury, and Disease Caused by Consumer Products[END_REF]. As such, we include economic pain as a possible type of pain.

Having discussed the manifestation of pain, through physical & bodily, mental injury (negative emotion), or economic loss, we now turn to developing a model to describe the links between product and service characteristics that can induce pain. In particular, we briefly review the scholarship on negative experiences with products on the assumption that those negative experiences may either be precursors to or indicators of pain induced by a product.

How Products Cause Pleasure or Pain

The expression of negative opinions toward products and services is a growing area of concern in product marketing, especially arising from the significant volume of social media content about products and services. A number of scholars have therefore attempted to understand how products and services can stimulate emotional responses in individuals. In design research, Desmet (2012) identified six main sources of positive emotions elicited through human-product interactions: a) the object; b) the meaning of the object; c) interaction with the object; d) activity facilitated by the object; e) the individual; and f) other individuals involved in interacting with the object. In interaction design, researchers investigate how an interactive product can create a positive 'user experience' [START_REF] Law | Modelling user experience -An agenda for research and practice[END_REF]. One of the most highly cited models identifies two types of product attributes perceived by users as causing a positive experience: pragmatic quality or the perceived utility and usability of the interactive product in supporting the individual to achieve personal behavioural goals; and, hedonic quality or the perceived ability of the interactive product to communicate values of importance to the individual or to facilitate personal development through stimulation, novelty, and challenge [START_REF] Hassenzahl | The Interplay of Beauty, Goodness, and Usability in Interactive Products[END_REF]. Figure 1 describes the causal relation between product attributes and positive user experience.

Figure 1 Causal model of positive user experience

While the absence of beauty and goodness will prevent a positive experience from occurring, it is unclear whether the model would predict pain. Nonetheless, the basic concept is that hedonic characteristics and pragmatic qualities cause a positive user experience. There are some problems extending this model into our model to describe situations that can induce pain. First, it is not directly obvious that the absence of hedonic characteristics could produce pain. Second, depending upon the subjective preferences of the individual, products that are too novel or too challenging to use, or require a significant investment of effort to learn to use properly and effectively, could constitute an annoyance --an affect-mental pain. Answering whether or not such an annoyance constitutes pain may depend upon the population of individuals who report the annoyance and whether this population is worth addressing. Finally, some pragmatic product characteristics may annoy a user or be a "pet peeve" (a minor annoyance that recurs across a number of products or situations) --such as individuals who do not like the touch screen interface on a cell phone. Whether this dislike constitutes pain is an empirical problem rather than an existential one. Our model aims to describe whether or not the characteristics of a situation could nonetheless be characterized as causing pain no matter how objectively minute.

Ontology development

Assumptions

In producing the ontology, we make the following assumptions:

The ontology is limited to describing a construct for pain induced by objects and their use within situations. It is not a general ontology of pain. The ontology is not intended to measure pain or the quality and intensity of pain perceived. The ontology excludes individual differences in the subjective experience of pain, i.e., pain-sensitive and pain-resistant individuals.

We must also necessarily delineate a system boundary as shown in Figure 2. The pain model does not extend beyond the system boundary defined by the operator, the recipient, and a bystander (defined below). While the pain caused can extend to the supply chain and to society, such as the social loss due to large-scale injuries, the current model does not include these conditions.

Figure 2 System boundary of pain model

Causal model

The concept is that the source of a pain produces a type of pain within a particular situation as shown in Figure 3. From Desmet (2012), we describe a) the object; b) the meaning of the object; c) interaction with the object; and, d) the activity facilitated by the object as sources of pain. We believe that these categories subsume the pragmatic and hedonic qualities of a product as suggested by [START_REF] Law | Modelling user experience -An agenda for research and practice[END_REF]. In turn, pain can be experienced by a) the individual and b) other individuals. We distinguish these other individuals with reference to whether the individuals were the ones who were actively engaged in using the source of the pain or whether they circumstantially happened to be present when the source of the pain caused injury. In considering mental pain, we consider all types of negative affect as described previously.

Figure 3 Ontology of pain model

Definitions of the semantic elements in the pain model follow, with examples taken from actual accidents recorded in the National Electronic Injury Surveillance Systesm (NEISS) (Division of Hazard and Injury Data Systems, 2004) are shown in Table 1.

Validation

Experiment

Ontology evaluation entails a number of criteria including consistency, completeness, conciseness, expandability, and sensitiveness [START_REF] Gómez-Pérez | Evaluation of ontologies[END_REF][START_REF] Vrandečić | Ontology evaluation[END_REF]. In this study, we performed evaluations for consistency. To test for consistency, we performed a study of inter-coder reliability to determine whether the definitions for the semantic elements could be consistently applied to the same data.

To test the ontology, we investigated the consistency of its application over the NEISS (Division of Hazard and Injury Data Systems, 2004). The NEISS records all hospital visits associated with an injury due to a consumer product. The database records in the "Comments narrative" a description of what the individual was doing when the injury occurred, the product involved, and the location of the incident. A sample narrative is, "70YOM FELL 6 FEET OFF A LADDER ONTO HEAD FRACTURED NECK" [70 year old male fell 6 feet off a ladder onto head and fractured neck]". Under the PxD ontology, the pain source is the Use ("fell off") of the "ladder", which produced Physical & Bodily Injury to his "neck" during Correct use to the Operator, a "70 year old man", under Expected conditions.

Table 1 Pain ontology with salient words emphasized

Intercoder reliability evaluation

One author coded a set of 50 randomly selected narratives and then recoded the set with a 24hour break period in between. Definitions and their interpretations were clarified and then the data set re-coded (with 24 hour breaks in between) until the coder reliability exceeded 0.8. Krippendorff's alpha [START_REF] Krippendorff | Reliability in Content Analysis[END_REF] is used for all calculations of coder reliability.

Narratives for which the codes were inconsistent across two coding were discussed with the other author to obtain agreement on their definition.

Another person uninvolved with the project was trained for one hour on the coding using a random set of 20 narratives. One author and the coder discussed the coding for each of the narratives until the coder felt confident on its use. The trained coder then coded independently the same set of 50 narratives coded by the author. One author and the coder reviewed their codes and discussed discrepancies. Discrepancies were discussed so that the author and the trained coder could agree upon consistent definitions for each of the categories and exemplars for each definition. Finally, one author and the trained coder coded another 50 randomly selected narratives. Krippendorff's alpha was calculated using SPSS [START_REF] Hayes | Answering the Call for a Standard Reliability Measure for Coding Data[END_REF] for each category, as shown in Table 2. The overall inter-coder reliability was 0.8937. For the category of Condition, there were only two discrepancies out of 50 cases. However, given 50 cases and only two possible values per case, even one discrepancy would lead to a value of =.6598. Thus, the low value for this category can be attributed to the single category and small number of scale points. Nonetheless, the overall inter-coder reliability is sufficiently high to accept the coding as reliable [START_REF] Lombard | Content Analysis in Mass Communication: Assessment and Reporting of Intercoder Reliability[END_REF]. It can therefore be concluded that the definitions for the pain situations is likely to be clear enough such that with appropriate training, any individual would be able to use it to code situations in which pain occurs.

Conclusion

This paper introduced a model and ontology to describe customer pain induced by a product. The model was shown to be consistent through an inter-coder reliability study based on actual data of pains as recorded by a US-based database of injuries reported to emergency rooms. While the model focuses on products, it should be readily extensible to processes, which are more likely to produce mental pain rather than physical or bodily pain. In addition, the model is silent about the magnitude of the pain. The judgement on the value of pain minimization could be handled using the concept of value buckets [START_REF] Yannou | Observations From Radical Innovation Projects Considering the Company Context[END_REF].

The purpose of the model is not to code pain for the sake of coding. Rather, its aim is to assist designers in uncovering the factors that may cause pains so that those situations could be designed-out, where appropriate. From a design management perspective, the follow-on matter is what designers and companies should do with insights into causes and situations of pain. One way forward is to utilize the pathways through the pain-driven design ontology as usage scenarios [START_REF] Yannou | Set-based design by simulation of usage scenario coverage[END_REF]. The design strategy in this case would be to identify a product that best covers the usage scenario space that produces, e.g., the most number of bodily & physical pain.

Table 2 Intercoder reliability

 2

	Category	alpha ()
	Source	.9403
	Type	1
	Situation	.8462
	Condition	.4844
	Target	.8657

Acknowledgements


Guangshuo Chen

Aline Carneiro

Carlos Sarraute

Towards an Adaptive Completion of Sparse Call Detail Records for Mobility Analysis (Highly focused technical solution)

Keywords: Call detail records, user mobility, human trajectories, location boundaries

Call Detail Records (CDRs) are a primary source of whereabouts in the study of multiple mobility-related aspects. However, the spatiotemporal sparsity of CDRs often limits their utility in terms of the dependability of results. In this paper, driven by real-world data across a large population, we propose two approaches for completing CDRs adaptively, to reduce the sparsity and mitigate the problems the latter raises. Owing to high-precision sampling, the comparative evaluation shows that our approaches outperform the legacy solution in the literature in terms of the combination of accuracy and temporal coverage. Also, we reveal those important factors for completing sparse CDR data, which sheds lights on the design of similar approaches.

I. INTRODUCTION

In the past decades, the proliferation of personal mobile devices makes Call Detail Records (CDRs) a very promising source of location information [START_REF] Naboulsi | Large-scale mobile traffic analysis: a survey[END_REF]. Collected by mobile network operators for billing purposes, CDRs document the details about when, where and how mobile phone subscribers generate voice calls or text messages, usually across remarkably large populations. The rich information from CDRs has led to a dramatic increase in mobility-related studies, such as identifying important locations [START_REF] Isaacman | Identifying important places in peopleś lives from cellular network data[END_REF], optimizing paging in cellular networks [START_REF] Zang | Mining call and mobility data to improve paging efficiency in cellular networks[END_REF], and understanding dynamics of human mobility [START_REF] González | Understanding individual human mobility patterns[END_REF].

The sparsity of CDRs often has an adverse impact on the dependability of study results. Due to the bursty and irregular nature of the communication activities they capture, CDRs are habitually sparse in time, and thus may not record a user's whereabouts with a stable and consistent frequency. The incomplete mobility information from CDRs causes possible biases on characterizing mobility-related features [START_REF] Ranjan | Are call detail records biased for sampling human mobility?[END_REF], [START_REF] Hoteit | Filling the gaps: On the completion of sparse call detail records for mobility analysis[END_REF], [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF]. To deal with the sparsity, sometimes heavy filters have to be applied on CDRs to select users having enough mobility information [START_REF] Song | Limits of Predictability in Human Mobility[END_REF].

Data completion aims at filling spatiotemporal gaps in CDRs as much and accurate as possible. It is to locate users continuously in time by leveraging the information of users' instantaneous whereabouts. Though it does not fully conquer the sparsity, as locations logged by CDRs are usually incomplete [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF], data completion can relieve the temporal sparsity of CDRs and problems the latter raises. The legacy solution for completing sparse CDR data is to hypothesize that a cell tower location documented in a CDR is available and representative for a period (typically one hour) rather than only at a time instant when an activity happens, as used in [START_REF] Song | Limits of Predictability in Human Mobility[END_REF], [START_REF] Jo | Spatiotemporal correlations of handset-based service usages[END_REF]. This solution is actually a reflection of human nature, i.e., one tends to stay in the vicinity of her voice call places most of the time [START_REF] Ficek | Inter-Call Mobility model: A spatio-temporal refinement of Call Data Records using a Gaussian mixture model[END_REF].

A major drawback with the legacy solution is that it always expands all CDRs by the same period. One this point, previous findings have shown that using a fixed period at all time is inadequate. In the scenario of determining whether and when a mobile subscriber stays at home during the nighttime, Hoteit et al. [START_REF] Hoteit | Filling the gaps: On the completion of sparse call detail records for mobility analysis[END_REF] found that estimating the home period adaptively by historical CDRs outperformed the legacy solution with using a fixed period (10pm, 7am) in terms of accuracy. In the general scenario of completing CDRs during the daytime, we found that the legacy solution aforementioned might lead to a significant spatial error in our previous work [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF], which reported significantly that the spatial error was positively correlated with the cell size.

The studies above reveal the importance of having an adaptive approach for data completion for better accuracy. So far, however, there has been little discussion about this aspect. Although [START_REF] Hoteit | Filling the gaps: On the completion of sparse call detail records for mobility analysis[END_REF] proposed an adaptive solution for the scenario of identifying user's home, it is not a universal design and does not consider the environmental information like the cell size.

In this paper, we keep on focusing on data completion for CDRs. We explore (i) what can be extracted from CDRs as features for their completion, and (ii) which features are critical to the design of a universal adaptive approach. Our results contribute to the effort on reliable CDR data completion in the following ways.

• Our investigation is based on two real-world datasets.

Compared with previous GPS datasets, the dataset which we leverage as ground-truth still features high temporal resolution but covers movements of a larger number of users. Details are provided in Sec. II.

• We propose two adaptive approaches for completing sparse CDR data and assess their quality on hundreds of thousands of CDRs leveraging the ground truth information. They outperform the legacy solution: keep-ing a low spatial error and shortening uncompleted periods. Also, we shed light on the main features which are related to completing CDRs through learning realworld data. Details are provided in Sec. III.

Conclusions are finally discussed in Sec. IV.

II. DATASETS

We leverage two datasets collected from a major cellular operator in Mexico: the target dataset which is composed of CDRs, and the flow dataset to build ground truth information of user movements.

The target dataset contains CDRs of 36, 735 users recorded from April 1st to August 31st, 2015. On each of these days, CDRs are collected during [10am, 6pm], prevailing working hours. Each CDR provides the detailed information of a user's activity (i.e., a phone call or a text message), consisting of the involved devices (i.e., caller/callee or message sender/receiver, as anonymized identifiers), the activity time, the routing cell tower location, and the activity duration.

The flow dataset is composed of flows collected during [10am, 6pm] across the same population as the target dataset 1 . Each flow describes the lifecycle of a TCP or UDP session, and consists of the device identifier, the session time, and, particularly, the cell tower location where a session ends. Therefore each user has a fine-grained discrete trajectory of locations by her flows. Due to the operator's limits of privacy, we can only have three days of flows (July 19th, 20th and August 9th, 2015). On these days, we use the flows to construct continuous mobility information as ground truth. For that we complement each discrete trajectory by expanding each flow from a time instant to a continuous period, as illustrated in Fig. 2(1).

We plot the cumulative distribution function (CDF) of the number of records and the inter-event time in Fig. 1(a) and Fig. 1(b), respectively. The figures reveal that: (i) each user has far more flows than CDRs (95% of users have less than 10 calls but more than 200 flows); (ii) these flows spread the 8hour observing period with a dense temporal coverage (in 95% of cases, a user has two consecutive flows within 100 seconds, but only in 20% of cases, has two consecutive CDRs).

Overall, the flow dataset contains fine-grained mobility information. Its high temporal granularity ensures flows capture all handovers of cell towers in the observing period of each day, and thus supports the use of trajectories in this flow dataset as ground truth in our analysis.

III. CDR DATA COMPLETION

In this section, we propose two adaptive approaches for completing CDR data. The approaches are driven by real data and aim at filling temporal gaps of unknown locations between consecutive activities. We evaluate their performance 1 The following data pre-processing steps are carried out prior to our analysis, in order to guarantee that every user's movement satisfies an appropriate temporal granularity in the flow dataset. We first apply the recursive lookahead filter on each user's flows to tackle the undesirable effects of celltower oscillation [START_REF] Bayir | Mobility profiler: A framework for discovering mobility profiles of cell phone users[END_REF]. We then filter out those who have two consecutive flows within higher than 20 minutes. We refer the reader to [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF] for all details. 2) estimating a location boundary by the static approach: (1) Suppose five consecutive flows recorded at time t 1 , . . . , t 5 and at cell locations A, B, C, D. The two flows at t 1 and t 2 are merged as they are observed consecutively at the same cell A. Each transition between two cells is assumed to occur at the mid-time of corresponding consecutive flows. In this way, a continuous trajectory is built. [START_REF] Isaacman | Identifying important places in peopleś lives from cellular network data[END_REF] The static approach estimates a fixed-period location boundary (-t d , +t d ) attached with a CDR at time t CDR at the cell C, so as to assume the user remains at the cell C during this period, while actually she moves from the cell B to D, indicating that only a sub-period is accurate in this location boundary.

by comparing with the legacy solution introduced in Sec. I which we refer as the static approach in the following.

The static approach is to hypothesize that a user always stays in the corresponding cell during a period centered at the time of each CDR, as illustrated in Fig. 2(2). By the static approach, we present the idea of location boundary. Each location boundary contains a period corresponding to the completion of a CDR representing that a user's whereabout during this period. In the static approach, all CDRs are completed by symmetric location boundaries of the same size

(-t d , +t d ).
The period in a location boundary should be estimated according to the very situation of the activity. For instance, a location boundary deserves a large t d if the user is walking when making a call, but fits a small t d if she is on a highspeed train. The arbitrary determination of t d in the static approach leads to a significant spatial error in practice due to the complexity of a user's realistic behavior [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF].

With regards to this, we introduce two novel adaptive approaches, where t d (or t In the following, we introduce how a location boundary is estimated in the two adaptive approaches in Sec. III-A. After that, we compare all the three approaches from two perspectives: accuracy and coverage, discussed in Sec. III-B.

A. Determining adaptive location boundaries

Observable factors from CDRs: Leveraging a large number of a user's CDRs collected during the long-term observing period, we can learn her behavior for identifying a location boundary though the following factors in three categories:

1) Event-related factors: These are the metadata of a CDR, including the activity's time, type (call/message) and duration2 .

2) Long-term behavior factors: The radius of gyration (URg) of a user, the number of a user's locations (ULoc) appearing in the observing period, and the number of a user's active days (UDAY). These factors characterize a user by giving senses of (i) her long-term mobility and (ii) her habit on generating calls and text messages, computed by her CDRs produced during the 5-month observing period.

3) Location-related factors:

The first factor in this category is related to the cell size3 , i.e., the average call radius (CR). Since we have no knowledge of the actual cell coverage, we assume a homogeneous propagation environment and an isotropic radiation of power in all directions at each cell tower, so that we are able to roughly estimate each cell's CR using a composition of Voronoi cells extracted from CDRs which covers the area, as in [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF].

The rest of the factors describe the location where the activity happens regarding its importance to the user. For that, we learn from the algorithm presented by Isaacman et al. [START_REF] Isaacman | Identifying important places in peopleś lives from cellular network data[END_REF], which is designed to determine prominent locations which the user usually spends a large amount of time and/or visits frequently. Their algorithm firstly clusters all locations which appear in a user's trajectory of CDRs, and then identify for each cluster whether a cluster is important by measuring these observable factors derived from each cluster in the following, as used in our work: (i) the number of days on which any cell tower in the cluster was contacted (CDay); (ii) the number of days which elapse between the first and the last contact with any location in the cluster (CDur); (iii) the sum of the number of days cell towers in the cluster were contacted (CTDay); (iv) the number of cell towers the cluster (CTower); (v) the distance from the activity location to the centroid of the cluster (CDist).

Training and estimating: We develop our approaches by firstly training our model with a set of 65, 791 CDRs collected on August 9th. The other two days, i.e., July 19th and 20th, on which we have the ground-truth information are then utilized into the testing phase. In this way, the model is trained having as input the above described factors and the location boundaries extracted from the ground-truth information. More specially, given a CDR, its user and location, we derive a vector (x 1 , • • • , x n ) from the above factors, such as: (i) the categorial factor type is converted to two binary features by one-hot encoding; (ii) the time is converted to two separate time differences (in seconds) from the activity time to 10am and to 6pm; (iii) other factors are used as values they are.

In the sym-adaptive approach, this vector and the symmetric location boundary (-t d , +t d ) of each activity extracted from the ground-truth information are provided as input to a model trained by Gradient Boosting Regression Trees (GBRT) [START_REF] Friedman | Greedy function approximation: a gradient boosting machine[END_REF], under the squared loss function. Once the training phase is completed, the testing phase consists in estimating the interval (-t d , +t d ) by feeding the vector derived from 136, 562 CDRs collected on the other days (July 19th and 20th) into the regression formula.

In the asym-adaptive approach, the factors vector and the two limits of an asymmetric location boundary, i.e., t d , of each activity extracted from the ground-truth information are separately provided as input resulting in two trained GBRT models. The training phase is the same as in the sym-adaptive approach but works independently on the two models. Fig. 3 shows the relative importance of factors with respect to the estimation of a location boundary after the training phase of the adaptive approaches. This figure allows us drawing the following main conclusions, valid for both approaches.

• We notice the three most important factors: the activity's time, the cell radius, and the radius of gyration. This indicates that for a cell, how long a user stays inside mainly depends on its size, the precise time the activity occured, and the user's long-term mobility.

• Surprisingly, the activity's type is the most pointless factor, indicating that knowing whether a user generates a call or a message is useless in determining a location boundary.

B. Coverage and accuracy

To validate the location boundaries given by the three previous discussed completion approaches, we again use the CDRs collected on July 19th and 20th, on which we have the ground-truth information. Note that our approaches are applicable to CDRs on other days, though there is no ground-truth information. Hereafter, we use the days having ground-truth information only for comparison and validation reasons. For the static approach, t d is set to 5/15/30/45/60/90/120 minutes for location boundaries, respectively.

Additionally, since the goal of data completion is to fill the temporal gaps, we compare the three approaches in terms of accuracy and coverage. For that, we use the following measures to quantify the completed CDRs:

• Spatial error: The average cumulative error in the distance over time during a location boundary, quantifies accuracy in terms of activity.

• Coverage: The filling rate, defined as the ratio of the covered duration (i.e., the sum of the periods of a user's location boundaries) to the observing period, represents to what degree a user's CDRs are completed.

• Accuracy: The ratio of the accurate duration (i.e., the sum of all accurate sub-periods) to the covered duration (i.e., the sum of all periods) on a user's location boundaries, quantifies accuracy in terms of the user.

Intuitively, an ideal data completion approach should cover the observing period as much and precise as possible, i.e., satisfying high accuracy and coverage simultaneously.

Fig. 4(a)(b) plots the distribution of the spatial error over all location boundaries. It confirms that the spatial error increases as t d becomes larger when using the static approach, which is revealed in our earlier work [START_REF] Chen | Relevance of Context for the Temporal Completion of Call Detail Record[END_REF]. More importantly, the performance of the two adaptive approaches is nearly as good as the static approach with t d = 30 minutes in terms of the spatial error.

To further compare the approaches in terms of the combination of accuracy and coverage, we plot in Fig. 4(c)(d) the mean of accuracy versus mean of coverage over the observing users. We notice that using a large t d contributes to enhancing coverage with reducing accuracy as a price, indicating that the static approach cannot achieve high accuracy and high coverage together.

However, the two adaptive approaches show splendid performance. The sym-adaptive approach reaches the same level of accuracy as the static with t d = 30 minutes. Yet it has a better temporal coverage (approximately as good as the static with t d = 90 minutes). This indicates that the symadaptive approach can complete more time while it can still ensure accuracy. As to the asym-adaptive approach, it performs better in terms of coverage with losing a small degree of accuracy, compared with the sym-adaptive approach, and still outperforms the static.

Overall, we see a clear advantage of the sym-adaptive approach over the others in Fig. 4, as it achieves the best combination of fair accuracy and high temporal coverage.

IV. CONCLUSION

In this paper, we focused on data completion and proposed two novel data-driven approaches which utilized multiple factors of a user's behavior and determined location boundaries adaptively for completing CDRs. The comparative evaluation showed that the proposed approaches outperformed the legacy solution in the literature. Further research should be done to investigate a heuristic approach which does not rely on the contextual information.

5

 5 Number of records (flows or CDRs) per user

Fig. 1 .

 1 Fig. 1. (a) CDF of the number of CDRs (as dashed lines) and flows (as solid lines) per user. (b) CDF of the inter-event time between two consecutive CDRs (as dashed lines) and flows (as solid lines).

Fig. 2 .

 2 Fig.2. A demo of (1) constructing the ground truth and (2) estimating a location boundary by the static approach: (1) Suppose five consecutive flows recorded at time t 1 , . . . , t 5 and at cell locations A, B, C, D. The two flows at t 1 and t 2 are merged as they are observed consecutively at the same cell A. Each transition between two cells is assumed to occur at the mid-time of corresponding consecutive flows. In this way, a continuous trajectory is built.[START_REF] Isaacman | Identifying important places in peopleś lives from cellular network data[END_REF] The static approach estimates a fixed-period location boundary (-t d , +t d ) attached with a CDR at time t CDR at the cell C, so as to assume the user remains at the cell C during this period, while actually she moves from the cell B to D, indicating that only a sub-period is accurate in this location boundary.

  d ) in a location boundary is adaptively determined by its own CDR, unlike the static approach which uses a unified threshold. They are (i) the sym-adaptive approach makes a CDR into a symmetric location boundary as (-t d , +t d ) and (ii) the asym-adaptive approach makes a CDR into an asymmetric location boundary as (-t

Fig. 3 .

 3 Fig. 3. Relative Importance of features in determining accurate location boundaries.

Fig. 4 .

 4 Fig. 4. CDF of the spatial error of location boundaries computed on (a) Sunday and (b) Monday; mean of accuracy versus mean of coverage per user on (c) Sunday and (d) Monday, across the static, sym-adaptive and asym-adaptive approaches.

For this attribute, the duration of a text message is set to 0 second.

We exclude the antenna-related information such as the transmit power or RF propagation environment, which a mobile operator rarely provides.

ACKNOWLEDGMENT

The authors would like to thank GranData for providing the data used for the experiments. This work was supported by the EU FP7 ERANET program under grant CHIST-ERA-2012 MACACO.


L

  'archive ouverte pluridisciplinaire HAL, est destinée au dépôt   Distributed under a Creative Commons Attribution| 4.0 International License A molecular model for flow induced crystallization of polymers Arnaud Poitou, Amine Ammar To cite this version: Arnaud Poitou, Amine Ammar. A molecular model for flow induced crystallization of polymers. Macromolecular Symposia, 2002, Flow-Induced Crystallization of Polymers 185 (1), pp.243-255. 10.1002/1521-3900(200208)185:13.0.CO;2-S. hal-01004850