Sebastien Destercke
email: sebastien.destercke@hds.utc.fr

Olivier Strauss
email: olivier.strauss@lirmm.fr

Kolmogorov-Smirnov test for interval data

Keywords: Interval data, homogeneity test, approximation, p-box

In this paper, we are interested in extending the classical Kolmogorov-Smirnov homogeneity test to compare two samples of interval-valued observed measurements. In such a case, the test result is interval-valued, and one major difficulty is to find the bounds of this set. We propose a very efficient computational method for approximating these bounds by using a p-box (pairs of upper and lower cumulative distributions) representation of the samples.

Introduction

In many applications, the precise value of data may only be known up to some precision, that is it may be interval-valued. Common examples are censored data (e.g., censor limitations) or digital data. When performing statistical tests, ignoring this imprecision may lead to unreliable decisions. For instance, in the case of digital data, quantization can hide the information contained in the data and provide unstable decision.

It is therefore advisable to acknowledge this imprecision in statistical tests, if only to provide results robust to this imprecision. By robust, we understand tests that will remain cautious (i.e., will abstain to say something about the null hypothesis) if not enough information is available. However, treating this imprecision usually leads to an increased computational costs, as shown by various authors in the past [START_REF] Nguyen | Computing Statistics Under Interval and Fuzzy Uncertainty: Applications to Computer Science and Engineering[END_REF][START_REF] Otero | Bootstrap analysis of multiple repetitions of experiments using an interval-valued multiple comparison procedure[END_REF][START_REF] Hébert | Fuzzy multidimensional scaling[END_REF]. This means that developing efficient methods to compute statistics with interval data is a critical issue.

In this paper, we explore the extension of the Kolmogorov-Smirnov (KS) homogeneity test to interval data, and more precisely its computational aspects. To our knowledge, this aspect has not been considered in the past, even if some but not much works on the KS test with interval or fuzzy data exist [START_REF] Hesamian | Two-sample Kolmogorov-Smirnov fuzzy test for fuzzy random variables[END_REF][START_REF] Mballo | Decision trees on interval valued variables[END_REF]. Approximate and exact bounds that are straightforward to compute are provided in Section 3, while notations and reminders are given in Section 2.

In Section 4, we illustrate our results on a image based medical diagnosis problem. Indeed, in such problems a common task is to detect whether two regions of a quantized image have similar pixel distributions.

Preliminary material

Komogorov-Smirnov (KS) homogeneity test [START_REF] Conover | Practical non-parametric statistic[END_REF] is commonly used to compare two samples A = {a i |i = 1, . . . , n, a i ∈ R} and B = {b i |i = 1, . . . , m, b i ∈ R} of measurements to determine whether or not they follow the same probability distribution. Those samples are supposed to be independently drawn from a continuous one-dimensional realvalued probability distributions.

If F A (F B ) denote the empirical cumulative distributions built from A (B), that is if

F A (x) = #{a ∈ A|a ≤ x} n (1)
with #E the cardinal of a set E, then the KS test statistic KS is defined by:

KS(A, B) = sup x∈R |F A (x) -F B (x)|
Under the null hypothesis H 0 that the two-samples are drawn from the same distribution, the statistic β (n, m)KS(A, B) converges to the Kolmogorov distribution, with

β (n, m) = 1 n + 1 m .
Using the critical values of the Kolmogorov distribution, the null hypothesis can be rejected at level α if KS(A, B) > β (n, m)κ α . One common value of this rejection threshold is κ 0.05 = 1.36.

As this test makes very few assumptions about the samples (i.e., it is non-parametric) and aims at testing a complex hypothesis (with respect to, e.g., comparing two means), it requires in practice relatively large samples to properly reject the null hypothesis.

In this paper, we explore the case where observations are interval-valued, i.e., they correspond to two sets

[A] = {[a i , a i ]|i = 1, . . . , n} and [B] = {[b i , b i ]|i = 1, .
. . , m} of real-valued intervals. As recalled in the introduction and further explored in Section 4, such imprecision may be the result of some quantization process.

In the next section, we study the interval-valued statistic resulting from such data, and in particular provide efficient approximative (and sometimes exact) bounds for it, using the notion of p-box.

Kolmogorov-Smirnov Test with interval-valued data

Let us first introduce some notations. We will call selection of Given this, the imprecise Kolmogorov-Smirnov Test

[A] a set S [A] of values S [A] := {a i |i = 1, . . . , n, a i ∈ [a i , a i ]} where each a i is picked inside the interval [a i , a i ], i = 1, . . .

, n. We will denote by S ([A]) the set of all selections of [A]. To a selection S [A] corresponds an empirical cumulative distribution F S

[KS]([A], [B]) = [KS([A], [B]), KS([A], [B])]
is an interval such that

KS([A], [B]) = inf

S [A] ∈S ([A]), S [B] ∈S ([B]) sup x∈R |F S [A] (x) -F S [B] (x)|, (2)
KS([A], [B]) = sup S [A] ∈S ([A]), S [B] ∈S ([B]) sup x∈R |F S [A] (x) -F S [B] (x)|. (3)
Computing such values is not, a priori, a trivial task since the number of possible selections for both sets of intervals

Approximating p-box

A p-box [START_REF] Ferson | Constructing probability boxes and dempster-shafer structures[END_REF] [F, F] is a pair of cumulative distributions such that F(x) ≤ F(x) for any x ∈ R. The usual notion of cumulative distribution is retrieved when F = F, and a pbox usually describes an ill-known cumulative distribution that is known to lie between F and

F. That is, to a p-box [F, F] we can associate a set Φ([F, F]) of cumulative distributions such that Φ([F, F]) = {F|∀x ∈ R, F(x) ≤ F(x) ≤ F(x)}.
Here, we will use it as an approximating tool.

For a set of intervals [A], let us denote by S a and S a the particular selections S a = {a i |i = 1, . . . , n} and S a = {a i |i = 1, . . . , n}. Then, we define the p-box

[F [A] , F [A] ] ap- proximating [A] as F [A] := F S a and F [A] := F S a .
We have the following property

Proposition 1. Given a set of intervals [A], we have F ([A]) ⊆ Φ([F [A] , F [A] ])
Proof. Consider a given selection S A . For every a i in this selection, we have

a i ≤ a i ≤ a i .
Since this is true for every i = 1, . . . , n, this means that F S A is stochastically dominated3 by F S A and stochastically dominates F S A , i.e.

F S a (x) ≤ F S A (x) ≤ F S a (x), ∀x ∈ R
and as this is true for every selection S A , we have

F ([A]) ⊆ Φ([F [A] , F [A] ]
). To see that the inclusion is strict, simply note that F S A can only take a finite number of values, while cumulative distributions in Φ([

F [A] , F [A] ]
) can be strictly monotonous.

This shows that the associated (convex) set Φ([F [A] , F [A] ]) is actually a conservative approximation of F ([A]). The next example illustrates both the p-box [F [A] , F [A]

] and Proposition 1.

Example 1. Consider the case where we have 3 sampled intervals, with the three following intervals:

[a 1 , a 1 ] = [2, 7] [a 2 , a 2 ] = [6, 12] [a 3 , a 3 ] = [10, 16]
Figure 1 illustrates the obtained p-box and one cumulative distribution

( F) included in Φ([F [A] , F [A] ]). However, F is not in F ([A]
), since any empirical cumulative distribution obtained from a selection on 3 intervals can only takes its values in the set {0, 1 /3, 2 /3, 1}.

F [A] F [A]

Approximating KS([A], [B]) and KS([A], [B])

Consider two samples [A] and [B] and the associated p-boxes [F

[A] , F [A] ] and [F [B] , F [B] ].
We can now introduce the approximated imprecise KS Test [KS] = [KS , KS] such that:

KS([A], [B]) = sup x∈R max{|F [A] (x) -F [B] (x)|, |F [A] (x) -F [B] (x)|}, (4)
KS ([A], [B]) = sup x∈R D [A],[B] (x), (5)
with

D [A],[B] (x) =    0 if [F [A] (x), F [A] (x)] ∩ [F [B] (x), F [B] (x)] = / 0 min{|F [A] (x) -F [B] (x)|, |F [A] (x) -F [B] (x)|} otherwise
These approximations are straightforward to compute (if n + m intervals are observed, at worst they require 2n + 2m computations once the p-boxes are built). We also have the following properties:

Proposition 2. Given a set of intervals [A] and [B], we have KS([A], [B]) = KS([A], [B]) Proof. The value KS([A], [B]) is reached on x either for a pair {F [A] , F [B] } or {F [A] , F [B] }. As any pair F 1 , F 2 with F 1 ∈ Φ([F [A] , F [A] ]) and F 2 ∈ Φ([F [B] , F [B] ]) would have a KS

statistic lower than KS([A], [B]), and given the inclusion of Proposition 1, this means that KS([A], [B]) ≤ KS([A], [B]

). To show that they coincide, it is sufficient to note that all distributions

F [A] , F [A] , F [B] , F [B]
can be obtained by specific selections (i.e., the one used to build the p-boxes).

This shows that the upper bound is exact. Concerning the lower bound, we only have the following inequality:

Proposition 3. Given a set of intervals [A] and [B], we have KS ([A], [B]) ≤ KS([A], [B])

Proof. Immediate, given the inclusion of Proposition 1 and the fact that KS ([A], [B]) is the minimal KS statistics reached by a couple of cumulative distributions respectively in [F

[A] , F [A] ] and [F [B] , F [B] ]
And unfortunately this inequality will usually be strict, as shows the next example. Consider the following example:

[a 1 , a 1 ] = [1, 8] b 1 , b 1 = [2, 7] [a 2 , a 2 ] = [9, 15] b 2 , b 2 = [6, 12] b 3 , b 3 = [10, 16]
A simple look at Figure 2 allows us to see that KS ([A], [B]) = 0 in this case.

The inequality between KS ([A], [B]) and KS([A], [B]) can also be strict when KS ([A], [B]) = 0. It should be noted that the discrepancy between KS ([A], [B]) and KS([A], [B]) will decrease as the number of sampled intervals increases. Finally, a noticeable situation where KS ([A], [B]) will be an exact bound (KS ([A], [B]) = KS([A], [B])) is when [F [A] , F [A] ] and [F [B] , F [B] ] are disjoint, that is either F([A]) > F([B]) or F([A]) < F([B]).

F(x) F [A] F [A] F [B] F [B]
Fig. 2. P-boxes of Example 2

Decision making using an imprecise-valued test

One of the main features of this extension is that it provides a pair of (conservative) bounds KS ([A], [B]) and KS([A], [B]) rather than a precise value KS(A, B).

In contrast with usual tests that either reject or do not reject an hypothesis, this leads to three possible decisions: the answer to the test can be yes, no or unknown, the last one occurring when available information is insufficient.

In fact, interpreting this test is straightforward. Let γ = β (n, m)κ α be the significance level.

-

If KS ([A], [B]) > γ then we can conclude that there is no possible selections S [A] of [A] and S [B] of [B] such that KS(S [A] , S [B]
) ≤ γ and thus the hypothesis that the two-samples are drawn from the same distribution can be rejected at a level α.

-On the contrary, if KS([A], [B]) < γ then there is no possible selections S [A] of [A]

and

S [B] of [B] such that KS(S [A] , S [B]
) ≥ γ and thus the hypothesis that the twosamples are drawn from the same distribution cannot be rejected at a level α. -Otherwise, we will conclude that our information is too imprecise to lead to a clear decision about rejection.

This new test will therefore point out those cases where the data imprecision is too important to lead to a clear decision. As we shall see in the next section, it allows one to deal with quantization in a new way, namely it can detect when the disturbance or loss of information induced by the quantization makes the test inconclusive. It should be noted that, as KS([A], [B]) is an approximated lower bound, indecision may also be due to this approximation, yet experiments of the next section indicate that this approximation is reasonable.

Experimentation

The experimentation we propose is based on a set of medical images acquired by a gamma camera. In such applications, statistical hypothesis testing is often used to determine whether pixels distribution in two different regions of an image are similar or not. Physicians usually try to control the probability of making a decision leading to harmful consequences, and make it as low as possible (usually 0.05). The advantage of using a KS test in this case is that it makes very few assumption about the distribution. However, in such applications, it is quite common to deal with quantized information, i.e., real-valued information constrained to belong to a small subset of (integer) values. Since the KS test is designed to compare pairs of continuous distributions, it is necessary to ensure that the statistical test is robust with respect to the data model. Indeed, the value of the statistic computed from quantized data may differ markedly from the calculation based on original (non-quantized) but unavailable data.

Physicians would usually try to avoid a wrong decision, and prefer to acquire additional data when the actual data are not fully reliable. Thus, knowing that no decision can be taken based on the current set of data is a valuable piece of information.

We illustrate this weakness of the usual KS test with a set of medical images acquired by a gamma camera (nuclear medicine images) whose values are quantified on a restricted number of values. This experiment also highlights the ability of the extended KS test to avoid wrong decisions induced by quantization. It aims at mimicking real medical situations where the nuclear physician has to compare the distribution of values in two regions of interest in order to decide whether or not a patient has a specific disease.

The set of images is made of 1000 planar acquisitions of a Hoffman 2-D brain phantom (acquisition time: 1 second; average count per image 1.5 kcounts, 128 × 128 images to satisfy the Shannon condition), representing 1000 measures of a random 2D image (see Figure [START_REF] Hébert | Fuzzy multidimensional scaling[END_REF]). Due to the fact that nuclear images are obtained by counting the photons that have been emitted in a particular direction, pixel values in a nuclear image can be supposed to be contaminated by Poisson distributed noise. Due to the very short acquisition time, the images were very noisy, i.e. the signal to noise ratio was very low. More precisely, the average pixel value in the brain corresponded to a 69% coefficient of variation of the Poisson noise. Moreover, the number of different possible values to be assigned to a pixel was low and thus, within those images, the impact of quantization was high: pixel possible values were {0, 256, 512, 768, 1024, 1280, 1536, 1792, 2048}.

To obtain less noisy and less quantized images, we summed the raw images (see e.g. Figure [START_REF] Hesamian | Two-sample Kolmogorov-Smirnov fuzzy test for fuzzy random variables[END_REF]). The higher the number of summed images, the higher the average pixel value, and thus the higher the signal to noise ratio and the higher is the number of possible values for each pixel. When summing the 1000 raw images, we obtained the high dynamic resolution and high signal to noise ratio image depicted in Figure [START_REF] Mballo | Decision trees on interval valued variables[END_REF].a.

We use the KS test to decide whether the two regions depicted in Figures [START_REF] Mballo | Decision trees on interval valued variables[END_REF].b and (5).c can be considered as being similar or not (the null hypothesis). Considering the number of pixels in each region (n = 183, m = 226), the significance level for a p-value α = 0.05 is γ ≈ 0.1910. Testing the two regions with the reference image (Figure [START_REF] Mballo | Decision trees on interval valued variables[END_REF].a) provides the following values: KS(A, B) ≈ 0.2549, KS ([A], [B]) ≈ 0. 2505 KS([A],[B]) ≈ 0.2549, leading to conclude that the similarity of regions A and B should be rejected at a level 0.05, which can be considered as our ground truth.

We use the KS test for comparing the same regions but with 300 pairs of images that have been randomly selected in the set of 1000 original images. In that case, the classical test accepts the similarity of the two regions, while the imprecise test is inconclusive for each pairs: KS (

[A], [B]) < γ < KS([A], [B]
). We now do the same test with images having a higher dynamic obtained by summing p = 2, 3, . . . , 40 images. For each value of p, we count the number of times the classical test provides the right answer, i.e. reject the null hypothesis at level 0.05 (γ ≤ KS(A, B)). We then compute the ratio of this count over 300. For the extended KS test, we compute two ratios: the ratio of times when γ ≤ KS ([A], [B]), i.e. we must reject the null hypothesis at level 0.05, and the ratio of times when γ ≤ KS([A], [B]), i.e. we can reject the null hypothesis at level 0.05. We also compute the number of times where the test is inconclusive, i.e. KS ([A], [B]) < γ <

KS([A], [B]).

Figure ( 6) plots these ratio versus p, the number of summed images. On one hand, concerning the classical KS test, it can be noticed that depending on the quantification level, the answer to the test differs. In fact, when the number of pixel's possible values is low, the test concludes that H 0 cannot be rejected most of the time, leading to a decision that the two distributions are similar even though they are not. When p increases, so increases pixel's possible values and increases the ratio of correct answer. Thus, quantization has a high impact on the conclusions of a classical KS test.

On the other hand, concerning the extended KS test, it can be noticed that the null hypothesis can always be rejected. The impact of the quantization only affects the ratio of times when the null hypothesis must be rejected. Thus the impact of quantization here is much more sensible, in the sense that when quantization is too severe (information is too poor), the test abstains to make a decision. Also, in all cases, the test is either inconclusive or provides the right answer, and is therefore never wrong, which is what we could expect from a robust test. Fig. 6. Correct decision ratio with the classical test (black) and with the extended test (blue for can be rejected, red for must be rejected), superimposed with the proportion of times extended test is inconclusive (green).

Conclusions

In this paper, we have introduced efficient methods to approximate the bounds of the KS test with interval-valued data. We have demonstrated that the upper bound is exact while the lower bound is, in general, only a lower approximation. However, the experiments have shown that this is not too conservative approximation and still allows to take decision when enough information is available.

The obvious advantages of this paper proposal is its efficiency (computational time is almost linear in the number of sampled intervals), however we may search in the future for exact rather than approximated lower bounds. Since KS test result only depends on the ordering (i.e., ranking) of sampled elements between them, a solution would be to explore the number of possible orderings among elements of [A] and [B], or to identify the orderings for which the lower bound is obtained (the number of such orderings, while finite, may be huge).

Finally, it would also be interesting to investigate other non-parametric homogeneous tests, such as the Cramer-Von Mises one.

  [A] obtained by Eq. (1), and we denote by F ([A]) the (non-convex) set of such empirical cumulative distributions.

FFig. 1 .

 1 Fig. 1. P-box of Example 1

Example 2 .

 2 Consider the case where n = 2, m = 3 and where n i=1 [a i , a i ] = / 0, m i=1 [b i , b i ] = / 0. This means that, for every selection S [A] ∈ S ([A]) and S [B] ∈ S ([B]), we have that the empirical cumulative distributions F S [A] and F S [B] respectively takes at least one value in { 1 /2} and in { 1 /3, 2 /3}. This means that KS([A], [B]) = 0 (as every cumulative distributions coming from selections will assume different values), while it is possible in such a situation to have KS ([A], [B]) = 0.

Fig. 3 .

 3 Fig. 3. 6 acquisitions of the Hoffman 2-D brain phantom

Fig. 4 .Fig. 5 .

 45 Fig. 4. 6 images obtained by summing up 10 raw acquisitions of the Hoffman 2-D brain phantom

Recall that F 1 stochastically dominates F 2 if F 1 ≤ F 2 .

Acknowledgements

This work was partially carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program " Investments for the future" managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02)


Jana Trgalova

Anne Berit Fuglestad

Mirko Maracci

Hans-Georg Weigand

Introduction

Jana Trgalová

INTRODUCTION TO THE PAPERS OF WG15

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION TO WG15 TECHNOLOGIES AND RESOURCES IN MATHEMATICS EDUCATION INTRODUCTION

Technologies in mathematics education have been a topic of the working group since the CERME 1999. Since then, technologies as well as research in this field have significantly evolved. In particular, recent years have brought about an important shift in considering technologies within a wide range of resources for students, teachers and teacher educators. There have been new developments concerning software, hand-held technology, online classroom activities, but also more traditional geometry tools, curricular materials, textbooks etc. The introduction of the term "resource" in the title of the technology working group since the CERME 6 congress in 2009 reflects this shift [START_REF] Gueudet | Technologies and resources in mathematical education. Introduction to the Working group 7[END_REF].

Technologies and resources have become more abundant and accessible, bringing about some changes in math teaching and learning (Hoyles & Lagrange 2010). They offer new possibilities for representation, modelling, simulation and access to information and knowledge [START_REF] Tooke | Using Information Technology in Mathematics Education[END_REF], and suggest new ways of generating student engagement, motivation and creativity [START_REF] Passey | The Motivational Effect of ICT on Pupils[END_REF]. Teaching methods with technologies are increasingly focusing on problem-solving approaches [START_REF] Fuglestad | ICT for Inquiry in Mathematics: A Developmental Research Approach[END_REF]. Technologies also offer means of collaborating and sharing resources between students and teachers [START_REF] Trouche | Connectivity and Virtual Networks for Learning[END_REF]. But presently, despite the use of digital technologies in the public and business world, and the tremendous number of research and practical classroom papers, the use of technologies in mathematics education and the impact on the change of curricula are still limited (Hoyles & Lagrange, 2010).

The group organized the contributions around four main themes: (1) Design and use of technologies and resources, (2) Technologies, resources and teachers' professional development, (3) Students' learning with technologies and resources, and (4) Technology-based assessment in math teaching. Surprisingly, the latter does not appear as a matter of concern since all papers addressed the first three themes.

The group work was organized in two sub-groups discussing respectively the issues of technologies and resources in relation with teachers' professional development (theme 2), and the issues related to the design of technologies and resources and their impact on students' learning (themes 1 and 3). In the following, we give a brief overview of the group work according to the three above-mentioned themes.

THEMES IN WG15 1) Design and use of technologies and resources

Papers related to this theme concern on the one hand the design of innovative

2) Technologies, resources and teachers' professional development

Papers related to this theme investigate teachers' use of technologies and resources (both inside and outside the classroom), factors affecting their integration in classrooms, and issues related to teachers' professional development. These issues are addressed focusing on different aspects, from different perspectives and adopting different methodologies. The analysis of the teachers' practice in classroom is investigated at different levels of granularity, each one requiring the design and use of different methodological tools, from fine-grained analyses (Billington), through survey studies (Bretscher), to studies combining both qualitative and quantitative methods (Drijvers). As it clearly emerged, the integration of technologies and resources in the classroom poses a number of challenges to teachers, several of which can be described and discussed in terms of the Technological Pedagogical Content Knowledge that they need to develop (Fuglestad). From the instrumental perspective, some of these challenges can be related to the need for the teacher to transform an ICT tool both to a mathematical and a didactical instrument (Haspekian). All these considerations raise the issue of the teachers' professional development and how it impacts the teachers' use of technology. Studies concerning this issue regarded, for instance, the actual (Amado) or intended use (Pittalis) of technology by pre-service or in-service teachers, who attended specific training courses on technologies in math education. The issue of teachers' professional development is strictly related to the issue of the dissemination of research results among teachers. Developing communities involving both researchers and teachers is proposed as a means to trigger collaboration and communication among them and to foster the dissemination of research results (Lagrange).

3) Students' learning with technologies and resources

Papers Panhuizen). The task design is supported by two main theoretical approaches: variation theory [START_REF] Marton | Learning and Awareness[END_REF] and theory of didactical situations [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. Studies involving innovative pieces of software investigate their potentialities for mathematics learning (Ladel & Kortenkamp) as well as for organizing and exploiting the internal resources (Aldon). Another important issue discussed in the group was the question of assessment of students' competencies (Weigand): what kind of knowledge and skills developed by the student working with technologies do we want to assess and how can we assess it?

CONCLUSION

What lessons can be learnt from the work of the WG15? Although the three themes proposed to the working group appeared as strongly articulated, some important issues emerged for each of them. The design of technologies and resources seems to be driven by the consideration of their added-value for the teaching and learning mathematics and relies on the users' feedback, which becomes part of the design process. The group discussions allowed getting a deeper insight into the complexity of ICT integration in teachers' practices, which requires a double instrumental genesis in teachers: a personal one, yielding an instrument for teachers' math activity, and a professional one, yielding an instrument for math teaching. This raises the issue of teacher training focusing on the development of teachers' technological pedagogical content knowledge. The idea of teachers' communities sharing resources and practices in using ICT emerges as a powerful means to favour teachers' professional development. In studying students' learning with technologies, the instrumental approach becomes widely used to analyse student-tool interaction. It highlights a strong interconnectedness of mathematical and technological knowledge. However, the impact of technology on students' achievements appeared as very difficult to measure, mostly due to the lack of appropriate methodology.

What perspectives can be outlined for the future CERME conference? There is still a need to develop a comprehensive theoretical framework to support teachers in their integration of ICT to the benefit of students' learning and methodological tools to evaluate the impact of using ICT on students' learning and teachers' practices.

Emerging topics, such as communities of practice, quality of resources or best practices, require further theoretical and methodological development. Finally, some topics, which are under-represented, would deserve researchers' interest: use of "new" new technologies, such as interactive white board, mobile devices, Web 2.0, as well as designing ICT for students with special needs.

  technological tools or resources. Ladel & Kortenkamp design and develop a multitouch technology for learning numbers and operations, Sabra & Trouche study the design by a community of teachers of online secondary school math textbook. On the other hand, the use of existing technological tools and resources are addressed, such as the use by teachers of specific curricular material accompanying TI-Nspire (Persson) or of math textbooks (Özgeldi & Çakiroğlu), students' use of TI-Nspire technology (Aldon), of a spreadsheet (Tabach) or of graphic calculators (Consciência & Oliveira; Storfossen). The question of the design of specific tools for students with special needs, such as visually impaired students (Kohanová) or students with dyscalculia, was raised. In the group discussions, added value of technology appeared as the most important element determining what kind of tool to use, how and what for. Various roles have been assigned to technology in math education by the participants: fostering motivation, enhancing calculation, visualization, (guided)

exploration, hands-on experience on abstract models, allowing validation by providing feedback, but also helping teachers teach with systems providing individualized learning paths based on students' competence diagnosis.

  addressing issues related to this theme concerned both widespread technological tools, such as dynamic geometry software(Attorps et al.; Camacho & Santos), symbolic or graphic calculator (Consciência & Oliveira; Storfossen; Weigand), and more innovative tools like TI-Nspire handheld device (Aldon; Persson) or online games (Kolovou & van den Heuvel-Panhuizen). Some of these papers focus on student-tool interactions in order to study processes of students' appropriation of the tool (Consciência & Oliveira; Camacho & Santos) or students' transition between various tools (Tsitsos & Stathopoulou). These studies draw mostly on the instrumental approach[START_REF] Rabardel | People and technology -a cognitive approach to contemporary instruments[END_REF] as a theoretical framework specifically designed for studying teaching and learning phenomena involving technology. In other papers technologies are considered rather as a tool mediating mathematics learning. The choice of a tool and the design of appropriate tasks are the main concerns in these papers(Attorps et al.; Kolovou & van den Heuvel-


A Typology of the Unintended Consequences of Drug Crop Reduction (1)

Drug control policies and interventions, like any other policies and interventions, generate many unintended consequences. Most often, such consequences are mentioned without being defined or presented in a typology and they are rarely explained in terms of causality. This paper will stress how the existing work on the unintended consequences of drug control policies and interventions suffers from little or no definition and will then provide such a definition and a typology applied to three major interventions meant to achieve drug crop reduction: forced eradication, alternative development, and opium bans. In the end it will explain how a typology of unintended consequences can help to better understand the failure and even the counterproductivity of some interventions. Differentiating between direct and collateral unintended consequences allows us to better attribute the occurring of unintended consequences to a specific intervention and, or, to the intended consequence of the interventions.

The steady increase in global illegal opium production observed since the early 1970s (3) occurred in spite of the many efforts deployed by the international community to suppress or reduce illegal opium poppy cultivation worldwide. Illegal opium production has increased despite countless forced eradication campaigns and in spite of many crop substitution and alternative development programs.

It can even be argued that the increase of illegal opium production is partly due, among other obvious push and pull actors, to the counterproductivity of forced eradication campaigns and to the inadequacy of alternative development projects. In fact it is now widely agreed that the global prohibition of certain drugs and the war on drugs have largely failed to reach their stated goals, that is, a drug-free world (Chouvy, 2009) (4).

Drug control policies and interventions, like other policies and interventions, generate many unintended consequences, most of which can be held responsible for the overall failure of prohibition (which can also be blamed for generating seriously disruptive unintended consequences) but also for countless collateral damages in the social, economic, political, environmental, and human rights areas.

Alleged unintended consequences of drug control policies and interventions include crop displacement (one aspect of the so-called balloon effect), increased prices and production, worsened corruption, heightened armed violence (especially in the context of armed conflicts), weakened counterinsurgency, but also social unrest, ethnic insurgency, environmental degradation, increased deforestation, destruction of legal crops, increased poverty and debt, school dropouts, prostitution, human smuggling and trafficking, needle sharing and spread of blood-borne diseases, health issues from poor quality of drugs, substance switch, increased street crime and violence, deteriorated human rights, etc. [START_REF] Bertram | Drug War Politics: The Price of Denial[END_REF][START_REF] Chouvy | Opium. Uncovering the Politics of the Poppy[END_REF][START_REF] Costa | Making drug control 'fit for purpose': Building on the UNGASS decade. Statement of the Executive Director of the United Nations Office on Drugs and Crime, Fiftyfirst session of the Commission on Narcotic Drugs[END_REF][START_REF] Friesendorf | US Foreign Policy and the War on Drugs: Displacing the Cocaine and Heroin Industry[END_REF][START_REF] Heilmann | The international control of illegal drugs and the U.N. treaty regime: Preventing or causing human rights violations? SelectedWorks[END_REF][START_REF] Nadelmann | My Testimony to Congress on the War on Drugs[END_REF][START_REF] Reuter | The unintended consequences of drug policies[END_REF][START_REF] Tullis | Unintended Consequences. Illegal Drugs and Drug Policies in Nine Countries[END_REF]World Bank, 2004;etc.).

Most often, such unintended consequences are mentioned without being defined or presented in a typology. They are also rarely explained in terms of causality, which is highly problematic considering how often drug control policies and interventions focus on consequences rather than on causes (for example targeting opium production instead of poverty and food insecurity) and how easily consequences can turn into causes of further consequences. It is also a problem because most such unintended consequences are the subject of highly controversial and passionate debates, for example, between advocates and opponents of forced eradication. A typology of unintended consequences of drug control policies and interventions is therefore needed not only to better understand the chains of causality that may hinder these policies and interventions, but also to help foresee and prevent the unintended consequences they might generate, especially the most perverse and harmful ones. This is basically what Ethan Nadelmann called for in a 2010 testimony to the U.S. congress on the war on drugs when he asked that "federal agencies involved in the drug war devote a portion of their budgets to evaluating the efficacy and unintended consequences of their policies and programs" [START_REF] Nadelmann | My Testimony to Congress on the War on Drugs[END_REF]. A better understanding of unintended consequences is obviously a prerequisite to a better policy evaluation and, eventually, to better drug control policies: determining if a policy's consequences are intended or not, beneficial or harmful, or even perverse, is crucial. This article will first stress how the existing work on the unintended consequences of drug control policies and interventions suffers from little or no definition. It will then provide such a definition. A typology of the unintended consequences of drug control policies and interventions will follow and will be applied to three major interventions meant to achieve drug crop reduction: forced eradication, alternative development, and opium bans (coercion not to plant). In the end it will explain how a typology of unintended consequences can help better understand the failure and even the counterproductivity of some interventions. the stated goal of the international drug control regime is to regulate the legal production, trade and consumption of drugs on the one hand and, on the other hand, to achieve the suppression of illegal drug production, trafficking and possession mainly through criminal law, that is, through prohibition measures. Yet, as many have observed, "the system, being a prohibition regime focusing on control of the production of psychoactive substances that are deemed to be harmful, has over time developed side-effects that were not foreseen (at least in their magnitude) when it was established" (Heilmann, 2010: 32). These side-effects, or unintended consequences, have been denounced at length for decades.

Undefined unintended consequences

In

Yet, among the vast literature on drug issues very little material explains clearly what unintended consequences are and how they occur. LaMond Tullis' 1995 book on "Unintended Consequences.

Illegal Drugs and Drug Policies in Nine Countries", for example, provides no definition or typology of unintended consequences. Other texts, such as a 1988 article by Ethan Nadelmann, describe unintended consequences in more details and even give a sense of the complexity of the issue but fall short of defining them [START_REF] Nadelmann | Drug Policy: A Bad Export[END_REF]. Among the few authors that have attempted to define what unintended consequences are and how the concept applies to the field of illegal drug studies is Peter Reuter. In a 2009 exploratory report titled "The unintended consequences of drug policies", Reuter does not define what unintended consequences are but he raises a fundamental point when he explains that "an important distinction is between consequences that arise from prohibition itself, as opposed to those resulting from specific implementing programmes" (Reuters, 2009: 4). He then offers a valuable typology -which he terms taxonomy (5) -of unintended consequences, in which he details the mechanisms at work, the bearers of consequences, and the nature of harms (Reuter, 2009: 10). In the end, Reuter stresses a point of great importance when he writes that "almost all of the unintended consequences share one important characteristic; they are unmeasured" (Reuter, 2009: 10). Indeed, most of the so-called unintended consequences of the illegal drug industry or of drug control policies and interventions are unmeasured and unproven, something that is explained in part because "measuring the illegal drug industry raises innumerable conceptual, technical, and political issues" (Thoumi, 2005: 186; see also [START_REF] Reuter | Measuring Global Drug Markets. How good are the numbers and why should we care about them[END_REF]. In a text on unintended consequences, Antonio Maria Costa, a former Executive Director of the United Nations Office on Drugs and Crime, raises the case of such a consequence that is very often mentioned but never measured, the so-called balloon effect, or geographical displacement of a crop (Costa, 2008: 10).

Costa rightly explains that large such balloon effects can be inferred from various repeated correlations but that the causality between a squeeze in one place and a swell in another place can hardly be measured or proven. Correlation obviously does not necessarily imply causation, something that the Fundamental Problem of Causal Inferenceaccording to which it is impossible to observe the effect that would have happened if the determinant were exposed to another condition at the same timeonly makes more complicated [START_REF] Holland | Statistics and Causal Inference[END_REF][START_REF] Heckman | The Scientific Model of Causality[END_REF]. This is actually what Cornelius Friesendorf, the author of the most detailed and serious study of the balloon effect (a term he rightly rejects), stresses when he writes that "a heuristic, interpretative approach" is needed in order to grasp "the presence of various empirical black boxes, the presence of complex, dynamic interaction effects, the fuzzy nature of corruption and other phenomena contributing to displacement, and the difficulty of assigning causal weight to explanatory factors and processes" (Friesendorf, 2005: 67).

In his paper Costa lists five unintended consequences of what he terms "the control system and its application", that is, the prohibition regime on one hand, and drug control policies and interventions on the other hand, two very different things that he groups indiscriminately. The first unintended consequence he mentions, a "huge criminal black market", is clearly a consequence of the prohibition regime. The second unintended consequence is actually a consequence of the black market itself: what he calls the "policy displacement", that is, the focus on coercion and the allocation of public funds to law enforcement and public security rather than to public health. Geographical displacement, the third unintended consequence, is not a direct consequence of the prohibition regime but rather of drug production reduction interventions, especially bans and eradication. A fourth unintended consequence is "substance displacement", for example from opium to heroin: again, a consequence of repression, both at the production reduction and demand reduction levels. The fifth unintended consequence, according to Costa, is the exclusion and stigmatization of drug users, the causes of which are more difficult to ascertain. These unintended consequences, far from being exhaustive, differ at least according to their direct causes and, for the sake of clarity, should not be combined in a unique category. Where Costa is right, though, is when he writes that "unless we face these unintended consequences head-on, we will continue to be mesmerized by the many paradoxes of the drug problem" (Costa, 2008: 10-11). But before being faced head-on, unintended consequences first need to be defined and categorized according to their occurrence mechanisms and their causes. This is obviously no small task as Francisco Linares explains that, despite unintended consequences being "undoubtedly one of the key concepts in sociology" and despite the efforts of many scholars who have attempted its analysis (including Baert, Boudon, Elster, Merton, and Portes), "there is not a general explanatory model of the emergence of unintended consequences of social action" (Linares, 2009: 1). This essay obviously does not intend to offer such a general explanation. It is an initial attempt at better understanding the unintended consequences of drug control policies and interventions.

Toward a definition and a typology of unintended consequences

Strictly speaking, unintended consequences are consequences that were not planned for in a plan of action, whether the consequences are contrary or not to the actor's initial objectives. Of course, and as stressed by Daniel Little, an unintended consequence differs from an "unfortunate event" (6). In what is widely considered to be the seminal paper on unintended consequences, "The Unanticipated Consequences of Purposive Social Action" (1936), Robert Merton explained that, "rigorously speaking, the consequences of purposive action are limited to those elements in the resulting situation which are exclusively the outcome of the action, i.e., those elements which would not have occurred had the action not taken place" (Merton, 1936: 895). The unintended consequences of drug control policies should therefore be considered to be the direct and strict outcomes of drug control policies and interventions, that is, of the efforts produced to achieve the reduction or suppression of drug production, trafficking, and consumption.

Yet, Merton actually refers to unforeseen or unanticipated consequences in his paper, not to unintended consequences. He explains that "unforeseen consequences should not be identified with consequences which are necessarily undesirable (from the point of view of the actor)" and that the "intended and anticipated outcomes of purposive action, however, are always, in the nature of the case, relatively desirable to the actor, though they may seem axiologically negative to an outside observer" (Merton, 1936: 895). In fact, unintended consequences differ from unanticipated consequences (7) since outcomes that were not intended by purposive action can be foreseen or not: unintended consequences can be anticipated or not. Indeed, as Paul Helm explains, "a person might therefore not intend a particular consequence of his action but still foresee it. In saying that he might not intend a particular consequence it is implied not that the consequence is necessarily not intended but that it is neither intended nor unintended" (Helm, 1971: 51-52).

Moreover, an unintended consequence, whether foreseen or not, can also be harmful or beneficial, Adam Smith's "invisible hand" being a famous example of the latter. It can also be neutral but we will not concern ourselves with this specific case here. Daniel Little stresses that "we can't know all the possible results of an action undertaken". He then explains that the so-called law of unintended consequences implies that, "no matter how careful one is in estimating the probable consequences of a line of action, there is the residual possibility that the action will produce harmful unanticipated consequences that negate the purpose of the action" (what we will later call perverse unintended consequences). According to Little, the planner of an action "should design the plan so as to minimize avoidable bad consequences; then do a cost-benefit analysis to assess whether the value of the intended consequences outweighs the harms associated with the unintended consequences." Indeed, it is possible that "an undesired outcome is both unintended but also fully foreseen." In fact, some will argue that most of the unintended consequences of drug control policies seem to fall into that category. Most of them have been suspected for decades but keep occurring mostly as the result of the perpetuating, and often of the increase, of inadequate or failed policies and interventions. Such consequences also keep occurring partly because they are often ignored or, to use Merton's terminology, not recognized. Again, most unintended consequences can only be alleged because they are unmeasured and unproven.

But to make things a bit more complex, Helm writes: "An action can perfectly well have unintended consequences which are the effect not of any intended consequences of it but simply of the action itself. So an action may have one or a set of unintended consequences (set I) that is collateral to another consequence or set of consequences (set II), in the sense that set I may have an intended consequence as one of its necessary conditions, set II may not" (Helm, 1971: 52). It is important indeed to acknowledge that the "unintended consequences of intended actions" clearly differ from "the collateral consequences of intended actions" (Helm, 1971: 52).

At this point the understanding of causal mechanisms is essential. Considering how complex interaction effects and contingent conditions can prove and how they can change outcomes in unforeseeable ways, the choice of probabilistic causality over deterministic causality seems logical: one can only assume the likely, not the systematic, effect of an independent variable on a dependent variable. As Friesendorf rightly explains about crop displacementwhat he calls a side effect rather than an unintended consequence (8) -the fact that "numerous background conditions, idiosyncrasies, and explanatory factors may operate simultaneously or in a random order […] means that although displacement can be studied as a chronological sequence of events, the underlying causality of displacement cannot be studied in this manner" (Friesendorf, 2005: 44-45). Indeed, illegal drug production and drug control policies and intervention take place in a system of complex interactions that has the potential of generating many unintended and unforeseen consequences. As stressed by Robert Jervis and others, such complex systems are non-linear and cannot be "understood by adding together the units or their relations" [START_REF] Jervis | System Effects: Complexity in Political and Social Life[END_REF]Aoi et al., 2007: 11). Alejandro Portes also expresses skepticism about the routine implementation of a linear relationship, that is, the "straight arrow between the avowed goal of actors-individual or collective-and the achieved end-state" (Portes; 2000: 8) (9).

Direct and collateral unintended consequences (harmful, beneficial, and perverse)

A typology of unintended consequences is a prerequisite to their study and analysis, if only to avoid a few common mistakes, including an obvious yet not uncommon such mistake mentioned by Chiyuki Aoi et al.: "Unintended consequences need to be distinguished from a failure to achieve the intended consequences" (Aoi et al., 2007: 6). Indeed, an increase in opium poppy cultivation caused by forced eradication clearly differs from a cultivation increase being unhindered by failed forced eradication, or even from a renewed cultivation the year after a successful yet unrepeated forced eradication.

To begin with, one needs to distinguish between two main types of unintended consequences: the unintended consequences of an action (set II of Helm or direct consequences) and the unintended consequences of the intended consequence of the action (set I of Helm or collateral consequences).

Direct and collateral unintended consequences can of course be harmful, beneficial, or even neutral (for actors, institutions, and even the environment), but also foreseen or unforeseen. But they can also be perverse.

Perverse unintended consequences are consequences of an action (direct consequence) or of the intended consequence of an action (collateral consequence) that contradict the initial goal of a given purposive action: an increase in cultivation caused by forced eradication or some of its consequences (poverty and debt increase) for example. Here the temporal order and the bearers of the action obviously matter: since one consequence may easily become a cause, there are also proximate and ultimate, or rather distal (for causality chains may be endless), unintended consequences that occur very far down and, or, very late, in the causality chain. Indeed, forced eradication can be said to be successful in year 1 but counterproductive in year 2. Perverse unintended consequences obviously should matter the most to policy makers because they reveal the inefficiency or even the counterproductivity of a given policy and, or, intervention.

Yet, as Daase and Friesendorf write, "many unintended consequences do not fall into neat categories": "it is not always clear whether and when unintended consequences impact on policy initiators or others, and whether such consequences are positive or negative" (Daase, Friesendorf, 2010: 10). Still, unintended consequences being consequences that were not planned for in a plan of action, one should distinguish between two main types of unintended consequences: direct and collateral. Such a distinction basically allows us to differentiate the consequences of an action from the consequences of the intended consequence of an action: it can prove useful in assessing the efficiency or the inefficiency of a given policy and of its various interventions.

All public policies create unintended consequences of the direct and collateral types, whether they are beneficial, harmful, neutral, or perverse. Some of them can be foreseen, others cannot. Harms and benefits can be felt at various levels, in the short term or in the long term, and by various bearers: from individuals to various groups and subgroups, governments, nations, to national and international organizations. In the end, drug control policies can only be successful, that is, to achieve their intended consequences, if no perverse unintended consequence is generated or, in other words, if they do not end up being counterproductive. That is, also, when harmful unintended consequences are kept to a minimum so that further consequences, including perverse ones, are less likely (10).

Table 1. Example of various unintended consequences of forced eradication

Direct unintended consequences

Collateral unintended consequences The intended consequence of forced eradication (the action) of opium poppies is the suppression of the crop. The most obvious unintended consequences of successful forced eradication are crop displacement and, or, increased cultivation, as mentioned, for example, in a World Bank report on Afghanistan: "A key lesson is that eradication alone will not work and is likely to be counterproductive, resulting in perverse incentives for farmers to grow more drugs (e.g. in Colombia), displacement of production to more remote areas, and fuelling of violence and insecurity (Peru, Bolivia, Colombia), which in several cases forced the eradication policy to be reversed and led to adverse political outcomes" (World Bank, 2004: 87). Here, of course, causality can only be assumedsince it can hardly be scientifically provenon the basis of previous observations (correlations), and also of induction (11). To be more specific, crop displacement and increased cultivation can be said to

Action

From action to direct UCs, to intended consequence, and to collateral UCs be collateral perverse unintended consequences: the outcome is clearly contrary to the intended consequence of the initial action and is clearly an effect of that intended consequence (set I of Helm).

In this case it could be argued that we are also confronted with a foreseen or at least a foreseeable unintended consequence for such consequences have allegedly occurred so often and in so many countries that they could be said to be predictable (12). Now, the action of forcefully eradicating a crop might also stir armed violence as insurgents (in Afghanistan especially) often oppose the eradication teams (themselves often if not always protected by armed forces). Armed violence can be said to be an unintended consequence of the action itself but not of the crop having being eradicated (that is, the action's intended consequence) per se: it is a direct harmful unintended consequence (set II of Helm) that could easily be foreseen since armed violence often occurs to prevent the action from being carried out and can take place without any eradication being conducted (when violence achieves its goal of preventing eradication).

Environmental pollution and the destruction of food crops or other cash crops can be said to be other direct harmful unintended consequences when eradication is conducted through aerial spraying of chemicals: they can occur outside of the actual destruction of the drug crops and are then consequences not of the drug crop having been destroyed but of the action itself. Here, again, these consequences can easily be foreseen as such collateral damages have often been documented, especially in South America [START_REF] Messina | Defoliation and the war on drugs in Putumayo, Colombia[END_REF][START_REF] Vargas | The Anti-Drug Policy, Aerial Spraying of Illicit Crops and their Social, Environmental and Political Impacts in Colombia[END_REF].

Harmful unintended consequences, whether direct or collateral, are legion and many of them could again easily be foreseen since they have been documentedyet rarely measured or provenon numerous occasions and in various countries. To name but a few: poverty increase (the main driver of illegal opium production being poverty, eradicating poppy fields most often increases poverty); debt increase (when opium is sold ahead of the harvest and when eradication prevents the repayment of debts); food shortages (when opium is meant to be sold to buy food), school drop-outs (when opium sales allow families to send their children to school), etc. The eradication of an opium poppy crop will often deprive targeted farmers and seasonal workers from most of their income (and, or, ability to repay debts) and resources (access to land, work, credit) and can be said to be responsible of various coping strategies (unintended consequences) that include rural exodus, prostitution (especially in Burma), deforestation (as in Burma but also in Afghanistan), poaching (again, in Burma or in Laos), etc. [START_REF] Chouvy | Opium. Uncovering the Politics of the Poppy[END_REF][START_REF] Jelsma | Downward Spiral. Banning Opium in Afghanistan and Burma[END_REF].

It is important to note that if harmful unintended consequences of forced eradication do not always cause the occurring of perverse unintended consequences, the latter rarely occur outside of preceding harmful unintended consequences: depending on the context and on other factors, poverty and debt increase may or may not lead to crop displacement or increased cultivation. Causality is of course not systematic and some opium farmers may have other options than renew and increase cultivation. Crop displacement is also not systematic and is highly context-dependent and subject to many explanatory factors. What is more, harmful unintended consequences can take place ahead of, or after, perverse unintended consequences: some such consequences (poverty and debt increase for example, but also price hikes) can be caused by the intended consequence (eradicated crops) and be the immediate cause of crop displacement or cultivation increase (to cope with income loss), but other harmful unintended consequences can also be caused by the perverse unintended consequence (deforestation caused by crop displacement, decrease of food crop production caused by increased drug crops) at a later stage. Unfortunately, there are far less beneficial unintended consequences, beyond the price hikes benefiting some farmers, the unlikely development of alternative crops, the even more unlikely lowering of corruption, and targeted poppy cultivators and seasonal workers joining the military (as in Afghanistan).

In the end, forced eradication most often increases poverty and therefore reinforces the main driver of opium production without addressing its causes, including land scarcity, unequal land tenure arrangements, oversized households (Afghanistan) or lack of workforce (Burma, Laos), food insecurity, climatic vagaries, political upheavals, armed conflicts, etc. [START_REF] Chouvy | Opium. Uncovering the Politics of the Poppy[END_REF][START_REF] Ducourtieux | Shifting Cultivation and Poverty Reduction in the Uplands of Lao PDR: A Complex Issue[END_REF][START_REF] Mansfield | Diversity and Dilemma: Understanding Rural Livelihoods and Addressing the Causes of Opium Poppy Cultivation in Nangarhar and Laghman, Eastern Afghanistan. A Report for the Project for Alternative Livelihoods (PAL) in Eastern Afghanistan[END_REF]. Forced eradication is highly likely to be counterproductive, that is, to have perverse unintended consequences. Worse, forced eradication also generates many harmful unintended consequences, most of them collateral. We will see that while alternative development also generates harmful unintended consequences, most of them are direct unintended consequences.

Also, alternative development is much less likely to provoke perverse unintended consequences.

Unintended consequences of alternative development

Rural development, a constructive intervention often labeled as alternative development when implemented in regions of illegal agricultural drug production, sets goals that take much longer to be met and whose success is much more complicated to demonstrate than forced eradication. Causality chains and unintended consequences are far more complex to infer than with forced eradication, not only because development-related interventions have considerably evolved and diversified during the past four decades (unlike forced eradication) but also because rural development being a constructive intervention, it relies on a wide and complex range of actions rather than on single measures. Such actions range from introducing new seeds or new crops, along with improving agricultural techniques and tools, to providing or upgrading health and social equipments and varied infrastructure work.

Also, national strategies can be organized around various priority areas and pillars of action. They often involve a large number of actors, from national agencies to international bodies and local and foreign NGOs. Assessing the efficiency of rural development in suppressing or even reducing illegal agricultural drug crops in such complex contexts can obviously prove very difficult.

In fact, many observers agree that "in those places where lasting reductions in production have been seen, other possible influences on farmer decisions not to cultivate drug crops can be put forward as being equally likely causes for change. These include: overall economic growth (Thailand and Viet Nam), political change (Myanmar), increasing government access to formerly remote areas (Pakistan), social pressure (Lao PDR, Bolivia), subsidies (Thailand), and booming prices for alternative crops (coffee and cacao growing areas)" (UNODC, 2005: 9-10; see also [START_REF] Jelsma | Alternative Development And Drug Control: A Critical Assessment[END_REF].

Unintended consequences of alternative development have also proven very diverse depending on the type of rural development undertaken but also on the extremely varied local settings: the crop substitution programs implemented in Turkey and Thailand in the early 1970s differed from the integrated rural development first experimented with in Peru in the early 1980s, and from the alternative development approach devised in the 1990s and tested for example in Thailand and in Pakistan. Crop substitution programs have been widely documented both in Latin America and in Asia. In the highlands of Thailand such programs produced mixed results in the 1970s with harmful unintended consequences such as market gluts and decreasing prices, soil and stream pollution due to excess of chemical pesticides and fertilizers, etc. (Renard, 2001: 57-68) Eventually, the 2000s, marked by the unprecedented increase of Afghanistan's opium production, saw the emergence of the alternative livelihoods approach (13), the last step in a forty-year trend that saw development programs in drug production contexts become increasingly complex [START_REF] Mansfield | Alternative Livelihoods: Substance or Slogan? AREU Briefing Paper[END_REF].

Since the intended consequence of alternative development programs is the reduction or even the suppression of a drug crop, its perverse unintended consequences would presumably be the same compared with forced eradication, that is, crop displacement and, or, increased cultivation. Yet, provided that human security and state stability exist, successful alternative development projects are less likely to have such perverse unintended consequences: alternative development often fails for various reasons but rarely proves counterproductive, unlike forced eradication [START_REF] Chouvy | Opium. Uncovering the Politics of the Poppy[END_REF]. This is actually an example of when "unintended consequences need to be distinguished from a failure to achieve the intended consequences" (Aoi et al., 2007: 6). In the end, alternative development is more likely to provoke direct unintended consequences than collateral unintended consequences. This is because the intended consequence, the suppression of the illegal crop, is rarely obtained (for various reasons), but also because the suppression of an illegal crop is to be obtained by progressively substituting an economy to another one. There is obviously also much less risk of perverse unintended consequences occurring when crop suppression is obtained progressively through economic development than when it is achieved suddenly and without compensation through forced eradication (14). In fact, in the context of alternative development, crop displacement and cultivation increase phenomena are more likely to be the effect not of the intended consequence of the action (alternative development takes years if not decades to achieve drug production reduction) but of the effect of the action itself (the procurement of alternative livelihoods).

Such unintended consequences were suspected in various countries, at various times, and for various reasons, almost always in the short term (Chouvy, 2009: 178-187). In Afghanistan, for example, some alternative development projects are likely to have played a positive role in reducing opium production in targeted areas and for short periods (in Helmand in the late 2000s for example) but are also believed of having quickly spurred production around the project areas. In fact, when rural development projects are scarce and, or, insufficiently funded, opium production can be perceived as part of a means of attracting international aid. Alternative development being always carried out locally and often as pilot projects, it can act as an incentive for opium production in areas where no rural development, "alternative" or not, is implemented. Yet the failure of an alternative development project can also lead to the quick resurgence of a drug production, as has been the case in northern Laos in the late 2000s, when poppy cultivation largely resumed because, according to UNODC, of a lack of economic alternatives. The resurgence or even increase of cultivation can also happen despite, and not because of, the implementation of an alternative development project: most notably when alternative development takes place after forced eradication and not before.

Unintended consequences of opium bans

Opium bans (coercion not to plant) can also prove highly counterproductive, as the opium ban implemented by the Taliban in 2000 clearly showed (Chouvy, 2009: 150-152;Macdonald, 2007: 84).

Opium bans differ from eradication for they amount to the interdiction of cultivation, not to the forced destruction of standing crops: local (the Wa in Burma for example) or national authorities (the Taliban, the Pakistani and the Lao governments for example) issue an opium ban and opium farmers comply. Successful interdiction results from the use of authority and power while forced eradication is obtained by force, although threat of force (including eradication) use obviously makes bans more easily and widely respected. A consequence of the Taliban ban was the steep rise of opium prices in the country, which translated into a collateral perverse unintended consequence (15) of the ban (it was a consequence of the intended consequence of the ban), which was that poppy cultivation resumed with renewed vigor as early as 2002 as many opium farmers had to quickly repay their debts (aggravated by the ban) and took advantage of inflated opium prices. These two consequences actually kept driving Afghan opium production up during the following years, in spite of the repeated opium bans pronounced by the Karzai administration.

Such a cultivation and production increase had actually already occurred in Pakistan consecutively to the 1979 Prohibition Order: since the population had been warned in 1978 that opium poppy cultivation would be banned in 1980, the 1979 illegal crop became the largest in Pakistani history, something that was most likely expected and can only be said to have been a direct unintended consequence for it was not an effect of the intended consequence (rather an effect of the anticipated intended consequence) but an effect of the action (a forthcoming ban). Here the effective ban was largely held responsible, rightly or not, for a hike of the Afghan production, in the same way that the 2003 surge in cultivation (collateral perverse unintended consequence) in the Wa area of Burma was said to be linked to the opium ban enforced the same year in the neighboring Kokang region. Both the Kokang (2003) and Wa (2005) opium bans also had collateral harmful unintended consequences: assistance was then (and still is) clearly "insufficient to offset the impact of the opium bans, and to cover basic needs of ex-opium farmers" (Transnational Institute, 2005: 16). In fact, according to UNODC, "in Special Region 2 (Wa) […] where local authorities enforced an opium ban in 2005, farmers have lost up to 70% of their cash income" (UNODC, 2006: 15).

Of the importance of acknowledging and understanding unintended consequences

Better assessments of drug control policies and interventions are clearly needed to design policies and interventions that are more efficient, less harmful and, of course, less counterproductive. But a better assessment of drug control policies and interventions requires more than simply enumerating unintended consequences: it actually takes the developing of systematic ways of studying them [START_REF] Reuter | The unintended consequences of drug policies[END_REF]. And a more thorough study of unintended consequences first requires not confusing them with a failure to achieve the intended consequences [START_REF] Aoi | Unintended Consequences of Peacebuilding Operations[END_REF]. Then, as Peter Reuter makes clear, another "important distinction is between consequences that arise from prohibition itself, as opposed to those resulting from specific implementing programmes" [START_REF] Reuter | The unintended consequences of drug policies[END_REF](Reuter, : 2010)).

What is also required is discerning between the two main types of unintended consequences: direct and collateral, that is, between, the unintended consequences of an action and the unintended consequences of the intended consequence of an action [START_REF] Helm | Manifest and Latent Functions[END_REF]. Differentiating between direct and collateral unintended consequences makes a more precise assessment of actions and of their intended consequences possible. Since the reduction or suppression of illegal drug cultivation can be obtained through three different interventions, separately or concurrently, in various sequencing orders, and with different resources made available for each intervention, a better assessment of drug supply reduction at the production level implies analyzing complex interaction effects and contingent conditions that can change outcomes in unforeseeable ways. As stressed by Friesendorf, empirical black boxes, complex and dynamic interaction effects, corruption, and the "difficulty of assigning causal weight to explanatory factors and processes" makes the study of unintended consequences a very difficult undertaking (Friesendorf, 2005: 67). Yet, differentiating between direct and collateral (the reduction and, or, suppression of illegal drug cultivation) that are to be better selected to achieve drug control policy goals.

Specifying expected outcomes of drug control policies, on one hand, and, on the other hand, acknowledging errors from the past in order to avoid perverse but also harmful unintended consequencesand especially the foreseeable onesis key to a more efficient and less detrimental international drug control regime. The variety and complexity of unintended consequences clearly needs to be integrated in the design process of drug control policies in order to avoid

4)

Despite the post-1971 surge in illegal production of opium, coca, and maybe cannabis, partisans of the containment theory suggest that an "increase in the size and scope of the illicit drug industry would have been far greater in the absence of law enforcement" (Windle and Farrel, 2012: 874). Others adopt a more balanced approach: "The consolidation and expansion of the control regime in the 1960s, 1970s, and 1980s, to include prohibition against consumption, did not prevent renewed expansion of opiate consumption or the tendency toward mass markets and widespread distribution networks-nor does the adoption of the more stringent policies appear to have caused them" (Paoli, Greenfield, Reuter, 2012: 932).

5)

On the difference between typology and taxonomy, see In another text, Friesendorf, along with Christopher Daase, explains that it is better to speak of unintended consequences rather than of side effects or paradoxical effects: referring to Jervis they ask how one can say which effect was a main effect and which one was a side effect. (Daase, Friesendorf, 2010: 9) 9)

Portes actually offers a valuable typology of five alternatives that represent different end-states from those assumed by a purposive logic: (1) the real goal is not the apparent one;

(2) the real goal is not what the actors actually achieve; (3) the real goal emerges from the situation itself; (4) the original goal is real, but the end-state is contrary to its intent; (5) the original goal is real, but it is achieved by an unexpected combination of events (Portes, 2000: 8).

10) It was asked by two anonymous reviewers of this paper to refer to the literature on types of crime displacement [START_REF] Repetto | Crime Prevention and the Displacement Phenomenon[END_REF] and on the diffusion of benefits theory, according to which the geographical displacement of crime (crimes prevented in a given area are displaced into other areas), for example, happens along with the geographical diffusion of benefits (in addition to crime reduction or suppression in the target area there are further benefits in terms of reduced crime levels in other areas) [START_REF] Barr | Crime Placement, Displacement, and Deflection[END_REF][START_REF] Clarke | Diffusion of Crime Control Benefits: Observations on the Reverse of Displacement[END_REF]). Yet we doubt that an approach in terms of crime displacement and of diffusion is adapted to the subject under review, for various reasons: because opium farmers are not criminals (maybe offenders, but not by choice); because opium production is more an economic matter than a police or military matter; because crop displacement in rural areas of poor countries cannot decently be compared to crime displacement in cities of modern countries; and because the diffusion of benefits is very unlikely to take place in the case of forced eradication (low opportunity cost of planting opium poppy and having it eradicated, which means that cultivating wheat instead of opium poppy is generally not an option as few farmers have sufficient land) or in the case of alternative development (the history of AD is mostly a history of failed small scale projects and no so-called halo, bonus, free-rider or other multiplier effects can be envisaged).

11)

David Hume explains how inductionand not reasoningallows us to link these two propositions: "I have found that such an object has always been attended with such an effect, and I foresee, that other objects, which are, in appearance, similar, will be attended with similar effects". Hume then explains: "I shall allow, if you please, that the one proposition may justly be inferred from the other: I know, in fact, that it always is inferred. But if you insist that the inference is made by a chain of reasoning, I desire you to produce that reasoning." [START_REF] Hume | Enquiries Concerning the Human Understanding, and Concerning the Principles of Morals[END_REF][START_REF] Hume | Enquiries Concerning the Human Understanding, and Concerning the Principles of Morals[END_REF]: part II §29)

12)

Francisco Linares distinguishes weak from strong unintended consequences:

"Unintended consequences which are foreseeable (whether desirable or undesirable) are not "unexpected" in the same sense that unforeseeable ones are (that is why they are "weak"), because the causal mechanism in the second case is not usually clear" (Linares, 2009: 8).

13)

The alternative livelihoods approach has emerged in order to overcome the limitation of previous development approaches that have been "designed as a specific response to reductions in opium production" but did not address "the underlying structural and institutional reasons that have led to the growth of opium poppy cultivation in the first place" (Mansfield, 2007: 70).

14)

The speed of implementation that characterizes forced eradication actually explains, in part, its non durability.

15)

The rumor actually went that such an outcome was actually intended or at least that the Taliban planned on benefiting from the sale of opium stocks at increased prices.

  order to assess what the unintended consequences of drug control policies and interventions are, one first needs to define what unintended consequences are and, before that, assess what the stated goals of drug control policies are. According the 1961 Single Convention on Narcotic Drugs,

  Drug production reduction can be attempted in three different ways, independently or concurrently: coercion not to plant (when local or national authorities issue and implement a crop ban, out of authority), eradication (when standing crops are destroyed, forcefully or not), and economic development usually referred to as alternative development (when economic alternatives are provided to farmers involved in illegal agricultural production). Forced eradication of drug crops is the most coercive intervention aimed at achieving drug production reduction and provides a good example of the various unintended consequences that drug control policies and their interventions can generate.

  unintended consequences allows us to better attribute the occurring of unintended consequences to a specific intervention or to a specific set of interventions, on the one hand, and to the intended consequence of the interventions on the other hand.Harmful unintended consequences are of course to be avoided as much as possible, whether they occur before or after the suppression of illegal drug cultivation, in the short term or in the long term, and no matter who the bearers are. Cost-benefit analyses obviously matter to assess whether the value of the intended consequences outweighs the harms associated with the unintended consequences but drug control policies and interventions can still be successful despite causing serious harms to various bearers. Yet, harmful unintended consequences can foster perverse unintended consequence and compromise drug control policies altogether. Avoiding perverse unintended consequences, i.e. the counterproductivity of the actions undertaken or of their intended consequences, is of course a prerequisite to more efficient drug control policies. The fact that collateral perverse unintended consequences are much less likely to be generated by alternative development than by forced eradication shows that it is the interventions and not their basically identical intended consequences

  counterproductive drug control interventions. Yet, at this stage, further theoretical and empirical work is obviously needed to better understand how and why intended and unintended consequences of drug control policies and interventions are produced. What is needed is more complex and detailed typologies along with more accurate descriptions of the causality chains at work, something that obviously requires specific empirical research, for example on where, when, how, and why cultivation increases and crop displacements occur, or not.

	Notes
	1)	This article is part of the LINKSCH Research Project, funded by the European
	Commission. Thanks go to Laurent Laniel and David Mansfield for their comments on earlier
	drafts of this paper.
	2)	Geographer and research fellow at CNRS-Prodig, Paris, France. Author of Les
	territoires de l'opium (2002, Genève: Olizane), Yaa Baa (2004, Singapore: Singapore
	University Press), and Opium. Uncovering the Politics of the Poppy (2009 / 2010, London /
	Cambridge: I.B. Tauris / Harvard University Press).
	3)	Allegedly 1,066 tons in 1970, one year before the launch of the war on drugs, and,
	according to the United Nations Office on Drugs and Crime, at least 10,649 tons in 2009,
	including 2,895 tons of non-processed opium (UNODC, 2010). In 2012, without explanation,
	UNODC did not mention (or did not include in its estimate?) any production of non-
	processed opium, making its global estimate of 6,995 tons impossible to compare with the
	2009 output (UNODC, 2012).