Hugo Boujut

Vincent Buso

Guillaume Bourmaud

Jenny Benois-Pineau

Rémi Mégret

Yann Gaestel

Jean-François Dartigues

Egocentric vision IT technologies for Alzheimer disease assessment and studies

Keywords: Egocentric vision, wearable camera, activity monitoring, IADL, Alzheimer disease, dementia

Egocentric vision technology consists in capturing the actions of persons from their own visual point of view using wearable camera sensors. We apply this new paradigm to instrumental activities monitoring with the objective of providing new tools for the clinical evaluation of the impact of the disease on persons with dementia. In this paper, we introduce the current state of the development of this technology and focus on two technology modules: automatic location estimation and visual saliency estimation for content interpretation.

I. INTRODUCTION

The context of actual work is the multi-disciplinary research on Alzheimer disease [START_REF] Dovgalecs | Multiple Feature Fusion Based on Co-Training Approach and Time Regularization for Place Classification in Wearable Video[END_REF][START_REF] Karaman | Human Daily Activities Indexing in Videos from Wearable Cameras for Monitoring of Patients with Dementia Diseases[END_REF][START_REF] Szolgay | Detection of moving foreground objects in videos with strong camera motion[END_REF]. The goal here is to ensure an objective assessment of the capacity of patients in conducting IADLs (Instrumental Activities of Daily Living). Impairments in IADL are considered as the core of the diagnosis and the management of Alzheimer's disease. Thus an objective evaluation is crucial. Within the Dem@care FP7 EU 1 project several complementary approaches are combined. This project follows our previous study on the applicability of the technology and automatic recognition of activities ANR Blan IMMED project [START_REF] Dovgalecs | Multiple Feature Fusion Based on Co-Training Approach and Time Regularization for Place Classification in Wearable Video[END_REF][START_REF] Karaman | Human Daily Activities Indexing in Videos from Wearable Cameras for Monitoring of Patients with Dementia Diseases[END_REF] Based on a combination of several sensors (within a subset of: motion, presence, fixed or wearable cameras, audio…) automatic processing of the acquired data extracts low-level events, that are then analyzed and compared to provide a comprehensive representation of the activities of the monitored person. This information can then be shown to a clinician or medical staff as indices to better assess the status of a patient or fed into an automatic system for patient feedback. In this paper, we consider the use of a wearable camera worn on the shoulder as a tool that provides a privileged viewpoint on the IADLs. The complete analysis by a clinician of the videos acquired in such situations would help assessing the effect of the disease by providing a direct close-up observation of the IADLs when the patient accomplish them, either in an ecological situation at home 1 http://www.demcare.eu/ or during directed exercises in the Lab. This nevertheless requires the development of advanced automatic data processing algorithms, for two main reasons: first, the amount of data to be analyzed is tremendous and cannot be exploited efficiently without tools to detect, summarize and present in a suitable way only the most relevant information from the bulk of data; second, the raw video data may not be displayed completely as privacy issues would require the data to be either adequately filtered to select only a subset of the video or processed entirely automatically to produce aggregated descriptors. In [START_REF] Karaman | Human Daily Activities Indexing in Videos from Wearable Cameras for Monitoring of Patients with Dementia Diseases[END_REF] the framework for video monitoring with wearable camera was designed. The recording is realized at patient's home, i.e. in an ecological situation. The computer vision task consists in an automatic recognition and indexing of IADLs from a taxonomy proposed to patients in each recording. In this paper, we focus on two technology modules within this framework: automatic estimation of localization and visual saliency prediction of the patient .executing the IADL

II. MATERIAL AND METHODS

Wearable camera acquisition Hardware

The camera is fixed on a lightweight jacket worn by the person. It has to be positioned such that the field of view captures the instrumental activity zone in front of the person and context elements without being occluded by the shoulder. The acquisition set-up was studied by neuropsycho motriciens in order to ensure the least inconvenience for the patients. The currently used camera is a GoPro camera that records the video sequence using H.264 compression with a resolution of 1280x960p at 30 frames/s. Its very low weight (around 100g) is unnoticeable by the wearer. During capture, the video is stored on the onboard SDcard, which can then be processed and visualized on a computer with the analysis software.

Location estimation

For recognition of IADL and feed-back loop to the patients, the identification of patients' position in his home is of primarily importance. Location estimation from ambient cameras in an instrumented home is not sufficiently precise, due to the presence of occlusions by elements of the environments and caregivers. This is why location estimation from egocentric video is needed. FIGURE 1: Wearable camera device and matching support jacket. Location estimation can be done at different granularities: room [START_REF] Dovgalecs | Multiple Feature Fusion Based on Co-Training Approach and Time Regularization for Place Classification in Wearable Video[END_REF], place or 3D position of the patient [START_REF] Wannous | Place Recognition via 3D Modelling for Personal Activity Lifelog Using Wearable Camera[END_REF]. When the indoor environment can be modeled by a set of known places or a 3D model the 3D localization of the person can be estimated. The pipeline currently developed and used at University Bordeaux 1 is following the architecture described in [START_REF] Wannous | Place Recognition via 3D Modelling for Personal Activity Lifelog Using Wearable Camera[END_REF]. It combines interest point detection and matching with Structure from Motion reconstruction to produce 3D point-clouds from a set of up to hundreds of images.

Visual saliency for content interpretation

The content recorded with wearable cameras is a rich source of information for a medical practitioner for analysis of the behavior and the sequence of tasks of the patient executing IADLs. One of the important indicators of the patient's condition is the concentration on the activity being fulfilled. This indicator can be extracted from video, if the focus of interest of the patient differs from predicted one. Furthermore, we seek to limit the automatic analysis of the observed dynamic scene to the area of interest which is visually salient for the medical practitioner. For egocentric video the problem of visual saliency detection is posed in a new and challenging way. Two different persons are involved: the Actor wearing the video camera and the Viewer who is interpreting the video. Their visual saliencies are not the same. Indeed according to the physiological studies [START_REF] Prablanc | Optimal response of eye and hand motor systems in pointing at a visual target[END_REF][START_REF] Dorr | Variability of eye movements when viewing dynamic natural scenes[END_REF], the human gaze anticipates the motor action of limbs when fulfilling an activity. We study relation between the Actor's and Viewer's saliencies realized on a subjective saliency maps, obtained with eye-trackers on the basis of this biophysical fundings.

Video Data

For this work, a dataset containing the eye locations of the persons performing the actions (Actors) is needed in order to compare with the gaze coordinates of the people watching these actions on video (Viewers). Along with their paper [START_REF] Fathi | Learning to recognize daily actions using gaze[END_REF], the authors have publicly released the GTEA video and gaze dataset obtained with Tobii eye-tracking glasses. The videos are at a 15fps rate and a 640*480 pixel resolution. For the gaze location, two points per frame are recorded (30 samples per second). The subjects were asked to execute a specific IADL "preparing a meal" using different ingredients placed on the table in front of them. In total 17 videos of 4 min in average are available, performed by 14 different participants.

Extraction of visual saliency maps

We extract saliency maps from eye-tracker data (subjective maps), both for Actor, supplied with GTEA data set and for the Viewer, which we recorded in a psycho-visual experiment involving 22 subjects. To build the maps, the reference method [START_REF] Wooding | Eye movements of large populations: deriving regions of interest, coverage, and similarity using fixation maps[END_REF] was used, transforming sparse gaze locations into a dense map by adapted Gaussian filtering. An example of such a map is given in Figure 2. The relation between Actors' and Viewers' saliency maps we propose is based on the findings [START_REF] Prablanc | Optimal response of eye and hand motor systems in pointing at a visual target[END_REF][START_REF] Dorr | Variability of eye movements when viewing dynamic natural scenes[END_REF] that there exists a time shift around 500 ms between the gaze fixation on the visual target and beginning of the action. This is the Actor point of view. The Viewer is interested in the action thus his saliency map has to be shifted in time. Hence, our assumption is that given a saliency map of the Viewer, the prediction of the unknown Actor's saliency map is possible by a time shift. In the experiments we verify this hypothesis.

III. RESULTS

Location estimation

The 3D reconstruction approach was tested on data acquired within the Dem@care project in the CHU Nice Laboratory, which is used for assessment on patients. The The current approach is suitable to model selected parts of the environment, which can be reconstructed separately (see Figure 3). This illustrates the possibility to obtain a precise 3D localization of the person in a known environment, to be later merged with complementary information: gaze analysis, location of objects, instrumental actions with the hands…

Saliency models comparison

To check the hypothesis about the time-shift between visual saliency (gaze fixation) maps of Viewer and Actor in the egocentric video, we fulfilled the manual test first. The experience was done for 8 of the videos provided from GTEA dataset. The experimental distribution of Time-shift values showed that most of the actions are acknowledged by the viewer around 8 frames later than the actor (533ms corresponding to [START_REF] Prablanc | Optimal response of eye and hand motor systems in pointing at a visual target[END_REF][START_REF] Dorr | Variability of eye movements when viewing dynamic natural scenes[END_REF].) Then an automatic comparison of visual saliency maps between Viewers and Actors was realized. The similarity metrics used were NSS, AUC, and PCC [START_REF] Meur | Methods for comparing scanpaths and saliency maps: strengths and weaknesses[END_REF]. They express the correlation between maps. Statistical Wilcoxon test confirmed the hypothesis on the existence of such a timeshift. As for the values, the results with the AUC metric are given by Figure 4. They are displayed for different time-shifts between actors' (fixed) and viewers' (varying in time) saliency maps. The NSS and PCC comparison metrics scores behaved similarly. The computation of these three metrics clearly brings to the same conclusion: the actors' saliency maps show more correspondence with those of the viewers when the latter are considered with an advance (around 10 frames = 667ms). Hence the Actor saliency can be predicted from that one of the viewer. IV. DISCUSSION -CONCLUSION Hence in this paper we introduced a new an exciting approach in the use of IT for Alzheimer disease assessment and studies: the egocentric video. The results of our anterior studies confirmed that such video observations allow for objective assessment of patients IADL [START_REF] Karaman | Human Daily Activities Indexing in Videos from Wearable Cameras for Monitoring of Patients with Dementia Diseases[END_REF]. The proposed methods for automatic analysis , such as location estimation of the patient and prediction of visual saliency, that is of the focus of interest of the patient, allow for identifying important elements in patient's behavior when executing IADL: i) where he is and ii) where he is supposed to look. The proposed methods are being integrated in the whole automatic system of patients' behavior designed in the Dem@care project. Additional recordings are currently planned within the Dem@care project, in both controlled scenarios in Lab, as well as in Homes.

FIGURE 2 :

 2 FIGURE 2: Example of visual saliency map focused on the hand of the person wearing the camera.

FIGURE 3 .

 3 FIGURE 3. Reconstruction of a 3D model of the TV place of the CHUN@Lab environment from GoPro Images. The camera trajectory is shown in green dots (best viewed in color).

FIGURE 4 -

 4 FIGURE 4-Histogram of AUC scores between the actor's and viewer's saliency maps for different time-shifts

ACKNOWLEDGEMENT This research has been supported by national grant ANR-09-BLAN-0165-02 (IMMED project) and the European Community's program (FP7/2007-2013) under Grant Agreement 288199 (Dem@Care project).


Stéphane Vautier

How to State General Qualitative Facts in Psychology?

Keywords: Qualitative research, Measurement, Falsifiability

In what form should qualitative psychological statements have to be in order to be general and falsifiable? I show how basic qualitative measurement systems can be defined that allow for the detection and testing of general relational facts of the if-then form in a well-defined range of potential qualitative, static or dynamic, multivariate observations. The formal framework permits the pointing out of general research problems falling under the scope of empirical investigation.

Introduction

Empirical scientists search for empirical statements that are general in a sense that, supposedly, they can be fully specified. Moreover, empirical scientists praise falsifiable statements, because recognizing error gives an opportunity to revise existing knowledge [START_REF] Bachelard | La formation de l'esprit scientifique [The formation of the scientific mind[END_REF][START_REF] Popper | The logic of scientific discovery[END_REF]. Nevertheless, it does not seem straightforward to find examples of general and falsifiable qualitative statements in psychology. In the present paper, I investigate how general qualitative facts could be formulated in such a way that it is made clear how to check and improve their empirical validity, reserving random variables for interpretative purposes.

The first section criticizes the paradigm which consists of the routine statistical operationalization of intuitive claims. Usual quantitative intuitions expressed in socalled general hypotheses are not falsifiable without the help of operationalization. Operationalization of a 'general hypothesis' involves a double process: quantitative encoding, and statistical restatement [START_REF] Vautier | The operationalization of general hypotheses versus the discovery of empirical laws in Psychology[END_REF]. The second section lays bare the formal structure of the domain of observables, and focuses on the if-then rule as a special form of a general qualitative relational factual statement. In the third section practical S. Vautier Université de Toulouse-Le Mirail, Pavillon de la Recherche, 5 allées Antonio Machado, 31058 Toulouse cedex 9 E-mail: vautier@univ-tlse2.fr aspects of if-then rules, like their detection and what to do with falsifying cases, are outlined.

Problems with Common Trends in Psychological Research

Statistical Operationalization of Intuitive Claims

A problematic trend in psychological research consists of dealing with statements which do not fulfill the falsifiability criterion. For example, consider the following statement:

H : Smokers trying to stop smoking feel more anxious than smokers not trying. Asking in what sense could such a statement be valid, one realizes that there is no obvious answer; "to feel more anxious than" has no natural associated validity criterion. Nevertheless, the statement refers purportedly to a general empirical fact, since its applies to any smoker.

The statistical operationalization of that statement H would cause the investigator to estimate parameters of a random 'anxiety' sampling variable Y conditionally to an independent variable X assessing the status of a smoker. The experimental approach would randomly assign smokers to experimental (X = 1) vs. control (X = 0) conditions, whereas the observational approach would merely observe the statut of each smoker on the independent variable X. In both cases, the null hypothesis of equal means would be tested, and likely rejected-statement A :

µ(Y |X = 1) > µ(Y |X = 0), p < .05,
where the parameters to be estimated are defined with respect to normal density functions of the same variance in an ad hoc probability space; moreover, an effect size could be computed, indicating the magnitude of the mean effect-statement B, say: d = .3. Are A and B general statements? Statements A and B are not intended to characterize particular smokers at any time. It seems that statistical statements are specific to random variables defined with respect to a specific probability space, which cannot be falsified.

Moreover, the ANOVA approach to a statement like H has been considered a methodological paradox, as

In the physical sciences, the usual results of an improvement in experimental design, instrumentation, or numerical mass of data, is to increase the difficulty of the "observational hurdle" which the physical theory of interest must successfully surmount; whereas, in psychology and some of the allied behavior sciences, the usual effect of such improvement in experimental precision is to provide an easier hurdle for the theory to surmount ([3], p. 103). The paradox would vanish if the focus statement was represented by the null hypothesis instead of its alternative, that is, the numerical values of µ(Y |X) were explicitly hypothesized. Unfortunately, numerical values cannot be derived from quantitative intuitions like H .

Measurement Scales and Implicit Nominal Approximations

Another problematic trend in psychological research, which is closely related to the former, consists of the preference for quantitative-like formulations, despite a lack of evidence that there is something quantitative to be measured [START_REF] Essex | Between numbers and notions: A critique of psychological measurement[END_REF][START_REF] Michell | Normal science, pathological science and psychometrics[END_REF][START_REF] Michell | Is psychometrics pathological science?[END_REF]. Numerical encoding of 'anxiety' is paradigmatic. A set of questionnaire items is selected, and selfratings are then assigned a numerical value, yielding a composite test score, which is thought of as a more or less reliable and valid measurement of a 'relative position' on the 'anxiety' continuum-test scores allegedly vary on an interval scale. However, test scores do not even possess the property of representing completely ordered sets of responses.

One example is sufficient to highlight this point. Consider a very simple test Y composed of two dichotomic items a, b. Each item refers to a psychological process or concept allowing the ordering of the observed values:

0 i ≤ i 1 i , i = a, b. The set M(Y ) of the possible response vectors is M(Y ) = {(0 a , 0 b ), (0 a , 1 b ), (1 a , 0 b ), (1 a , 1 b )}.
Most pairs of response vectors in M(Y ) can be ordered by the relationship ≤ ab , which is defined as follows:

(x a ≤ a y a and x b ≤ b y b ) ⇔ (x a , x b ) ≤ ab (y a , y b ), (1)
where {x, y} = {0, 1}. However, ≤ ab is a partial order because it cannot order the pair {(0 a , 1 b ), (1 a , 0 b )}. Now, consider the usual test scores associated with the test Y . They are elements in the range of the aggregation function f , which assigns to any response vector of M(Y ) the algebraic sum of its numerical values. According to Stevens [START_REF] Stevens | On the theory of scales of measurement[END_REF], in order to use test scores as measurements, their numerical properties should reflect the empirical properties of the objects they represent, that is, the vectors y i of M(Y ), i = 1, . . . , 4. But f (y i ) does not reflect validly the empirical properties of the vectors in M(Y ). In particular, it is contradictory to state that the vectors y 2 = (0, 1) and y 3 = (1, 0) are distinct on the one hand, which means that they do not belong to the same nominal class, and to state, on the other hand, that f [(0, 1)] = 1 and f [(1, 0)] = 1 which means that (0, 1) and (1, 0) belong to the same nominal class.

From the representational point of view, test scores are replete with nominal approximations. Whether nominal approximations can be justified is not the point. The point is that, to the best of my knowledge, test developers do not justify nominal approximations. Hence, usual numerical psychological scales have no clear epistemological foundations. In the next section, I develop a qualitative measurement perspective which does not require the kind of nominal approximations implied by the "quantitative" interpretation of composite test scores, and which provides a clear meaning of the word "general".

3 Restrictive If-Then Rules in Multivariate Qualitative Measurement Systems

The Multivariate Qualitative Measurement System

A multivariate qualitative measurement system is composed of a vector

Z = (Z 1 , . . . , Z k )
of k ≥ 1 empirical variables, where at least one variable is qualitative, i.e., of the nominal or ordinal type. Notice that the term "variable" does not refer to a probability space, but is used to denote empirical values referring to an empirical attribute. If some variables of Z are quantitative, there is no loss of generality to treat them formally as qualitative-ordinal-variables. Each variable Z i , i ∈ {1, . . . , k}, is characterized by its set of possible values M(Z i ). Thus, values of observables range in the set M(Z), which is the Cartesian product of the sets M(Z i ):

M(Z) = M(Z 1 ) × • • • × M(Z k ). (2)
A multivariate qualitative measurement system may also comprise change variables. A change variable refers to a couple of (static) variables (Z i1 , Z i2 ) associated with two time points 1 and 2, and its values are vectors in M(Z i1 ) × M(Z i2 ). Such a change variable can be labeled a 2-time variable. More generally, one can introduce x-time variables in Z, where x denotes the number of time points.

The cardinality of M(Z) is given by

card[M(Z)] = k ∏ i=1 m i , (3)
where m i is the cardinality of M(Z i ). In practice, card[M(Z)] may be a huge number. I will elaborate on this point later.

The complete definition of the measurement system requires the definition of its domain, viz., the population U of individuals who are supposed to possess a value in M(Z) at time t ∈ T , where T denotes an appropriate time interval. 1 Coming back to the smokers' example, should John, who smokes no more than 10 cigarettes a year, be included in the population of 'smokers'? The inclusion problem can be restated by defining the population U as the population of animals; then, being or not being a 'smoker' pertains to the specification of Z, by introducing in Z the variable, say, Z 1 , which assigns a value to each individual with respect to what is meant by 'smoker'-Z 1 may be a Cartesian product of variables allowing the convenient specification of a 'smoker'.

Restrictive If-Then Rules

Given a well-defined measurement system [U, T, M(Z)], what kind of general facts can be formulated? I will focus on relational facts, that is, on statements specifying associations between an independent, multivariate variable X, and a dependent, possibly multivariate variable Y . Thus, the vector Z decomposes as the couple Z = (X, Y ), and the codomain M(Z) decomposes as the Cartesian product M(Z) = M(X)×M(Y ).

Various kinds of relational facts can be defined with respect to the measurement system [U, T, M(X) × M(Y )]. I will focus on factual statements having the form of restrictive if-then rules. Restrictive if-then rules in [U, T, M(X) × M(Y )] refer to observations performed on any individual at any time in an appropriate time interval t ∈ T . They express formally as

∃ α(X) ⊂ M(X), β (Y ) ⊂ M(Y ), ∀ (u, t) ∈ U × T, X[(u, t)] ∈ α(X) ⇒ Y [(u, t)] ∈ β (Y ), (4)
which reads at least a strict subset α(X) of values on X and a strict subset α(Y ) of values on Y exist, such that for any pair of an individual u observed at time t in the appropriate time interval T , falling in the class associated with the subset α(X) implies falling in the class associated with the subset α(Y ). 2Such rules can be noted R β α . They are general in the sense that they are supposed to be valid for any (u, t) ∈ U × T . They are restrictive in the sense that they apply to strict subsets of values α(X) and β (Y ). The conditions α(X) allow the specification of a subpopulation of individuals and possible experimental conditions affecting their responses on Y .

For example, consider the statement "each time a smoker tries to stop smoking, she or he get anxious." Let Z = (X 1 , X 2 , Y ), where X 1 takes the value 1 when u is a smoker, 0 otherwise, X 2 takes the value 1 when "u's trying to stop smoking" is valid, 0 otherwise, and Y takes the value 1 when u endorses the item questionnaire "I feel anxious", and 0 otherwise. X = (X 1 , X 2 ). Thus, expressing the statement as the restrictive if-then rule R 1

(1, 1) yields

R 1 (1, 1) : ∀ (u, t) ∈ U × T, X(u, t) = (1, 1) ⇒ Y (u, t) = 1, (5)
which makes it a falsifiable statement, provided (i) the decision rules used to assign a value in M(Z) to any individual are objective-objective rules are rules which yield rater-invariant measurements in M(Z)-, and (ii) there is no logical impossibility to observe a counterexample.

Falsification of a Restrictive If-Then Rule

To falsify R 1 (1, 1) , it is sufficient to find one counterexample, that is, one occurrence of the form

¬R 1 (1, 1) : ∃ u ∈ U, ∃ t ∈ T, X(u, t) = (1, 1) and Y (u, t) = 0. ( 6
)
Whether formal falsification should entail the withdrawal of the statement in the set of "interesting" statements will be discussed later.

The preceding development gives the form of a restrictive if-then empirical rule on [U, T, M(X) × M(Y )]. Suppose that an investigator is able to state the restrictive if-then empirical rule R

β i : ∃ i ∈ {1, . . . , card[M(X)]}, ∃ β (Y ) ⊂ M(Y ), ∀ (u, t) ∈ U × T, X(u, t) = x i ⇒ Y (u, t) ∈ β (Y ). (7)
R β i is falsified by ∃ (u, t) ∈ U × T, X(u, t) = x i and Y (u, t) ∈ M(Y )\ β (Y ). (8)
However, considering a given data sample, R β i cannot be falsified if no (u, t) can be observed such that X(u, t) = x i . Whether some (u, t) can be observed such that X(u, t) = x i is an empirical issue. Reporting the evidenced validity of a given restrictive if-then rule requires two empirical data: (i) the number n of relevant cases that have been observed, and (ii) the number e of exceptions that have been discovered. Thus, a given if-then rule R β i , which is a theoretical, and general statement about observables, can be associated with its empirical counterpart R

β i (n, e).
Obviously, being able to report general facts is a prerequisite to being able to explain them. The next section focuses on data analysis oriented toward the finding of potentially interesting findings in an observed contingency table associated with the measurement codomain.

Practicing Restrictive If-Then Rules

Given a measurement system [U, T, M(X) × M(Y )], the contingency table associated with a relevant data sample is useful to detect two kinds of interesting regularities, viz., sufficient conditions, and necessary conditions.

Detection of Potential Sufficient Conditions

Let the rows and columns of the observed contingency table be associated with the values x i on X, and the values y j on Y , respectively, where i = 1, . . . , card[M(X)] and j = 1, . . . , card[M(Y )]. Let n, n i• , n • j , and n i j denote the sample size, the frequency of the row i, the frequency of the column j, and the frequency of the cell i j, respectively.

Potential restrictive if-then rules correspond to interesting rows in the contingency table. An interesting row i is defined as a row such that

n i• > 0, ∃ j ∈ {1, . . . , card[M(Y )], n i j ≈ 0, (9)
where the sign ≈ means that the cell frequency n i j can be viewed as no quantity or negligible quantity of exceptions falsifying R j i . The sufficient condition associated with Y (u, t) = y j is X(u, t) = x i .

The screening of the entire contingency table associated with the data may provide a series of findings of the form R j i (n i• , n i j ), where n i j ≈ 0, that are eligible for introduction into the list of basic nomothetic statements relevant to M(X) × M(Y ). Whether such general empirical statements are of substantive import is a theoretical issue.

To illustrate, consider the following observed contingency table associated with the bivariate independent variable X = (X 1 , X 2 ) and the univariate dependent variable Y . 0 1 00 1 4 01 1 0 10 6 0 11 2 0 (10) A potential restrictive if-then rule is R 0 ¬(0, 0) ; in other words, a potential sufficient condition associated with Y (u, t) = 1 is negatively formulated as X(u, t) = (0, 0).

Detection of Potential Necessary Conditions

The data also suggests a necessary condition associated with Y (u, t) = 1, that is, X 2 (u, t) = 0. This necessary condition can be stated as the following restrictive ifthen rule:

X 2 (u, t) = 0 ⇒ Y (u, t) = 1, (11)
which is equivalent to

Y (u, t) = 1 ⇒ X 2 (u, t) = 0. ( 12
)
More generally, necessary conditions associated with a given subset δ (Y ) of states on Y are states γ(X) on X such that

Y (u, t) ∈ δ (Y ) ⇒ X(u, t) ∈ γ(X). (13)
The systematic detection of necessary conditions on X associated with a given state on the dependent variable is out of the scope of the present paper.

Discussion

Psychological research that rests on so-called general hypotheses operationalized through ad hoc probability spaces is unlikely to discover general relational empirical facts, if the word "general" means that what is stated is valid for any (u, t) ∈ U × T [START_REF] Vautier | The operationalization of general hypotheses versus the discovery of empirical laws in Psychology[END_REF]. Furthermore, a basic requirement of scientific inquiry is making clear what can be observed, which is hardly achieved with pseudo-quantitative measurements. Starting from a domain of observables based on a finite set of independent and dependent possibly multivariate qualitative attributes, I have shown that empirical statements having the form of restrictive if-then rules are general statements. Actually, restrictive if-then rules define special relations (M(X), M(Y ), G), where G is a graph such that there is at least one possible row with at least one impossible cell. The words "possible" and "impossible" refer to observations (u, t) drawn in U × T . The range within which restrictive if-then rules can be stated and falsified is explicit. An investigator interested in finding potential restrictive if-then rules in the empirical world of interest only has to search for "gaps" in actualized rows of the observed contingency table associated with M(X) × M(Y )-actualized "gaps" suggest impossible cells in possible rows. Also, restrictive if-then rules can serve to define necessary conditions associated to a given state β (Y ) on the dependent variable Y .

Sufficient or necessary conditions are theoretical statements because they are supposed to be valid for any case (u, t) which can be observed (measured). Contrary to what happens in physics, psychology stumbles over identity of people, which could be understood as a sufficient condition for making generality impossible. It is noteworthy that the formulation displayed in Equation 4 allows for different levels of generality. Levels of generality are partially ordered because they refer to conjunctions of restrictions on the codomain of X. For example, adding an identity variable in X allows for the selection of a specific person in U. Thus, the generality of sufficient or necessary conditions that are restricted to a given person lies on T . Then, generality is possible as soon as T is not a point.

Restrictive if-then rules state a necessity principle: if some conditions α(X) are fulfilled on X, it is necessary that other conditions β (Y ) be fulfilled on Y too. Thus, one can wonder whether it is reasonable to expect valid restrictive if-then rules in psychology. There are some reasons to expect that psychological if-then rules may be formulated with a high degree of corroboration: robust effects do exist in psychology.

Although one counterexample suffices to falsify a restrictive if-then rule, it would be unwise to automatically deny the interest of that rule. As the result of a measurement process, falling in a given observational class may be considered a fallible observation. Thus, exceptions may result from measurement error. However, suspecting measurement error should yield the investigator to repeat his or her observation rather than to interpret automatically the exception as the effect of randomness. Counterexamples can be analyzed according to their epistemic value and their practical impact in decision settings.

From the epistemic viewpoint, counterexamples may be trivial or non-trivial. Trivial counterexamples are those that could be easily eliminated if additional, excluding conditions were taken into account, yielding a more detailed characterization of (u, t) at the level of the independent variables-and a new if-then rule-, or those that could be eliminated if problems in measurement turned out to explain the exceptions. Non-trivial counterexamples are those cases which cannot be eliminated within the current state of knowledge. Can unknown excluding independent variables be discovered? Can the tuning of the dependent variable be improved in order to eliminate the exceptions?

From the practical viewpoint, a rule of type R β α (see Equation 9) allows one to predict Y (u, t) ∈ β (Y ) given X(u, t) ∈ α(X). Let n be the number of observed cases falling in the class α(X), and e the number of relevant counterexamples. The predictive interest of the rule R β α (n, e) is threatened by the ratio e/n, which gives the proportion of counterexamples to the rule. This ratio can be used as an estimate of a probability to be wrong when predicting Y (u, t) ∈ β (Y ) given X(u, t) ∈ α(X), if the probability space underlying such an interpretation is specified appropriately.

Sufficient or necessary conditions associated with a restrictive if-then rule may be interpreted in terms of causality. However, causal interpretations require available general facts. An important task of the scientific endeavor in psychology is to find corroborated general facts, and facts of the form of restrictive if-then rules are potential candidates. If a given measurement system [U, T, M(Z)] turns out to be unproductive, that is, if everything is possible in M(Z), the scientific challenge becomes: find another form of general facts that can be stated in [U, T, M(Z)], or find a new observational domain M(Z ) within which restrictive if-then rules can be found.

When measurements refer to x time points, t may be defined as a convenient vector in T x .

In the sequel, the notation X[(u, t)] will be simplified by the notation X(u, t).

Acknowledgements I am grateful to Didier Dubois and Henri Prade for their helpful comments on a previous version of the present paper.