G Vilaplana

M Grandemange

M Gohlke

O Cadot
email: cadot@ensta.fr

Global mode of a sphere turbulent wake controlled by a small sphere

Keywords: Sensitivity, Control, Global mode, Turbulent wake

A small control sphere is displaced in a cross flow plane downstream of a main sphere of diameter D in a uniform flow at Re = 33000. The wake is studied by means of 4 fixed hot wire probes. It is shown that without the control sphere (reference case), vortex loops are shed from only one side of the sphere at a Strouhal number St = 0.19. This reference wake structure has a planar symmetry defining an azimuthal phase θ W that is observed to be random in time. The secondary smaller sphere is treated as a local disturbance of the reference wake. It is found that the shedding occurs closer to the symmetry axis for the control sphere placed at the center of the wake than for the natural case. When the control sphere is off-centered, a subharmonic at half the natural frequency appears and the azimuthal phase becomes imposed by the control sphere position. A pure subharmonic mode is observed when the control sphere reaches the separated shear, suggesting alternative vortex loops shedding from both sides of the wake.

Introduction

Recent theoretical efforts have allowed the retrieval of the structural sensitivity of the global mode of two-dimensional wakes in the laminar regime [START_REF] Hill | A theoretical approach for analyzing the restabilization of wakes[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF][START_REF] Luchini | Structural sensitivity of the finiteamplitude vortex shedding behind a circular cylinder[END_REF] )obtained experimentally by [START_REF] Strykowski | On the formation and suppression of vortex sheddingat low Reynolds numbers[END_REF] and very recently in the turbulent regime [START_REF] Meliga | Sensitivity of 2-d turbulent flow past a d-shaped cylinder using global stability[END_REF] obtained in the experiment of [START_REF] Parezanović | Experimental sensitivity analysis of the global properties of a 2D turbulent wake[END_REF]. For three dimensional cases, the theory applied to axisymmetric flows in the laminar regime (Meliga et al., 2009b) predicts a zone of high sensitivity in the recirculation bubble, particularly around the separatrix. Besides this pioneering work, theoretical prediction for three dimensional turbulent wakes still remains an open issue. On the experimental side, [START_REF] Grandemange | On experimental sensitivity analysis of the turbulent wake from an axisymmetric blunt trailing edge[END_REF] examined the sensitivity of a three dimensional symmetric blunt body in the turbulent regime using small objects in the wake. In this experiment, the presence of the supports that hold the axisymmetric body produce a bi-stable wake, where each state is not axisymmetric and has an m=1 azimuthal periodicity. Depending on their geometry, the small objects select the azimuthal phase or force the axisymmetry of the wake.

The goal of the present experiment is to preserve the axisymmetry of the body as much as possible. We have chosen the academical case of the sphere that is well documented in the literature. The dynamics of the separated flow over spheres undergoes several transitions associated with different wake regimes as the Reynolds number increases [START_REF] Sakamoto | A study on vortex shedding from spheres in a uniform flow[END_REF][START_REF] Szaltys | Nonlinear evolution of instabilities behind spheres and disks[END_REF]. At very low Reynolds numbers, the flow is steady and axisymmetric. A first bifurcation is reported for Re ≈ 210 : a steady azimuthal mode m = 1 appears. A lift force is generated and a pair of counterrotating vortices develops downstream moving the wake off the streamwise axis so that the flow loses its axisymmetry but keeps a planar symmetry. The plane of symmetry defines the azimuthal phase of the mode. A second bifurcation occurs at a higher Reynolds number Re ≈ 277 : the wake starts oscillating but preserves the same planar symmetry and azimuthal phase. Becoming turbulent, the preference toward this azimuthal phase vanishes and the flow becomes statistically axisymmetric [START_REF] Mittal | Symmetry properties of the transitional sphere wake[END_REF]. The mixing layer instabilities in the close wake degenerate into large scale vortex loops developing from the end of the recirculation bubble ; the wake oscillates randomly [START_REF] Taneda | Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106[END_REF][START_REF] Sakamoto | A study on vortex shedding from spheres in a uniform flow[END_REF] and may have a helical structure highly coherent in space [START_REF] Pao | Vortex structure in the wake of a sphere[END_REF][START_REF] Berger | Coherent vortex structures in thewake of a sphere and a circular disk at rest and under forced vibrations[END_REF][START_REF] Yun | Vortical structures behind a sphere at subcritical Reynolds numbers[END_REF][START_REF] Cannon | Observations of large-scale structures in wakes behind axisymmetrical bodies[END_REF]. This unsteady global mode (dominant mode m = 1) is reported at 0.1 < St < 0.2 depending on Reynolds numbers.

The present work explores the global mode modification of the turbulent wake due to the presence of a secondary smaller sphere, treated as a local disturbance in its close wake. The impact of the controlled sphere is reported through sensitivity analysis of the global spatial structure and frequency of the wake. To our knowledge, such an approach has not been carried out so far. Wakes interaction have been investigated for identical spheres [START_REF] Schouveiler | Interactions of the wakes of two spheres placed side by side[END_REF][START_REF] Yoon | Flow-induced forces on two nearby spheres[END_REF] in the laminar regime around Re = 300. They lead to different patterns in the vortex shedding and Strouhal numbers, that do not result from a disturbance approach.

The article is organized as follows. First, the experimental set-up and measurements are presented in section 2. Section 3 analyzes the wake of the main sphere, especially the global mode structure and the topology of the shedding are clarified. Section 4 is devoted to results of the disturbed wake with the small control sphere. Eventually, discussion and concluding remarks are presented in the final section.

Experimental set-up

The main sphere of diameter D = 40 mm is placed in a feedback wind tunnel having a uniform velocity of U 0 = 12.5 m/s with a free stream turbulence level of 0.5%. The diameter D and inlet velocity U 0 are used to build non-dimensional values marked with an asterisk. The Reynolds number is Re = U 0 D/ν = 33000 for the main sphere. The cross section of the wind tunnel is a square of dimensions 0.45 m×0.45 m and the sphere is attached from its center to each corner of the test section by four metal wires of diameter 0.5 mm as depicted in Fig. 1. A system of tension at the edge of the frame allows to stretch the wires. The tension is adjusted in order to avoid any observable or hearable vibrations of the wires and the sphere. This technique of fixation is commonly used for studies of three dimensional bluff body wakes, see for instance [START_REF] Cannon | Observations of large-scale structures in wakes behind axisymmetrical bodies[END_REF]; [START_REF] Howe | Aerodynamic lift and drag fluctuations of a sphere[END_REF]; [START_REF] Jang | Piv analysis of near-wake behind a sphere at a subcritical reynolds number[END_REF]; [START_REF] Sakamoto | The formation mechanism and shedding frequency vortices from a sphere in uniform shear-flow[END_REF].

The control sphere has a diameter of d = 12 mm= 0.3D. The surface ratio between the main sphere and the control sphere is sufficiently small, (d/D) 2 = 0.09 to consider the control sphere as a disturbance for the main sphere wake. The control sphere is also attached by four 0.5 mm metal wires from its center to the corners of a squared moving frame of dimensions 0.26 m × 0.26 m. They are sufficiently stretched to avoid any vibrations. The frame is made by four metal profiles with a cross section of 3 mm×15 mm with smooth edges. The upper part of the frame is fixed to a 10 mm rod driven by a two-axis displacement console standing on the top floor of the test section area. The passage is made air proof by adding heavy sliding sheets between the open part of the roof and the rod. Let x c , y c , z c be the coordinates of the center of the control sphere whose origin is taken at the center of the main sphere. The corresponding polar coordinates will be referred to as r c and θ c . The control sphere is positioned at a fixed downstream distance z c = 0.9D and displaced by steps of 1 mm in the area of the xOy plane: ∆x c × ∆y c = 1.25D × 1.25D.

Four fixed hot wire anemometers measure the velocity in a downstream transversal plane at z * = 2 (Fig. 1b). The four measurement points denoted by L, T, R and B are located respectively at (x = -0.35, y = 0), (0, 0.35), (-0.35, 0) and (0, -0.35) in dimensionless quantities. The letters are referring to the left, top, right and bottom part of the main sphere seen from the backview in Fig. 1(a). The hot wires are all oriented in such a way to be sensitive to the velocity component comprised in the azimuthal plane θ (see Fig. 1a). The four velocity time series are simultaneously recorded using Labview software and a National Instrument acquisition board at a sampling frequency of 1 kHz. We denote by u i (t) the measurements obtained with the i = [L, R, T, B] hot wire anemometers and δu i (t), the velocity fluctuation about the mean. For specific positions of the control sphere, a flying hot wire probe, sensitive to the (yOz) velocity component is used to explore the wake in a transversal plane z * = 2. In this case the velocity measurement is simply referred to as u(t) and its standard deviation to u rms .

Reference wake

We first study the case of the main sphere without the control sphere, for which velocities are recorded during 600 s. The time series are very intermittent as shown in Fig. 2(a). One can observe on each signal, time windows of varying durations having lack of fluctuations. The averaged spectra performed over 1 s duration all present a dominant frequency at f 1 = 60 Hz. The frequency f 1 , that we will refer to as the mode 1 frequency corresponds to a Strouhal number of 0.19. Another, but less significant characteristic frequency is observed around f 2 = 27 Hz which might correspond to a subharmonic of the mode 1 frequency. We computed the coherence functions defined as :

γ ij (f ) = ũi (f )ũ cc j (f ) |ũ i (f )| |ũ j (f )| , ( 1
)
where ũi (f ) is the Fourier transform of u i (t), "cc" stands for complex conjugate and the averaging ... is performed over 1s duration. The coherence is computed between the left and right probes (ij) = (LR) and between the top and bottom probes (ij) = (T B). For simplicity, we will call γ LR the horizontal correlation and γ T B , the vertical correlation. The coherence functions, modulus and phase are displayed in Fig. 3(a). The modulus of the coherence presents a peak around 60 Hz with a corresponding phase shift of π for both the horizontal and vertical coherence functions. The mode 1 can be then viewed as a combination of two equivalent asymmetric oscillations, a vertical one and an horizontal one. However, these oscillations cannot result from an alternative vortex shedding between both side of the wake as commonly observed for the 2D Kármán street. This is demonstrated in the following by considering the time dependant mean fluctuation on each probe over a period T :

δu 2 irms (t) = 1 T t+T t [u i (t) - 1 T t+T t u i (t)dt] 2 dt, i = [L, T, R, B]. (2)
The joint probability distribution PDF(δu Lrms , δu Rrms ) calculated over 600 s is shown in Fig. 3

Wake with control sphere

For each position of the control sphere in the transversal plane z * = 0.9, the four velocity signals captured by the L, T, R, and B probes are recorded during 200 s. At each position, we compute the time dependant fluctuation δu irms (t) as defined in Eq. 2 with T = 0.025 s. We expect that the barycenter of these four fluctuation signals will locate instantaneously the maximum of the shedding in the transversal plane z * = 2. The barycenter (x * F (t), y * F (t)) is defined as : We measure the azimuthal phase of the shedding θ W from the mean position of the fluctuation barycenter and its radial position r W , as the most probable position of the fluctuation barycenter. For the reference wake, we find r * W = 0.3.

x * F (t) = 0.
The control sphere acts as a m = 1 disturbance except for the position at the center (x * c , y * c ) = (0, 0). The azimuthal phase θ W vs. the position of the control sphere is plotted in Fig. 5(a). We can see that the phase θ W is related to the sphere polar angle as θ W ≈ θ c + π implying that the shedding always occurs on the opposite side of the control sphere location. b) is the same measurements when the control sphere is removed. It takes into account the obstruction of the moving holding system only (frame and crossed wires see Fig. 1). The 5 contours range from 0.02m 2 /s -2 to 0.1m 2 /s -2 with an interval contour levels of 0.02m 2 /s -2 . probes) with four axis of symmetry at θ = n π 4 with n = 1...4. On the diagonals, the radial position of the shedding is not very sensitive to the control sphere position. This effect can be explained by the alignment of the wire holding the main sphere with the wire holding the control sphere (see Fig. 1). This special configuration of the wires might create a disturbance as important as the control sphere itself imposing a radial position close to the natural reference case r * W = 0.3. The largest sensitivity is observed on the horizontal and vertical axis. Figure 5(c) and (d) restrict the sensitivity analysis to the control sphere displaced on an horizontal path (x * c > 0, y * c = 0). Figure 5(c) displays the radial position of the wake r * W . Figure 6(a) is the auto power spectrum density | ũ L (f, x * c )| 2 . We only show the spectrum recorded by the probe L located on the opposite side of the control sphere path (i.e. where the shedding occurs) which actually contains all the spectral information. One can see in Fig. 5(c) that the effect of the control sphere is almost to bring the shedding closer to the symmetry axis compared to the reference case. The closest positions are obtained for the centered control sphere (r * W ≈ 0.15) and off-centered at r * c ≈ 0.37 (r * W ≈ 0.07). It is also for these two positions that the energy of the fluctuations is more concentrated in the spectra of Fig 6(a) because the shedding is also closer to the measurement points. The two energy concentrations points are around the fundamental frequency f 1 = 55.5(±1.5) Hz for the centered control sphere and around a lower frequency f 2 = 38(±1.5) Hz for off-centered control sphere. The modes are pure only for these two cases. Between these two positions, 0.1 < r * c < 0.35 the radial position r W is similar to that of the reference case and both the fundamental and the low frequency are present. The same measurements have been performed in Fig 6(a) but with a removed control sphere. In that case we can see that the effect of the control sphere supporting system (frame and crossed wires) on the the global frequency is negligible since the natural frequency is retrieved whatever the supporting system position is.

(c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
We show in the sequence in Fig. 7(a-f), the joint probability distribution of the time dependant mean fluctuation δu irms (t) (Eq. 2 with T = 0.025 s) measured by the right and left probe for different positions of the control sphere. The centered position exhibits the same behavior as for the natural reference wake (Fig. 3) meaning that vortex loops are shed from one side of the wake with a random azimuthal phase. Once the control sphere is slightly shifted (Fig. 7b-d pure mode at frequency f 2 is shown in (Fig. 7e,k). The barycenter of the fluctuation (see Fig. 7e) is actually moving back to the center because the fluctuations are no longer recorded by only the left probe, but simultaneously by the left and right probe (see Fig. 7k). This observation associated with the dominance of the subharmonic mode is consistent with an alternate shedding between the side of the control sphere and the opposite side.

Finally, the spatial distribution of the rms fluctuations in the plane z * = 2 measured with the flying probe is shown in Fig. 8 for the two extreme positions r * c = 0 (Fig. 8a) and r * c = 0.35 (Fig. 8b). We can actually check that for the centered position, the fluctuations are closer to the wake center compared to the reference case in Fig. 4(b). For the off-centered case we can see two maxima of fluctuations; around the edge of the control sphere and on the opposite side, around the edge of the main sphere.

Discussions and conclusions

When the control sphere is at the center of the wake of the main sphere, we find similar global properties to that of the reference case, with a pure mode having a frequency close to St = 0.19 as described in the sketch of Fig. 4(a) and a random azimuthal phase. The remarkable effect is the rapprochement of the shedding with the center of the wake (r * W ≈ 0.15 with centered control sphere and 0.3 for the reference case). In the cases of the off-centered control sphere, the corresponding disturbance which has an m = 1 wavenumber sets the turbulent wake at a fixed azimuthal phase. This result generalizes to an axisymmetric situation, the previous observation of [START_REF] Grandemange | On experimental sensitivity analysis of the turbulent wake from an axisymmetric blunt trailing edge[END_REF] for which only two azimuthal phase were observable due to the properties of the natural wake. It was also found that depending on the downstream distance of the perturbation to the body, the shedding could be either on the same side of the steady perturbation or on the opposite side. In the present experiment we only explore one downstream location z * c =0.9. For radial positions such that r * c < 0.3 the shedding occurs on the opposite side of the control sphere location θ W = θ c + π. We do not exclude that for different downstream positions, azimuthal phase of θ W = θ c could be observed. In any cases, the fixation of the azimuthal phase by a m = 1 steady disturbance is a property known for laminar wake that induces the so-called imperfect bifurcation (Meliga et al., 2009a). This major effect avoids the investigation of a complete experimental sensitivity analysis of the global mode using a steady disturbance as in [START_REF] Parezanović | Experimental sensitivity analysis of the global properties of a 2D turbulent wake[END_REF], since it seems impossible to place the disturbance anywhere else than in the plane of planar symmetry of the mode. The control sphere also modifies the spectral content of the fluctuations and a clear transition between two modes having a factor two in frequency is observed. At the centered position, the global frequency is close to that of the natural wake. A subharmonic, at half the global frequency appears as soon as the control sphere is off-centered. Once the radial position of the sphere reaches r * c = 0.3 only the subharmonic mode is observed (Fig. 5d). We can explain the factor 1/2 in the frequency jump by the appearance of a new shedding pattern, for which vortex loops are not shed from mostly one side of the wake as for the reference wake, but alternatively from the edge of the control sphere and the edge of the main sphere. These two positions can be repaired by the two maxima of velocity fluctuations in Fig. 8(b). To illustrate this interpretation, we show instantaneous velocity fields in Fig. 9 measured with a particle image velocimetry. For the reference wake in Fig. 9(a), the vortex loops are observable at the top of the field by the periodic over-velocities in red color. In the second field in Fig. 9(a), over-velocities are distributed alternatively at the top and at the bottom of the wake.

It is noteworthy to mention that the subharmonic frequency is also observable in Fig. 2(b) for the reference wake but at a low amplitude. The alternate vortex loops shedding mode should also exists for the natural wake, meaning that the wake might switch during small duration from the single sided vortex loops shedding (mode 1) to the alternative vortex loops shedding (mode 2). The control sphere plays a role in the selection of the dominant mode.

We also observed slight but significant variations of the frequencies of the modes in Fig. 5(b). Whatever the mode, the frequency is increased when the control sphere approaches the inner part of the separated layers. This seems to be consistent with the cylinder turbulent wake analysis of [START_REF] Parezanović | Experimental sensitivity analysis of the global properties of a 2D turbulent wake[END_REF] who also observed an increase of the global frequency when the control cylinder approaches the inner part of the shear layer. Same arguments can probably be used: the separated shear may reattach on the control sphere which concentrates the vorticity compared to that of the natural case. This thinner shear will lead to a higher frequency of the global mode. For larger values of the radial position (r * c > 0.45 in Fig. 5d), the control sphere must widen the wake which tends to decrease the frequency.

Figure 1 :

 1 Figure 1: Sketch of the experimental set-up. Backview (a), the flow comes from behind. Four hot wire probes (not on scale) denoted by L for left, R for right, T for top and B for bottom (directions are related to the backview) are symmetrically positioned 2D downstream from the center of the main sphere (b).

Figure 2 :

 2 Figure 2: Time series (a) and averaged autopower spectra (b) of the four velocity measurements u i (t) with i = [L, T, R, B] (as depicted in Fig. 1(b)) for the reference wake without control sphere. Except u L (t) in (a), each velocity measurement has been vertically shifted down by a multiple of 10 m/s.

Figure 3

 3 Figure 3: (a): Modulus |γ| (bottom) and phase φ (top) of the horizontal γ LR (thick line) and vertical coherence function γ T B (thin line). Joint probability density function (b), PDF(δu Rrms , δu Lrms ), see text for definitions of the variables δu Lrms and δu Rrms .

  (b) for T = 0.025 s corresponding to 1.5 shedding at f 1 = 60 Hz. If alternate shedding should occur between the left and right probe, they should record the same averaged velocity fluctuations over 1.5 period and then a single peak on the diagonal of the joint distribution would appear in Fig.3(b). Instead, two symmetric peaks are observed revealing that the shedding is always either located on one side of the wake or on the other side. It confirms the shedding wake structure proposed byMittal et al.

Figure 4 :Figure 5 :

 45 Figure 4: Sketch (a) of the vortex shedding at Strouhal number 0.19. Map of velocity fluctuations u rms (x * , y * ) in a plane z * = 2 for (b) natural reference case and (c) with a steady m = 1 disturbance introduced by a vertical rod placed above the summit of the sphere. The grayscale in (b) and (c) are identical with a contour interval of 0.2 m/s.

Figure 6 :

 6 Figure 6: Auto power spectra | ũ L (f, x * c )| 2 δf measured by the L probe vs. the position of the control sphere (a) on the horizontal path (y * c = 0).(b) is the same measurements when the control sphere is removed. It takes into account the obstruction of the moving holding system only (frame and crossed wires see Fig.1). The 5 contours range from 0.02m 2 /s -2 to 0.1m 2 /s -2 with an interval contour levels of 0.02m 2 /s -2 .

Figure 7 :

 7 Figure 7: Top raw (a-f) : joint distribution P DF (δu Rrms , δu Lrms ). Bottom raw (g-l) : joint distribution of the barycenter of the fluctuations P DF (x * F , y * F ). Each column corresponds to a position of the control sphere displaced on an horizontal path (y * c = 0, x * c ≥ 0): (a,g) x * c = 0; (b,h) x * c = 0.07; (c,i) x * c = 0.15; (d,j) x * c = 0.28; (e,k) x * c = 0.35; (f,l) x * c = 0.5. The empty circles represent the main and the control spheres.

Figure 8 :

 8 Figure 8: Map of velocity fluctuations u rms (x * , y * ) in a plane z * = 2 for (a) centered control sphere and (b) off-centered at r * c = 0.35. Grayscale and contour interval are identical to Fig 4(b) and (c). The circular white dotted lines in (a) and (b) mark the control sphere.

Figure 9 :

 9 Figure 9: Modulus of the instantaneous velocity measured with Particle Image Velocimetry, (a) without control sphere, (b) control sphere at x * c = 0.35.


Marie Cottrell
email: cottrell@univ-paris1.fr

Patrice Gaubert
email: gaubert@univ-littoral.fr

Efficient estimators : the use of neural networks to construct pseudo panels

Keywords: pseudo-panels, Kohonen map, measurement error, AIDS model

Pseudo panels constituted with repeated cross-sections are good substitutes to true panel data. But individuals grouped in a cohort are not the same for successive periods, and it results in a measurement error and inconsistent estimators. The solution is to constitute cohorts of large numbers of individuals but as homogeneous as possible. This paper explains a new way to do this: by using a selforganizing map, whose properties are well suited to achieve these objectives. It is applied to a set of Canadian surveys, in order to estimate income elasticities for 18 consumption functions..

Introduction

The need for panel data is synthesized by Baltagi [START_REF] Baltagi | Econometric Analysis of Panel Data[END_REF] who enumerates the main advantages as follow:

to control for the individual heterogeneity to obtain more information, more variability, less collinearity between variables, more degrees of freedom and more efficiency to study more precisely the dynamics of adjustment to identify and to measure some effects which are not detectable when using cross-sections data or time series alone and few other more. According to Verbeek [START_REF] Verbeek | A Guide to Modern Econometrics[END_REF] estimators based on panel data are more accurate and more robust for an incomplete specification. The lack of this kind of data or their inadequacy to be used for specific studies (for instance the PSID and consumption behaviour) leads to the construction of pseudo panels from repeated cross sections. It has been showed by Deaton [7] that the estimators obtained this way possess the same properties as those obtained from true panel data. Nevertheless, some specific problems arise with construction of pseudo panels, as a result of the grouping of individuals to constitute the cohorts. The fact that the individuals are not the same in two successive observations of the same cohort result in inconsistent estimators. It may be analyzed as a problem of measurement error. Deaton has proposed, in his seminal paper, a treatment of the resulting bias. Verbeek and Nijman [START_REF] Verbeek | Minimum MSE estimation of a regression model with fixed effects and a series of cross-sections[END_REF], analyzing carefully the different aspects of this problem, specially the asymptotic properties, show that this solution is practically inappropriate and define the conditions leading to consistent estimators.

The paper is organized as follows. In a first part the factors influencing the properties of the estimators are presented according to the results obtained by Verbeek et al. to define the conditions to be respected in the construction of the cohorts of a pseudo panel. In a second part a Kohonen map is used on a set of Canadian surveys and the cohorts obtained are studied in their main characteristics. The first one is the typology of consumption behaviour obtained, which explains why these cohorts are well suited according to the conditions defined by Verbeek et al. The last part shows how to use these cohorts in the estimation of a demand system according to the AIDS specification. The income elasticities obtained for 18 consumption functions are computed. Two of them are compared with those computed with a true panel (PSID) on a very similar period of time.

The need for panel data and pseudo panels : properties and problems

2.1

Panel data estimators

(1) Definitions. The most important aim using panel data is the correction of the endogeneity bias linked to the heterogeneity of individual behaviors. In the case of consumption behavior it has been shown (Gaubert [START_REF] Gaubert | Cellulage de données d'enquêtes de consommation par une méthode neuronale[END_REF], for instance) that very different behaviors may be identified in a given population, induced by factors most of the time not present in the estimated equation, resulting in biased estimators. These properties are presented with the most simple model 1,..., ; 1,...,

it it it y x u i N t a b = + + = = T ) 2 s
for an observation i at the period t. The term u it may be specified using two ways:

Model I (fixed individual effect) it i it u m n = +
where μ i is a non-random non-observable characteristic and ν it a random variable with the usual properties ( ).

( ) and the μ i and the ν it are independent.

Two tests are used, one to verify the existence of individual effects (Fisher type), and one to choose the correct model between I and II (Hausman type, see Hausman [START_REF] Hausman | Specification test in econometrics[END_REF]).

(2) Pseudo panels. Due to the lack of true panel data, Deaton [START_REF] Deaton | Panel Data from Time Series of Cross-Sections[END_REF] has demonstrated that it is possible to use repeated cross-sections of a population (with completely different individuals from one sample to the next one) to construct pseudo panels and obtain estimators with the same properties as those obtained with true panel data.

(3) According to Deaton cohorts are constituted by grouping the individuals of a survey using a variable which does not change across time, like the birth date, to ensure that this characteristic may be used to group the same subpopulation on repeated cross-sections. This variable may be combined with a few other ones in order to obtain a better definition of the cohorts. Each cohort is created using the same characteristics with a set of cross-sections. This leads to the time dimension of this new unit obtained by averaging the different variables over all the individuals belonging to the cohort.

Problems

Of course the individuals are not the same in a given cohort c created in two consecutive surveys. Even the number of individuals is changing, so the "fixed" effect obtained by averaging μ i over the individuals is changing with time for the same cohort c. Moreover, this effect is correlated with the x it in a major part of the economic relations represented with this kind of model, resulting in the inconsistency of random effect estimators. Two problems have to be distinguished (see Deaton [START_REF] Deaton | Panel Data from Time Series of Cross-Sections[END_REF] and essentially Verbeek [START_REF] Verbeek | Minimum MSE estimation of a regression model with fixed effects and a series of cross-sections[END_REF]).

(1) A measurement error causes the inconsistency of the estimators. Deaton already explained it but defining a clearly inappropriate corrected procedure of estimation: the required asymptotic property of the estimators in the time dimension is, obviously, never encountered.

(2) A loss of efficiency is due to the grouping of individuals: the estimation on grouped data leads to a loss of efficiency, as it is known since Cramer [START_REF] Cramer | Efficient grouping, regression and correlation in Engel curve analysis[END_REF], and the question of how to constitute efficiently the groups has been extensively treated by Haitovsky [START_REF] Haitovsky | Regression estimation from grouped observations[END_REF] for a general purpose and by various authors later in specific cases. According to these authors the reduction of both problems may be obtain being very careful when constituting the cohorts.

The two objectives pursued in order to obtain consistent and efficient estimators are important but the construction of pseudo panels is one of the cases where there is a trade-off between them.

Minimizing the within variance of cohort means, relatively to the variance of measurement error, is achieved by grouping very similar individuals in each cohort, and reducing the number of individuals accepted in each one. The definition of numerous cohorts is in favour of more precise estimators, but the small numbers of individuals in each one implies that the computed mean is a poor estimator of the true mean of the population.

Conversely, the reduction of measurement error obtained with large cohorts results in heterogeneous ones, and a few number of units on which the model have to be estimated.

Then the solution is to use a set of variables presenting a good adequacy with the studied phenomenon in order to constitute homogeneous cohorts relatively to the significant variables used to describe it. While doing this, and depending on the total number of individuals surveyed in the repeated cross-sections, it is necessary to control for the number of cohorts and the number of individuals grouped in each one.

The constitution of cohorts with the simple crossclassification, obtained using a small number of qualitative variables like age (date of birth), education and so on, is certainly hazardous relatively to the precise sources of bias. A specific technique seems to be more appropriate.

Cohorts defined with a Kohonen map

The aim of this application is to construct cohorts having the following properties: to be defined using factors quite stable over time in order to link reasonably the successive observations of each cohort to be strongly homogeneous relatively to the phenomenon studied (here the consumption behaviour) and as different as possible between them to obtain precise estimators to include a number of individuals large enough to allow the use of asymptotic reasoning on the obtained estimators.

The data

We use

Variables used to make the classification

(1) To construct cohorts as homogeneously as possible, relatively to the model we have to estimate, and at the same time, to obtain cohorts as different as possible between them, the best indication is to use the significant variables of these behaviours: a cohort constituted mainly with households sharing more or less the same order of preferences seems to present the first property required to construct correctly the pseudo panel. So, the principal variables used are the budget shares defining the consumption behaviour of each household. This structure is not as independent over time as is the date of birth (in Deaton's cohorts): the shares are varying over the period of observation, according to the Engel laws, but the period of time is not too long to make the hypothesis that these behaviours are only slightly varying, as it is verified when the different resulting cohorts are analysed according to the studies cited above about the efficient method to group data, the explanatory variables of the model are included in the input data space of the algorithm 1 , with a special treatment of age. This means that the algorithm is used to construct groups using quantitative variables, the qualitative ones being used only to interpret the groups obtained. (2) Age of the individuals has to be treated in a specific way: if it is used to construct the classification like the other variables, the result is that the classes obtained are very homogeneous considering all the variables used, including age. This is a main difference if we compare these cohorts with those produced with the Deaton's method. The problem is that this group is constituted of individuals who share the same consumption behaviour at the same age. So a part of the dynamic process of consumption, the fact that people reach a level of consumption in some good at different steps of their life cycle, is concealed. Constraining the group to have a common age does presumably produce a kind of cohort with an averaged behaviour leading to measures similar to the ones obtained on a simple cross-section.

To avoid this, dummy variables are created to represent classes of age and the limits of these classes are corrected 2 in the successive crosssections to conform with the idea of cohorts as defined by Deaton.

A more simple treatment of age is tried simultaneously: controlling for age varying only 1 Besides of that the model estimated include a time variable in order to capture the effect of changing environment as well as changing tastes. 2 4 years more in 1986 and 6 more in 1992.

with the inclusion of this variable in the specification of the estimated model.

Classification and constitution of the cohorts

The map constructed is a grid of 64 nodes. The performance of the Kohonen algorithm to reveal very differentiated behaviours of consumption has been presented in a former study (Cottrell et al., 2000).

( 1) Some examples are presented to verify the quality of the classification to create groups which have very different behaviours. These behaviours may be briefly described, and related to qualitative variables characterizing the household and its composition: for the whole sample the main functions are "food at home" (15 %) and "housing" (18 %), then "personal transport" (8 %), "clothing" (6.6 %) and "leisure activities" (6.2 %); the shares of the 13 other functions are between 2 and 4 %. class 1 has a consumption dominated by vehicles expenditures, it is constituted with households of two adults older and receiving higher resources than the population average class 4 is made of households of two adults older than the preceding ones and with a high level of expenditures devoted to the transfers class 13, with households of one adult of middle age and one child, is characterized by a high level of "Food away from home" and "Housing" class 22 with 2 adults older than the average is dominated by "Health" expenditures classes 24 and 32 with one or two old and poor adults are devoting most of their incomes to "Food at home" and "Housing" conversely, class 57, with households of two adults and two children, younger and richer than the average, have a consumption behaviour dominated by "Education" expenditures. These are only a few examples to enlighten on the quality of the output produced by the neural algorithm.

(2) More precisely the differences between consumption behaviours may be measured in order to verify that the main objective defined when constructing cohorts is reached. Distances between classes. The Mahalanobis distances between the nodes may be computed using the code vectors at the end of the iterative process.

They may be represented on a grid similar to the Kohonen map3 using polygons which express the distance between a node and its 8 closest neighbours: the more the polygon is far from the contours of the cell the more distant is the node from the corresponding neighbours. This map shows clearly the significant differences between the nodes, even with the use of neighbouring during the whole process of construction of the map 4 . Within and total variances. Another way to measure the quality of this classification is to measure the share of total variance computed for the whole sample resulting in within variance when the cohorts are defined: the smaller this variance relatively to total variance, the more homogeneous the cohorts.

A non-parametric test may be computed, the Wilks test, on the distribution of the budget shares over the 64 groups obtained. A comparison between this measure obtained with the Kohonen cohorts, and the one produce by a Deaton-like construction (age combined with level of education and region of residence) on the same set of Canadian surveys and with the same number of groups is presented (Table 2) For instance class 4 is constituted of two adults older than the average, they have a high budget share for "Transfers". This constitutes a cohort, C i thereafter and C it for the observation of this cohort at the period t, of 181 individuals observed in 1982, 158 individuals in 1986 and 139 in 1992. These 3 groups have something common: a specific consumption behaviour which produces their assignment to this class which is closer to their own behaviour than any other. The size of each class varies across the map, but there are only 12 classes with less than 300 individuals, meaning that these cohorts gather less than 100 individuals for one of the surveys or more. Only 7 cohorts have between 150 and 225 observations. All the others are more numerous, with a number of observations in each C it greater than 100. According to the computations produced by Verbeek et al. about the size of the bias, these cohorts seem to present the right properties to obtain consistent and efficient estimators.

In the following we work with this pseudo panel which consists of 64 "statistical" individuals measured three times.

Application : consumption functions

(1) Our pseudo panel of consumption expenditures produced by the neural treatment of 3 crosssections leads to the estimation of demand functions using an AIDS specification, as it has been defined by Deaton and Muellbauer [START_REF] Deaton | An Almost Ideal Demand System[END_REF].

According to Banks et al. [START_REF] Banks | Quadratic Engel curves and consumer demand[END_REF] quadratic terms are added in order to capture some non-linearities which appear to be significant for some of the j functions (QUAIDS). Differentiated prices for each survey are not available, and the price variables have to be removed from the model. The effect of changing prices as well as the effect of changing environment will be taken into account through the use of a time effect. Due to the classical measurement error attached to the household's incomes, the total expenditure has been substituted as an instrument. It is well known that this instrument may be itself affected by a measurement error, being the sum of items diversely concerned by this type of error. The hypothesis used here is that this error is cancelled by grouping the data into cohorts.

As is usual for this type of estimation, 2 control variables are added: the age of the household's head and the family equivalized size. As a result, the equation 5 where w is the budget share (of one function) for the individual i (the mean of the cohort C i ) at the period t, y is the total expenditure, age the age of the household's head, size the size of the household using the Oxford scale; year is time dummy and μ is the individual effect.

For the AIDS specification, the β 2 term is removed.

(2) Constructing the variables of the statistical individuals 6 . We estimate this equation at the cohorts' level not on the individual values: for each class produced by the classification we compute the mean of every variable used in the model.

Because the dependent variable is the share of expenditure in one good relatively to total expenditure, the computation of the mean of each other variable has to be weighted using a factor All the variables used in the model are computed this way, as a weighted mean of the values measured at the individual level in each cohort. The heteroscedasticity introduced by this construction has to be removed: this is done by pre-multiplying each variable by the inverse of the square root of the factor inflating the residual variance, that is

2 1 ( ( ) it ht h C g Î å .
The model is estimated for the observations constituted by these transformed values computed for the 64 cohorts observed over 3 periods.

The AIDS and QUAIDS specifications are systematically used and tested to identify the one in adequacy with the data. At the same time, models I and II (fixed or random individual effects) are successively tested, even if the fixed effect specification is generally preferred due to the correlation between the effect and the explanatory variables. The Hausman test is used to indicate the more convenient.

(3) The elasticities. The elasticities computed from the estimated parameters (Table 3) present a very good level of accuracy For two functions, "food at home" and "food away from home", a comparison may be tempted: income elasticities have been obtained using the same specification for a similar period of time (1985-87) using true panel data on American families (PSID) 7 .

The numbers are very close: respectively 0.24 and 0.80 with PSID data, 0.23 and 0.89 with this pseudo panel.

Conclusions

The construction of a pseudo panel from repeated cross-sections using a neural network like the Self-Organizing Map appears to be a means to overcome the major drawbacks attached to the classical pseudo panels.

It produces cohorts with a great homogeneity relatively to the phenomenon studied, depending on the variables chosen to constitute the input data. As the principle of this technique is to transform multidimensional data into a structure compressing this information while preserving the essential, that is the initial topology, the result corresponds to the aim if the variables used are pertinent. The only limit to the quality of the pseudo panel obtained is the number of individuals available in each survey: in order to use asymptotic reasoning to evaluate the estimators and to obtain accurate estimations the number of cohorts must be greater than 50 and the size of each cohort has to be at least 100 observations. This determines the minimum size of the surveys, considering that the algorithm produces reasonably balanced classifications, but there is some variation in size between the classes produced. The combination of a set of pertinent variables in the input data of the algorithm gives the opportunity to use qualitative variables to construct the cohorts, even if the estimated model is not suited to include them with a fixed effect specification.

Fig. 1 .

 1 Fig. 1. The Kohonen Map: a representation of the code vectors.

Fig. 2 .Fig. 3 .

 23 Fig. 2. The Kohonen Map: the distances between the classes .

Table 2

 2

	-Share of within variance relatively to total
	variance with neural and Deaton-like cohorts
		SOM	Deaton-
			like
	Alcohol-Tobacco	48.95	94.59
	Food at home	51.12	85.55
	Food away from	50.03	96.49
	home
	Housing	47.20	86.59
	maintenance
	Communication	77.22	94.68
	Others	47.10	98.13
	Transfers	34.25	87.19
	Education	25.26	89.81

Table 3

 3

	-Total expenditures elasticities
	Functions	Class. with age as	Age only in the
		dummies	model
		Elasticity Student Elasticity Student
	Alcohol-Tobacco	0.730 7.686	0.879 11.459
	Food at home	0.474 10.170	0.419 9.153
	Food away from home	1.275 17.118	1.185 13.709
	Housing maint.	0.636 8.006	0.384 3.992
	Communication	0.846 13.171	0.779 11.701
	Others	0.974 14.234	1.075 16.568
	Transfers	1.316 14.475	1.410 10.671
	Education	1.258 19.353	1.460 14.104
	Clothing	0.963 17.182	1.004 13.214
	Housing	0.905 15.674	0.990 13.183
	Leisure	1.258 30.941	1.246 19.584
	Furniture	1.005 14.610	0.948 10.040
	Health	1.045 13.754	1.142 11.281
	Security	1.322 27.175	1.262 11.645
	Personal care	0.848 13.406	0.855 16.396
	Personal transport	0.908 9.554	0.857 7.806
	Public transport	1.091 10.483	0.768 5.949
	Vehicles	1.898 8.988	1.919 7.661
	For 10 functions over 18, the adequate specification
	is AIDS, and for 6 functions the Hausman test
	rejects the error component form.
	These results are consistent with the general ideas on
	necessary goods (elasticity significantly lower than
	1) and dynamic goods (goods that budget share
	grows with the income).

See Appendix Fig.2.

Except the last iteration which is usually executed with a neighbouring distance reduced to one : only the winner node have its code vector adapted.

For each function, the subscript of which being omitted in order to facilitate the reading.

See a detailed presentation of the transformation in Cardoso et al.[START_REF] Cardoso | Estimation de lois de consommation sur un pseudo-panel d'enquêtes de l'Insee[END_REF]

The programs used to construct the network and to realize some further statistical treatments may be obtained at http://samos.univ-paris1.fr .

Acknowledgements

The data is obtained from Statistics Canada through Pr. Simon Langlois, Université Laval (Québec). The authors thank François Gardes for numerous suggestions and comments, and participants of Paris I seminar, LEMMA seminar and 9th International Conference of ACSEG (2002) for their comments. 7 See Gardes et al. [10] : total expenditures elasticities have to be multiplied by a factor of 0.7, the income elasticity of total expenditures.