Experimental assessment of a PCM to air heat exchanger storage system for building ventilation application

Keywords: Heat exchanger, PCM, experimentation, heat release, dimensionless correlation - Thermal Energy Storage Systems

This paper presents a heat exchanger prototype containing PCM material designed to provide a 1kW heating power during 2 hours (i.e. energy storage of 2kWh). The exchanger is tested in a closed-loop wind tunnel, used to provide constant airflow rates with temperature changes selected so that the PCM is allowed to melt, then to solidify. Temperature and air velocity measurements are achieved for eight airflow rates and the heating power is estimated. The second objective of this work is to provide detailed results suitable with the validation of numerical models. Therefore, the geometry of the exchanger is detailed and the uncertainty associated with the heating power measurement is given. Results show that enough energy is stored in the exchanger, yet the heating power is lower than 1kW before 2 hours because of the use of a of constant airflow rate during the tests. On the other hand, the global behaviour of the heat exchanger is rather simple, which gives confidence in using a simplified approach in further work and help with optimising the current prototype.

HIGHLIGHTS

• The objective is to withdraw the heat pump from the grid during the two hours of electrical peak power • Design of a new heat exchanger with PCM integrated in a air ventilation system

• Laboratory experiment to analyze the heat exchanger efficiency

• Dimensionless analyses to have general conclusions on the system efficiency

COVER LETTER

Dear Editor of Applied Thermal Engineering

The paper has been revised based on the reviewers comments. A list of answers to the reviewers is also enclosed. I feel that the quality of the paper has been enhanced by our revision. Then, I hope that this new submission will fit to the publication requirements of Applied Thermal Engineering.

I look forward to hearing from you.

Best regards, Joseph VIRGONE Professor of Université Lyon1

Experimental assessment of a PCM to air heat exchanger storage system for building ventilation application Labat M. 1,3 , Virgone J. *,1,2 , David D. 1,2 , Kuznik F 1,3

1 Université de Lyon, CNRS 2 Université Lyon 1, CETHIL UMR 5008, F-69622, Villeurbanne, France .K

INTRODUCTION

Nowadays, thermal energy storage systems (TESS) are essential to rationalize the energy management and the use of environmental energy potential. Concepts including phase change materials (PCM) are technically mature for building applications and innovations [START_REF] Mehling | Heat and Cold Storage with PCM: An up to Date Introduction Into Basics and Applications, Heat and Mass Transfer[END_REF][START_REF] Tyagi | Pcm thermal storage in buildings: a state of art[END_REF][START_REF] Waqas | Phase change material (pcm) storage for free cooling of buildings: a review[END_REF][START_REF] Halawa | Thermal performance analysis of a phase change thermal storage unit for space heating[END_REF][START_REF] Dubovsky | Analytical model of a pcm-air heat exchanger[END_REF][START_REF] Vakilaltojjar | Analysis and modelling of a phase change storage system for air conditioning applications[END_REF][START_REF] Borderon | Full-scale study of a building equipped with a multi-layer rack latent heat thermal energy storage system[END_REF][START_REF] Zalba | Free-cooling of buildings with phase change materials[END_REF][START_REF] Dolado | Characterization of melting and solidification in a real-scale pcm air heat exchanger: experimental results and empirical model[END_REF][START_REF] Vakilaltojjar | Phase Change Thermal Storage System for Space Heating and Cooling[END_REF][START_REF] Dolado | Experimental validation of a theoretical model: uncertainty propagation analysis to a pcm-air thermal energy storage unit[END_REF]. Most of the concepts are dedicated to heating purposes, with either a passive or an active point of view. On the other hand, studies reviewed in [START_REF] Osterman | Review of pcm based cooling technologies for buildings[END_REF] aim at reducing the cooling loads and some examples are presented in [START_REF] Lazaro | PCM air heat exchangers for free cooling application in buildings: Experimental results of two real-scale prototypes[END_REF][START_REF] Mosaffa | Thermal performance of a multiple pcm thermal storage unit for free cooling[END_REF][START_REF] Borderon | Intégration des matériaux à changement de phase comme système de régulation dynamique en rénovation thermique[END_REF].

In this study, the general idea is to use a TESS to influence the distribution of the electrical demand for a typical dwelling, heated by an electrical system. In several countries, the electricity rate is significantly varying during the day, along with the energy demand. Therefore, such system presents an economical interest. In our study, the system is a heat exchanger containing PCM and integrated in the ventilation network of the building. It is designed to be function with the electrical heating system, for example a heat pump, and used to store energy when the electricity rate (or the whole electricity demand) is low. When the electricity rate increases, the heating system is shut down and heat is released from the exchanger. This general concept is also known as electrical peak loads shaving and is detailed in [START_REF] Virgone | Peak power reduction using a PCM storage coupled with a heat pump in a ventilation system[END_REF]. The concept is:

-Phase (a): storage of heat process is carried during energy off-peak period. The heating system runs and the heated air is passing through the heat exchanger. The material phase is changing from solid to liquid and required to absorb heat from the airflow.

-Phase (b): release of heat process is carried during energy peak period. The heating system stops and the cold air is passing through the heat exchanger. The material phase is changing from liquid to solid and required to release heat to the airflow.

Sufficient energy has to be stored in order to replace the use of the heating system, that is to say, to maintain comfortable indoor conditions. Of course, this requirement depends strongly on the building of interest. However, this study does not focus neither on a specific building nor a ventilation network, but aims at verifying the feasibility of this general concept. In our study, the specific requirements are the release of a heating power of 1kW during 2 hours (i.e. 2kWh). Both, the latent and sensible heats are used for energy storage/release in order to decrease the system size.

In the paper, a heat exchanger built at our laboratory is presented. It is tested in a wind-tunnel facility so its heating power is measured under dynamic conditions representative of its integration in a real ventilation network. The main objective is to understand how the system behaves and to verify the required heating power. The second objective of this work is to provide detailed results suitable with the validation of numerical models. This need is clearly mentioned in [START_REF] Lopez | Numerical modelling and experimental validation of a PCM to air heat exchanger[END_REF], where the authors collected data from several experiments and underlined the lack of detailed measurements.

The first section of this paper deals with the prototype built at our laboratory and the PCM material. In the second section, the experimental facility and the earliest characterisation tests are presented. Finally, the heating power and the cumulative energy are presented and discussed.

PROTOTYPE DESIGN METHODOLOGY

In the first part of this section, the thermal properties of the PCM are described. In the second and third parts, the design of the PCM heat exchanger is explained.

PCM properties

The selected material was paraffin based. Its commercial name is Microtek 37D1 and its thermal properties 1 http://www.microteklabs.com/pdfs/MPCM-37D%20Product%20Data%20Sheet.pdf were measured in our laboratory. Results are presented in Table 1. The effective specific heat was measured on a 0.0086g sample, using the DSC method (Differential Scanning Calorimeter) at a 1°C.min -1 heating rate (see figure 1). During the melting, the phase change effects are observed when the temperature is growing from 25 to 35°C. A heat flux peak occurs at 34.5°C.

During the solidification, the phase change effects are observed between 33°C and 22C. The heat flux peak occurs at 31°C at the secondary peak is observed at 24°C. Still, the values of the latent heat obtained with both experiments are in good agreement (3%).

Design of the PCM containers

The PCM is distributed into 34 plane-parallel aluminium containers (dimensions: 0.018m × 0.200m × 1.000m). Each container was made of two identical cavities filled with PCM. Each cavity was sealed with aluminium sidings (0.016m × 0.016m × 0.048m. The cavities were partially filled and air vents were placed at the top of the aluminium sides to prevent PCM leakages due to the density changes (Figure 2). The total mass of empty containers was 18.53kg, and the total mass of PCM was measured at 27.88kg.

Design of the heat exchanger

The heat exchanger was made of the 34 containers and cooling fins (the total mass of the cooling fins was 4.69kg). Containers were pilled up by two, which results in 17 blocks (0.200m × 0.018m × 1.000m). The blocks were separated by fins of identical cross-sections (see Figure 3). The fins awere designed so that the air layer which is separating two blocks is divided into 50 channels (0.004m × 0.018m × 1.000m). The whole assembly was surrounded with aluminium plates and the overall cross-section was 0.200m × 0.630m. Finally, the exchanger was insulated using a 0.040m thick layer of polystyrene and mineral wool (i.e. thermal resistance of about 1m 2 .K/W).

Aluminium corner Fastening strap

Cooling fins PCM container

EXPERIMENTAL SET-UP AND CHARACTERISATION TESTS

The main objective is to estimate the heating power of the heat exchanger; it can be evaluated with:

( ) ( )

A A P V A T C Q T P ∆ ⋅ ⋅ ⋅ = , & ρ (1)
Where the air density ρ A is given by:

) ( 15 . 273 293 . 1 K T A A ⋅ = ρ (2)
The uncertainty on the measured heating power u(P) is estimated assuming there is no correlation between the different parameters. It leads to equation [START_REF] Waqas | Phase change material (pcm) storage for free cooling of buildings: a review[END_REF] given in [START_REF] Eurachem/Citac | Quantifier les incertitudes dans les mesures analytiques[END_REF]. Therefore, the accuracy depends on both the estimation of the airflow rate Q V and of the temperature difference ∆T A . As a consequence, these two parameters have to be estimated carefully.

( )

∑         ⋅ ∂ ∂ = 2 ) ( i i x x P P u σ (3)
The heat exchanger is placed into a closed-loop wind tunnel with a constant airflow rate. Constant temperatures are applied at the inlet (20°C and 45°C); the PCM will completely melt and solidify. The heating power is estimated during the solidification of the PCM. To give a better characterisation of the heat exchanger, this experiment is repeated for 8 different airflow rates, ranging from 300 to 900m

3 .h -1 .
In the first part of the section, the experimental facility and the instrumentation are presented. Second, both temperature and airflow measurements are discussed for the lowest and the highest airflow rates. Finally, pressure drop measurements are detailed.

Description of the air-supply system

The experimental facility includes a complete HVAC system on the bottom part, and a testing cavity on the top part. The heat exchanger is placed in the testing cavity (Figure 4 and Figure 5): it is 2.5 m long and its section is 1m x 1m. In order to control the airflow rate inside the exchanger, both an upstream and a downstream section are added.

The upstream section is made of two parts and its overall length is 1m. In the first part, the cross-section is gradually reduced to the targeted cross-section (slope: 30°). In the second part, the cross-section re mains constant so an established airflow can be obtained. The downstream section is a 0.4m long element with a constant cross-section in order to produce a uniform airflow. This allows achieving temperature measurements at the outlet of the exchanger.

Instrumentation

First, the temperature difference is measured by the mean of 5 differential thermocouples fixed on nylon ropes in the upstream and in the downstream sections. Their positions are given in Figure 6. These thermocouples were all calibrated in laboratory and their uncertainty is ±0.3°C. Three T-type thermocouples are added in the downstream section to measure the absolute air temperature.

The second key parameter in the heating power calculation is the airflow rate, which is monitored by the airsupply system. However, air leakages can happen, which may result in a lower airflow rate. Consequently, the heat exchanger and its setting within the experimental facility are designed in order to be as airtight as possible:

• Fastening techniques are preferred to drilling techniques to gather the different part of the prototype;

• The exchanger is tightened on the downstream section: aluminium corners are used to increase the contact surface (see Figure 3) and a rubber strip with identical dimensions is screwed onto (not visible in Figure 3). Two fastening straps are passing through the aluminium rails, then wrapped around the downstream section and stretched; • Aluminium tape is used to enhance the sealing for both the upstream and the downstream section.

Still, leakages are happening and have to be taken into account. A unidirectional hot wire anemometer is set inside the upstream section in order to monitor the air velocity. It is sensitive to air speeds ranging from 0.05 to 2.50m.s -1 but not to the airflow direction. Preliminary tests are described in the next section. Other measurements are achieved to give a much complete understanding of the heat exchanger behaviour:

• A pressure difference device is connected to both the upstream and the downstream sections, close to the position of the temperature sensors and allows measuring the pressure difference. • 32 thermocouples are used to measure temperatures at various locations inside the heat exchanger.

For the sake of clarity, only three of them will be discussed. They are located directly in the container placed in the middle of the exchanger.

The number and accuracy of the sensors are summarized in Table 2.

Estimation of the global airflow rate

The hot-wire anemometer is used to measure the air velocity in the upstream section when the heat exchanger is not installed. It is moved successively at nine different locations (see Figure 6). These measurements are achieved for two airflow rates (300 and 900m ).

Besides, commercial CFD software STAR CCM is used to simulate the airflow in the upstream section. The air volume is meshed with an unstructured mesh containing more than 500,000 polyhedral cells. This mesh is refined close to the sidings, so the 0.01m boundary layer is composed of 10 cells. The boundary conditions are:

• A homogeneous airflow at the inlet;

• A zero-pressure condition at the outlet;

• Adiabatic wall condition for other boundaries.

Reynolds number is calculated in the cross-section located 0.1m before the heat exchanger (the exchanger is not simulated); its value is higher than 10 4 for velocities measured at 300m , which means the airflow is turbulent. Therefore, a classical k-epsilon turbulence model is selected. Figure 7 presents the velocity profiles computed at middle-height in the cross-section where the sensors are installed. First, these results show that the design of the upstream section allows obtaining an established flow. This is very convenient as the global airflow rate can be estimated from few measurements. Second, the comparison with airflow measurements showed that simulation results were consistently higher than measured velocities (the average difference is 9%), which can be the consequence of leakages.

About the heating power calculation, it is assumed the leakages are resulting in a uniform decrease of the air velocity (∆v = 9%), and so as for the airflow rate. Consequently, the airflow rate can be estimated from a single air velocity measurement as presented in equation 4. The coefficient K represents the ratio of the velocity measured in the middle the cross section to the average velocity (K = 0.89). A similar approach is presented in [START_REF] Falk | Ventilated rainscreen cladding: A study of the ventilation drying process[END_REF] and leads to values ranging from 0.6 to 0.9, depending on the duct geometry and on the airflow rate.

( )

v K S v Q Meas V ∆ - ⋅ ⋅ ⋅ = 1 (4)

Pressure drop measurements

Pressure drop is measured during the experiments: it rises up to 93 ± 3Pa at 900m . Results are plotted against the measured airflow rate computed with equation 4 in Figure 8. Finally, the airflow rate is related to pressure difference by a power law:

n V p C Q ∆ = . & (5)
The fitted values are C = 22.518m .Pa -1 and n = 0.8148. A very good agreement is obtained: 91% of the measurements are included in the ± 3% range. Somehow, this gives confidence in the airflow rate calculation. These results can be used to design the integration of the heat exchanger in the ventilation network and calculate the related energy increase. First, the temperature change at the inlet is not immediate. This results from the regulation of the closed loop system. Still, temperature measurements achieved directly in the PCM show a regular behaviour. Moreover, the temperature drop from 45°C to 20°C is rather quick and should not affect significantly the study of the heating power.

After a quick decrease from 45 to 35°C, a stable-temperature-period is observed in the material. It lasts 18min at 900m . Then, temperature starts to decrease again. At 300m

3 .h -1
, a second stable-temperature-period is observed at 30.5°C, which indicates that the solidification of the material is ongoing.

Finally, measured temperatures in the PCM at 900m 3 .h -1 are equal to the inlet temperature 1h30min after the temperature change, meaning the heat release is finished. Therefore, it is obvious that the heat exchanger does not release 1kWh during 2h for this experiment. However, at 300m

3 .h -1
, the heat release is not complete two hours after the temperature change.

Temperature difference measurements during solidification

Temperature difference between the inlet and the outlet of the heat exchanger are presented in Figure 10. The highest temperature differences are obtained at the beginning of the experiment and remain constant for about 15min. Then, it decreases rather regularly to zero.

The temperature difference is not homogeneous over the cross-sections; the observed difference rises up to 4°C and the standard deviation is 1.3°C for the two experiments. Lower values are measured close to the sides while the highest values are measured at the bottom. These differences cannot be easily explained. For example, the standard deviation is lower than 0.1°C when the exchanger is neither storing nor releasing heat. Consequently, lower temperature differences measured close to the sides cannot be explained by a defective insulation of the exchanger. On the other hand, observed differences could be the consequence of non-uniform velocities over the cross-section or a non-uniform solidification over the heat exchanger height.

RESULTS AND DISCUSSION

Estimation of the heating power and of the cumulative energy

The heating power P is calculated as presented in equation 1. The airflow rate Q V is calculated as presented in equation 4 and the temperature difference ∆T A is the average value of the 5 measurements. Results are presented in Figure 11. For the sake of clarity, only few measurements are plotted because of the quick variation of the temperature difference (Figure 10). As the airflow rate remains constant over a single experiment, the heating power follows closely the measured temperature difference (Figure 10). At 900m , the heating power rises quickly to 2770 ± 157W for 15min, then decreases regularly to reach 500 ± 95W at 1h15min. At 300m , the maximum heating power is lower (1620 ± 225W) because of the lower airflow rate. It reaches 1025 ± 187W at 30min. Still it remains higher than 522 ± 157W during 2h15min. Consequently, the initial goal (obtaining a minimum 1kW heating power during 2 hours) is not reached for both cases.

The heating power is integrated over the heat release period (measurements with a period of 1mn were used for this calculation). This represents the cumulative heat released by the heat exchanger. The uncertainty is estimated using the approach presented in (3) and results are presented in Figure 12 for airflow rates tested. First, the final value of the cumulative energy is almost the same for each experiment: the average value is 2.34kWh and the standard deviation is 0.07kWh. This means the exchanger has stored enough energy to meet the heating demand.

Second, the standard deviation is twice lower than the measurement uncertainty (± 0.15kWh) and the calculated cumulative energy varies little with the airflow rate. This means the methodology and the instrumentation are relevant. Still, it is observed the computed total energy is slowly decreasing as tests are conducted (from the lowest to the highest airflow rate). This is the direct consequence from PCM leakages happening at the sidings of the containers; almost 0.8kg of PCM was collected from leakages (3% of the total mass). The impact of these mass losses is not influencing in the discussion of the results. If the integration of such an exchanger in a real building was considered, this would become a major issue.

Finally, it is obvious that the heat release is faster at a higher airflow rate. However, results are nearly the same for the highest airflow rates (804 and 901m

General behaviour

To give an idea of the general heat exchanger behaviour, it is convenient to use dimensionless numbers.

Here, the idea is to show the impact of an increasing airflow rate on the dynamic behaviour of the heat exchanger. Therefore, it is relevant to use the Fourier and the Stanton numbers as presented in equations 6. The Stanton number measures the ration of the heat transferred into a fluid by convection to the thermal capacity of the fluid. The Fourier number characterizes the ration of the diffusive transport rate (by conduction) to the storage rate. For the Stanton number calculation, the convective heat transfer coefficient h c has to be estimated. Here, it is determined based on the geometric considerations presented in [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF]. Because of the cooling fins, the airflow rate between two containers is laminar for each airflow rate. Therefore, the convective heat transfer coefficient is constant with a value of h c = 37W.m . As a result, the Stanton number variation depends only on the air velocity.

Fourier number is computed for the second half of the heat release, that is to say when the liquid fraction is lower than 0.5. In this case, it is assumed that the material properties for the solid state (see Table 1) are representative of the whole material. Heat transfer is assumed to be symmetric in a single container, so the characteristic length e is equal to half of the total thickness (e = 0.009m). Finally, t r is selected to correspond to a given percentage of the total energy released. Fourier number is plotted against the inverse value of the Stanton number in Figure 13.

Figure 13: Comparison of Fourier and Stanton numbers for different percentage of heat release

The Fourier number is decreasing when the inverse Stanton number is increasing, meaning that a higher airflow rate is accelerating the heat release. An asymptotic behaviour is observed when the inverse Stanton number is higher than 200. A very good agreement is obtained between measurements and a general exponential equation 7. The fitted values of coefficients A1, A2 and A3 are presented in Table 3.

( )

1 3 exp 2 1 - ⋅ ⋅ + = St A A A Fo (7)

Outlook

Although PCM solidification is neither homogeneous nor easy to predict, the global behaviour of the heat exchanger is rather simple. This is an interesting result for numerical modelling purposes as this allows considering a simplified approach. Furthermore, measurements and geometry details presented in this paper are suitable with the validation of numerical models (similar measurements are used in [START_REF] Lopez | Numerical modelling and experimental validation of a PCM to air heat exchanger[END_REF]). Once validated, the model could be used to optimise the heat exchanger: this could help with reaching the initial objective (obtaining a 1kW heating power during 2 hours).

However, using a constant airflow rate does not seem to be a suitable solution. Instead, a variable airflow rates should be considered. However, it should be kept in mind that this experimental study has been achieved for one single cooling fins geometry. Cooling fins can be used to modify the behaviour of the heat exchanger. Indeed, this element has several impacts:

• Heat transfer is increased because of the increased surface;

• The pressure loss is increased, which is a major issue in the design of a ventilation network; • The airflow is laminar in this case, because of the small dimensions of the channels. It will become turbulent for larger channels, which will affect both the pressure loss and heat transfer. Also, this represents a limit in the validation of numerical models.

Therefore, it sounds promising to study much in detail the airflow rate management.

CONCLUSION

In this paper, a heat exchanger containing more than 27kg of PCM is presented and tested. It is designed to store enough energy to replace a 1kW heat pump during 2 hours. The exchanger is placed within an air-supply system used to produce a constant airflow rate with temperature changes.

The main objective of the experimental tests is to determine the heating power during heat release. For this reason, earliest tests are conducted to ensure accurate estimations of both the airflow rate and temperature differences. The experimental set-up allows the evaluation of the energy stored/released in the system with an accuracy of 6 %. But, the initial objective could not be obtained for constant airflow rates varying from 300 to 900m . However, the energy stored and released is higher than 2kWh. Then, An optimization of the heat transfers is necessary to reach the objectives.

Although quick and non-homogeneous heat transfer is observable within the PCM (for example natural convection may occur), the global behaviour of the heat exchanger is rather simple. Therefore, an exponential relationship between the Fourier and the Stanton numbers could be fitted. This indicates considering a simplified numerical model is relevant for further work. Then, it could be used to optimise the current prototype and reach the initial objective, with using a variable airflow rate and other designs of cooling fins for example. The data presented in this paper will also be used to validate a numerical modelling of a PCM to air heat exchanger.

Figure 2 :

 2 Figure 2: Scheme of a single container filled with PCM

Figure 3 :Figure 4 :

 34 Figure 3 : External view the heat exchanger front

Figure 5 :

 5 Figure 5: Scheme of the exchanger installed in the air-supply system

Figure 6 :

 6 Figure 6: Position of the sensors installed in the upstream section (0.1m away from the heat exchanger)

Figure 7 :

 7 Figure 7: Velocity profiles computed with commercial CFD software for 300 m 3 .h -1 and 900m 3 .h -1

Figure 8 :

 8 Figure 8: Pressure drop measurements for the experiments and fitted power law 3.5. Temperature measurements within the PCM during solidification Temperature measurements in the PCM during the solidification at 300 and 900m 3 .h-1 are presented in Figure9. Two sensors are located 0.1 m away from the inlet and the outlet and the third one is located at middle-length (0.5 m). The time reference (t = 0 h) corresponds to the airflow temperature change at the inlet.

Figure 9 :

 9 Figure 9: Temperature measurements during the solidification period at 300m 3 .h -1 (left) and 900m 3 .h -1

Figure 10 :

 10 Figure 10: Temperature difference measurement during solidification at 300m 3 .h -1 (left) and 900m 3 .h -1

Figure 11 :

 11 Figure 11: Measured heating power after the temperature change (from 45 to 20°C)

Figure 12 :

 12 Figure 12: Cumulative energy released by the system after the temperature change from 45 to 20°C for 900m 3 .h -1 (left) and for the airflow rates tested (right)

Table 1 : Thermal properties of the material

 1

				Solid state	Liquid state
	Density (kg.m	-3 )		870	779
	Thermal conductivity (W.m	-1 .K -1 )	0.24	0.18
	Heat capacity (kJ.kg -1 .K -1 )		2.398 (10C)	2.206 (33C)
	Melting heat (kJ.kg -1 )		222

Figure 1: Equivalent heat capacity of the Microtek 37 from DSC measurement

Table 2 : Description of the instrumentation

 2

	Physical parameter	Number	Accuracy
	Temperature difference (°C)	5	± 0.3
	Air velocity (m.s	-1 )	1	± 0.08
	Pressure difference (Pa)	1	± 0.5 % of the value
	Temperature (°C)	32	± 0.5

Table 3 : Coefficient values used in (8) fitted with the measurements

 3

		tr 50%	tr 66%	tr 75%	tr 90%
	A1	2,39E+00	3,05E+00	3,71E+00	5,16E+00
	A2	1,43E+01	1,72E+01	2,12E+01	2,49E+01
	A3	-2,07E-02	-1,68E-02	-1,73E-02	-1,70E-02

.h -1), which means an asymptotic behaviour has been reached.

ACKNOWLEDGMENTS

This work has been possible with the support of the French ANR project STOCKAIR2 with the reference: ANR-2010-STKE-010.


Definition. Let I be the unit interval. The family is defined for all (u, v) ∈ I 2 by,

C θ,φ (u, v) = uv + θ[max(u, v)]φ(u)φ(v).
where φ and θ are differentiable I → IR functions (vanishing at most on isolated points).

Theorem. C θ,φ is a copula if and only if φ and θ satisfy the following conditions:

• boundary conditions: φ(0) = 0 and (φθ)(1) = 0,

• θ is non increasing on I,

• φ ′ (u)(θφ) ′ (v) ≥ -1 for all 0 ≤ u ≤ v ≤ 1.
Remark. The family can be split in two sub-families according to θ(1) = 0 or φ(1) = 0.

Cécile Amblard & Stéphane Girard

Measure of association.

Let (X, Y ) a random pair with joint distribution H(x, y) = C(F (x), G(y)). Spearman's Rho: probability of concordance minus the probability of discordance of two random pairs with respective joint cumulative law C(F, G) and F G.

ρ = 12 1 0 1 0 C(u, v)dudv -3.
In the case of C = C θ,φ , we have

ρ θ,φ = 12 Φ 2 (1)θ(1) - 1 0 Φ 2 (t)θ ′ (t)dt , where Φ(t) = t 0 φ(u)du.
Remark.

• If θ(1) = 0, then ρ θ,φ ≥ 0. • If θ is a constant function, then ρ θ,φ = 12θΦ 2 (1).

Upper tail dependence.

The upper tail dependence coefficient is defined as

λ = lim t→1 P(F (X) > t|G(Y ) > t) = lim u→1 C(u, u) 1 -u , where C is the survival copula, i.e. C(u, v) = 1 -u -v + C(u, v).
In the case where C = C θ,φ , we have

λ θ,φ = -φ 2 (1)θ ′ (1).
Remark.

• If φ(1) = 0, then λ θ,φ = 0.

• If θ is a constant function, then λ θ,φ = 0.

Cécile Amblard & Stéphane Girard

2. First sub-family, the case θ(1) = 0.

Examples.

• Fréchet upper bound. Choosing φ(x) = x and θ

(x) = (1 -x)/x yields C θ,φ (u, v) = M (u, v) = min(u, v). • Independent copula. θ(x) = 0 yields C θ,φ (u, v) = Π(u, v) = uv. • Cuadras-Augé family: φ(x) = x and θ(x) = x -α -1, 0 ≤ α ≤ 1 yields C θ,φ (u, v) = min(u, v) α (uv) 1-α = M α (u, v)Π 1-α (u, v),
which is the weighted geometric mean of M and Π.

Remark.

• θ(1) = 0 and θ ′ (u) ≤ 0 imply θ(u) ≥ 0 for all u ∈ I.

• 0 ≤ ρ θ,φ ≤ 1 -→ Modelling of positive dependences.

• Lower (0) and upper bounds (1) of ρ θ,φ and λ θ,φ are reached respectively by the Π and M copulas.

Dependence properties: definitions.

Assume X and Y are exchangeable. X and Y are

• Positively Quadrant Dependent (PQD) if P(X ≤ x, Y ≤ y) ≥ P(X ≤ x)P(Y ≤ y)
for all (x, y).

• Left Tail Decreasing (LTD) if P(Y ≤ y|X ≤ x) is non-increasing in x for all y.

• Right Tail Increasing (RTI) if P(Y > y|X > x) is nondecreasing in x for all y.

• Stochastically Increasing (SI) if P(Y > y|X = x) is nondecreasing in x for all y.

• Left Corner Set Decreasing (LCSD) if P(X ≤ x, Y ≤ y|X ≤ x ′ , Y ≤ y ′ ) is non-increasing in x ′ and y ′ for all (x, y).

• Right Corner Set Increasing (RCSI) if P(X > x, Y > y|X > x ′ , Y > y ′
) is nondecreasing in x ′ and y ′ for all (x, y).

Theorem. X and Y are:

• PQD iff φ(u) has a constant sign on I.

• LTD or LCSD iff either {φ(u)/u is non increasing and ∀u ∈ I, φ(u) ≥ 0} or {φ(u)/u is non decreasing and ∀u ∈ I, φ(u) ≤ 0}.

• RTI or RCSI iff φ(u)/(1 -u) and θ(u)φ(u)/(1 -u) are monotone.

• SI iff either {φ and θφ are concave and ∀u ∈ I, φ(u) ≥ 0} or {φ and θφ are convex and ∀u ∈ I, φ(u) ≤ 0}. In this case, we restrict ourselves to a constant function θ, i.e.

θ(x) = θ ∈ [-1, 1].
Theorem. C θ,φ is a copula if and only if φ and θ satisfy the following conditions:

• boundary conditions: φ(0) = 0 and φ(1) = 0,

• |φ ′ (x)| ≤ 1 for all x ∈ I, • |φ(x)| ≤ min(x, 1 -x), for all x ∈ I.
Examples.

• φ(x) = min(x, 1 -x): upper bound of the above theorem,

• φ(x) = x(1 -x): Farlie-Gumbel-Morgenstern family of copulas (Morgenstern, 1956), which contains all copulas with both horizontal and vertical quadratic sections (Quesada-Molina, Rodríguez-Lallena, 1995)

• φ(x) = x(1 -x)(1 -2x): symmetric copulas with cubic sections (Nelsen et al, 1997), • φ(x) = π -1 sin(πx). Cécile Amblard & Stéphane Girard 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Upper bound, Farlie-Gumbel-Morgenstern, cubic sections, sinus.

Cécile Amblard & Stéphane Girard

Measure of association. The Spearman's Rho can be rewritten as:

ρ θ,φ = 12θ I φ(u)du 2 ,
and it follows that -3/4 ≤ ρ θ,φ ≤ 3/4 for all θ ∈ [-1, 1]. Similar bounds hold for the Kendall's Tau: -1/2 ≤ τ θ,φ ≤ 1/2. Upper tail dependence. ρ θ,φ = 0.

Dependence properties. Similar to the previous family in the case θ > 0.

Symmetry properties: definitions.

• X is symmetric about a if (X -a) and (a -X) are identically distributed (id).

• X and Y are exchangeable if (X, Y ) and (Y, X) are id.

• (X, Y ) is marginally symmetric about (a, b) if X and Y are symmetric about a and b respectively.

• (X, Y ) is radially symmetric about (a, b) if (X -a, Y -b) and (a -X, b -Y ) are id. • (X, Y ) is jointly symmetric about (a, b) if the pairs (X -a, Y -b), (a -X, b -Y ), (X -a, b -Y ) and (a -X, Y -b) are id.
Theorem. In the C θ,φ family:

• If X and Y are id then X and Y are exchangeable.

Besides, if (X, Y ) is marginally symmetric about (a, b) then:

• (X, Y ) is radially symmetric about (a, b) if and only if either ∀u ∈ I, φ(u) = φ(1 -u) or ∀u ∈ I, φ(u) = -φ(1 -u).
• (X, Y ) is jointly symmetric about (a, b) if and only if ∀u ∈ I, φ(u) = -φ(1 -u).

Cécile Amblard & Stéphane Girard

4. Inference procedures.

Assumptions.

• We restrict ourselves to the second sub-family, with constant function θ:

C(u, v) = uv + θφ(u)φ(v).
-→ Estimation of θ (scalar) and φ (univariate function).

-→ Identifiability problem: (θ, φ) and (αθ, φ/ √ α) yield the same copula for all α > 0.

• We focus on the PQD case: θ > 0 and φ has a constant sign.

Under these assumptions, the family can be rewritten

C(u, v) = uv + ψ(u)ψ(v),
where

ψ(x) = √ θ|φ(x)|.
-→ The estimation of C reduces to the estimation of ψ (positive univariate function).

Estimation of ψ 1) Preprocessing:

• {(x i , y i ), i = 1, . . . , n} a sample of (X, Y ) from the cdf H(x, y) = C(F (x), G(y)).

• Rank transformations: u i = rank(x i )/n and v i = rank(y i )/n.

{(u i , v i ), i = 1, . . . , n} an approximate sample from the copula C(u, v).

• Pseudo-observations {w i = max(u i , v i ), i = 1, . . . , n} from C(w, w) = w 2 + ψ(w).

2) Projection estimate: linear combination of basis functions:

{e k , k ≥ 1} ψ(w) = k≥1
a k e k (w), w ∈ I.

Choice of the set of functions:

• no orthogonality condition,

• boundary conditions e k (0) = e k (1) = 0 for all k ≥ 1 so that ψ(0) = ψ(1) = 0.

Cécile Amblard & Stéphane Girard

3) Optimization problem: Define

• w 1,n ≤ • • • ≤ w n,n
, the ordered pseudo-observations,

• M and

M ′ two matrices M i,k = e k (w i,n ), M ′ i,k = e ′ k (w i,n ), k ≥ 1, i ∈ {1, . . . , n}, • a and b two vectors b i = (i/(n + 1) -w 2 i,n ) 1/2 , a i unknown, i ∈ {1, . . . , n}. Definition of the estimator. • ψ(w i,n ) = C(w i,n , w i,n ) -w 2 i,n ≃ i/(n + 1) -w 2 i,n for i = 1, . . . , n can be rewritten min a M a -b 2 , • ψ(w i,n ) ≥ 0 can be rewritten M a ≥ 0, • | ψ(w i,n )| ≤ 1 can be rewritten -1 ≤ M ′ a ≤ 1.
-→ Constrained least-square problem.

Estimation of the Spearman's rho

Recall that Another solution: adapt the nonparametric estimator of the Kendall's Tau introduced in (Genest, Rivest, 1993)

ρ θ,φ = 12θ I φ(u)du 2 = 12 I ψ(u)du
to obtain ρNP = 6 n(n -1) n i=1 n j=1 1{u j < u i , v j < v i } - 3 2 ,

Estimation of high probability regions

Definition. The α-quantile of the copula C is defined by

Q α = inf{λ(S) : P(S) ≥ α, S ⊂ I 2 }, 0 < α ≤ 1,
where λ is the Lebesgue measure on I 2 . Partitions.

{I k , k = 1, . . . , N } be the equidistant N -partition of I, K k,ℓ = I k × I ℓ the associated N × N grid. Denote δ k,ℓ ∈ {0, 1}, (k, ℓ) ∈ {1, . . . , N } 2 . Estimator: Qα = k,ℓ K k,ℓ 1{δ k,ℓ = 1}.
Optimization problem. The δ k,ℓ are defined by min

N k=1 N ℓ=1 δ k,ℓ ,
under the constraints δ k,ℓ ∈ {0, 1} and

N k=1 N ℓ=1 δ k,ℓ P (K k,ℓ ) ≥ α,
where P (K k,ℓ ) is an estimation of the probability P (K k,ℓ ).

Cécile Amblard & Stéphane Girard

Algorithm.

• First step: sort the P (K k,ℓ ) in decreasing order to obtain the sequence Pτ , τ = 1, . . . , N 2 .

• Second step: Computation of the number of subsets of the partition:

J = min j, j τ =1
Pτ ≥ α .

• Third step: selection of the J first subsets:

δ k,ℓ = 1 if 1 ≤ τ (k, ℓ) ≤ J,
Estimation of P (K k,ℓ ). Two solutions:

• Semi-parametric estimate based on ψ

PSP (K k,ℓ ) = 1 N 2 + ψ k N -ψ k -1 N ψ ℓ N -ψ ℓ -1 N .
• Nonparametric estimate

P NP (K k,ℓ ) = 1 n n i=1 1{(u i , v i ) ∈ K k,ℓ }.
Cécile Amblard & Stéphane Girard

Simulation results.

Numerical experiments on the family of copulas C k generated by the set of functions

∀k ≥ 1, ψ k (x) = 1 -x k + (1 -x) k 1/k , x ∈ I. • When k = 1, C 1 : uniform distribution on I 2 . Spearman's Rho ρ 1 = 0. • When k → ∞, ψ k (x) → ψ ∞ (x) = min(x, 1 -x) for all x ∈ I.
C ∞ : mixture of two uniform distributions on the squares [0, 1/2] 2 and [1/2, 1] 2 with mixing parameter 1/2. Spearman's Rho ρ ∞ = 3/4 (the maximum value in the sub-family).

• When 1 < k < ∞, bivariate distribution "interpolating" between the two previous ones.

Chosen basis of functions:

e s,ℓ (x) = sin π 2 (2 s+1 x -ℓ) 1{2 s+1 x ∈ [ℓ, ℓ + 2]},
s is a scale parameter, ℓ is a location parameter. 6. Real data. n = 225 countries, two variables: X, the life expectancy at birth (years) in 2002 of the total population and Y , the difference between the life expectancy at birth of women and men. http://www.odci.gov/cia/publications/factbook/. According to the PQD test proposed in (Scaillet, 2004), these data are PQD.

Outline 1 .

 1 Definition and basic properties. 2. First sub-family, the case θ(1) = 0. 3. Second sub-family, the case φ(1) = 0. 4. Inference procedures. 5. Simulation results. 6. Real data. Cécile Amblard & Stéphane Girard 1. Definition and basic properties.

  sub-familyCécile Amblard & Stéphane Girard 3. Second sub-family, the case φ(1) = 0.

2 .

 2 Replacing ψ by ψ yields the following semi-parametric estimator: introduced β k = I e k (u)du.

1

  Estimation of high probability regions Q α from C 4 . Red: α = 0.25, green: α = 0.5, yellow: α = 0.75. Top left: simulated sample, top right: nonparametric estimate, bottom left: semiparametric estimate, bottom right: semiparametric estimate with the true function ψ, (n = 500).

1

  Estimation of high probability regions Q α from C 8 . Red: α = 0.25, green: α = 0.5, yellow: α = 0.75. Top left: simulated sample, top right: nonparametric estimate, bottom left: semiparametric estimate, bottom right: semiparametric estimate with the true function ψ, (n = 500).

1

  Estimation of high probability regions Q α from real data. Red: α = 0.25, green: α = 0.5, yellow: α = 0.75. Top left: real data, top right: real data after rank transformation, bottom left: nonparametric estimate, bottom right: semiparametric estimate.

  Estimation of the generating function and of the Spearman's Rho (ρ k ). The mean value of the estimates ρSP and ρNP are evaluated on 100 repetitions.Estimation of high probability regions Q α from C 2 . Red: α = 0.25, green: α = 0.5, yellow: α = 0.75. Top left: simulated sample, top right: nonparametric estimate, bottom left: semiparametric estimate, bottom right: semiparametric estimate with the true function ψ, (n = 500).

		1							1
		0.8							0.8
		0.6							0.6
	0.5	0.4						0.5	0.4
	0.45	0.2						0.45	0.2
	0.3 0.35 0.4	0.9 1 k ρ k × 10 -2 mean( ρSP ) × 10 -2 mean( ρNP ) × 10 -2 0.3 0.35 0.4 1 0 0.81 0.18 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 0	1
	0.25	2 1	0.8 42.5			0.25 43.0	1			41.2
	0.2	4 0.8	0.7 66.4			0.2 65.8	0.8			64.3
	0.15	6 0.6	0.6 71.2			0.15 70.6	0.6			68.8
	0.1	8 0.4	0.5 72.8			0.1 72.1	0.4			70.2
	0.05	0.2	0.4					0.05	0.2
	0.1 0.2 0.3 0.6 True functions ψ k (x), k ∈ {2, 4, 8} -Estimated functions ψk (x), k ∈ {2, 4, 8}, n = 100. 0 0.1 0.2 0.3 0.4 0.5 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 0
			0	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

Cécile Amblard & Stéphane Girard


Alexandru Mardale

Romanian Complex Adnominal Prepositional Phrases

Introduction

Romanian displays at least two types of adnominal (i.e. attached to a noun) complex prepositional phrases (PPs) (i.e. formed by two and rarely three simple prepositions). The first one involves deverbal or processual (and generally) uncountable matrix nouns -it is illustrated in (i) below, while the second one involves various types of countable matrix nouns -it is illustrated in (ii):

(i) coborâtul de pe deal 1 (type A) descent-the of on hill (the descent off the hill) (ii) copilul de pe stradă (type B) child-the of on street (the child from the street) Our purpose in this presentation is to examine the conditions that determine such constructions, as well as to propose an analysis that shows at the same time their common properties and their differences. We will insist on the second type, which is particular among Romance.

Type A

Below there are some examples for this type of construction:

(1) a. venirea de la Paris coming-the of at Paris (the return from Paris) b. un vânt din deşert (din is a contracted preposition from de and în) a wind of-in desert (a wind from the desert) c. ieşirea de sub anonimat exit-the from under anonymity (coming into prominence) d. plimbatul pe sub poduri walking-the on under bridges (walking under the bridges) e. o excursie pe la mănăstiri a trip on at monasteries (a trip to the monasteries) f. alergatul printre obstacole (printre, contracted form of p(r)e and între) running-the on-between obstacles (running through obstacles) Among the properties of this kind of construction, we notice that the prepositions that can introduce it (as P1) are especially de (1a-c) and pe (1d-f). De is considered here as an ablative preposition, expressing the separation from a point or place of origin (cf. plecarea de la Paris (the depart from Paris)). Pe may express the approximation and/or the location (cf. jucatul pe sub masă (the playing under the table )). There is no constraint regarding the occurrence of the second (or third) preposition (P2 or P3). Thus, different prepositions may occur there (la 'at ', în 'in', sub 'under', peste 'over', lângă 'near', între 'between', etc.).

At the same time, we have noticed in the introduction that these complex NPs are attached to deverbal or processual uncountable nouns and, in this respect, they have the same form and function as in the verbal structure they come from (cf., a.o., Chomsky 1970 With regard to the noun complement introduced by P2, we notice here that it is generally a DP2 (i.e. a noun with a determiner), as in (3) below, but under certain circumstances, it may as well be an NP (i.e. a noun without a determiner), as in (4) below:

(3) a. un vânt din deşertul african / *din deşert african (a wind from the African desert) b. o excursie pe la mănăstirile din Moldova / ?? pe la mănăstiri din Moldova (a trip to the monasteries from Moldavia) (4) alergatul printre obstacole naturale / ?? obstacolele naturale (the running through natural obstacles)

Parallel with this distinction, we notice that the constructions having a constituent like P2+DP in an adnominal position may alternate with an adverb (as in (5) below), while those having a constituent like P2+NP may not (6):

(5) a. venirea de la Paris => venirea de acolo (the arrival from Paris) (the arrival from there) b. mersul pe sub poduri => mersul pe aici (the walk under the bridges) (the walk (around) here) (6) alergatul printre obstacole => *alergatul pe acolo3 (the running through obstacles) (the running over there)

The contrast that we have just described may be also explained if we take into account the type of denotation of the adnominal complement. To be more precise, the constituents spelled out as DPs (cf. ( 5)) denote places (thus, a specific type of individual), hence their ability to alternate with adverbs. On the other hand, the constituents spelled out as NPs (cf. ( 6)) denote properties (thus, they are not referential), hence the impossibility to alternate with adverbs. Furthermore, we stress here that certain complex PPs under discussion have the possibility to occur in their simple form (i.e. without P1), as in (7) below. In such cases, the interpretation of the construction is not the same, namely we switch from an origin-type reading (with complex P, (7a-b)) to a destination-type reading (with simple P, (7a'-b'))4 : Taking into account the different properties examined up to this point, we can propose an analysis of complex PPs of type A, represented as in the following structure:

(8) DP 1 D 1 NP 1 N 1 PP 1 P 1 PP 2 P 2 DP 2 / NP 2 a. o sosire de la Paris (an arrival from Paris) b.
un vânt din (de + în) deşert (a wind from the desert)

Starting from this representation, we may -in order to conclude on complex PPs of type Aput forth the following analysis elements:

(i) they are introduced by two ordinary (i.e. lexical) simple prepositions. In other words, lexical P1 takes as a complement a PP introduced by another lexical P2. The latter may take as a complement a DP or an NP;

(ii) the status of the complex adnominal PP is different according to the type of noun to which it attaches: if the latter is a deverbal noun (8a), the PP functions as an argument; on the other hand, if it is not a deverbal noun (8b), the PP functions as an adjunct.

Type B

The following examples illustrate this type of complex PPs:

(9) a. vecinul de la parter neighbour-the of at ground floor (the neighbour from the ground floor) b. fotografia de pe raft photo-the of on shelf (the photo on the shelf) c. More particular, these are adnominal prepositional adjuncts whose salient property -that makes them different from the previous type (cf. A above) -is to be introduced by de, as the only element permitted as P1:

(10) a. *vecinul pe la parter neighbour-the on at ground floor b. *evenimentele pe la 1848 events-the on at 1848

We note that de has to be present even if its complement is not a PP, but an adverb:

(11) a. casa de la mare => casa de acolo (the house from the seaside) (that house over there) b. cartea de pe raft => cartea de aici (the book on the shelf) (this book over here) c. evenimentele din 1989 => evenimentele de atunci (the events of 1989) (the events back then) d. progresele din prezent => progresele de acum (the progress from the present) (the nowadays progress)

Another difference from the previous type, coming more precisely from their status as adjuncts, is that these complex PPs are not sensitive to the nature of the noun to which they attach. They may, as a consequence, occur with different types of nouns: relational (9a), iconic (9b), of event (9c), deverbal (9d), of objects (9e-f), etc. Semantically (lexically) speaking, such adnominal PPs express the placement in time or in space:

(12) a. cărţile de (the city which lies between the lakes) Furthermore, we notice the existence of certain de constructions that are ambiguous, allowing for an analysis of either type A or type B. Thus, the examples as those provided under (13) below receive a priori the ablative reading (i.e. expressing the source) or the adnominal reading (i.e. expressing location):

(13) a. vinul din Italia (the wine from Italy) b. vinul din pivniţă (the wine from/in the basement) c. florile de la munte (the mountain flowers / the flowers from the mountains)

In such situations, the context helps remove the possible ambiguity. More precisely, the interpretation varies depending on whether we suppose the presence of a motion verb in the structure. As such, in (14) -where there is no motion verb -the complex PP receives a type A interpretation, while in (15) it will get a type B interpretation due to the presence of the motion verb a aduce 'to bring':

(14) Vinul din Italia se vinde bine peste tot (The wine (produced in/coming) from Italy sells well everywhere) (15) A adus multe vinuri din călătorie. Vinul din Italia a fost mai apreciat decât cele din alte părţi (He brought many wines from his trip. The wine (that was brought) from Italy was more appreciated that the ones (brought) from other places)

It is also important to emphasise that all adverbials of type B cannot be introduced by de. In fact, the latter cannot occur when they attach to uncountable deverbal nouns, more precisely in the following situations:

(i) when the noun to which they attach denotes a complex event (i.e. having overtly realised arguments):

( Finally, if the adnominal PP allows for a generic reading describing a certain type of event (i.e. without describing a relation between the event and a specific placement), de is ruled out:

(18) a. studiul la bibliotecă /*de la bibliotecă (the study at the library) b. dansul pe masă / *de pe masă (the dance on the table )   (ii) when the noun to which they attach is a non-specific indefinite in intensional (19a) or generic (20) contexts. In fact, the presence of de in these constructions triggers a specific reading (19b), while its absence is correlated to a non-specific reading (19c):

(19) a. Vrea o casă la mare (reading: a house being (built) at the seaside) (S/he wants a house at he seaside) b. Vrea o casă de la mare (reading: a certain house that is at the seaside) (S/he wants a (certain) house at the seaside) c. Vrea o casă anume la mare (S/he wants a certain house at the seaside) (20) O casă la mare e mai scumpă decât una la munte (A house at the seaside costs more that a house in the mountains) (iii) when the noun to which they attach is a definite generic in the plural:

(21) a. Casele la mare sunt în general locuinţe de vacanţă (The houses at the seaside are generally holiday residences) b. Casele de la mare sunt în general locuinţe de vacanţă (the same translation as ( 21a))

In such a context, de may in fact occur (as in (21b)), while keeping the generic (thus, nonspecific) reading. The absence of de is moreover correlated to an additional constraint: the meaning expressed by the adnominal PP has to be an essential condition for the expressed generalisation. This explains why de cannot be absent in the following example ((22a) vs. ( (iv) de cannot occur either in case the noun to which it attaches is in a predicative position (i.e. after the copula), more precisely in a definition or characteristic structure ((23) vs. ( 24)) :

(23) a. Burdwan este o localitate în India (Burdwan is a town in India) b. Burdwan este o localitate din (de + în) India (the same translation as in (23a))

But:

(24) a. *Indranil est un prieten în India (Indranil is a friend in India) b. Indranil este un prieten din (de + în) India (Indranil is a friend from India)

To sum up the conditions of the occurrence of de with type B, the generalisation that we can put forth is that it has in fact to occur in structures having a specific interpretation.

Another noteworthy aspect while examining this kind of structure is the (categorial) status and the function of de. In fact, we have seen that de occurring with type B is different from the previously examined one with type A, in the sense that it does not have the properties of an ordinary preposition (cf. also Mardale 2009, Tănase-Dogaru 2011 for Romanian, Fábregas 2012 for similar constructions in Spanish). Furthermore, we keep in mind that de from type B is traditionally interpreted as a sort of nominal copula (GALR 2005[START_REF] Fábregas | Multiple Genitives and the Distinctness Condition: the Case of the Spanish DP, ms, accessed on 21 st of july 2012[END_REF], GBLR 2010[START_REF] Nedelcu | Prepoziţii complexe[END_REF][START_REF] Ţenchea | Relations spatiales et temporelles exprimées par les compléments du nom et de l'adjectif en roumain et en français. L'expression de l'espace et du temps en français : quelles formes pour quels sens ?[END_REF], whose role is similar to that of the relative care 'that/which/who', which can be used to paraphrase the structure (cf. also (12) above):

(25) un vecin de la etajul 1 = un vecin care locuieşte la etajul 1 (a neighbour from the first floor) (a neighbour that lives at the first floor)

We also note that this de cannot occur in the corresponding verbal structures, but -as we have seen until now -only to introduce adjuncts in a nominal context:

(26) a. Am pus cartea pe raft / *de pe raft. Cartea de pe raft / *pe raft a fost interesantă (I put the book on the shelf. The book from the shelf was interesting.) b. A construit o casă la mare / *de la mare. Casa de la mare / *la mare a costat foarte mult (He has built a house at the seaside. The house at the seaside cost a lot.)

In a Romance contrastive perspective, we emphasise that this kind of construction is not to be found in languages of the same family. To answer the question of the categorial status and the function of de occurring with type B, we can rely on the analysis of Rubin op. cit. The special properties of de in Romanian (and, by extension, of de in Chinese and na in Tagalog) have made this author consider them not as lexical categories (i.e. as genuine prepositions), rather as a sort of functional elements (cf. also [START_REF] Van Riemsdjik | Functional Preposition[END_REF] whose occurrence is strictly constrained.

In fact, as we have seen until now, de is the only one that can occur in such position (i.e. that cannot alternate with other prepositions (cf. (10) above)), it has no lexical meaning and its occurrence is restricted to the adnominal context, its role being to introduce locative or temporal modifiers (PPs); moreover, its presence leads to a specific reading of the structure as a whole.

Based on such specific properties of de, we can consequently consider them as the realisation (i.e. lexicalisation) of a functional category, namely Mod (as Modifier). Considering this analysis, the structures in which it occurs shall be represented as follows:

Concluding remarks

In this contribution, we have examined the constructions involving two types of adnominal complex PPs in Romanian. Despite their partial formal homonymy, we have shown, on the one hand, that they are PPs formed with two lexical prepositions expressing the separation from a point or place of origin (type A -plecarea de la mare 'the leaving from the seaside').

They generally attach to deverbal nouns and may function as arguments or as adjuncts of the latter. On the other hand, we have seen that they are PPs formed with a functional element (namely de) and a lexical preposition expressing time or location (type Bşedinţa de la prânz 'the midday reunion', ziarul de pe birou 'the newspaper on the desk'). They occur with different types of nouns and function exclusively as adjuncts of the latter. We have also noticed that they are subject to numerous constraints of occurrence (notably the specific interpretation) and, furthermore, that this is a peculiar construction among Romance.

Works cited

  for a general discussion on deverbal nouns and Stan 2003 on Romanian): (2) a. venirea (Mariei) de la Paris <= Maria vine de la Paris (Mary's arrival from Paris) (Mary arrives from Paris) b. plimbatul (Mariei) pe sub poduri <= Maria se plimbă pe sub poduri (Mary's walk under the bridges) (Mary walks under the bridges)

  (7) a. venirea de la Paris ≠ a'. venirea la Paris (the arrival from Paris) (the arrival to Paris) b. mersul pe sub poduri ≠ b'. mersul sub poduri (the walk under the bridges) (the walk underneath the bridges)

  )): (22) a. *Clădirile în New York au multe etaje (The buildings in New York have a lot of storeys) b. Clădirile din (de + în) New York au multe etaje (The buildings from New York have a lot of storeys)

  (the organisation of the OG in London in 2012) b. semnarea acestei convenţii la Paris / *de la Paris (the signing of this convention in Paris)On the other hand, if the head noun denotes a simple event (i.e. without an overtly realised argument), we have to use de if the adnominal PP has a specific interpretation:

	(17) a. atentatele de la 11 septembrie / *la 11 septembrie
	(the September 11 attacks)
	b. revoluţia din (de + în) 1907 din România / *în 1907 în România
	(the revolution of 1907 in Romania)

16) a. organizarea Jocurilor Olimpice la Londra în 2012 / *de la Londra din 2012

  Here are some Romance examples showing this impossibility: (27) a. J'ai mis le bol sur la table. Le bol sur la table / *de sur la table est encore chaud (French) b. J'ai une réunion dans une semaine. La réunion dans une semaine / *de dans une semaine me stresse (28) a. Ho messo la tazza sul tavolo. La tazza sul tavolo / *di sul tavolo è ancora calda (Italian) b. Ho una riunione fra una settimana. La rinunione della settimana / *di della settimana mi fanno innervosire (29) Puse el bol sobre la mesa. El bol sobre la mesa / *de sobre la mesa todavía está caliente (Spanish)It is however attested for languages such as Chinese and Tagalog (cf.[START_REF] Rubin | The Structure of Modifiers[END_REF], and these under similar conditions as in Romanian. To be more precise, the de elements in Chinese and na in Tagalog occur exclusively in an adnominal adjunct context, to introduce PPs in constructions with specific interpretation.The examples (30a) and (31a) show in fact that de and na -the same with de in Romanian (see (26) above) -are absent in locative verbal structures, while (30b) and (31b) show that they are obligatory in the corresponding nominal structures:

	(30) a. na yiben shu zai zhuozi-shang (Chinese)
	that-CL book on table-high
	(the book is on the table)
	b. na yiben zai zhuozi-shang de shu
	that-CL on table-high DE book
	(the book on the table)
	(31) a. Nasa probinsya ang bahay (Tagalog)
	in-the province TOP house
	(the house is in the country)
	b. (Binili niya) ang bahay na nasa probinsya
	bouthgt he TOP house NA in-the countries
	((he has bought) the house in the country)

* This paper is part of a larger project, A Reference Grammar of Romanian, conducted and edited by Carmen Dobrovie-Sorin and Ion Giurgea. During its elaboration, it has benefitted of their precious comments and observations. I am grateful to them. 1 Romanian displays a particular phenomenon regarding the use of prepositions and the definite article (which is enclitic in Romanian), namely the former is ruled out. This phenomenon takes place when the noun phrase preceded by a preposition does not have any other constituents, and this occurs for the definite reading (examples (i) and (ii) above). On the other hand, if a noun preceded by a preposition has another constituent (regardless of its nature), the definite article is obligatory: (i') coborâtul de pe dealul mare descent-the of on hillthe big (the descent of the big hill), (ii') copilul de pe strada mea child-the of on street-the my (the child from my street).

We adopt here the terminology and distinctions made within the framework of the so-called X-bar theory (N.[START_REF] Chomsky | Remarks on nominalization[END_REF], R. Jackendoff 1977), which considers that any (syntactic) category -regardless of its nature (lexical or functional) -may allow for a three-level projection (X°, the basic level, which corresponds to the head of the projection; X', the intermediary level, which bears the head and its possible arguments; X'' (or XP), the maximal level, which is the head of the construction, its arguments, specifiers and possible adjuncts). In this view, the DP label shall be understood as the maximal projection of the D category (as Determiner), which obligatorily entails an NP argument (which is the maximal projection of the N category (as Noun)). Moreover, the NP projection is rather different from the DP, since the former -unlike the latter -does not entail a D.

Note that this is a possible example, but with a different meaning.

This fact is also due to the syntactic and semantic properties of motion verbs from which the deverbal nouns are derived, and it is not and exclusive property of the prepositions.