Michel Busquet
email: michel.busquet@obspm.fr

Patrice Barroso

Thierry Melse

Daniel Bauduin

Miniature shock tube for laser driven shocks

Keywords: shock tube, laboratory astrophysics, radiative precursor

We describe in this paper the design of a miniature shock tube (smaller than 1 cm3) that can be placed in a vacuum vessel and allows transverse optical probing and longitudinal backside XUV emission spectroscopy. Typical application is the study of laser launched radiative shocks, in the framework of what is called "laboratory Astrophysics".

planet core Equation of State, and Radiative Shocks which is the matter of the present paper. Radiative Shocks [START_REF] Zeldovich | Physics of Shock Waves and High Temperature Hydrodynamic Phenomena[END_REF] are super-critical shocks, where the emitted radiation is large enough to launch a radiation driven ionization wave (called radiative precursor) in front of the density jump and to cool down the post-shock region. In the Universe, such radiative shocks are found in accretion shock of matter falling on Young Stellar Objects, in proto-stellar disks, T Tauri stars or binary systems, as well as in outflows into circumstellar medium from Supernovae and Supernovae Remnants.

Scaling laws have been derived by different authors [START_REF] Zeldovich | Physics of Shock Waves and High Temperature Hydrodynamic Phenomena[END_REF], [START_REF] Mihalas | Foundations of radiation Hydrodynamics[END_REF]. They end in experiments on laser facilities where a miniature shock tube, filled with a low pressure high atomic number gas, is placed inside an enclosure pumped down to a low pressure (P<10 -4 torr) so the laser at high irradiance can propagate until the "piston" of the miniature shock tube. The challenge is to control dimensions to the required values but to allow probing of the shock in these tiny assay. Various means to diagnose the created shock and its radiative precursor have been used by different teams: image of the self-emission of a solid foil impacted by the precursor, [3] interferometric measurement, [4], [5] x-ray radiography [6] and Thomson scattering. [7] During the experiment we performed on the PALS facility [8], we used interferometric measurement of the electronic density in the precursor for a large traveling distance (up to 6 mm), thanks to the targets we specially designed for this experiment and that we describe in this paper. We first present in the next section the general setup of the experiment and the requirements for the target. In Sec.3 we describe in detail these targets, a sub-centimetric assembly of a miniature shock tube, the flyer (or "piston"), the holding base and the pump and fill circuit. The piston is to be launched by irradiation from a high-power laser (0.5 TW). We also describe a variant of these targets designed for backside spectroscopy that minimizes the perturbations brought by the joint of the shock tube to the base. This variant has allowed us to demonstrate the feasibility of XUV spectroscopy of radiative shocks.

2-Experimental setup and shock tube requirements

Target requirements are given by the experimental setup (Fig. 1). The radiative shock is created in a miniature shock tube, closed at one end by a thin 10 microns plastic foil, to be acting as the piston. A powerful laser is focused on the plastic foil, smoothed with a Phase Zone Plate [9] that gives a focal spot of approximatively 550 microns FWHM, with 90% of energy in a diameter of 700 microns. Thus a 0.7 mm width of the canal is well suited. Larger focal spot would mean lower intensity, and lower shock velocity. We used the frequency tripled laser of the PALS facility [10] delivering ~ 150 J in 0.35 ns at !=438 nm. The power of the laser pulse is half a TW and the laser intensity impacting the foil is around or above 10 14 W/cm 2 , which means that the target has to be placed in a vacuum chamber (below P=10 -4 torr) for the laser to reach the plastic foil. Laser ablation of the plastic accelerates the foil by rocket effect, ending in a piston velocity of a few 10 km/s when using a pressure of ~ 0.2 bar in the tube. The shock tube has to be filled with Xenon (or a mixture of air and Xenon) at a fraction of atmosphere (typically P=0.2 bar), as high Z material is required for a radiative shock to develop in front of the hydrodynamic shock created by a solid piston at such a velocity. The radiative precursor is probed by visible light interferometry, which allows measuring electron density [4], [8]. Also, we would like to measure the XUV emission in front of the shock, leading to close the back end of the tube with an XUV window. This backside XUV window, as well as the front side plastic foil has to withstand a differential pressure of up to 1 atm, in both directions :

-vacuum outside the tube (placed in the vacuum chamber) and up to 0.5 bar in the tube, when firing the laser -or ambient pressure outside the tube, and zero pressure inside the tube when purging it from air before filling it with Xenon.

To observe long time behavior (up to 100 ns) a length of 6 mm is required as the average velocity of the precursor might reach 60 km/s. The target will be placed in one arm of a Mach-Zehnder interferometer, thus optical quality side windows covering the whole length of the tube is obviously required.

Figure 1 schematic experimental layout

In summary we want to make an assembly (or "target") with -a tube of down to 0.7 mm transverse size -and 6 mm length,

-with optical quality windows on 2 sides of the canal, which shall then be of square section, -a 10 microns thin plastic foil at one end -and an XUV window at the other, -an appropriate pump and fill circuit with low perturbation on the shock propagation inside the tube -and a convenient base plate for holding the target.

-material of the side walls should be controlled, as precursor propagation is sensitive to their albedo [11] The experiment needs also that a 0.5 microns gold layer is deposited on the backside of the plastic foil to block x-rays coming from the laser interaction region.

A functional layout of the target is presented in Fig. 2 upper and lower sides: bulk aluminum lateral sides: glass windows transparent to the interferometric probe beam In targets made for the only purpose of backside XUV spectroscopy, we want to further minimize perturbation to the shock propagation until the very end of the tube. An internal circular section can be used and will ease numerical simulation of the radiative precursor.

We successfully build such targets, packing in less than 1 cm3 assembly all of the required features. We now give details on the target design in the following sections.

3-description of the targets for interferometric probing.

The targets are made from micro-machined aluminium. They consist in a 2 mm thick, 12x12mm square base plate holding a two-prong "fork" (Fig. 3). Optical quality BK7 glass windows are glued on the side of the prongs. The two prongs, that have a thickness of 0.7 mm, and the two glass windows glued on them are delimiting a 6 mm long 0.7 x 0.7 mm square channel, that will be our shock tube.

Wall material can be controlled by sputtering various components on both the aluminium piece and the inner side of the windows before assembling. For the experiment performed on the PALS facility in 2007 [8], we used either aluminium (10 nm coating) or gold (15 nm coating over a 2 nm coating of Chromium). Regarding the propagation of the low temperature (< 30 eV) radiative shock, these coating are thick enough to play the role of bulk material. For the interferometric probing, the transmission of the coated pair of windows is above 10%, and can be managed by unbalancing the intensity in the two arms of the Mach-Zehnder interferometer. The high-speed high-precision saw used to cut the glass windows produces square ends suitable to glue the plastic foil on top of them, with a vacuum tight connection.

A 0.6 mm hole is drilled in the center of the base plate for the two purpose of -allowing back side XUV observation (it will be covered by a vacuum tight XUV transmitting window glued on the back of the base plate)

-creating a pump and fill circuit of the shock tube in connection with two holes drilled in the 2 mm thickness (Fig. 4) of the base plate, like secret passages in the walls of a medieval castle.

Two fill tubes (1mm in diameter) will be glued inside these holes, connected to a small 1cm 3 pressure ballast tank (Fig. 5) that will allow the target to hold the required pressure for a couple of hours even if there were small leaks, or if there is gas releasing materials.

On one of the two prongs, we machine a set of grooves for magnification reference. For a good XUV transmission, the backside XUV window can be made of a thin foil (0.1 to 0.2 !m) of either Aluminium, SiC or Si 3 N 4 . In the last 2 cases, a 0.5 mm thick wafer is used, chemically etched on a surface of 1 x 1 mm to the required thickness. For Al, we use grid-supported foils of the type classically used for scanning electron microscopy (SEM). As said previously, the channel is to be filled with 0.05-0.5 bar gas and has to withstand a 1 bar differential pressure in both directions. This directs the choice to polystyrene for making the front-end thin foil, not too much stiff (as it would break under pressure) and not too much elastic (as it would deflate when pumping the tube and inflate once placed inside the vacuum chamber). Polystyrene can also bear the high temperature encountered in the process of sputtering the thin gold overcoat required to stop x-rays originated from the laser interaction.

The target is to be placed in one arm of a Mach-Zehnder interferometer. The output of the interferometer is imaged on the entrance slit of a visible light streak camera. An example of time resolved interferogram of shock propagation in the tube is shown in Fig. 6, demonstrating the possibilities of the target described in this paper.

The shock tube can even be made with a smaller section, down to 0.3 x 0.3 mm 2 with stepped prongs and staggered windows on the sides.

Figure 6. Time resolved interferogram of the shock tube in action. The whole 6 mm length is imaged (horizontal axis) and the time observation last 100ns (vertical axis).

4-description of "blind" targets for backside spectroscopy.

At the back end of the tube, pumping hole and base plate steps result in a complex geometry and will produce perturbation to the propagation of the radiative shock after the first 6 mm. Though it is not too big a problem for time resolved observation, the XUV spectrographic recordings we performed (through the central hole in the base plate) are time integrated, and would suffer from this perturbation.

We thus design a different target with no lateral windows, In these "blind" targets, the channel is made from a cylindrical tube, encased in the base plate with flattened end to create the pump and fill circuit (Fig. 7). A 0.1-0.2 mm spacing, between the end of the tube and the backside window, creates a damping expansion chamber, where the shock weakens before reaching the XUV window.

Preliminary results obtained with a low sensitivity, slitless, XUV spectrograph are shown in Fig. 8 with different conditions (wall material of Alumina or gold, gas fill of pure Xe or 50-50 mixture of Air and Xenon). The sharp cut-off observed at 17.5nm results from the L-edge of the Aluminum filter.

Without slit in front of the spectrograph, the spectral resolution is given by the source size. A resolution of !/"! around 100 is obtained. Also, as we were working on the filter pack and the settings of the spectrograph, sensitivity varies from shots to shots. In the future, we shall use our high sensitivity, high spectral resolution and medium spatial resolution XUV spectrograph recently made for the purpose of direct measuring of wall albedo. [12] Figure 7 -3D cutout and photograph (courtesy of F.Thais) of a "blind target". The flattened end of the encased cylinder and the spacing between the window and the cylinder allows a pumping circuit (black thick line).

Figure 8 -2 recorded spectra with different conditions (wall material and gas fill). The spectral resolution !/"! around 100 is given by the source size as we used a slitless spectrograph. Note that the spectrograph filtering and setting was changed between shots, thus sensitivity varies from shots to shots.

-TARGET OPERATION AND CYCLE TIME

The minimum time between two shots on such high energy (>100 J ) laser facility is between 20 and 40 minutes, due to laser operation requirements (cooling down of the laser components, laser chain alignment, capacitor bank charging, …). On the other hand opening and pumping down the vacuum chamber is around 10 minutes, and target alignment may also reach 10 minutes, depending on the viewing system implemented around the vacuum chamber. So the time required (a couple of minutes) to fill the target with the appropriate fill gas (or mixture) at the chosen pressure add no extra delay, as it can be done during the vacuum operating cycle. The piston and the windows are destroyed during the laser pulse or in the first few milliseconds after, but the metal casing can be analyzed "postmortem" to check the quality of the laser pointing. Digital micro-machining of the socket and gluing the windows allow excellent reproducibility of the targets. On the PALS laser facility, aging of the Iodine active medium of the laser is now the main origin of the variability (contamination comes from glass cells, pumps and tubing). However, we have found that shots are fairly reproducible during the same day.

-CONCLUSIONS AND PERSPECTIVES

We successfully build targets, assembling in less than 1 cm3 all of the required features, and achieved miniature shock tubes of 6 mm length and down to 0.7 mm transverse dimension, with a design allowing to go down to 0.3 mm. Two essential features are 1) the possibility to place the miniature shock tube inside a vacuum chamber and 2) the optical quality of the tube allowing transverse probing in visible light. We are currently working on the feasibility of XUV transverse probing of modified targets.

Used on the PALS laser facility, [8] these targets have allowed to probe electron density in the shock tube for time up to 100 ns after the laser pulse. Recording of backside XUV emission with spectral resolution has also been demonstrated. Specific experiment to measure the wall albedo at a temperature of 10-30 eV is described elsewhere. [12] With laser facility of the 1 kJ class, and/or with a different design of the piston, shock velocities above 150 km/s should be within reach with the same target design.

Figure 2

 2 Figure 2 layout of the shock tube shows its required functions

Figure 3 -

 3 Figure 3 -3D drawing of the target, the two glass side windows and the gilded plastic foil. The XUV window is glued on the back of the base plate.

Figure 4 .

 4 Figure 4. Cutout of the target, mounted with the 4 windows (note the thinned wafer glued on the back), shows the pump and fill circuit (black lines).

Figure 5 .

 5 Figure 5. Target is prepared on the pumping system. Note the 1 cm3 ballast tank. Inset shows details of the glued assemblage (courtesy of F.Thais).

-ACKNOWLEDGMENTS

We thank O. Madouri (LPN) and V.Petitbon (IPN) for providing us with the SiC windows and the polystyrene thin films. The interferometer was built in collaboration with Ouali Acef (Observatoire de Paris) and Frédéric Thais (CEA/DSM). We acknowledge the participation of Frédéric Thais, Chantal Stehlé, Guillaume Loisel, Norbert Champion, Michaela Kozlova, Tomas Mocek, Bedrich Rus and his team, during the experiment performed at PALS under the P.I.ship of Chantal Stehlé and Michel Busquet [13]. We obtained financial supports from the Access to Research Infrastructures activity in the Sixth Framework Programme of the EU (contract RII3-CT-2003-506350 LaserLab Europe), from the RTN JETSET (contract MRTN-CT-2004 005592), from CEA/DSM (Saclay), from Observatoire de Paris and from the French National Programme of Stellar Physics (PNPS), CNRS-PICS 4343. This research was partially supported by the Czech Ministry of Education, Youth and Sports (project LC528). We also acknowledge financial support from the OPTON LASER INTERNATIONAL Corporation.


Charles Bouveyron

Gilles Celeux

Stéphane Girard

Intrinsic Dimension Estimation by Maximum Likelihood in Isotropic Probabilistic PCA

Keywords: Probabilistic PCA, isotropic model, dimension reduction, intrinsic dimension, maximum likelihood, asymptotic consistency

A central issue in dimension reduction is choosing a sensible number of dimensions to be retained. This work demonstrates the surprising result of the asymptotic consistency of the maximum likelihood criterion for determining the intrinsic dimension of a dataset in an isotropic version of Probabilistic Principal Component Analysis (PPCA). Numerical experiments on simulated and real datasets show that the maximum likelihood criterion can actually be used in practice and outperforms existing intrinsic dimension selection criteria in various situations. This paper exhibits and outlines the limits of the maximum likelihood criterion. It leads to recommend the use of the AIC criterion in specific situations. A useful application of this work would be the automatic selection of intrinsic dimensions in mixtures of isotropic PPCA for classification.

Introduction

The analysis of high-dimensional data has become an important problem in statistical learning and dimension reduction has a central place in such settings. Among all existing methods, Principal Component Analysis (PCA) [START_REF] Jolliffe | Principal Component Analysis[END_REF] and its probabilistic version (PPCA) [START_REF] Tipping | Mixtures of probabilistic principal component analyzers[END_REF][START_REF] Tipping | Probabilistic Principal Component Analysis[END_REF] are two popular techniques. A central issue in dimension reduction is choosing a sensible number of dimensions to be retained. We refer to [START_REF] Camastra | Data dimensionality estimation methods: a survey[END_REF] for a review on this topic. Two kind of approaches have been proposed in the last decades for intrinsic dimension estimation.

Local methods. The local approach estimates the topological dimension (defined as the basis dimension of the tangent space of the data manifold) from the information contained in sample neighborhoods. Fukanaga-Olsen's algorithm [START_REF] Fukunaga | An algorithm for finding intrinsic dimensionality of data[END_REF] consists of estimating the rank of the variance matrix computed locally on a Voronoi tessellation. In [START_REF] Bruske | Intrinsic dimensionality estimation with optimally topology preserving maps[END_REF], the Voronoi tessellation is computed thanks to a topology representing network. The algorithms proposed by Pettis et al. [START_REF] Pettis | An intrinsic dimensionality estimator from nearest-neighbor information[END_REF] and Verver-Duin [START_REF] Verveer | An evaluation of intrinsic dimensionality estimators[END_REF] are based on the analysis of the distances from one point to its nearest neighbors. The main limitation of local approaches is their sensitivity to outliers.

Global methods. The global approach consists of unfolding the whole dataset into a linear subspace. The estimated intrinsic dimension is then the dimension of the resulting subspace. Such methods can be divided into three subfamilies.

• Projection methods: The lower dimensional subspace can be estimated by minimizing some projection errors. Examples of such approaches include PCA [START_REF] Jolliffe | Principal Component Analysis[END_REF] sometimes associated with Cattell's scree test [START_REF] Cattell | The scree test for the number of factors[END_REF] and its non linear extensions based either on auto-associative models [START_REF] Karhunen | Representation and separation of signals using nonlinear PCA type learning[END_REF][START_REF] Chalmond | Nonlinear modeling of scattered multivariate data and its application to shape change[END_REF] or Mercer kernels [START_REF] Schölkopf | Nonlinear component analysis as a kernel eigenvalue problem[END_REF]. Multidimensional scaling type algorithms aim at finding the projection which (locally) preserve the distances among data. Recent methods include LLE [START_REF] Roweis | Nonlinear dimensionality reduction by locally linear embedding[END_REF] and ISOMAP [START_REF] Tenenbaum | A global geometric framework for non linear dimensionality reduction[END_REF].

• Fractal-based methods: These techniques rely on the assumption that the dataset is generated by a dynamic system. Their goal is to estimate the dimension of the attractor associated to this dynamic system. For instance, [START_REF] Kegl | Intrinsic Dimension Estimation Using Packing Numbers[END_REF] addresses this problem through the estimation of the boxcounting dimension and some heuristic methods are introduced in [START_REF] Camastra | Estimating the intrinsic dimension of data with a fractal-based method[END_REF].

Most of these methods are designed for low-dimensional datasets since their complexity grows exponentially with the dimension.

• Model-based methods: The use of a parametric model permits to derive a maximum likelihood (ML) estimator of the intrinsic dimension. For instance, in [START_REF] Levina | Maximum Likelihood Estimation of Intrinsic Dimension[END_REF], the number of points in a small sphere is modeled by a Poisson process. We also refer to [START_REF] Mackay | Comments on 'maximum likelihood estimation of intrinsic dimension[END_REF] for a bias correction of the previous ML estimator. In a similar spirit, [START_REF] Fan | Intrinsic dimension estimation of manifolds by incising balls[END_REF] uses a polynomial regression based on a uniformity assumption. Several methods are based on a Bayesian approach: Minka [START_REF] Minka | Automatic choice of dimensionality for PCA[END_REF] proposes a direct calculation of the Laplace approximation of the marginal likelihood while the Bayesian Information Criterion (BIC) [START_REF] Schwarz | Estimating the dimension of a model[END_REF] is an asymptotic approximation of it. In [START_REF] Fraley | Bayesian regularization for normal mixture estimation and model-based clustering[END_REF], a regularized BIC is introduced where the likelihood is evaluated at the maximum a posteriori estimator instead of the maximum likelihood estimator. This criterion is used in [START_REF] Nyamundanda | Probabilistic principal component analysis for metabolomic data[END_REF] to select the dimensionality in PPCA with covariates. We also refer to [START_REF] Bishop | Bayesian PCA[END_REF][START_REF] Everson | Inferring the eigenvalues of covariance matrices from limited, noisy data[END_REF][START_REF] Rajan | Model order selection for the singular value decomposition and the discrete Karhunen-Loève transform using a Bayesian approach[END_REF] for alternative approximations of the evidence. The underlying idea is that the likelihood is an increasing function of the complexity and thus of the dimensionality as well. This remark motivated the authors to use penalized likelihood criteria.

In this paper, a constrained version of PPCA, called isotropic PPCA, is considered. This model could appear as a restrictive model but it can be useful in specific situations. In particular, it has been proved to be efficient for classification problems in high dimension [START_REF] Bouveyron | Robust supervised classification with mixture models: Learning from data with uncertain labels[END_REF] where parsimonious models are desirable. This paper demonstrates the surprising result that the maximum likelihood criterion is asymptotically optimal in the case of the isotropic PPCA model, the complexity of the model being not an increasing function of the dimensionality. The ML criterion is compared in different situations on simulated and real data to two classical model selection criteria, AIC [START_REF] Akaike | A new look at the statistical model identification[END_REF] and BIC [START_REF] Schwarz | Estimating the dimension of a model[END_REF], to the empirical scree-test of Cattell [START_REF] Cattell | The scree test for the number of factors[END_REF], and to the model-based methods [START_REF] Fan | Intrinsic dimension estimation of manifolds by incising balls[END_REF], [START_REF] Levina | Maximum Likelihood Estimation of Intrinsic Dimension[END_REF], and [START_REF] Minka | Automatic choice of dimensionality for PCA[END_REF]. This paper is organized as follows. Section 2 introduces an isotropic version of probabilistic PCA and considers the estimation of its parameters. Section 3 focuses on the intrinsic dimension estimation and demonstrates that the maximum likelihood method can be used for this task in the context of the isotropic PPCA model. Section 4 illustrates on simulations and real datasets the behavior of the proposed approach in different situations and Section 5 gives some concluding remarks.

Isotropic Probabilistic PCA

In this section, after having recalled the Probabilistic PCA (PPCA) model, it is reformulated using an eigenvalue decomposition. An isotropic version of PPCA is then introduced and inference aspects are addressed.

Factor Analysis, Probabilistic PCA and Extreme Component Analysis

The Factor Analysis model [START_REF] Bartholomew | Latent Variable Models and Factor Analysis[END_REF][START_REF] Basilevsky | Statistical Factor Analysis and Related Methods[END_REF] links linearly a p-dimensional random vector y to a d-dimensional Gaussian vector x of latent variables:

y = Hx + µ + ε.
The p × d factor matrix H relates the two random vectors and µ ∈ R p is a fixed location parameter. When d < p, the latent vector x provides a parsimonious representation of y. In this context, d is interpreted as the intrinsic dimension of y and is thus the parameter of interest in this study. Without loss of generality, it can be assumed that x ∼ N (0, I d ). If, moreover, the noise ε is supposed to be Gaussian ε ∼ N (0, Ψ), where Ψ is a p × p variance matrix, and independent from x, then we end up with a Gaussian distribution for the observations y, i.e. y ∼ N (µ, Σ) where:

Σ = HH t + Ψ. (1)
In such a case, the model parameters can be estimated by maximum likelihood even though an iterative procedure is involved. To overcome this practical difficulty, one can assume an isotropic noise Ψ = bI p with b > 0. This model is referred to as the Probabilistic PCA model [START_REF] Tipping | Probabilistic Principal Component Analysis[END_REF] or to as the Sensible PCA model [START_REF] Roweis | EM algorithms for PCA and SPCA[END_REF]. The variance matrix of y can be also simplified as:

Σ = HH t + bI p .
In [START_REF] Williams | Products of Gaussians and Probabilistic Minor Component Analysis[END_REF] method, the converse assumption is made. Finally, the two approaches are unified in the Extreme Component Analysis (XCA) method [START_REF] Welling | Extreme Components Analysis[END_REF] where the noise ε is supposed to be orthogonal to the columns of H. This assumption yields Ψ = b(I -H(H t H) -1 H t ) in (1) and thus the eigenvalues of Σ are h 1 2 , . . . , h d 2 and b. Since no assumption is made on their relative magnitudes, PPCA and PMCA may be interpreted as particular cases of XCA.

Isotropic Probabilistic PCA

Similarly, it may be of interest in specific contexts, such as high-dimensional classification, to consider an isotropic factor matrix. In this case, the matrix H can be rewritten as H = √ a -bV with a > b and where V is a p × d matrix such that V t V = I d . Thus, the variance matrix of the observation y is given by: Σ = (ab)V V t + bI p .

Let U be a p × (pd) matrix such that Q := [V, U ] is an orthogonal p × p matrix containing p eigenvectors of Σ. Introducing ∆ = Q t ∆Q the diagonal matrix of eigenvalues, an alternative, and more intuitive, parametrization of

Σ is Σ = Q∆Q t .
Moreover, the matrix ∆ associated with the isotropic PPCA model has the following form:

∆ =             a 0 . . . 0 a 0 0 b 0 . . . . . . 0 b                d        (p -d)
with a > b. Let us emphasize that, since H is supposed to have only two different eigenvalues, the assumption a > b is made without loss of generality and thus this model can also be interpreted as an isotropic XCA model. The isotropic PPCA model is parametrized by µ, Q, a, b and d. A graphical representation of the isotropic PPCA model is given by Figure 1. As it can be observed on Figure 2 which illustrates the model in a 3-dimensional space, such a model assumes that the distribution is spherical and modelled by a within the d-dimensional latent subspace where the data actually live. The d-dimensional latent subspace is spanned by the d first columns of Q which control the orientation of the subspace whereas µ locates the subspace in the original space. The isotropic PPCA model supposes as well that the variance of noise can be modelled outside the latent subspace with a unique parameter b. Finally, it should also be noted that the mixture model introduced in [START_REF] Bouveyron | High Dimensional Discriminant Analysis[END_REF] is a mixture of isotropic PPCA applied to discriminant analysis, i.e. each class is modelled by a specific isotropic PPCA model.

Inference for isotropic PPCA

Before focusing on the estimation of the intrinsic dimension d, the inference on model parameters for the isotropic PPCA model is considered. In the case of the isotropic PPCA model, the parameters to be estimated are µ, a, b, U and V . As in the classical Gaussian framework, the maximum likelihood strategy is retained for parameter estimation. Denoting by n the number of observations, the log-likelihood associated with the isotropic PPCA model is: where W is the empirical variance matrix:

- 2 n log(L) = d log(a) + (p -d) log(b) + 1 a d j=1 v t j W v j + 1 b p-d j=1 u t j W u j , (2)
W = 1 n n ℓ=1 (x ℓ -μ)(x ℓ -μ) t , μ = 1 n n ℓ=1 x ℓ .
The estimation of the matrices U and V is similar to the estimation of H in the context of the actual PPCA model (see [START_REF] Tipping | Mixtures of probabilistic principal component analyzers[END_REF] for further details). For a given value of d, the ML estimator of the transformation matrix V is the matrix containing the eigenvectors associated with the d largest eigenvalues of the empirical variance matrix W . Similarly, the ML estimator of U is the matrix containing the eigenvectors associated with the pd smallest eigenvalues of W . Using this eigenvalue decomposition of W in (2), we obtain

- 2 n log(L) = d log(a) + (p -d) log(b) + 1 a d j=1 λ j + 1 b p j=d+1 λ j , (3)
where λ j is the jth eigenvalue of W . It follows that

â = 1 d d j=1 λ j and b = 1 (p -d) p j=d+1 λ j .
As one can observe, ML estimates of parameters a and b are respectively the means of the largest and smallest eigenvalues of the empirical variance matrix. From a numerical point of view, such estimates should be more stable than eigenvalue estimates when the number of observations is small compared to the data dimension p. Furthermore, it is not necessary in practice to compute the (pd) smallest eigenvalues of W since b can be computed as b = (tr(W )da) /(pd).

Estimation of the intrinsic dimension by maximum likelihood

In this section, we focus on the estimation of the intrinsic dimension d * . First the following proposition is proved.

Proposition: The maximum likelihood of the actual intrinsic dimension d * is asymptotically unique and consistent in the case of the isotropic PPCA model.

Proof: Since d is an integer parameter, it is possible to compute the likelihood for each value of d = 1, ..., p -1 and to select the value associated to the largest likelihood. From Equation (3), the maximized log-likelihood of the isotropic PPCA model can be written at the optimum θ = (μ, â, b, Û , V ) as

- 2 n log(L( θ, d)) = d log(â) + (p -d) log( b) + tr(W ) b + 1 â - 1 b d j=1 λ j . Since â = 1 d d j=1 λ j , b = 1 (p-d)
p j=d+1 λ j and tr(W ) = p j=1 λ j , the loglikelihood reduces to:

- 2 n log(L( θ, d)) = d log(â) + (p -d) log( b) + p.
Consequently, the maximization of the likelihood is equivalent to the minimization of φ n (d) = d log(â) + (pd) log( b). Asymptotically, as the number of (independent) observations n tends to infinity, W converges almost surely (a.s.) to Σ. As a consequence of Lemma 2.1 [START_REF] Tyler | Asymptotic Inference for Eigenvectors[END_REF], which holds for all symmetric matrix, λj converges a.s. to a if j ≤ d * and λj converges a.s. to b if j > d * . Two cases can arise.

Situation d ≤ d * . In this case, â → a and b → 1 p-d [(d * -d)a + (p -d * )b] a.s. when n → ∞. Consequently, φ n (d) → φ(d) a.s. where φ(d) = d log(a) + (p -d) log (d * -d) (p -d) a + (p -d * ) (p -d) b ,
or equivalently Similarly to the first situation, we can write

φ(d) -p log(a) p -d * = (p -d) (p -d * ) log 1 + (p -d * ) (p -d) b a -1 = δ log 1 + γ δ , with δ = (p-d) (p-
φ(d) -p log(b) d * = d d * log 1 + d * d a b -1 = δ log 1 + γ δ , with δ = d d * and γ = a b -1.
Again, the study of φ(d) reduces to the study of ψ(δ) = δ log 1 + γ δ where δ ≥ 1 and γ ≥ 0. Remarking that ψ is a strictly increasing function on [1, +∞) for all γ ≥ 0, its minimum is reached for δ = 1 and therefore the minimum of φ on [d * , p] is reached for d = d * . As a conclusion, we have proved that the likelihood associated with the model has asymptotically a unique maximum for the actual intrinsic dimension d * of the data.

From this proposition, it is deduced that the maximum likelihood criterion can be used to estimate d * in the context of the isotropic PPCA model. Usually, as for instance for the general PPCA model, model selection criteria using the maximum likelihood need an additional penalty term because the maximum likelihood of a model is asymptotically a non decreasing function of the number of model parameters. However, for isotropic PPCA, the proposition states that the likelihood is asymptotically maximum for the intrinsic dimension d * of the data. Therefore ML criterion is a good candidate to estimate the intrinsic dimension of a dataset in the isotropic PPCA framework. The reason why ML can be used to estimate d * for the isotropic PPCA model is the perfect duality between the subspace spanned by the eigenvectors associated with the d largest eigenvalues of W and the supplementary noise subspace with dimension (pd). Because of the symmetry between a and b occurring in the isotropic PPCA model, the number of parameters to be estimated is the number of model parameters, for fixed d, is

ν(d) = p + 2 + min{d(p -(d + 1)/2), (p -d)(p -(p -d + 1)/2)}.
Consequently, the complexity of the isotropic PPCA model does not increase strictly with d: it increases between 1 and the actual intrinsic dimension d * and decreases between d * and (p -1). Other criteria of the form ML + pen(n) where pen(n) is a penalty such that pen(n)/n tends to 0 as n tends to infinity are other consistent criteria to estimate this intrinsic dimension. In the following such criteria as AIC and BIC are compared with ML criterion in a numerical experiment.

Numerical Experiments

This section presents numerical experiments on simulated and real datasets in order to highlight the main features of different intrinsic dimension estimation methods in the context of the isotropic PPCA model. The maximum likelihood criterion, for which we have demonstrated the asymptotic consistency, is compared in the following to two penalized likelihood criteria (AIC and BIC), an empirical criterion (Cattell's scree-test), the MleDim method of [START_REF] Levina | Maximum Likelihood Estimation of Intrinsic Dimension[END_REF], the Laplace approach of [START_REF] Minka | Automatic choice of dimensionality for PCA[END_REF] and the cross-validated likelihood. For the sake of simplicity, the following experiments will be set up according to two parameters: α = n/p and β = d * a/[(pd * )b]. The parameter α controls the estimation conditions through the ratio between the number of observations and the dimension of the observation space. The second parameter, β, controls the signal to noise ratio through the condition number a/b of the variance matrix. We recall that AIC and BIC respectively penalize the log-likelihood by the quantities ν(M) and ν(M) log(n)/2 where ν(M) is the number of independent parameters (complexity) of the used model M. The scree-test of Cattell is an empirical method which compares the differences between consecutive eigenvalues with a fixed threshold for finding a breakdown point in the eigenvalue scree. We refer respectively to [START_REF] Levina | Maximum Likelihood Estimation of Intrinsic Dimension[END_REF] and [START_REF] Minka | Automatic choice of dimensionality for PCA[END_REF] for details on the MleDim and Laplace approaches. Finally, for all the following experiments, the parameters p, b and d * will remain fixed to the values p = 50, b = 1 and d * = 20.

An introductory example

The first experiment aims to show the behavior of the three likelihoodbased criteria (ML, AIC and BIC) according to the signal to noise ratio β for a fixed value of α. The simulated model for this experiment is the isotropic PPCA model. The parameter α has been set to 5 which means that the estimation conditions are favorable. Figure 3 shows the eigenvalue scree (left panels) and the behavior of the three likelihood-based criteria (from left to right, ML, AIC and BIC) for different values of β. The first row of Figure 3 considers an easy situation where the eigenvalue scree has a clear breakdown point between relevant and irrelevant dimensions and all criteria succeed in finding the correct intrinsic dimension d * = 20. The second row presents a slightly more difficult situation for which ML and AIC still succeed in determining d * whereas BIC penalizes too much the likelihood and fails in determining d * . Finally, the last row focuses on a difficult situation where there is no elbow in the eigenvalue scree. In this case, AIC and BIC fail in estimating d * by proposing d = 1. Conversely, ML slightly overestimates the actual value of d * by proposing d = 28. It should be noticed that, in the dimension reduction framework, slightly overestimating the intrinsic dimension is preferable to underestimating it because the probability of discarding relevant dimensions is lower.

Influence of the signal to noise ratio

The second experiment focuses on the influence of the signal to noise ratio (parameter β) on the intrinsic dimension estimation with the seven studied dimension selection methods and this for different values of α = n/p. In order not to favour the likelihood-based methods, ML, AIC and BIC, the simulated model used in this numerical experiment will not be the isotropic PPCA model but a uniform model with d * dimensions of variance a/12 and (pd * ) dimensions of variance b/12 is used. The results have been averaged from 50 independent simulated datasets. Figure 4 (a)-(c) aims to compare the behavior of criteria based on the isotropic PPCA model with state of the art criteria according to β for three values of α = 1, 2, 3. Among the criteria based on the isotropic PPCA, ML, AIC, BIC and the 10 fold cross-validated likelihood (CV Lik.) are studied. The scree-test of Cattell, the MleDim criteria [START_REF] Levina | Maximum Likelihood Estimation of Intrinsic Dimension[END_REF] and the Laplace approximation of the integrated likelihood [START_REF] Minka | Automatic choice of dimensionality for PCA[END_REF] stand for the state of the art criteria. From this figure, it appears that, for (c) α = 1 and β = 1, ..., 30 large values of α and β, all the methods perform rather well except MleDim. When α decreases to 1, AIC, CV Lik., Laplace and ML are clearly more efficient than the other criteria. ML could be recommended since it slightly overestimate d * while the other ones underestimate it. We also experimented the adaption of the Laplace criterion to the isotropic PPCA model. As expected, it behaves slightly better than BIC but remains outperformed by AIC and ML.

Influence of the n/p ratio

We now focus on the influence of the n/p ratio (parameter α) on the intrinsic dimension estimation with the seven studied criteria for different values of the signal to noise parameter β. Again, the results have been averaged from 50 independent simulated datasets. Figure 5 shows the behavior of the criteria ML, AIC, BIC, Cattell, MleDim, Laplace and CV Lik. according to α for three values of β = 1, 2, 3. When there is a clear breakdown point in the eigenvalue scree (Figure 5 (a)-(b)), all criteria are efficient for a large range of α values except, as previously, MleDim. It should be however noticed that the scree-test of Cattell and BIC appear again to be less efficient than the other criteria. When there is no clear breakdown point in the eigenvalue scree (Figure 5 (c)), BIC and MleDim fail in estimating d * whatever the α value is. Cattell, AIC, Laplace and CV Lik. tend to underestimate d * whereas ML only slightly overestimates it. This study demonstrates as well that the task of estimating the intrinsic dimension of a dataset is extremely difficult when α and β are both close to 1.

Application to supervised classification

Supervised classification offers the ability to numerically evaluate the performance of the studied methods on real data (for which the actual intrinsic dimension is unknown) through the correct classification rate. We selected six datasets on the UCI Machine Learning Repository (http://archive.ics.uci-.edu/ml/ ): Abalone, Glass, Satellite, Sonar, USPS and Wine datasets. We also considered the Maldi dataset [START_REF] Alexandrov | Biomarker discovery in MALDI-TOF serum protein profiles using discrete wavelet transformation[END_REF], coming from mass-spectrometry, which has less observations than measured variables. Indeed, this kind of situations has become a recurrent and challenging scenario in several scientific fields such as, for instance, Biology or Medecine. The USPS dataset has been modified to focus on discriminating the three most difficult classes to be classify, namely the classes of the digits 3, 5 and 8. This dataset has been called USPS 358. The second and the third columns of Table 1 give respectively the number of observations and the number of dimensions of the six datasets. Each dataset was randomly split into a learning set of 90% of the observations and a validation set made of the remaining observations for simulating a difficult classification situation. The intrinsic dimension d was then selected with each dimension selection method before designing the classifier using quadratic discriminant analysis (QDA) on the d first principal (c) β = 1 and α = 1, ..., 50 components (classical PCA). The correct classification rate was computed afterward on the validation set. The results have been averaged from 50 repetitions of the experimental setup. This experimental setup has been applied to the criteria based on the isotropic PPCA model (ML, AIC, BIC and Iso-Laplace) and to the Laplace criterion based on the standard PPCA model. Table 1 reports the average correct classification rate in percentage of each dimension selection method for the seven datasets. The standard deviations are also provided and the average selected dimensions are given into parentheses. The main point is that isotropic criteria always select dramatically less dimensions than the non isotropic model while providing similar or better classification results. Among the criteria based on the isotropic PPCA model, ML is always similarly or more efficient than the others and provides, in addition, more stable results since its variance is usually less than the one of the other criteria. In particular, ML performs better than other criteria on the Maldi dataset for which the N/p ratio is lower than 1.

Discussion

The present work focused on the estimation of the intrinsic dimension d which controls in PPCA the number of parameters to be estimated. This problem can be regarded as a model selection problem. From this point of view, it can be thought of as surprising to propose the maximum likelihood (ML) as a model selection criterion since in most situations this criterion could be expected to increase with the model complexity. Because of the duality between the two subspaces considered with our model, it is not surprising that the ML criterion is able to find the actual intrinsic dimension of the data without requiring an additional complexity penalty.

The theoretical result of Section 3 ensures that the ML criterion is consistent to estimate the actual intrinsic dimension of the isotropic PPCA model. In practice, the sample size n is finite and can be small in regard to p. Thus, it could happen that the sample variability leads to a ML criterion whose maximum is attained for a larger dimension than the actual intrinsic dimension d * . In such cases and especially for small n, a slight penalty term, as the AIC penalty term, could be desirable to select a proper intrinsic dimension. Figure 6 displays a summary of the recommendations that could be given from our experience. As it can be seen on this figure, it appear that AIC can outperform ML criterion when n/p < 5 for a moderate signal to noise ratio. Finally, when n/p < 2, no method performs well in selecting d and, in this case, it could be recommended to compare the dimensions selected by AIC and ML and to choose d on an empirical ground when the AIC and ML selected dimensions differ. And, in that purpose, the recommendations provided by Figure 6 could be helpful.

The theoretical result exhibited in this work should have interesting applications in methods related to or based on the isotropic probabilistic PCA model. On the one hand, the intrinsic dimension selection approach proposed in this work could be used to approximately determine the intrinsic dimension in PPCA, PMCA and XCA. Indeed, although the maximum likelihood estimate is not asymptotically consistent for the PPCA, PMCA and XCA models, it could provide a first approximation of the intrinsic dimension for those models. On the other hand, the isotropic PPCA model has been used in [START_REF] Bouveyron | High Dimensional Discriminant Analysis[END_REF], in a supervised classification framework, for modelling and classifying the data of K classes in different subspaces with specific intrinsic dimensions d * k , k = 1, ..., K, estimated through an empirical strategy. In such a context, the BEC criterion [START_REF] Bouchard | Selection of generative models in classification[END_REF] could be also used since it is a penalized-likelihood criterion taking into account the classification goal. Alternatively, the asymptotic optimality of the ML criterion for the isotropic PPCA model should allow this classification method to efficiently determine the intrinsic dimension d * k of each class using the ML criterion avoiding numerical problems when α or/and β go close to 1. Furthermore, the low and insensitive computational cost of the likelihood-based methods is a great advantage over the other dimension selection methods while incorporated in iterative processes such as the EM algorithm.

Figure 1 :

 1 Figure 1: Graphical representation of the isotropic PPCA model.

Figure 2 :

 2 Figure 2: The isotropic PPCA model: a controls the variance in the latent subspace E spanned by the d first columns of Q, µ locates the subspace in the original space and b controls the variance outside E.

  d * ) and γ = b a -1. Thus, the study of φ(d) reduces to the study of ψ(δ) = δ log 1 + γ δ where δ ≥ 1 and γ ≤ 0. Since ψ is a strictly increasing function on [1, +∞) for all γ ≤ 0, its minimum is reached for δ = 1 and therefore the minimum of φ on [1, d * ] is reached for d = d * . Situation d ≥ d * . Here, â → 1 d (d * a + (dd * )b) and b → b a.s. when n → ∞. It leads to φ n (d) → φ(d) a.s. where φ(d) = d log d * d a + dd * d b + (pd) log(b).

Figure 3 :

 3 Figure 3: Dimension selection on data simulated according to the isotropic PPCA model with ML, AIC and BIC. The data were simulated with p = 50, b = 1 and the actual intrinsic dimension is d * = 20.

  (a) α = 3 and β = 1, ..., 30

Figure 4 :

 4 Figure 4: Average selected dimension according to the signal to noise ratio β for fixed values of α. The data were simulated according to a uniform distribution (see text for details) with p = 50 and the actual intrinsic dimension is d * = 20.

  (a) β = 3 and α = 1, ..., 50

Figure 5 :

 5 Figure 5: Average selected dimension according to the n/p ratio α for fixed values of β. The data were simulated according to a uniform distribution (see text for details) with p = 50 and the actual intrinsic dimension is d * = 20.

Dataset

Figure 6 :

 6 Figure 6: Recommended criteria for intrinsic dimension selection according to the n/p and signal to noise ratios in the context of the isotropic PPCA model.

  contrast to the general Factor Analysis model, all parameters µ, b and H benefit from closed form estimators. It is assume, without loss of generality, that the columns h 1 , ..., h d of H are orthogonal, i.e. H t H is diagonal and h 1 , . . . , h d are eigenvectors of Σ associated to the eigenvalues

	h 1	2 + b, . . . , h d	2 + b. Consequently, the d eigenvalues associated to the
	latent subspace are always larger than the eigenvalue b (with multiplicity
	p -d) associated to the noise subspace. In contrast, in the Probabilistic Minor Component Analysis (PMCA)

Table 1 :

 1 Classification results on real-world datasets: reported values are average correct classification rates computed on validation sets. Standard deviations are also provided as well as average selected dimensions which are given into parentheses.

		obs.	dim.	ML	BIC	AIC	IsoLaplace	Laplace
	Abalone	4177	8	54.1±1.9 (2)	54.1±1.9 (2)	54.1±1.9 (2)	54.1±1.9 (2)	53.0±2.0 (7)
	Glass	214	9	53.2±10.8 (6) 53.2±10.8 (6) 53.2±10.8 (6)	53.2±10.8 (6)	54.1±11.1 (8)
	Wine	178	13	98.1±3.1 (5)	96.5±4.0 (3)	96.8±3.9 (4)	96.5±3.9 (3)	98.6±2.6 (9)
	Satellite	6435	36	85.1±1.3 (8)	85.1±1.4 (8)	85.1±1.4 (8)	85.1±1.4 (8)	85.7±1.2 (33)
	Sonar	208	60	79.0±8.0 (25) 78.6±8.2 (20) 78.9±8.1 (21)	78.5±8.2 (20)	77.8±9.0 (56)
	USPS 358 2248	256	98.1±1.0 (81) 98.2±1.0 (57) 98.1±1.0 (73) 98.1± 1.0 (63)	95.6±1.6 (255)
	Maldi	112	3268 95.1±6.1 (32) 90.9±10.0 (2)	93.6±8.8 (8)	90.1±10.1 (1)	90.1±10.1 (1)


Fang-Yu Hu

Ph D Candidate

Interaction between urban spatial organization and flood risk, Taipei case study

Keywords: Urban spatial organization, flood, climate change, non-structural solution, Taipei

Nowadays, the dikes and high walls constructions in the water shore are the solution to protect from flooding in urban areas. However, the rapid and dense urbanization also the climate change indicate the danger of this solution. The urban planning recent research proposes to preserve and to make urban space for water in order to contribute on flood risk management. By this concept, water could be considered as one of urban key components which trace over urban contemporary forms/patterns. Our research questions the relation or the interaction between the urban space and the flood risk. These analyses contribute to improve the actual prevention and management of flood risk in urban areas. We are especially interested in the spatial urban planning, considered as an item, as an arrangement also as a built organization, and its relation with flooding risk. We take the city of Taipei as our case study, and we plan to compare it with the other case studies of CORFU (Collaborative Research on Flood Resilience in urban area, FP7).

INTRODUCTION

During recent years the natural catastrophes, especially flood-related, seem to be getting more and more frequent and destructive. Moreover, they are widely broadcast. The latest research demonstrates that these catastrophes are related to the buildings' locations and to their built forms, especially in the cities, there are numerous constructions situated along the river or along the coastline, and sometimes both. This paper is based on triple findings as follows.

Firstly, the urban population is keeping increasing since 19 th century. According to United Nations Human Settlements Programme (UN-HABITAT, 2008), a half of world population lives in the cities. The total urban population is expected to double from 2 to 4 billion over the next 30-35 years (United Nations, 2006). Further, the trend of rapid urban growth throughout the mi-20 th century in developed world has shifted to the transition and developing countries of Asia, Latin America and Africa with accelerated speed and enormous scale than ever. An unwanted side effect of this rapid urbanisation process is the increased susceptibility towards flooding by modifications of urban land use: the dense constructions of buildings and roads change the hydrological cycle in urban area. The construction of impervious surface in urban area may increase flood frequency and aggravate the severe damage of flash flood. According to G. Hollis (G.E. Hollis, 1998), over 50 percent of impervious surface in a drainage basin could increase the flood frequency: we observe that the frequency of severe flooding events occurs more often than the historical records. Also, the impervious surface of urbanisation supports the thunderstorm activity by retaining heat in land surface created urban heat island (UHI) effect (WHO, 2009).

Secondly, climate change may cause floods to occur more frequently and severely in urban area because of irregular events such as heavy precipitation, storm, storm surge and the rise of sea level (IPCC, 2008). The frequency of heavy precipitation would probably increase (Brunel and Pitte, 2010). Finally, in accordance with United Nations estimation, only 5 percent of new development afoot in the world's expanding cities is planned (Gentleman, 2007). Those unplanned urban expansion, such as slum settlement or shanty town, are often formed in the vulnerable to flood-related disaster zones. Asian cities, especially the cities in southern Asian regions, are most vulnerable to flooding events cause by summer monsoon or typhoons. There are several metropolis of these regions, where more than 2 million dwellers with their slum population comprising 43 percent of the city population (Adikari and al., 2010).

The urban crises (sanitary, economic, environmental crises, etc.) stimulated, in different period, the reflection on architectural and urban forms (Choay, 1965;Cerda, 1979, Sitte, 1980). Facing the flood-related disaster, the question is which types of urban spatial structures could better meet the new environmental challenge of climate change from now on. Nowadays, the dikes and high walls constructions along the water shore are the solution to protect from flooding in urban areas. However, the rapid and dense urbanization also the climate change indicate the danger of this solution. The urban planning recent research proposes to preserve and to make urban space for water in order to contribute on flood risk management (MSW programme, 2004; "Room for the River", 2006). By this concept, water could be considered as one of urban key components which trace over urban contemporary forms/patterns.

Our research questions the relation or the interaction between the urban space and its hydrology, or more precisely speaking, between the architectural also urban forms and the flood risk. These analyses contribute to improve the actual prevention and management of flood risk in urban areas. We are especially interested in the urban space, considered as an item, as an arrangement (Lévy and al., 2003) also as a built organization (Eleb and al., 1989), and its relation with flood risk. We take Taipei city, capital of Taiwan, as a principal case study and it might be compared with another case study cities of CORFU research programme.

MATERIAL & METHODS

Brief history of Taipei city

Taipei is the capital and the centre city of the largest metropolitan of Taiwan. Situated at the northern tip of the island, Taipei is located on Dan-shui River, and around 25km to Pacific Ocean. Taipei city is located in the valley, relatively narrow, of the Kee-lung and Xin-dian rivers, which joint to form the Dan-shui River along the urban western border (Figure 1.). The central areas on the western side of the municipality are the low-lying terrain (about 5 to 10 m above the sea level); The slope upward to the south and east and especially to the north, where it reaches 1,120 meters at Qi-xing Mount, the highest (extinct) volcano in Taiwan. Its dwellers started to settle in the junction of Dan-shui, Kee-lung and Xin-dian Rivers, port of Wan-hua (old name: Mankah), where became the commercial centre since 1723. In 1860, the northern port of Da-dao-cheng became the commercial centre because of the displacement of the confluence of 3 rivers. Taipei was a distant port of Chinese continent during the Qing dynasty. After French military attack in 1882, Qing dynasty decided to build a fortification city between these 2 existing ports.

The choice of city emplacement depends on the principles of Feng-Shui-Xue (風水學), as every Chinese city. After the annexation of Taiwan by Qing dynasty in 1885, Taipei took progressively its political and commercial important role in the island. In 1886, Qing dynasty made Taipei became the provincial capital. When the Japanese acquired Taiwan in 1895 after the First Sino-Japanese War, they retained Taipei as the capital of the island. During Japanese colonisation (1895-1945), the city was transformed by the principals of colonisation city: the destruction of Chinese city fortification and its urban structure, the installation of urban sanitation system without long-term perspective, etc. The city expands progressively towards east hill areas.

Since 1949, the city was densified rapidly as arrival of Republic China government and it reached the first 1 million dwellers in the early 1960s. After 1967, the economic dynamism attracted the population towards northern island and Taipei became a city with 2 million dwellers in the mi-1970s. Until 1980, the city expanded towards east and north; a series of urban projects were launched. The urban transportation system was also elaborated, for example, the establishment the underground rail network, the construction of expressway and of the metro network. The city spread out over hills in the north and in the south, especially in the Nei-hu and Wen-shan districts. The population stopped to increase rapidly since mi-1990s but Taipei is still one of the highest density cities in the world. In 2010, the city composed by 2,62 million habitants in the surface around 272 km². The city is denser in the east of Dan-shui River because there is riskier for the flooding events in the west of the river because of the low-lying terrain (less than 2,3 metres above sea level).

Flood concern and city's protection

The urban densification increases the proportion of urban impervious surfaces and it raises the flooding risk. The flooding risk is elevated because of the geographical condition: the city is situated a low-lying terrain with dense hydrological network (Figure 2.). Furthermore, Taipei has humid sub-tropical climate. The annual average temperature is around 23,2 °c; the annual average precipitation in plain area and in the mountain is around 3 000 mm and 4 500 mm. The precipitation might increase during the rainy and typhoon season, form May to October. According historical records, the typhoon could attack the island 3 or 4 times per year and could bright precipitation over 100 mm per hour. In these circumstances, the concordance of different phenomena: the narrow river mouth, high tide and obstruction on the riverbed by sludge, aggravate the situation. (1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996) and it was divided into 3 phases: the construction of dikes and high walls, the erection of floodway channel and the Kee-lung river rectification (Figure 3.). Firstly, the construction of dikes and high walls along the Dan-shui river (high walls are around 13 meter high), along the Kee-lung and Xin-dan rivers and other streams (high walls are around 6 meter high). The length of dike built is 109 141 meter. Secondly, the creation of pluvial water evacuation system in case of heavy precipitation (78,5 mm/hr) and the installation of more than 80 pomp stations in urban low-lying terrains try to protect against flooding. Afterwards, the construction of Er-chong floodway channel along Dan-shui and Da-han rivers measures 7,7 meter length 450 meter width. This strip of land along the river could evacuate pluvial water; it is also composed by 83 sports grounds, several bicycle paths and leisure parks. Finally, the Keelung river rectification between Nan-gan and Shi-lin districts shorten the river trail in order to reduce the flooding. This river rectification project also allows installing a new science business park and residential district with the surface of 238 hectare. However, during Nari typhoon passing in 2001, this solution caused the inundation in the upstream area, so the flooding risk was not really reduced. Nari typhoon passing in 2001, during 49 hours, indicated certain points deserved to be studied facing on the heavy precipitation events. During this typhoon passing, whole island of Taiwan was attacked and several cities inundated. According to National Metrology Office, Taipei suffered from the heaviest precipitation, hit the record, during 24 hours in the recent one hundred years: 425 mm. In the whole island, the damage was evaluate about 40 billion NT dollars (0,95 billion euros). In Taipei, the dikes collapses of Kee-lung river have caused the inundation in 16 underground metro stations; the traffic service recovered 6 months later. Also, the roads of centre city were inundated and water poured into the buildings and basements. The city government was aware of the limit of those protections against flooding, and tried to establish an integrated flood risk prevention and management program since 2005.

With its geographical and climatic conditions and its structure experiences against flooding, Taipei city is an interesting case study for our research focused on the evolution between urban form and its hydrology. Further, the dense and rapid urbanization in Taipei modifies the hydrological cycle in the urban area and it aggravates the flood risk.

RESULTS AND DISCUSSION

The geographers specialised in flood risk, for instance, Helga-Jane Scarwell and Richard Lagnier consider two flood risk factors endanger the urban areas (Scarwell and al., 2004). Firstly, the flood risk management concentrated merely on the protections against natural hazard for a long time. The occurrence of flood hazard and flood event depends on seasonal or interannual variations, and climate fluctuation; it also relies on the land use evolution in the catchment area. These two factors allow analysing finer the flood risk management. The notion "vulnerability", enlarges its small-scale dimension: property and human life. It encompasses the whole ecosystem. They propose that the vulnerability should evaluate its material measure and also in immaterial. Thus, the vulnerability study issue is to make actual material damage inventory and to put the geohistorical analysis into perspective. By identification of flood risk favourable factors, the urban water management should be reorganized by taking into account its shortage and its excess, also its social and cultural aspects. It is necessary, according to Helga-Jane Scarwelle and Richard Lagnier, to develop an integrated global approach about urban water problems and all environmental related matters.

By mean of territorial planning measures, the urban water management related issues couldn't be neglected. André Guillerme, hydrologic engineer and historian, shows that henceforth it is necessary to focus on both of logics: integrated urban water management and territorial planning for sustainable development (Guillerme, 1997). We question how could the former assumption influence the future territorial planning politics? Valérie November, geographer and specialist in urban spatial risk, proposes to think about the "urban risk" as a rational entity. It possesses its own consistency of the urban interactions, and it should be reckoned with territorial/urban planning and with the urban risk management. Certain approaches define the risks by spatial dimension: drawing flood-prone area maps, pointing out the polluted zones or indicating insecure buildings. However, it seems that the spatial risk is more complicated. By Valérie November, the urban risk, especially natural or environmental, isn't only a threat to the city and its dwellers, but it's also one of the essential components of urban structure and its urbanisation evolution. Therefore, it is required to translate the relations of urban natural hazards, for example the flood hazards, by urban spatial development. She develops a geographical theory, "Spatiality of risk" (November, 2008), based on Anthropological theory of Bruno Latour (Latour, 2004) and on Economic theory of Joseph Schumpeter (Schumpeter, 1951). The objective is to realize how the risks emerge and what the relation between the risks and its affected areas is, by urban planning documents analysis also by target groups' and victim's survey. This approach presents not only material and spatial risk dimensions, but also social, political and economic ones. Those dimensions are the major factors of vulnerabilities and risks evolution. According to the result of these surveys, Valérie November found that the coexistence of natural and social risks would reinforce the precariousness of the place. Thus, Valérie November interrogates about the following questions: If is it just at random that different risks accumulated in one place? Why do people choose to live in the risk-prone zone and why do they neglect quickly the threat of risk occurred previously? Moreover, one of the case studies of flood risk named "Lully Effect" (November, 2009) shows that the prevention and management of flood risk in local scale is as much important as the one in territorial scale.

Chris Zevenbergen and William Veerbeek, researchers in Delft University of Technology, emphasize on the necessity to establish a spatial approach of flood resilience in urban development strategy, particularly facing the climate change (Zevenbergen and al., 2008). At first, the analysis of case studies spatial adaptations in different scales (drainage basin, city and building) leads to identifying the interactions between them. Then, the development of a spatial and temporal approach could be set up. In their point of view, the project UFM Dordrecht in Netherland is a convincing spatial adaptation example in multi-scales.

CONCLUSIONS

Based on above mentioned researches in different fields, the objective of our research is to investigate the relation or, more precisely, the interaction between the forms of urbanisation and the impacts of flood risk. We attempt to distinguish the different architectural and urban forms, which seem more adapted to the local climate and hydrology, also to the means of conceiving the urban spaces. We pay particularly attention to the cultural resolutions since there aren't solely the technical solutions. Furthermore, we seek to analyze their capacity of adaptation, of transformation and of renovation with rapid urbanisation tendency, especially in the high-density cities.

Our primary research hypothesis proposes to consider the flood risk as one of key urban structure components of contemporary city. A research centred on the urban evolution of the relation between urban forms and its hydrology, also of the interaction between flood disaster events' impact and urbanization forms could be a key solution of urban mutation facing on the climate change and flooding disaster caused by it. Moreover, the review upon the urban planning and design methods by taking the flood risk in consideration in urban spatial organisation could assist us with a further comprehension about the impact of urban space recomposition in contemporary city.

Our research consists to review the literature concerning this subject: the relation between urban pattern/form and its hydrology, especially the occurrence of flooding events. We reference the recent research results of EU program, such as MITCH, IMPACT, FloodMan, FLOODsite and ERA-NET CRUE. Even though those research programs have concentrated on the structural solution against flooding, certain results of these programs were applied in urban projects in European cities. It is useful to analyse concretely how the urban patter/form adapts to local climatic and hydrological conditions.

Our research then carries on the case study: Taipei. Our analysis consists, foremost, the comprehension about the evolution of urban structure and its hydrology from 1950 to 2010, significant urban development period (Figure 4.). For analysis the relation between water and the urban spatial organisation, the analysis criteria would be established.

Figure 1 :

 1 Figure 1: Location of Taipei city and its 12 districts source: Department of Urban Development, Taipei City Government

Figure 2 :

 2 Figure 2: Location of rivers, the dikes and pomp stations source: Water Resources Agency, Ministry of Economic Affairs

Figure 3 :

 3 Figure 3: Kee-ling river rectification source: Department of Urban Development, Taipei City Government

Figure 4 :

 4 Figure 4: Urban Morphology Analysis

• ADIKARI R., OSTI R. and NORO T. ( 2010), Flood-related disaster vulnerability: an impending crisis of megacities in Asia. In: Journal of Flood Risk Management,vol. 3,number 3,. • BRUNEL Sylvie and PITTE Jean-Robert (Edited by) (2010), Le ciel ne va pas nous tomber sur la tête, JC Lattès (ed.), Paris, 353 p. -2011-Hu-Paper-2011-06-30.doc

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