text:
Douglas M Nark

Michael G Jones

Estelle Piot

Assessment of Axial Wave Number and Mean Flow Uncertainty on Acoustic Liner Impedance Eduction

A key parameter in designing and assessing advanced broadband acoustic liners to achieve the current and future noise reduction goals is the acoustic impedance presented by the liner. This parameter, intrinsic to a specific liner configuration, is dependent on sound pressure level and grazing flow velocity. Current impedance eduction approaches have, in general, provided excellent results and continue to be employed throughout the acoustic liner community. However, some recent applications have indicated a possible dependence of the educed impedance on the direction of incident waves relative to the mean flow. The purpose of the current study is to investigate this unexpected behavior for various impedance eduction methods based on the Pridmore-Brown and convected Helmholtz equations. Specifically, the effects of flow profile and axial wavenumber uncertainties on educed impedances for upstream and downstream sources are investigated. The uniform flow results demonstrate the importance of setting a correct Mach number value in obtaining consistent educed impedances for upstream and downstream sources. In fact, the consistency of results over the two source locations was greatly improved by a slight modification of the uniform flow Mach number. In addition, uncertainty in educed axial wavenumber was also illustrated to correlate well with differences in the educed impedances, even with modified uniform flow Mach number. Finally, while less straightforward than in the uniform flow case, it appears that modification of the mean flow profile may also improve consistency of results for upstream and downstream results when shear flow is included.

I. Introduction

Aircraft noise reduction requirements continue to become more stringent with increases in air traffic. As fan noise produced by the engine is a dominant noise source for subsonic aircraft, significant effort has been directed toward the reduction of this component both at its source and along transmission/radiation paths. Acoustic liners have been particularly effective at reducing fan noise transmission/radiation. However, the introduction of advanced fan designs and shorter engine nacelles has highlighted a need for novel acoustic liner designs that provide increased fan noise reduction over a broader frequency range.

A key parameter in designing and assessing advanced broadband acoustic liners to achieve the aforementioned noise reduction goals is the acoustic impedance presented by the liner. This parameter, intrinsic to a specific liner configuration, is dependent on sound pressure level and grazing flow velocity. As discussed by Jones and Watson, 1 a number of methods have been developed to educe acoustic impedance in the presence of grazing flow. These range from direct measurement approaches (e.g., in situ microphones/probes, Laser Doppler Velocimetry) to indirect methods based on the combination of a duct propagation code and measurements within the hardwall sections (upstream/downstream or opposite) of a liner mounted in the flow duct. The indirect approaches have generally been based on three sets of equations: the convected Helmholtz equation, limited to uniform background flow, and the linearized Euler equations and the Pridmore-Brown equation, which include shear-flow effects. [START_REF] Watson | Impedance Eduction in the Presence of Shear Flow[END_REF][START_REF] Jones | Effects of Flow Profile on Educed Acoustic Liner Impedance[END_REF] The methods based on the convected Helmholtz and linearized Euler equations assume a finite-length acoustic liner mounted in a rectangular waveguide. In this case, the educed impedance value is iterated upon until the acoustic pressures predicted by a duct propagation code match the acoustic pressures measured at a number of microphone locations to within an acceptable tolerance. Alternatively, the method based on the Pridmore-Brown equation [START_REF] Pridmore-Brown | Sound Propagation in a Fluid Flowing Through an Attenuating Duct[END_REF] assumes an infinite-length acoustic liner mounted in a rectangular waveguide. This approach, which is well suited to cases where a single mode is dominant over a significant portion of the liner, uses amplitude and phase decay rates over the axial extent of the liner to directly compute the liner impedance.

Use of these three impedance eduction approaches implementing the Ingard 5 -Myers [START_REF] Myers | On the Acoustic Boundary Condition in the Presence of Flow[END_REF] boundary condition has, in general, provided excellent results and they continue to be employed throughout the acoustic liner community. However, some recent applications [START_REF] Renou | Failure of the Ingard-Myers boundary condition for a lined duct: An experimental investigation[END_REF][START_REF] Watson | A Comparative Study of Four Impedance Eduction Methodologies Using Several Test Liners[END_REF][START_REF] Weng | Impedance eduction in the presence of turbulent shear flow using the linearized Navier-Stokes equations[END_REF] have indicated a possible dependence of the educed impedance on the direction of incident waves relative to the mean flow. This unexpected behavior for locally reacting liners has led to the proposal of alternative boundary conditions and to the questioning of the hypotheses underlying the classical impedance eduction methods (see for instance Zhou and Bodén, [START_REF] Zhou | A systematic uncertainty analysis for liner impedance eduction technology[END_REF] Spillere et al., [START_REF] Spillere | On the effect of boundary conditions on impedance eduction results[END_REF] Bodén et al. [START_REF] Bodén | Comparison of the effect of flow direction on liner impedance using different measurement methods[END_REF] ). The purpose of the current study is to investigate the effects of flow profile and axial wavenumber uncertainties on impedance eduction methods based on the Pridmore-Brown and convected Helmholtz equations for upstream and downstream sources. A description of the duct geometry and test samples used in this investigation is provided in Section II. Details on the impedance eduction approaches are offered in Section III. Results are provided in Section IV, and concluding remarks regarding some of the more significant results and further areas of interest are presented in Section V.

II. Description of Test Configurations

A. Flow Duct

Over the last three decades, the NASA Langley Research Center has developed a number of test rigs for the evaluation of acoustic liners with and without grazing flow. For this study, the GFIT (see Fig. 1) was selected due to its extensive use in previous impedance eduction work [START_REF] Watson | A Comparative Study of Four Impedance Eduction Methodologies Using Several Test Liners[END_REF][START_REF] Watson | Comparison of a Convected Helmholtz and Euler Model for Impedance Eduction in Flow[END_REF][START_REF] Watson | Impedance Eduction in Ducts with Higher-Order Modes and Flow[END_REF][START_REF] Jones | Explanation of Anomalous Behavior Observed in Impedance Eduction Techniques Using Measured Data[END_REF] and to limit the presence of higher modes in the analysis. The GFIT, where the surface of the test liner forms a portion of the upper wall of the flow duct, has a cross-sectional geometry of 2.0" (50.8 mm) wide by 2.5" (63.5 mm) high and allows evaluation of acoustic liners with lengths from 2.0" (50.8 mm) to 24.0" (609.6 mm). The test liner is assumed to have an unknown, but uniform, normalized impedance, ζ . Throughout this paper all impedances are normalized by the characteristic impedance, ρ 0 c 0 , of the air flowing in the duct (left to right in Fig. 1). Acoustic drivers mounted upstream or downstream of the liner sample form the source section for the exhaust or inlet modes, respectively. For this study, these drivers are used to generate tones (one frequency at a time) over a frequency range of 400 to 3000 Hz in 200 Hz increments, at source levels (peak total SPL measured near the leading edge of the liner) of 120 and 140 dB, and at centerline Mach numbers of 0.0, 0.3, and 0.5 (measured via pitot-static probe located approximately one inch downstream of the downstream drivers). Fifty-three (53) microphones flush-mounted in the lower wall (opposite the liner, see Fig. 1b) are used to measure the acoustic pressure field over the axial extent of 40 in (1.016 m) beginning 8.0 in (0.203 m) upstream of the liner leading edge.

B. Liner Configurations

Two liner configurations having a length (treated area) of 16 in (0.406 m) are considered in the current study (see Fig. 2). The first liner contains numerous ceramic tubes with a depth of 3.25 in (82.55 mm) and a diameter of 0.025 in (0.635 mm). These tubes combine to provide a surface porosity of 57%. This liner is chosen because it is very linear, i.e., its impedance is independent of source SPL. The impedance of this liner is also insensitive to changes in the tangential flow velocity. The second liner consists of a perforate facesheet bonded to a honeycomb core, which is in turn bonded to a solid backplate. The perforate facesheet has a sheet thickness, t, of 0.025 in (0.635 mm), a hole diameter, d h , of 0.039 in (0.991 mm), and a porosity of 15%. The core depth, d, is 1.5 in (38.1 mm) and the impedance of this conventional perforate-over-honeycomb liner is weakly dependent on changes to the source SPL and tangential flow velocity. Data was acquired in the GFIT for both liners using two sources (upstream and downstream of the liner), two source SPLs (120, 140 dB), and three tangential flow velocities (Mach 0.0, 0.3, 0.5). Also, each test condition was repeated eleven times, such that the results can be used for measurement uncertainty analysis.

In addition to acoustic data, 2D flow profiles were acquired upstream and downstream of the test section. These profiles were acquired with each liner installed, as well as a hardwall insert, such that the effects of the liners could be assessed. A total pressure tube was traversed across the duct in 9 horizontal sweeps, where each sweep included 31 locations in the vertical dimension, for a total of 279 total pressures at each axial location. These total pressures were combined with a static pressure measurement in the same axial plane to determine the corresponding mean flow velocities (or Mach numbers). These results were used to compute 1D flow profiles using the 31 points acquired at the spanwise centerline (sweep 5 of 9) location, and were also used to compute 2D flow profiles, such that the effects of mean flow profile on the impedance eduction process could be evaluated. In addition, these flow profiles were used

III. Impedance Eduction Approach

For the current investigation, multiple indirect impedance eduction approaches are used to educe the liner impedance from the measured, multimodal, acoustic pressure field over the test liners. Initially, a method (labeled PBE) based on the Pridmore-Brown equation in the form

P mn (y) + 2k mn M o (y)P mn (y) k -k mn M o (y) + (k -k mn M o (y) 2 -k 2 mn - mπ H 2 P mn (y) = 0 (1)
and implemented by Watson et al. [START_REF] Watson | Performance of Kumaresan and Tufts Algorithm in Liner Impedance Eduction with Flow[END_REF] is utilized. Here, the superscript denotes the ordinary derivative with respect to the normal direction, y, and P mn (y) and k mn denote the acoustic pressure and associated axial propagation constant for a given horizontal (m) and vertical (n) normal duct mode. The governing equation (Eq. 1) is also subject to the boundary conditions

P mn (0) = 0 (2) P mn (H) = - ik ζ [1 -k mn M o (H)/k] 2 P mn (H) (3)
where Eq. 2 specifies a rigid lower wall and Eq. 3 specifies a treated upper wall using the the Ingard-Myers boundary condition. Under the assumption of an infinite-length acoustic liner mounted in a rectangular waveguide, the axial propagation constants are obtained from measured data. Then, Eqs. 1 and 2, along with specification of P mn (0), may be used to numerically determine P mn (H) and P mn (H). Subsequently, Eq. 3 may be used to educe the unknown impedance, ζ , of the test liner:

ζ = - ik [1 -k mn M o (H)/k] 2 P mn (H) P mn (H) . (4)
Note that this procedure may be used for either sheared (M o (y)) or uniform (M o = 0) mean flows. The latter condition will be considered first to isolate the effects of the specified mean flow magnitude.

In addition to the aforementioned PBE approach, an impedance eduction method (labeled CHE) that solves the convective Helmholtz equation using a finite element method [START_REF] Watson | Comparison of a Convected Helmholtz and Euler Model for Impedance Eduction in Flow[END_REF] is also used. Here, a finite-length acoustic liner is assumed to be mounted in a rectangular waveguide and the full GFIT test section (see Figure 1) is modeled. The Ingard-Myers impedance boundary condition is incorporated in the CHE analysis in weak form and cubic Hermite polynomials are used as the basis functions. The unknown normalized impedance, ζ , is educed by an iterative process that determines the resistance, θ , and reactance, χ, that reproduce the acoustic wall pressures at the microphone locations. This is achieved by minimizing the objective function

F(θ , χ) = nwall ∑ n=1 ||{R di f }|| L2 , {R di f } =            p mic (x 1 ) -p pred (x 1 ) p mic (x 2 ) -p pred (x 2 )
. . .

p mic (x nwall ) -p pred (x nwall )            ( 5
)
where nwall is the number of microphone locations (i.e., 53 for this investigation, as mentioned in Section II) and ||{R di f }|| L2 denotes the L 2 -norm of the complex residual error vector between the microphone data (p mic ) and predicted value (p pred ).

IV. Results and Discussion

A. Uniform Flow

Based on measured GFIT data, the eduction methodologies described in Section III were used to educe the impedance and/or axial wavenumber of the test liners for each test condition. Employing the repeated test points, 95% confidence intervals were also generated where appropriate. However, before identifying some the effects of axial wave number uncertainty, results for single instances provide excellent examples of the aforementioned source location (upstream or downstream) effects. Single instance educed impedances using the PBE approach with upstream and downstream source (120 dB) measurements for cases with no mean flow (M o = 0.0) are first presented in Figures 3 and4. Specifically, the educed normalized resistance and reactance values for the CT57 liner are provided in Figure 3. As expected, the results for the upstream (U) and downstream (D) compare very well in the absence of mean flow effects. In addition, the resistance results illustrate an interesting behavior of the CT57 liner, as a large increase in resistance is observed around the antiresonance ( f ≈ 2100 Hz). It should be noted that similar results were obtained for the higher source level in this and other instances. Therefore, all subsequent results are presented for the same 120 dB source level. Figure 4 shows the source location effects to be similarly negligible for the GE03 liner in the absence of mean flow. However, when a uniform flow (centerline Mach number of 0.3, M ave = 0.256) is included by setting M o = M ave in Eq. 4, the situation changes. As seen in Figure 5, the educed impedances for the CT57 liner with upstream and downstream sources are quite different. The discussion in Section I alluded to this type of unexpected behavior, which has increased interest in impedance boundary conditions and the effects of flow profile and axial wavenumber uncertainties on educed impedances using such indirect methods. As the measurements in the GFIT are obviously obtained in the presence of a three-dimensional sheared mean flow, the value of the input uniform Mach number appears to be a reasonable starting point for further investigation. In general, when uniform flow is considered, an average Mach number based on the measured profiles mentioned in Section II is used. However, the actual value used may be subject to interpretation. For example, Eversman and Gallman [START_REF] Eversman | Impedance Eduction with an Extended Search Procedure[END_REF] proposed including the uniform flow Mach number as an optimization parameter in their impedance eduction process. In addition, the actual GFIT lower wall microphone locations are not located on the duct centerline. Thus, it may be more appropriate to use a uniform mean flow Mach number value that is less than the overall average. In light of this, the value of the input uniform Mach number was progressively reduced to identify general trends in the educed impedance values. If proven to be an effective modification, this would ideally be a more explicit initial step in the eduction process. However, a more ad hoc approach, in which the input Mach number was reduced until the CT57 reactance curves showed consistent antiresonance values (i.e., negative going zero crossing at f ≈ 2100 Hz) seemed appropriate for this preliminary investigation. The results are presented in Figure 6 To further investigate this behavior, the same Mach number value, M = 0.236 (M*0.92), was used in the eduction process for the GE03 liner and the results are shown in Figure 7. While the impedance curves are more benign than the CT57 case, it is clear that the educed impedances are much more consistent with the reduced Mach number, particularly at higher frequencies. In fact, with the original Mach number value the reactance curve for the downstream source (D, M*1.00) contains a nonphysical flattening (even downward trend) above 1600 Hz. This feature is not present for the upstream source (U, M*1.00) and the two curves clearly diverge. However, with the reduced Mach number (M*0.92), the curves coalesce to a physically realistic result. While the reduction in Mach number provides more consistent results for upstream and downstream sources, there are specific frequencies at which anomalies remain. The differences in reactance values at 800 Hz for the CT57 liner in Figure 6 and at 1200 Hz for the GE03 liner in Figure 7 are examples. In light of this, the 95% confidence intervals on the axial wavenumbers discussed above were used to explore this further. For example, the real and imaginary parts of the axial wavenumber (with 95% confidence intervals denoted by error bars) educed for the CT57 liner at a centerline Mach number of 0.3 are presented in Figure 8. It can be seen that wavenumber values are generally quite consistent. However, there are large uncertainties in the values at 800 Hz for a downstream source. Referring back to Figure 6, this appears to contribute to anomalous reactance values for a downstream source regardless of input Mach number.

To further illustrate this behavior, the real and imaginary parts of the axial wavenumber (with 95% confidence intervals denoted by error bars) educed for the GE03 liner at a centerline Mach number of 0.3 are presented in Figure 9. Again, the wavenumber values are fairly consistent with large uncertainties at 1200 and 1400 Hz for the downstream source. There is also some increased uncertainty at 1600 Hz for the upstream source. However, these appear to be a lesser concern, as the associated confidence intervals are much smaller than those for the downstream anomalies. This is further supported by the results in Figure 7, which show anomalies in the resistance and reactance curves at 1200 and 1400 Hz. Thus, it appears that uncertainties in the wavenumber may be associated with erroneous impedance values. To close the discussion of impedance eduction with uniform flow, results for the two test liners using the CHE approach are also presented. Recall that this method entails numerical solution of the convected Helmholtz equation and does not rely on the axial wavenumber as input. Instead, the uniform mean flow Mach number and lower wall pressure measurements are provided and candidate impedance values are iterated upon until the objective function (based on the difference between measured and predicted wall pressures) is reduced to a specified tolerance. Therefore, the uniform flow Mach number and measured lower wall pressures (also used to educe axial wavenumbers in the PBE approach) were used as input. The results for the CT57 liner are provided in Figure 10 using the same nomenclature as that for the PBE approach. Similar to the PBE results in Figure 6, the modified input Mach number (M*0.92) leads to much greater consistency between the upstream and downstream source results. Again, the peaks in the resistances near antiresonance are well aligned, as are the overall reactance curves. Some increased discrepancies are visible around 800 Hz that may coincide with the wavenumber uncertainty in this frequency range. There are also considerable anomalies above 2600 Hz. However, this is consistent with previous results for this approach, as this implementation assumes plane waves in the hardwall sections and higher order modes are cuton above 2600 Hz in the GFIT at this Mach number.

Similar conclusions may be drawn from the GE03 results shown in Figure 11. Again, the educed impedances are much more consistent with the reduced Mach number. In addition, with the original Mach number value the reactance curve for the downstream source (D, M*1.00) contains an even more pronounced nonphysical flattening above 1600 Hz. This feature is not present for the upstream source (U, M*1.00) and the two curves clearly diverge even more than the PBE case. However, with the reduced Mach number (M*0.92), the curves again coalesce to a physically realistic result. It is also interesting to note slight anomalies near 1200-1400 Hz that may be linked to wavenumber

B. Shear Flow

Although space constraints preclude much more than a cursory investigation of the above observations related to the PBE approach using shear flow, a brief consideration of flow profile scaling may be of interest. As mentioned in Section II, 2D flow profiles were acquired upstream and downstream of the test section from which centerline (1D) profiles and average Mach number values could be extracted. As an example, flow profile measurements for a centerline Mach number of 0.3 are shown in Figure 12. In order to use this information in the shear flow PBE approach (see Eqs. 1-3), a fit of the extracted centerline (z = 1.0") mean flow profile (for the appropriate liner) of the form

M o ( ỹ) = λ 1 (1 -ỹ) ln 1 + λ 2 ỹ -λ 3 ỹ2 + λ 4 ỹ ln 1 + λ 5 (1 -ỹ) -λ 6 (1 -ỹ) 2 (6)
is first computed. Here, ỹ = y /H and λ n are constant coefficients determined by minimizing the L 2 -norm of the error between the 'fit' and measured values over the range of measurement locations. The functional form of Eq. 6 has been used in previous studies and provided results that match the measured data quite well. However, alternative forms may be equally (or more) valid and would obviously lead to different values of M o . Exploration of the effects of alternate forms of the mean flow profile fit is certainly of interest, and this is intended to be the focus of future studies. In this work, a simple sensitivity study is employed to gain insight into some of the effects of flow profile shape. The Mach number profile obtained from Eq. 6 was used to perform preliminary eduction with the PBE shear flow approach and, similar to the uniform flow cases, upstream and downstream results were inconsistent. Therefore, while not expected to be as applicable as in the uniform flow cases, a similar ad hoc approach was used in which the mean flow profile was scaled until the CT57 reactance curves showed consistent antiresonance values (i.e., negative going zero crossing at f ≈ 2100 Hz). The applicability of this approach may be suspect in this application for a number of reasons, including a mismatch in the centerline Mach number and modification of M o by only a constant factor. However, it may generate some basic insights into general flow profile modifications that lead to consistent results for upstream and downstream source locations. Bearing this in mind, the results of this cursory study are provided in Figure 13 from which one may glean a few observations of interest. First and foremost, the upstream and downstream results may be made more consistent by a simple scaling of the profile. However, a large scaling factor was required, which is contrary to the behavior for the uniform flow case. This is likely due, in large part, to the inclusion of M o as an additional parameter and will be investigated in future work. Additionally, based on the behavior of the reactance near 800 Hz, the uncertainty in wavenumber appears to have a similar effect as in the uniform flow case. Overall, this brief analysis provides some indication that, while certainly more complex than in the uniform flow cases, modification of the mean flow profile offers some hope to improving consistency in educed impedance values for upstream and downstream source locations.

V. Concluding Remarks

This paper describes the testing and subsequent eduction processes for two acoustic liners in the GFIT using upstream and downstream acoustic sources. The uniform flow results for the PBE and CHE approaches demonstrate the importance of setting a correct Mach number value in obtaining consistent educed impedances for upstream and downstream sources. In fact, the consistency of results over the two source locations was greatly improved by slightly decreasing the uniform flow Mach number from the initial average value. In addition, the effect of wavenumber uncertainty was also illustrated as large uncertainties for the downstream source correlated well with differences in the educed impedances, even with modified uniform flow Mach number. Finally, while less straightforward than in the uniform flow case, it appears that modification of the mean flow profile may also improve consistency of results for upstream and downstream results when shear flow is included in the PBE method.

Future work will include the establishment of a rigorous method for selecting the uniform flow Mach number for use in the aforementioned indirect impedance eduction methods. Exploration of the effects of alternate forms of the mean flow profile, and M o in particular, for shear flow cases will also be considered in subsequent studies.

  (a) GFIT schematic showing upstream and downstream drivers (exhaust and inlet modes, respectively). Red dashed vertical lines represent upstream and downstream axial locations for mean flow profile measurements. (b) GFIT test section with flush-mounted microphones and liner.

Figure 1 :

 1 Figure 1: Sketches of the Grazing Flow Impedance Tube (GFIT).

  (a) Ceramic tubular (CT57) liner. (b) Conventional, perforate-over-honeycomb (GE03) liner.

Figure 2 :

 2 Figure 2: Sketches of CT57 and GE03 test liners.

  (a) Normalized Resistance (b) Normalized Reactance

Figure 3 :

 3 Figure 3: Educed impedance for the CT57 liner at Mach 0.0. Results for upstream and downstream source locations are represented with red circles and blue squares, respectively.

  (a) Normalized Resistance (b) Normalized Reactance

Figure 4 :

 4 Figure 4: Educed impedance for the GE03 liner at Mach 0.0. Results for upstream and downstream source locations are represented with red circles and blue squares, respectively.

Figure 5 :

 5 Figure 5: Educed impedance for the CT57 liner at a centerline Mach number of 0.3. Results for upstream and downstream source locations are represented with red circles and blue squares, respectively.

  as the uniform Mach number was decreased from M o = 0.256 (M*1.00) to a value of M o = 0.236 (M*0.92), with an intermediate value included to illustrate the trend. As seen in the figure, the educed impedance values for upstream (U, M*0.92) and downstream (D, M*0.92) sources are quite consistent. The peak in the resistances near antiresonance are well aligned, as are the overall reactance curves. (a) Normalized Resistance (b) Normalized Reactance

Figure 6 :

 6 Figure 6: Educed impedance for the CT57 liner at a centerline Mach number of 0.3. Results for upstream and downstream source locations are represented with closed and open symbols, respectively.

Figure 7 :

 7 Figure 7: Educed impedance for the GE03 liner at a centerline Mach number of 0.3. Results for upstream and downstream source locations are represented with closed and open symbols, respectively. Note that impedance values at 400 and 600 Hz for the downstream source are not included due to anomalous behavior.

  (a) Upstream Source: Re[k mn ] (b) Upstream Source: Im[k mn ] (c) Downstream Source: Re[k mn ] (d) Downstream Source: Im[k mn ]

Figure 8 :

 8 Figure 8: Real and imaginary parts of the complex axial wavenumber (k mn ) educed for the CT57 liner at a centerline Mach number of 0.3. Error bars represent the 95% confidence intervals based on 11 datasets.

  (a) Upstream Source: Re[k mn ] (b) Upstream Source: Im[k mn ] (c) Downstream Source: Re[k mn ] (d) Downstream Source: Im[k mn ]

Figure 9 :

 9 Figure 9: Real and imaginary parts of the complex axial wavenumber (k mn ) educed for the GE03 liner at a centerline Mach number of 0.3. Error bars represent the 95% confidence intervals based on 11 datasets.

Figure 10 :

 10 Figure 10: Educed impedance for the CT57 liner at a centerline Mach number of 0.3. Results for upstream and downstream source locations are represented with closed and open symbols, respectively.

Figure 11 :

 11 Figure 11: Educed impedance for the GE03 liner at a centerline Mach number of 0.3. Results for upstream and downstream source locations are represented with closed and open symbols, respectively.

  (a) Upstream axial location (b) Downstream axial location

Figure 12 :

 12 Figure 12: Measured flow profiles at axial locations upstream and downstream of the GE03 liner with a centerline Mach number of 0.3. Grid lines represent the 9 horizontal and 31 vertical measurement locations.

Figure 13 :

 13 Figure 13: Educed impedance for the CT57 liner with a shear flow profile and centerline Mach number of 0.3. Results for upstream and downstream source locations are represented with closed and open symbols, respectively.

of 12 American Institute of Aeronautics and Astronautics

of 12 American Institute of Aeronautics and Astronautics

of 12 American Institute of Aeronautics and Astronautics

of 12 American Institute of Aeronautics and Astronautics

of 12 American Institute of Aeronautics and Astronautics

Acknowledgments

The authors would like to thank Martha Brown, Brian Howerton, and Max Reid for obtaining the GFIT data used in this study. This work is also part of the NASA/ONERA collaborative agreement on aircraft noise reduction. The NASA portion of this research was funded by the Advanced Air Transport Technology (AATT) Project of the NASA Advanced Air Vehicles Program (AAVP).


domain: math math.math-st qfin

text:
Arthur Charpentier

Stéphane Mussard
email: stephane.mussard@unimes.fr

Téa Ouraga

Principal Component Analysis: A Generalized Gini Approach

Keywords: Generalized Gini, PCA, Robustness

A principal component analysis based on the generalized Gini correlation index is proposed (Gini PCA). The Gini PCA generalizes the standard PCA based on the variance. It is shown, in the Gaussian case, that the standard PCA is equivalent to the Gini PCA. It is also proven that the dimensionality reduction based on the generalized Gini correlation matrix, that relies on city-block distances, is robust to outliers. Monte Carlo simulations and an application on cars data (with outliers) show the robustness of the Gini PCA and provide different interpretations of the results compared with the variance PCA.

Introduction

This late decade, a line of research has been developed and focused on the Gini methodology, see [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF] for a general review of different Gini approaches applied in Statistics and in Econometrics.1 Among the Gini tools, the Gini regression has received a large audience since the Gini regression initiated by [START_REF] Olkin | Gini regression analysis[END_REF]. Gini regressions have been generalized by [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF] in different areas and particularly in time series analysis. [START_REF] Shelef | A Gini-based methodology for identifying and analyzing time series with non-normal innovations[END_REF] and [START_REF] Carcea | A Gini autocovariance function for time series modeling[END_REF] investigated ARMA processes with an identification and an estimation procedure based on Gini autocovariance functions. This robust Gini approach has been shown to be relevant to heavy tailed distributions such as Pareto processes. Also, [START_REF] Shelef | A Gini-based unit root test[END_REF] proposed a unit root test based on Gini regressions to deal with outlying observations in the data.

In parallel to the above literature, a second line of research on multidimensional Gini indices arose. This literature paved the way on the valuation of inequality about multiple commodities or dimensions such as education, health, income, etc., that is, to find a real-valued function that quantifies the inequality between the households of a population over each dimension, see among others, [START_REF] List | Multidimensional Inequality Measurement: A Proposal[END_REF], [START_REF] Gajdos | Multidimensional generalized Gini indices[END_REF], [START_REF] Decancq | Weights in Multidimensional Indices of Well-Being: An Overview[END_REF]. More recently, [START_REF] Banerjee | A multidimensional Gini index[END_REF] shows that it is possible to construct multidimensional Gini indices by exploring the projection of the data in reduced subspaces based on the Euclidean norm. Accordingly, some notions of linear algebra have been increasingly included in the axiomatization of multidimensional Gini indices.

In this paper, in the same vein as in the second line of research mentioned above, we start from the recognition that linear algebra may be closely related to the maximum level of inequality that arises in a given dimension. In data analysis, the variance maximization is mainly used to further analyze projected data in reduced subspaces. The variance criterion implies many problems since it captures a very precise notion of dispersion, which does not always match some basic properties satisfied by variability measures such as the Gini index. Such a property may be, for example, an invariance condition postulating that a dispersion measure remains constant when the data are transformed by monotonic maps. 2 Another property typically related to the Gini index is its robustness to outlying observations, see e.g. [START_REF] Yitzhaki | Concentration indices and concentration curves[END_REF] in the case of linear regressions. Accordingly, it seems natural to analyze multidimensional dispersion with the Gini index, instead of the variance, in order to provide a Principal Components Analysis (PCA) in a Gini sense (Gini PCA).

In the field of PCA, [START_REF] Baccini | A L 1 norm PCA and a heuristic approach[END_REF] and [START_REF] Korhonen | Ordinal principal component analysis theory and an application[END_REF] are among the first authors dealing with a ℓ 1 -norm PCA framework. Their idea was to robustify the standard PCA by means of the Gini Mean Difference metric introduced by [START_REF] Gini | Variabilità e mutabilità, Memori di Metodologia Statistica[END_REF], which is a city-block distance measure of variability. The authors employ the Gini Mean Difference as an estimator of the standard deviation of each variable before running the singular value decomposition leading to a robust PCA. In the same vein, [START_REF] Ding | R 1 -PCA: rotational invariant L1-norm principal component analysis for robust subspace factorization[END_REF] make use of a rotational ℓ 1 norm PCA to robustify the variancecovariance matrix in such a way that the PCA is rotational invariant. Recent PCAs derive latent variables thanks to regressions based on elastic net (a ℓ 1 regularization) that improves the quality of the regression curve estimation, see [START_REF] Zou | Sparse Principal Component Analysis[END_REF].

In this paper, it is shown that the variance may be seen as an inappropriate criterion for dimensionality reduction in the case of data contamination or outlying observations. A generalized Gini PCA is investigated by means of Gini correlations matrices. These matrices contain generalized Gini correlation coefficients (see [START_REF] Yitzhaki | Gini's Mean difference: a superior measure of variability for non-normal distributions[END_REF]) based on the Gini covariance operator introduced by [START_REF] Schechtman | A Measure of Association Based on Gini's Mean Difference[END_REF] and [START_REF] Schechtman | A family of correlation coefficients based on the extended Gini index[END_REF]. The generalized Gini correlation coefficients are: (i) bounded, (ii) invariant to monotonic transformations, (iii) and symmetric whenever the variables are exchangeable. It is shown that the standard PCA is equivalent to the Gini PCA when the variables are Gaussians. Also, it is shown that the generalized Gini PCA may be realized either in the space of the variables or in the space of the observations. In each case, some statistics are proposed to perform some interpretations of the variables and of the observations (absolute and relative contributions). To be precise, an U -statistics test is introduced to test for the significance of the correlations between the axes of the new subspace and the variables in order to assess their significance. Monte Carlo simulations are performed in order to show the superiority of the Gini PCA compared with the usual PCA when outlying observations Gini correlation index. contaminate the data. Finally, with the aid of the well-known cars data, which contain outliers, it is shown that the generalized Gini PCA leads to different results compared with the usual PCA.

The outline of the paper is as follows. Section 2 sets the notations and presents some ℓ 2 norm approaches of PCA. Section 3 reviews the Ginicovariance operator. Section 4 is devoted to the generalized Gini PCA. Section 5 focuses on the interpretation of the Gini PCA. Sections 6 and 7 present some Monte Carlo simulations and applications, respectively.

2 Motivations for the use of Gini PCA In this Section, the notations are set. Then, some assumptions are imposed and some ℓ 2 -norm PCA techniques are reviewed in order to motivate the employ of the Gini PCA.

Notations and definitions

Let N * be the set of integers and R [R ++ ] the set of [positive] real numbers. Let M be the set of all N ×K matrix X = [x ik ] that describes N observations on K dimensions such that N ≫ K, with elements x ik ∈ R, and I n the n × n identity matrix. The N × 1 vectors representing each variable are expressed as x •k , for all k ∈ {1, . . . , K} and we assume that x •k = c1 N , with c a real constant and 1 N a N -dimensional column vector of ones. The K × 1 vectors representing each observation i (the transposed ith line of X) are expressed as x i• , for all i ∈ {1, . . . , N }. It is assumed that x •k is the realization of the random variable X k , with cumulative distribution function F k . The arithmetic mean of each column (line) of the matrix X is given by x•k (x i• ). The cardinal of set A is denoted #{A}. The ℓ 1 norm, for any given real vector x, is

x 1 = K k=1 |x •k |, whereas the ℓ 2 norm is x 2 = ( K k=1 x 2 •k ) 1/2 . Assumption 2.1. The random variables X k are such that E[|X k |] < ∞ for all k ∈ {1, . . .

, K}, but no assumption is made on the second moments (that may not exist).

This assumption imposes less structure compared with the classical PCA in which the existence of the second moments are necessary, as can be seen in the next subsection.

Variants of PCA based on the ℓ 2 norm

The classical formulation of the PCA, to obtain the first component, can be obtained by solving

ω * 1 ∈ argmax {Var[Xω]} subject to ω 2 2 = ω ⊤ ω = 1, (1)
or equivalently

ω * 1 ∈ argmax ω ⊤ Σω subject to ω 2 2 = ω ⊤ ω = 1, (2)
where ω ∈ R K , and Σ is the (symmetric positive semi-definite) K ×K sample covariance matrix. [START_REF] Mardia | Multivariate Analysis[END_REF] suggest to write

ω * 1 ∈ argmax K j=1 Var[x •,j ] • Cor[x •,j , Xω] subject to ω 2 2 = ω ⊤ ω = 1. With scaled variables 3 (i.e. Var[x •,j ] = 1, ∀j) ω * 1 ∈ argmax K j=1 Cor[x •,j , Xω] subject to ω 2 2 = ω ⊤ ω = 1. ( 3
)
Then a Principal Component Pursuit can start: we consider the 'residuals', X (1) = X -Xω * 1 ω * ⊤ 1 , its covariance matrix Σ (1) , and we solve

ω * 2 ∈ argmax ω ⊤ Σ (1) ω subject to ω 2 2 = ω ⊤ ω = 1.
The part Xω * 1 ω * ⊤ 1 is actually a constraint that we add to ensure the orthogonality of the two first components. This problem is equivalent to finding the maxima of Var[Xω] subject to ω 2 2 = 1 and ω ⊥ ω * 1 . This idea is also called Hotelling (or Wielandt) deflation technique. On the k-th iteration, we extract the leading eigenvector

ω * k ∈ argmax ω ⊤ Σ (k-1) ω subject to ω 2 2 = ω ⊤ ω = 1,
where [START_REF] Saad | Projection and deflation methods for partial pole assignment in linear state feedback[END_REF]). Note that, following [START_REF] Hotelling | Analysis of a complex of statistical variables into principal components[END_REF] and [START_REF] Eckart | The approximation of one matrix by another of lower rank[END_REF], that it is also possible to write this problem as min X -X ⋆ subject to rank[ X] ≤ k where • ⋆ denotes the nuclear norm of a matrix (i.e. the sum of its singular values)4 .

Σ (k-1) = Σ (k-2) -ω * k-1 ω * ⊤ k-1 Σ (k-1) ω * k-1 ω * ⊤ k-1 (see e.g.
One extension, introduced in d' Aspremont et al. (1920), was to add a constraint based on the cardinality of ω (also called ℓ 0 norm) corresponding to the number of non-zero coefficients of ω. The penalized objective function is then

max ω ⊤ Σω -λcard[ω] subject to ω 2 2 = ω ⊤ ω = 1,
for some λ > 0. This is called sparse PCA, and can be related to sparse regression, introduced in [START_REF] Tibshirani | Regression shrinkage and selection via the LASSO[END_REF]. But as pointed out in [START_REF] Mackey | Deflation methods for sparse PCA[END_REF], interpretation is not easy and the components obtained are not orthogonal. Gorban et al. (2007) considered an extension to nonlinear Principal Manifolds to take into account nonlinearities. Another direction for extensions was to consider Robust Principal Component Analysis. [START_REF] Candes | Robust Principal Component Analysis?[END_REF] suggested an approach based on the fact that principal component pursuit can be obtained by solving min X -X ⋆ + λ X 1 .

But other methods were also considered to obtain Robust PCA. A natural 'scale-free' version is obtained by considering a rank matrix instead of X. This is also called 'ordinal' PCA in the literature, see [START_REF] Korhonen | Ordinal principal component analysis theory and an application[END_REF]. The first 'ordinal' component is

ω * 1 ∈ argmax K j=1 R[x •,j , Xω] subject to ω 2 2 = ω ⊤ ω = 1 (4)
where R denotes some rank based correlation, e.g. Spearman's rank correlation, as an extention of Equation (3). So, quite naturally, one possible extension of Equation (2) would be

ω * 1 ∈ argmax ω ⊤ R[X]ω subject to ω 2 2 = ω ⊤ ω = 1
where R[X] denotes Spearman's rank correlation. In this section, instead of using Pearson's correlation (as in Equation ( 2) when the variables are scaled) or Spearman's (as in this ordinal PCA), we will consider the multidimensional Gini correlation based on the h-covariance operator.

3 Geometry of Gini PCA: Gini-Covariance Operators

The first PCA was introduced by [START_REF] Pearson | On Lines and Planes of Closest Fit to System of Points in Space[END_REF], projecting X onto the eigenvectors of its covariance matrix, and observing that the variances of those projections are the corresponding eigenvalues. One of the key property is that X ⊤ X is a positive matrix. Most statistical properties of PCAs (see [START_REF] Flury | Beyond the Pearson Correlation: Heavy-Tailed Risks, Weighted Gini Correlations, and A Gini-Type Weighted Insurance Pricing Model[END_REF] or [START_REF] Anderson | Asymptotic Theory for Principal Component Analysis[END_REF]) are obtained under Gaussian assumptions. Furthermore, geometric properties can be obtained using the fact that the covariance defines an inner product on the subspace of random variables with finite second moment (up to a translation, i.e. we identify any two that differ by a constant). We will discuss in this section the properties of the Gini Covariance operator with the special case of Gaussian random variables, and the property of the Gini correlation matrix that will be used in the next Section for the Gini PCA.

The Gini-covariance operator

In this section, X = (X 1 , • • • , X K ) denotes a random vector. The covariance matrix between X and Y , two random vectors, is defined as the inner product between centered versions of the vectors,

X, Y = Cov(X, Y ) = E[(X -E[X])(Y -E[Y ]) T ].
(5)

Hence, it is the matrix where elements are regular covariances between components of the vectors, Cov(X, Y ) = [Cov(X i , Y j )]. It is the upper-right block of the covariance matrix of (X, Y ). Note that Cov(X, X) is the standard variance-covariance matrix of vector X.

Definition 3.1. Let X = (X 1 , • • • , X K ) be
(X) = Cov(X, h(X)) is called h-Gini covariance matrix.
Since h is a non-decreasing mapping, then X and h(X) are componentwise comonotonic random vectors. Assuming that components of X are identically distributed is a reasonable assumption in the context of scaled (and centered) PCA, as discussed in footnote 3. Nevertheless, a stronger technical assumption will be necessary: pairwise-exchangeability. Definition 3.2. X is said to be pairwise-exchangeable if for all pair (i,

j) ∈ {1, • • • , K} 2 , (X i , X j ) is exchangeable, in the sense that (X i , X j ) L = (X j , X i ).
Pairwise-exchangeability is a stronger concept than having only one vector with identically distributed components, and a weaker concept than (full) exchangeability. In the Gaussian case where h(X k ) = Φ(X k ) with Φ(X k ) being the normal cdf of X k for all k = 1, . . . , K, pairwise-exchangeability is equivalent to components identically distributed.

Proposition 3.1. If X is a Gaussian vector with identically distributed components, then X is pairwise-exchangeable.

Proof. For simplicity, assume that components of X are N (0, 1) random variables, then X ∼ N (0, ρ) where ρ is a correlation matrix. In that case

X i X j ∼ N 0 0 , 1 ρ ij ρ ji 1 , with Pearson correlation ρ ij = ρ ji , thus (X i , X j ) is exchangeable.
Let us now introduce the Gini-covariance. [START_REF] Gini | Variabilità e mutabilità, Memori di Metodologia Statistica[END_REF] introduced the Gini mean difference operator ∆, defined as:

∆(X) = E(|X 1 -X 2 |) where X 1 , X 2 ∼ X, and X 1 ⊥ ⊥ X 2 , (6)
for some random variable X (or more specifically for some distribution F with X ∼ F , because this operator is law invariant). One can rewrite:

∆(X) = 4Cov[X, F (X)] = 1 3 Cov[X, F (X)] Cov[F (X), F (X)]
where the term on the right is interpreted as the slope of the regression curve of the observed variable X and its 'ranks' (up to a scaling coefficient). Thus, the Gini-covariance is obtained when the function h is equal to the cumulative distribution function of the second term, see [START_REF] Schechtman | A Measure of Association Based on Gini's Mean Difference[END_REF].

Definition 3.3. Let X = (X 1 , • • • , X K ) be a collection of K identically distributed random variables, with cumulative distribution function F . Then, the Gini covariance is ΓC F (X) = Cov(X, F (X)).
On this basis, it is possible to show that the Gini covariance matrix is a positive semi-definite matrix.

Theorem 3.1. Let Z ∼ N (0, 1). If X represents identically distributed Gaussian random variables, with distribution N (µ, σ 2 ), then the two following assertions hold:

(i) ΓC F (X) = σ -1 Cov(Z, Φ(Z))Var(X). (ii) ΓC F (X) is a positive-semi definite matrix. Proof. (i) In the Gaussian case, if h is the cumulative distribution func- tion of the X k 's, then Cov(X k , h(X ℓ )) = rσ • Cov(Z, Φ(Z)), where Φ is the normal cdf, see Yitzhaki & Schechtman (2013), Chapter 3. Observe that Cov(X k , h(X k )) = σ • Cov(Z, Φ(Z)), if h is the cdf of X k . Thus, λ := Cov(Z, Φ(Z)) yields: ΓC F ((X k , X ℓ )) = λ σ ρσ ρσ σ = λσ 1 ρ ρ 1 = λ σ Var((X k , X ℓ )). (ii) We have Cov(Z, Φ(Z)) ≥ 0, then it follows that C F ((X k , X ℓ )) ≥ 0: x ⊤ C F (X)x = x ⊤ Cov(Z, Φ(Z)) σ Var(X)x ≥ 0, which ends the proof. Note that ΓC F (X) = Cov(X, -F (X)) = ΓC -F (X)
, where F denotes the survival distribution function.

Definition 3.4. Let X = (X 1 , • • • , X K ) be a collection of K identically dis- tributed random variables, with survival distribution function F . Then, the generalized Gini covariance is GΓC ν (X) = ΓC -F ν-1 (X) = Cov(X, -F ν-1 (X)), for ν > 1.
This operator is related to the one introduced in Yitzhaki & Schechtman ( 2003), called generalized Gini mean difference GM D ν operator. More precisely, an estimator of the generalized Gini mean difference is given by:

GM D ν (x •ℓ , x •k ) := - 2 N -1 νCov(x •ℓ , r ν-1 x •k ), ν > 1,
where

r x •k = (R(x 1k ), . . . , R(x nk )
) is the decumulative rank vector of x •k , that is, the vector that assigns the smallest value (1) to the greatest observation x ik , and so on. The rank of observation i with respect to variable k is:

R(x ik ) := N + 1 -#{x ≤ x ik } if no ties N + 1 -1 p p i=1 #{x ≤ x ik } if p ties x ik . Hence GM D ν (x •ℓ , x •k ) is the empirical version of 2νΓC ν (X ℓ , X k ) := -2νCov X ℓ , F k (X k ) ν-1 .
The index GM D ν is a generalized version of the GM D 2 proposed earlier by [START_REF] Schechtman | A Measure of Association Based on Gini's Mean Difference[END_REF], and can also be written as:

GM D 2 (X k , X k ) = 4Cov X k , F k (X k ) = ∆(X k ).
When k = ℓ, GM D ν represents the variability of the variable x •k itself. Focus is put on the lower tail of the distribution x •k whenever ν → ∞, the approach is said to be max-min in the sense that GM D ν inflates the minimum value of the distribution. On the contrary, whenever ν → 0, the approach is said to be max-max, in this case focus is put on the upper tail of the distribution x •k . As mentioned in [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF], the case ν < 1 does not entail simple interpretations, thereby the parameter ν is used to be set as ν > 1 in empirical applications. 5Note that even if X k and X ℓ have the same distribution, we might have

GM D ν (X k , X ℓ ) = GM D ν (X ℓ , X k ), as shown on the example of Figure 1. In that case E[X k h(X ℓ )] = E[X ℓ h(X k )] if h(2) = 2h(1) (this property is never- theless valid if h is linear). We would have GM D ν (X k , X ℓ ) = GM D ν (X ℓ , X k ) when X k and X ℓ are exchangeable. But since generally GM D ν is not sym- metric, we have for x •k being not a monotonic transformation of x •ℓ and ν > 1, GM D ν (x •k , x •ℓ ) = GM D ν (x •ℓ , x •k ). Figure 1: Joint distribution of a random pair (X k , X ℓ ) such that E[X k h(X ℓ )] = E[X ℓ h(X k )], with non-exchangeable components X k L = X ℓ .

Generalized Gini correlation

In this section, X is a matrix in M. The Gini correlation coefficient (Gcorrelation frown now on), is a normalized GM D ν index such that for all ν > 1, see [START_REF] Schechtman | A family of correlation coefficients based on the extended Gini index[END_REF],

GC ν (x •ℓ , x •k ) := GM D ν (x •ℓ , x •k ) GM D ν (x •ℓ , x •ℓ ) ; GC ν (x •k , x •ℓ ) := GM D ν (x •k , x •ℓ ) GM D ν (x •k , x •k ) , with GC ν (x •k , x •k ) = 1 and GM D ν (x •k , x •k ) = 0, for all k, ℓ = 1, . . . , K.
Following [START_REF] Schechtman | A family of correlation coefficients based on the extended Gini index[END_REF], the G-correlation is well-suited for the measurement of correlations between non-normal distributions or in the presence of outlying observations in the sample.

Property 3.1. -Schechtman and Yitzhaki ( 2013):

(i) GC ν (x •ℓ , x •k ) ≤ 1. (ii) If the variables x •ℓ and x •k are independent, for all k = ℓ, then GC ν (x •ℓ , x •k ) = GC ν (x •k , x •ℓ ) = 0. (iii) For any given monotonic increasing transformation ϕ, GC ν (x •ℓ , ϕ(x •k )) = GC ν (x •ℓ , x •k ). (iv) If (x •ℓ , x •k ) have a bivariate normal distribution with Pearson correlation ρ, then GC ν (x •ℓ , x •k ) = GC ν (x •k , x •ℓ ) = ρ. (v) If x •k and x •ℓ are exchangeable up to a linear transformation, then GC ν (x •ℓ , x •k ) = GC ν (x •k , x •ℓ ).
Whenever ν → 1, the variability of the variables is attenuated so that GM D ν tends to zero (even if the variables exhibit a strong variance). The choice of ν is interesting to perform generalized Gini PCA with various values of ν in order to robustify the results of the PCA, since the standard PCA (based on the variance) is potentially of bad quality if outlying observations drastically affect the sample.

A G-correlation matrix is proposed to analyze the data into a new vector space. Following Property 3.1 (iv), it is possible to rescale the variables x •ℓ thanks to a linear transformation, then the matrix of standardized observation is,

Z ≡ [z iℓ ] := x iℓ -x•ℓ GM D ν (x •ℓ , x •ℓ ) . ( 7
)
The variable z iℓ is a real number without dimension. The variables x •k are rescaled such that their Gini variability is equal to unity. Now, we define the N × K matrix of decumulative centered rank vectors of Z, which are the same compared with those of X:

R c z ≡ [R c (z iℓ )] := [R(z iℓ ) ν-1 -rν-1 z •ℓ ] = [R(x iℓ ) ν-1 -rν-1 x •ℓ ].
Note that the last equality holds since the standardization ( 7) is a strictly increasing affine transformation. 6 The K × K matrix containing all Gcorrelation indices between all couples of variables z •k and z •ℓ , for all k, ℓ = 1, . . . , K is expressed as:

GC ν (Z) := - 2ν N (N -1) Z ⊤ R c z . Indeed, if GM D ν (Z) ≡ [GM D ν (z •k , z •ℓ )],
then we get the following.

Proposition 3.2. For each standardized matrix Z defined in (7), the following relations hold:

GM D ν (Z) = GC ν (X) = GC ν (Z). (8) GM D ν (z •k , z •k ) = 1, ∀k = 1, . . . , K. (9) Proof. We have GM D ν (Z) ≡ [GM D ν (z •k , z •ℓ )] being a K × K matrix.
The extra diagonal terms may be rewritten as,

GM D ν (z •k , z •ℓ ) = - 2 N -1 νCov(z •k , r ν-1 z •ℓ ) = - 2 N -1 νCov x •k -x•k 1 N GM D ν (x •k , x •k ) , r ν-1 z •ℓ = - 2 GM D ν (x •k , x •k ) νCov(x •k , r ν-1 z •ℓ ) N -1 - νCov(x •k 1 N , r ν-1 z •ℓ ) N -1 = GM D ν (x •k , x •ℓ ) GM D ν (x •k , x •k ) = GC ν (x •k , x •ℓ ).
Finally, using the same approach as before, we get:

GM D ν (z •k , z •k ) = - 2 N -1 νCov(z •k , r ν-1 z •k ) = GM D ν (x •k , x •k ) GM D ν (x •k , x •k ) = GC ν (x •k , x •k ) = 1. By Property 3.2 (iv), since r x •k = r z •k , then GC ν (x •k , x •ℓ ) = GC ν (z •k , z •ℓ ). Thus, GM D ν (Z) = - 2ν N (N -1) Z ⊤ R c z = GC ν (X) = GC ν (Z),
which ends the proof.

Finally, under a normality assumption, the generalized Gini covariance matrix GC ν (X) ≡ [GM D ν (X k , X ℓ )] is shown to be a positive semi-definite matrix.

Theorem 3.2. Let Z ∼ N (0, 1). If X represents identically distributed Gaussian random variables, with distribution N (µ, σ 2 ), then the two following assertions holds:

(i) GC ν (X) = σ -1 Cov(Z, Φ(Z))Var(X). (ii) GC ν (X) is a positive semi-definite matrix.
Proof. The first part (i) follows from [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF], Chapter 6. The second part follows directly from (i).

Theorem 3.2 shows that under the normality assumption, the variance is a special case of the Gini methodology. As a consequence, for multivariate normal distributions, it is shown in Section 4 that Gini PCAs and classical PCA (based on the ℓ 2 norm and the covariance matrix) are equivalent.

Generalized Gini PCA

In this section, the multidimensional Gini variability of the observations i = 1, . . . , N , embodied by the matrix GC ν (Z), is maximized in the R K -Euclidean space, i.e., in the set of variables {z •1 , . . . , z •K }. This allows the observations to be projected onto the new vector space spanned by the eigenvectors of GC ν (Z). Then, the projection of the variables is investigated in the R N -Euclidean space induced by GC ν (Z). Both observations and variables are analyzed through the prism of absolute and relative contributions to propose relevant interpretations of the data in each subspace.

The R K -Euclidean space

It is possible to investigate the projection of the data Z onto the new vector space induced by GM D ν (Z) or alternatively by GC ν (Z) since GM D ν (Z) = GC ν (Z). Let f •k be the kth principal component, i.e. the kth axis of the new subspace, such that the N × K matrix F is defined by

F ≡ [f •1 , . . . , f •K ] with R c f ≡ [r ν-1 c,f 1 , . . . , r c c,f
•K ] its corresponding decumulative centered rank matrix (where each decumulative rank vector is raised to an exponent of ν -1). The

K × K matrix B ≡ [b •1 , . . . , b •K ] is the projector of the observations, with the normalization condition b ⊤ •k b •k = 1, such that F = ZB. We denote by λ •k (or 2µ •k ) the eigenvalues of the matrix [GC ν (Z) + GC ν (Z) ⊤ ]. Let the basis B := {b •1 , . . . , b •h } with h ≤ K issued from the maximization of the overall Gini variability: max b ⊤ •k GC ν (Z)b •k =⇒ [GC ν (Z) + GC ν (Z) ⊤ ]b •k = 2µ •k b •k , ∀k = 1, . . . , K.
Indeed, from the Lagrangian,

L = b ⊤ •k GC ν (Z)b •k -µ •k [1 -b ⊤ •k b •k ],
because of the non-symmetry of GC ν (Z), the eigenvalue equation is,

[GC ν (Z) + GC ν (Z) ⊤ ]b •k = 2µ •k b •k , that is, [GC ν (Z) + GC ν (Z) ⊤ ]b •k = λ •k b •k . ( 10
)
The new subspace {f •1 , . . . , f •h } such that h ≤ K is issued from the maximization of the Gini variability between the observations on each axis f •k .

Although the result of the generalized Gini PCA seems to be close to the classical PCA, some differences exist.

Proposition 4.1.

Let B = {b •1 , . . . , b •h } with h ≤ K be the basis issued from the maximization of b ⊤ •k GC ν (Z)b
•k for all k = 1, . . . , K, then the following assertions hold:

(i) max GM D ν (f •k , f •k ) = µ •k for all k = 1, . . . , K, if and only if r ν-1 c,f •k = R c z b •k . (ii) b •k b ⊤ •h = 0, for all k = h. (iii) b •k b ⊤ •k = 1, for all k = 1, . . . , K. Proof. (i) Note that GM D ν (f •k , f •k ) = -2ν N (N -1) (f •k -f ) ⊤ r ν-1 c,f •k , where r ν-1 c,f •k is the kth column of the centered (decumulative) rank matrix R c f . Since f •k = Zb •k and f = ( f•1 , . . . , f•K ) = 0 then: 7 b ⊤ •k GC ν (Z)b •k = - 2ν N (N -1) b ⊤ •k Z ⊤ R c z b •k (11) = - 2ν N (N -1) b ⊤ •k Z ⊤ r ν-1 c,f •k (by r ν-1 c,f •k = R c z b •k ) = - 2ν N (N -1) f ⊤ •k r ν-1 c,f •k = GM D ν (f •k , f •k ).
Then, maximizing the multidimensional variability b ⊤

•k GC ν (Z)b •k yields from (10): b ⊤ •k [GC ν (Z) + GC ν (Z) ⊤ ]b •k = b ⊤ •k λ •k b •k ⇐⇒ b ⊤ •k GC ν (Z)b •k + b ⊤ •k GC ν (Z) ⊤ b •k = λ •k .
7 We have:

f•k = 1/N N i=1 f ik = 1/N N i=1 z ⊤ i• b •k = 1/N N i=1 z ⊤ i• b •k = 0. Since b ⊤ •k GC ν (Z)b •k = b ⊤ •k GC ν (Z) ⊤ b •k , then b ⊤ •k GC ν (Z) ⊤ b •k = λ •k /2 = µ •k ,
and so

GM D ν (f •k , f •k ) = λ •k for all k = 1, . . . , K.
The results (ii) and (iii) are straightforward.

Discussion

Condition (i) shows that the maximization of the multidimensional variability (in the Gini sense) b ⊤ •k GC ν (Z)b •k does not necessarily coincide with the maximization of the variability of the observations projected onto the new axis f •k embodied by GM D ν (f •k , f •k ). Since in general, the rank of the observations on axis f •k does not coincide with the projected ranks, that is,

r ν-1 c,f •k = R c z b •k , then, max b ⊤ •k GC(Z)b •k = GM D ν (f •k , f •k ).
In other words, maximizing the quadratic form b ⊤

•k GC(Z) ν b •k does not systematically maximize the overall Gini variability GM D ν (f •k , f •k ). However, it maximizes the following generalized Gini index:

GGM D ν (f •k , f •k ) := - 2ν N (N -1) b ⊤ •k Z ⊤ R c z b •k = - 2ν N (N -1) f ⊤ •k b ⊤ •k (R c z ) ⊤ .
In the literature on inequality indices, this kind of index is rather known as a generalized Gini index, because of the product between a variable f •k and a function Ψ of its ranks, Ψ(r

f k ) := b ⊤ •k (R c z ) ⊤ , such that: GGM D ν (f •k , f •k ) = - 2ν N (N -1) f •k Ψ(r f k ).
Yaari (1987) and subsequently [START_REF] Yaari | A Controversial Proposal Concerning Inequality Measurement[END_REF] proposes generalized Gini indices with a rank distortion function Ψ that describes the behavior of the decision maker (being either max-min or max-max).8

It is noteworthy that this generalized Gini index of variability is very different from Banerjee (2010)'s multidimensional Gini index. The author proposes to extract the first eigenvector e •1 of X ⊤ X and to project the data X such that s := Xe •1 so that the multidimensional Gini index is G(s) = s ⊤ Ψ(r s ), with r s the rank vector of s and with Ψ a function that distorts the ranks. Banerjee (2010)'s index is derived from the matrix X ⊤ X. To be precise, the maximization of the variance-covariance matrix X ⊤ X (based on the ℓ 2 metric) yields the projection of the data on the first component f •1 , which is then employed in the multidimensional Gini index (based on the ℓ 1 metric). This approach is legitimated by the fact that G(s) has some desirable properties linked with the Gini index. However, this Gini index deals with an information issued from the variance, because the vector s relies on the maximization of the variance of component f •1 . Alternatively, it is possible to make use of the Gini variability, in a first stage, in order to project the data onto a new subspace, and in a second stage, to use the generalized Gini index of the projected data for the interpretations. In such as case, the Gini metric enables outliers to be attenuated. The employ of G(s) as a result of the variance-covariance maximization may transform the data so that outlying observations would capture an important part of the information (variance) on the first component. This case occurs in the classical PCA. This fact will be proven in the next sections with Monte Carlo simulations. Let us before investigate the employ of the generalized Gini index GGM D ν .

Properties of GGM D ν

Since the Gini PCA relies on the generalized Gini index GGM D ν , let us explore its properties. Proposition 4.2. Let the eigenvalues of GC ν (Z) + GC ν (Z) ⊤ be such that

λ 1 = µ 1 /2 ≥ • • • ≥ λ •K = µ •K /2. Then, (i) GGM D ν (f •k , f •k ) = GGM D ν (f •k , f •ℓ ) = GGM D ν (f •ℓ , f •k ) = 0, for all ℓ = 1, . . . , K, if and only if, λ •k = 0. (ii) max k=1,...,K GGM D ν (f •k , f •k ) = µ 1 . (iii) min k=1,...,K GGM D ν (f •k , f •k ) = µ •K .
Proof. (i) The result comes from the rank-nullity theorem. From the eigenvalue Equation (10), we have:

b ⊤ •k GC ν (Z)b •k = λ •k /2 = µ •k .
Let f be the linear application issued from the matrix GC ν (Z). Whenever λ •k = 0, two columns (or rows) of GC ν (Z) are collinear, then the dimension of the image set of

f is dim(f ) = K -1. Hence, f •k = 0. Since b ⊤ •k GC ν (Z) ⊤ b •k = GGM D ν (f •k , f •k ) for all k = 1, . . . , K, then for λ •k we get: b ⊤ •k GC ν (Z) ⊤ b •k = GGM D ν (f •k , f •k ) = λ •k /2 = µ •k = 0. On the other hand, since f •k = 0, it follows that GGM D ν (f •k , f •ℓ ) = 0 for all ℓ = 1, . . . , K. Also, if f •k = 0 then the centered rank vector r c f •k = 0, and so GGM D ν (f •ℓ , f •k ) = 0 for all ℓ = 1, . . . , K.
(ii) The proof comes from the Rayleigh-Ritz identity:

λ max := max b ⊤ •1 [GC ν (Z) + GC ν (Z) ⊤ ]b 1 b ⊤ 1 b 1 = λ 1 . Since b ⊤ 1 GC ν (Z)b •1 = λ 1 /2 and because b ⊤ 1 GC ν (Z)b 1 = GGM D ν (f 1 , f 1
), the result follows. (iii) Again, the Rayleigh-Ritz identity yields:

λ min := min b ⊤ •K [GC ν (Z) + GC ν (Z) ⊤ ]b •K b ⊤ •K b •K = λ K . Then, b ⊤ •K GC ν (Z)b •K = GGM D ν (f •K , f •K ) = λ •K /2. The index GGM D ν (f •k , f •k
) represents the variability of the observations projected onto component f •k . When this variability is null, then the eigenvalue is null (i). In the same time, there is neither co-variability in the Gini sense between f •k and another axis f

•ℓ , that is GGM D ν (f •k , f •ℓ ) = 0.
In the Gaussian case, because the Gini correlation matrix is positive semidefinite, the eigenvalues are non-negative, then GGM D is null whenever it reaches its minimum.

Proposition 4.3. Let Z ∼ N (0, 1) and let X represent identically distributed Gaussian random variables, with distribution N (0, ρ) such that Var(X k ) = 1 for all k = 1, . . . , K and let γ 1 , . . . , γ K be the eigenvalues of Var(X). Then the following assertions holds:

(i) Tr[GC ν (X)] = Cov(Z, Φ(Z))Tr[Var(X)]. (ii) µ k = Cov(Z, Φ(Z))γ k for all k = 1, . . . , K. (iii) |GC ν (X)| = Cov K (Z, Φ(Z))|Var(X)|. (iv) For all ν > 1: µ k Tr[GC ν (X)] = γ k Tr[Var(X)]
, ∀k = 1, . . . , K.

Proof. From Theorem 3.2:

GC ν (X) = σ -1 Cov(Z, Φ(Z))Var(X).
From [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] (Chapter 26), when Z ∼ N (0, 1),

Cov(Z, Φ(Z)) = 1 2 √ π ≈ 0.2821.
Then the results follow directly.

Point (iv) shows that the eigenvalues of the standard PCA are proportional to those issued from the generalized Gini PCA. Because each eigenvalue (in proportion of the trace) represents the variability (or the quantity of information) inherent to each axis, then both PCA techniques are equivalent when X is Gaussian:

µ k Tr[GC ν (X)] = γ k Tr[Var(X)]
, ∀k = 1, . . . , K ; ∀ν > 1.

The R N -Euclidean space

In classical PCA, the duality between R N and R K enables the eigenvectors and eigenvalues of R N to be deduced from those of R K and conversely. This duality is not so obvious in the Gini PCA case. Indeed, in R N the Gini variability between the observations would be measured by GC ν ( Z) := -2ν

N (N -1) (R c z ) ⊤ Z, and subsequently the idea would be to derive the eigenvalue equation related to R N ,

[GC ν ( Z) + GC ν ( Z) ⊤ ] b •k = λ •k b •k .
The other option is to define a basis of R N from a basis already available in R K . In particular, the set of principal components {f 1 , . . . , f •k } provides by construction a set of normalized and orthogonal vectors. Let us rescale the vectors f •k such that:

f •k = f •k GM D ν (f •k , f •k ) . Then, { f 1 , . . . , f •k } constitutes an orthonormal basis of R K in the Gini sense since GM D ν ( f •k , f •k ) = 1
. This basis may be used as a projector of the variables z •k onto R N . Let F be the N × K matrix with f •k in columns. The projection of the variables z •k in R N is given by the following Gini correlation matrix:

V := -2ν N (N -1) F ⊤ R c z ,
whereas it is given by 1 N F ⊤ Z in the standard PCA, that is, the matrix of Pearson correlation coefficients between all f •k and z •ℓ . The same interpretation is available in the Gini case. The matrix V is normalized in such a way that V ≡ [v kℓ ] are the G-correlations indices between f •k and z •ℓ . This yields the ability to make easier the interpretation of the variables projected onto the new subspace.

Interpretations of the Gini PCA

The analysis of the projections of the observations and of the variables are necessary to provide accurate interpretations. Some criteria have to be designed in order to bring out, in the new subspace, the most significant observations and variables.

Observations

The absolute contribution of an observation i to the variability of a principal component f •k is:

ACT ik = f ik Ψ(R(f ik )) GGM D ν (f •k , f •k ) .
The absolute contribution of each observation i to the generalized Gini Mean Difference of f

•k (ACT ik ) is interpreted as a percentage of variability of GGM D ν (f •k , f •k ), such that N i=1 ACT ik = 1. This provides the most impor- tant observations i related to component f •k with respect to the information GGM D ν (f •k , f •k ).
On the other hand, instead of employing the Euclidean distance between one observation i and the component f •k , the Manhattan distance is used. The relative contribution of an observation i to component f •k is then:

RCT ik = |f ik | f i• 1 .
Remark that the gravity center of {f 1 , . . . , f •K } is g := ( f1 , . . . , f•K ) = 0. The Manhattan distance between observation i and g is then K k=1 |f ik -0|, and so

RCT ik = |f ik | f i• -g 1 .
The relative contribution RCT ik may be interpreted rather as the contribution of dimension k to the overall distance between observation i and g.

Variables

The most significant variables must be retained for the analysis and the interpretation of the data in the new subspace. It would be possible, in the same manner as in the observations case, to compute absolute and relative contributions from the Gini correlation matrix V ≡ [v kℓ ]. Instead, it is possible to test directly for the significance of the elements v kℓ of V in order to capture the variables that significantly contribute to the Gini variability of components f

•k . Let us denote U ℓk := Cov(f •ℓ , R c z •k ) with R c z •k the (decumulative) centered rank vector of z •k raised to an exponent of ν -1 and U •ℓ := Cov(f •ℓ , R c f •ℓ ).
Those two Gini covariances yield the following U -statistics:

U ℓk = U ℓk U •ℓ = v kℓ .
Let U 0 ℓk be the expectation of U ℓk , that is [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF], U ℓk is an unbiased and consistent estimator of U 0 ℓk . From Theorem 10.4 in [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF], Chapter 10, we asymptotically get that √ N (U ℓk -U 0 ℓk ) a ∼ N . Then, it is possible to test for:

U 0 ℓk := E[U ℓk ]. From Yitzhaki &
H 0 : U 0 ℓk = 0 H 1 : U 0 ℓk = 0.

Let σ2

ℓk the Jackknife variance of U ℓk , then it is possible to test for the null under the assumption N → ∞ as follows:9 U ℓk σℓk a ∼ N (0, 1).

The usual PCA enables the variables to be analyzed in the circle of correlation, which outlines the correlations between the variables z •k and the components f •ℓ . In order to make a comparison with the usual PCA, let us rescale the U -statistics U ℓk . Let U be the K × K matrix such that U ≡ [U ℓk ], and u •k the k-th column of U. Then, the absolute contribution of the variable z •k to the component f •ℓ is:

ACT kℓ = U ℓk u •k 2 .
The measure ACT kℓ yields a graphical tool aiming at comparing the standard PCA with the Gini PCA. In the standard PCA, cos 2 θ (see Figure 2 below) provides the Pearson correlation coefficient between f 1 and z •k . In the Gini PCA, cos 2 θ is the normalized Gini correlation coefficient ACT k1 thanks to the ℓ 2 norm. It is worth mentioning that the circle of correlation does not provide the significance of the variables. This significance relies on the statistical test based on the U -statistics exposed before. Because ACT depends on the ℓ 2 metric, it is sensitive to outliers, and as such, the choice of the variables must rely on the test of U 0 ℓk only.

Monte Carlo Simulations

In this Section, it is shown with the aid of Monte Carlo simulations that the usual PCA yields irrelevant results when outlying observations contaminate the data. To be precise, the absolute contributions computed in the standard PCA based on the variance may lead to select outlying observations on the first component in which there is the most important variability (a direct implication of the maximization of the variance). In consequence, the interpretation of the PCA may inflate the role of the first principal components. The Gini PCA dilutes the importance of the outliers to make the interpretations more robust and therefore more relevant. The mean squared errors of the eigenvalues are computed as follows:

M SE λ k = 1,000 i=1 (λ oi k -λ k ) 2 1, 000 ,
where λ oi k is the eigenvalue computed with outlying observations in the sample. The MSE of ACT et RCT are computed in the same manner.

We first investigate the case where the variables are highly correlated in order to gauge the robustness of each technique (Gini for ν = 2, 4, 6 and variance). The correlation matrix between the variables is given by: ρ =     1 0.8 0.9 0.7 0.8 1 0.8 0.75 0.9 0.8 1 0.6 0.7 0.75 0.6 1

   
As can be seen in the matrix above, we can expect that all the information be gathered on the first axis because each pair of variables records an important linear correlation. The repartition of the information on each component, that is, each eigenvalue in percentage of the sum of the eigenvalues is the following.

Eigenvalues

Gini The first axis captures around 82% of the variability of the overall sample (before contamination). Although each PCA method yields the same repartition of the information over the different components before the contamination of the data, it is possible to show that the classical PCA is not robust. For this purpose, let us analyze Figures 3a-3d below that depict the MSE of each observation with respect to the contamination process described in Algorithm 1 above.

On the first axis of Figure 3a, the absolute contribution of each observation (among 500 observations) is not stable because of the contamination of the data, however the Gini PCA performs better. The MSE of the ACTs measured during to the contamination process provides lower values for the Gini index compared with the variance. On the other hand, if we compute the standard deviation of all these MSEs over the two first axis, again the Gini methodology provides lower variations (see Table 2). Let us take now an example with less correlations between the variables in order to get a more equal repartition of the information on the first two axes.

ρ =    

1

-0.5 0.25 0.5 -0.5 1 -0.9 1 0.25 -0.9 1 -0.25 0.5 0 -0.25

1    
The repartition of the information over the new axes (percentage of each eigenvalue) is given in Table 3. When the information is less concentrated on the first axis (55% on axis 1 and around 35% on axis 2), the MSE of the eigenvalues after contamination are much more important for the standard PCA compared with the Gini approach (2 to 3 times more important). Although the fourth axis reports an important MSE for the Gini method (ν = 6), the eigenvalue percentage is not significant (1.56% Let us now have a look on the MSE of the absolute contributions of each observation (N = 500) for each PCA technique (4a-4b). We obtain the same kind of results, with less variability on the second axis. In Figures 4a-4b, it is apparent that the classical PCA based on the ℓ 2 norm exhibits much more ACT variability (black points). This means that the contamination of the data can lead to the interpretation of some observations as significant (important contribution to the variance of the axis) while they are not (and vice versa). On the other hand, the MSE of the RCTs after contamination of the data, Figures 4c-4d, are less spread out for the Gini technique for ν = 4 and ν = 6, however for ν = 2 there is more variability of the MSE compared with the variance. This means that the distance from one observation to an axis may not be reliable (although the interpretation of the data rather depends on the ACTs). 4 indicates that the PCA based on the variance is less stable about the values of the ACTs that provide the most important observations of the sample. This may lead to irrelevant interpretations.

Application on cars data

We propose a simple application with the celebrated cars data (see the Appendix). 10 The dataset is particularly interesting since there are highly correlated variables as can be seen in the Pearson correlation matrix given in Table 5. Also, the dataset is composed of some outlying observations (Figure 5): Ferrari enzo (x 1 , x 2 , x 5 ), Bentley continental (x 2 ), Aston Martin (x 2 ), Land Rover discovery (x 5 ), Mercedes class S (x 5 ), Smart (x 5 , x 6 ).

The overall information (variability) is partitioned over six components (Table 6). two axes are sufficient to project the data, the Gini PCA and the standard PCA yield the same share of information on each axis. However, we can expect some differences for absolute contributions ACT and relative contributions RCT . The projection of the data is depicted in Figure 6, for each method. As depicted in Figure 6, the projection is very similar for each technique. The cars with extraordinary (or very low) abilities are in the same relative position in the four projections: Land Rover Discovery Td5 at the top, Ferrari Enzo at the bottom right, Smart Fortwo coupé at the bottom left. However, when we improve the coefficient of variability ν to look for what happens at the tails of the distributions (of the two axes), we see that more cars are distinguishable: Land Rover Defender, Audi TT, BMW Z4, Renault Clio 3.0 V6, Bentley Contiental GT. Consequently, contrary to the case ν = 2 or the variance, the projections with ν = 4, 6 allow one to find other important observations, which are not outlying observations that contribute to the overall amount of variability. For this purpose, let us first analyze the correlations between the variables and the new axes in order to interpret the 7 to 10. Some slight differences appear between the Gini PCA and the classical one based on the variance. The theoretical Section 4 indicates that the Gini methodology for ν = 2 is equivalent to the variance when the variables are Gaussian. On cars data, we observe this similarity. In each PCA, all variables are correlated with Axis 1 and weight with Axis 2. However, when ν increases, the Gini methodology allows outlying observations to be diluted so that some variables may appear to be significant, whereas they are not in the variance case. Tables 8 and9 (ν = 4, 6) show that Axis 2 is correlated to speed (not weight as in the variance PCA). In this respect the absolute contributions must describe the cars associated with speed on Axis 2. Indeed, the Land Rover discovery, a heavy weight car, is no more available on Axis 2 for the Gini PCA for ν = 2, 4, 6 (Figures 8, 9

eigenvalues in % Gini ν = 2 Gini ν = 4 Gini ν = 6

Conclusion

In this paper, it has been shown that the geometry of the Gini covariance operator allows one to perform Gini PCA, that is, a robust principal component analysis based on the ℓ 1 norm.

To be precise, the variance may be replaced by the Gini Mean Difference, which captures the variability of couples of variables based on the rank of the observations in order to attenuate the influence of the outliers. The Gini Mean Difference may be rather interpreted with the aid of the generalized Gini index GGM D ν in the new subspace for a better understanding of the variability of the components, that is, GGM D ν is both a rank-dependent measure of variability in [START_REF] Yaari | The Dual Theory of Choice Under Risk[END_REF] sense and also an eigenvalue of the Gini correlation matrix.

Contrary to many approaches in multidimensional statistics in which the standard variance-covariance matrix is used to project the data onto a new subspace before deriving multidimensional Gini indices (see e.g. Banerjee ( 2010)), we propose to employ the Gini correlation indices (see [START_REF] Yitzhaki | The Gini Methodology. A Primer on a Statistical Methodology[END_REF]). This provides the ability to interpret the results with the ℓ 1 norm and the use of U -statistics to measure the significance of the correlation between the new axes and the variables.

This research may open the way on data analysis based on Gini metrics in order to study multivariate correlations with categorical variables or discriminant analyses when outlying observations drastically affect the sample.

Figure 2 :

 2 Figure 2: Circle of correlation

Algorithm 1 :

 1 Monte Carlo Simulation Result: Robust Gini PCA with data contamination 1 θ = 1 [ θ is the value of the outlier] ; 2 repeat 3 Generate a 4-variate normal distribution X ∼ N , N = 500 ; 4 Introduce outliers in 1 row of X: X o ji := θX ji with j = 1, . . . , 4 [for a random row localization]; 5 For each method (Variance and Gini), the ACT and RCT are computed for the axes 1 and 2 on the contaminated matrix X o ; 6 until θ = 1000 [increment of 1]; 7 return Mean squared Errors of eigenvalues, ACT and RCT ;

Figure 3 :

 3 Figure 3: ACT 1 , ACT 2 , RCT 1 and RCT 2 Gini ν = 2 Gini ν = 4 Gini ν = 6 Variance Axis 1 6.08 6.62 7.41 12.09 Axis 24.07 5.12 13.37 2.98

Figure 4 :

 4 Figure 4: ACT 1 , ACT 2 , RCT 1 and RCT 2

Figure 5 :

 5 Figure 5: Box plots

Figure 6 :

 6 Figure 6: Projections of the cars

  , 10). Note that the red line in the Figures represents the mean share of the information on each axis, i.e. 100%/24 cars = 4.16% of information per car.

Figure 7 :

 7 Figure 7: Variance ACTs

Figure 9 :

 9 Figure 8: Gini ACTs (ν = 2)

Table 1 :

 1 Eigenvalues and their MSE

		ν = 2	Gini ν = 4 Gini ν = 6	Variance
	eigenvalues (%)	81.65341	82.31098	82.28372	81.11458
	Axis 1
	MSE	12.313750 12.196975 12.221840 15.972710
	eigenvalues (%)	10.90079	10.47317	10.46846	11.35471
	Axis 2
	MSE	11.478541 11.204504 10.818344 10.688924
	eigenvalues (%)	5.062329	4.996538	5.088865	5.112817
	Axis 3
	MSE	2.605312	2.608647	2.799180	4.687323
	eigenvalues (%)	2.383476	2.219311	2.15896	2.417897
	Axis 4
	MSE	1.541453	1.826055	3.068596	2.295100

Table 2 :

 2 Standard deviation of the MSE of the ACTs on the two first axis

	Gini ν = 6 Variance

  ).

	eigenvalues (%)	Gini ν = 2 Gini ν = 4 Gini ν = 6	Variance
	eigenvalues (%)	55.3774	55.15931	54.96172	55.08917
	Axis 1
	MSE	17.711023 14.968196 12.745760	38.929147
	eigenvalues (%)	35.8385	35.86216	35.8745	36.06118
	Axis 2
	MSE	14.012198 16.330350 18.929923	30.948674
	eigenvalues (%)	7.227274	7.345319	7.527222	7.329535
	Axis 3
	MSE	4.919686	4.897820	5.036241	6.814252
	eigenvalues (%)	1.556831	1.633214	1.636561	1.520114
	Axis 4
	MSE	1.149770	7.890184	14.047539 1.438904

Table 3 :

 3 Eigenvalues and their MSE

Table 4 :

 4 Standard deviation of the MSE of the ACT on the two first axes As in the previous example of simulation, Table

		Gini ν = 2 Gini ν = 4 Gini ν = 6 Variance
	Axis 1	7.50	7.86	8.46	11.92
	Axis 2	0.62	0.75	0.77	11.24

Table 5 :

 5 capacity x 1 power x 2 speed x 3 weight x 4 width x 5 length x 6 Correlation matrix

	x 1	1.000	0.954	0.885	0.692	0.706	0.663
	x 2	0.954	1.000	0.933	0.528	0.729	0.663
	x 3	0.885	0.933	1.000	0.466	0.618	0.578
	x 4	0.692	0.528	0.466	1.000	0.477	0.794
	x 5	0.706	0.729	0.618	0.477	1.000	0.591
	x 6	0.663	0.663	0.578	0.794	0.591	1.000

Table 7 :

 7 Correlations Axes / variables (significance 5%)

	Gini (ν = 2)	capacity	power	speed	weight	width	length
	correlation -0.974	-0.945	-0.872	0.760	-0.933	-0.823
	Axe 1
	U -stat	-56.416 -25.005 -10.055 -4.093 -24.837 -12.626
	correlation -0.032	-0.241	-0.405	0.510	0.183	-0.379
	Axe 2
	U -stat	-0.112	-0.920	-1.576	2.897 0.526	1.666
	Gini (ν = 4)	capacity power speed	weight width	length
	correlation 0.982	0.948	0.797	0.858	0.952	0.888
	Axe 1
	U -stat	8.990	8.758 4.805 9.657 8.517 8.182
	correlation -0.021	0.207	0.516	-0.279 -0.147	-0.246
	Axe 2
	U -stat	-0.095 0.817	2.299 -1.773 -0.705 -1.200

Table 8 :

 8 Correlations Axes / variables (significance 5%)

	Gini (ν = 6)	capacity power	speed	weight	width	length
	valeurs -0.781	-0.759	-0.598	-0.730	-0.755	-0.701
	Axe 1
	U -stat	-4.036 -3.903 -3.137 -3.125 -3.882 -3.644
	valeurs 0.019	-0.170	-0.570	0.153	0.125	0.218
	Axe 2
	U -stat	0.089	-0.734 -1.914 0.734	0.569	0.906

Table 9 :

 9 Correlations Axes / variables (significance 5%, 10%)

	Variance	capacity power	speed	weight width	length
	valeurs 0.962	0.923	0.886	0.756	0.801	0.795
	Axe 1
	U -stat	11.802 11.322 10.866 9.282 9.825 9.752
	valeurs -0.126	-0.352	-0.338	0.575	-0.111	0.504
	Axe 2
	U -stat	-0.307 -0.855	-0.821	1.396 -0.269 1.223

Table 10 :

 10 Correlations Axes / variables (significance 5%, 10%)

See Giorgi (2013) for an overview of the "Gini methodology".

See Furman & Zitikis (2017) for the link between variability (risk) measures and the

In most cases, PCA is performed on scaled (and centered) variables, otherwise variables with large scales might alter interpretations. Thus, it will make sense, later on, to assume that components of X have identical distributions. At least the first two moments will be equal.

but other norms have also been considered in statistical literature, such as the Froebenius norm in the Eckart-Young theorem, or the maximum of singular values -also called 2-(induced)-norm.

In risk analysis ν ∈ (0, 1) denotes risk lover decision makers (max-max approach), whereas ν > 1 stands for risk averse decision makers, and ν → ∞ extreme risk aversion (max-min approach).

By definition GM D ν (x •ℓ , x •ℓ ) ≥ 0 for all ℓ = 1, . . . , K. As we impose that x •ℓ = c1 N , the condition becomes GM D ν (x •ℓ , x •ℓ ) > 0.

Strictly speaking[START_REF] Yaari | The Dual Theory of Choice Under Risk[END_REF] and[START_REF] Yaari | A Controversial Proposal Concerning Inequality Measurement[END_REF] suggests probability distortion functions Ψ : [0, 1] → [0, 1], which does not necessarily coincide to our case.

As indicated by[START_REF] Yitzhaki | Calculating Jackknife Variance Estimators for Parameters of the Gini Method[END_REF], the efficient Jackknife method may be used to find the variance of any U -statistics.

An R markdown for Gini PCA is available: https://github.com/freakonometrics/GiniACP/ Data from Michel Tenenhaus's website (see also the Appendix): https://studies2.hec.fr/jahia/webdav/site/hec/shared/sites/tenenhaus/acces anonyme/home/fichier excel/auto 2004.xls

Appendix cars capacity x 1 power x 2 speed x 3 weight x 4 width x 5 length x 6 Citroën C2 1.1 Base