turbulent eddies convecting within the boundary layer of an aerofoil are scattered into sound at the trailing edge .this , turbulent boundary layer trailing edge noise , is a major source of broadband noise for an aerofoil in a uniform flow .it is the dominant noise source for large wind turbines  . for installed fans ,trailing edge noise corresponds to the minimum achievable noise level  .although this paper focuses on trailing edge noise , rotors and propellers are subject to several other noise sources  . some of these are broadband , for example leading edge noise due to upstream turbulence , tip vortex induced noise and stall noise  .others are tonal , as in the case of : steady loading in the reference frame of the rotor ( ,  ) ; periodic unsteady loading produced by stationary distortions in the flow .  trailing edge noisecan be predicted in the time domain by solving the ffowcs - williams & hawkings ( fwh ) equation ( ) , as demonstrated by  .however , this type of prediction requires numerical differentiations and the calculation of retarded times that are computationally expensive .most importantly , the roles of blade geometry and operating conditions become clearer with the frequency domain formulations that are the subject of this paper .another key advantage of the frequency domain formulation is that it is more suited for a statistical description of the noise sources .one of the most successful frequency domain formulations was developed by amiet ( ) .assuming knowledge of the pressure fluctuations travelling towards the trailing edge , amiet s formulation gives an analytical expression for the power spectral density in the far field .this analytical expression makes it efficient and attractive for fast turn - around applications .it has been applied to numerous applications including low speed fans ( ) , helicopters ( ) and wind turbines ( ) .although amiet s approach was initially restricted to high frequencies relative to the chord ( ) , it has been extended to lower frequencies ( ) by  .the effect of rotation on trailing edge noise modelling was analysed in and for a rotor in a uniform flow .the first step is to estimate the instantaneous power spectral density radiating from the blade , while it is located at a particular azimuthal angle around the hub .secondly , the power spectra are averaged in time .the final expression involves a doppler factor of the form , where is the source frequency and the frequency of the observed sound .the exponent takes the value 1 in , 2 in and -2 in .it is clear that an independent formulation is needed to explain these discrepancies and identify the correct form of schlinker and amiet s theory .this paper presents a new formulation of trailing edge noise for small rotating blade elements ( section [ cha : kim - george - approach ] ) .it is analogous to the formulations developed by  for ducted fans , although cascade effects are not taken into account .the derivation starts with goldstein s ( ) expression for loading noise and uses the green s function of , which incorporates the effect of mean flow , to derive an expression for pressure in the frequency domain .this expression is then used to derive the instantaneous psd and the time averaged psd .the new formulations generalize the compact source solution of  , who used the expression of  for a rotating point load .they incorporate an accurate description of the blade rotation , hence of the effect of blade acceleration ; this effect is not taken into account in schlinker and amiet s formulation where the blade is assumed to be in uniform rectilinear motion .the time averaged formulation can therefore serve to explore the validity of schlinker and amiet s approach .the derivation of the new formulation was inspired by  who first compared schlinker and amiet s model with a solution of the fwh equation .their formulation of schlinker and amiet s model differed significantly from that of  but was not accompanied by an alternative derivation .also , they neglected the effect of mean flow and spanwise wavenumber .both effects are taken into account in our new formulation .we revisit schlinker and amiet s theory and present simple derivations for some key steps that help to obtain the correct result ( section [ sec : amiets - appr - rotat ] ) .of particular interest is the derivation showing how a theory derived for a stationary aerofoil can be generalized to an aerofoil in uniform motion .finally , the results are validated by comparing the new formulation to schlinker and amiet s for multiple test cases , including a wind turbine , a cooling fan and an aircraft propeller ( section [ sec : results ] ) .) are moving at mach in a quiescent medium .the source locations are at current time and at emission time.,height=170 ]  ) are moving at mach in a quiescent medium .the source locations are at current time and at emission time.,height=170 ]   consider the problem of trailing edge noise radiation in a uniform flow from a stationary rotor towards a stationary observer .although the results of this section will be applicable to propeller blades or fans , the rotor is described as a wind - turbine whereby the blade is being rotated by the incoming flow ( figure [ fig : rotor_reference_frame ] ) . to derive the pressure field in the frequency domain we start by expressing the pressure field in the time domain . from goldstein ( ) , loading noise can be expressed as where is the interval of time over which sound is emitted ( will be allowed to become infinitely large ) , is the blade planform and a free field green s function that satisfies the convected wave equation where ( with in the minus -direction as shown in figure [ fig : rotor_reference_frame ] ) .compared to alternative methods , a convected green s function simplifies the derivation by taking care of sound waves convection by the flow . derived an expression for that is analogous to the free field green s function , where , , and denote respectively the ( present ) source position , the observer position , the reception time and the emission time , and where the phase radius and amplitude radius are defined as ,\end{aligned}\ ] ] where . the geometric interpretation of and ( )is given in figure [ fig : garrick_watkins ] , for the equivalent problem of a rotor and observer moving at mach in a quiescent medium .the green s function takes a simple form in the far field ( ) when the source position is expressed in cylindrical coordinates and the observer position in spherical emission coordinates , where corresponds to in figure [ fig : garrick_watkins ] for ( present source at the hub ) : where the subscript means that the variable has been divided by the doppler factor .the above expressions can be obtained by applying taylor s approximation to equation  about the origin of the coordinate system , retaining terms up to order 0 for and 1 for , and by converting the observer position to emission coordinates .   since amiets trailing edge noise theory is expressed in the frequency domain , we seek a general expression for pressure in the frequency domain . taking the fourier transform over observer time of  yields where the fourier transform pair is defined as   the frequency domain green s function is obtained by taking the fourier transform over of  and incorporating the far field approximation .the result is further decomposed into a series of azimuthal modes ( see appendix [ sec : far - field - greens ] ) to simplify the computation of the gradient , where we have introduced the axial wavenumber and the radial wavenumber .  from figure[ fig : blade_coordinates](b ) , the lift lies in the plane ( ) and can be expressed in terms of the magnitude and the pitch angle .the dot product can be expressed in terms of the gradients of in the azimuthal and axial directions that are easily derived from . substituting the expression for the dot product into  yields a harmonic pressure of the form } { \int_{-t}^{t } l e^{-i(\omega \tau - n \gamma)}\ , d\tau}\ , d\sigma},\end{gathered}\ ] ] where is the magnitude of a modal wavenumber orthogonal to the blade planform .we now turn to the rotational motion of the source .we will assume that the pitch change axis ( pca ) of the blade is at the trailing edge ( te ) and that the trailing edge alignment and face alignment ( ) are 0 .in other words , lean and sweep are both neglected .the geometry of the blade is illustrated in figure [ fig : blade_coordinates ] . from figure[ fig : blade_coordinates](b ) the azimuthal angle of a source within the blade planform is .if tends to infinity , the time integral in  reduces to : the effect of the blade rotation is to select , for each azimuthal mode , the frequency from the spectrum of the blade loading , so   finally , the lift is given by amiet s theory ( detailed in section [ sec : trailing - edge - noise ] ) : for every spanwise wavenumber , a gust of amplitude travelling at convection speed and scattered at the trailing edge , gives rise to a pressure jump along the chord , where is the blade response function ( amiet , ) and is the chordwise source position measured from the trailing edge ( as shown in figure [ fig : blade_coordinates ] ) .following amiet , the gust amplitude is a function of angular frequency and radial wavenumber . here, we express the amplitude as a function of chordwise wavenumber and radial wavenumber .since the chordwise wavenumber of the gust equals , it is easy to show that .the blade loading due to trailing edge noise is therefore given by as illustrated in figure [ fig : blade_coordinates ] , the source coordinates can be expressed in terms of as the phase , where is a chordwise acoustic wavenumber : note how the acoustic wavenumbers in blade coordinates are obtained by rotating by angle around the pca .after expressing the lift in terms of amiet s blade response function using  and since in , we find that the impact of the lift on the pressure field is measured through the acoustically weighted lift defined as for which an analytical expression will be provided in equation .the frequency domain pressure at the observer location can be expressed as where .we now seek an expression for the instantaneous ( wigner - ville ) spectrum  for direct comparison with schlinker and amiet s theory ( equation ) .an autocorrelation function for the pressure at the observer position can be defined as ,\ ] ] where denotes the expected value and the complex conjugate .since the blade element is rotating around the axis , this function is periodic in with period .the instantaneous spectrum is defined as the fourier transform of over .an equivalent definition is where $ ] is a cross - correlation in the spectral domain .equation  states that the instantaneous spectrum is the inverse fourier transform over frequency of .   from ,the spectral cross - correaltion is given by ^ 2 }  \sum_{n=-\infty}^{+\infty } \sum_{n'=-\infty}^{+\infty } e^{i ( n - n ' ) ( \gamma_o - \pi/2 ) } \times \\{\int_{r_h}^{r_t }   { \int_{r_h}^{r_t }    { { \int_{-\infty}^{+\infty }     { { \int_{-\infty}^{+\infty }      \frac{k_l c}{u_c } \frac{k_l ' c'}{u_c ' } e^{i(k_r r - k_r ' r ' ) } \psi_l^ * \psi_l ' j_n ( k_r r ) j_{n'}(k_r ' r ' ) \\     e[p^\star(k_x , k_r ) p(k_x ' , k_r ' ) ]    \ , dk_r ' } }   \ , dk_r } }  \ , dr ' } \ , dr},\end{gathered}\ ] ] where the primed wavenumbers and non - primed wavenumbers are associated respectively with frequency and .we will now derive two approximate expressions and for , by simplifying the integrals over and respectively .these two expressions will then be combined to give an expression for that has hermitian symmetry ; this is necessary for the instantaneous psd to be real .  from  ( appendix d ) , if we assume that the flow is approximately 2d and that it is uniform across the span ( strip theory assumption ) then the two gusts are correlated only when their wavenumbers are equal : and ( i.e. , from the definition of below equation , if ) , and we have = u_c ' \ , \delta(k_r ' - k_r ) \ , \delta\left(\varpi - ( n ' - n)\omega\right )  \phi_{qq}(k_x ' , k_r'),\ ] ] where is the wavenumber spectrum of the aerofoil surface pressure produced by the turbulence .the above equation assumes that the convection velocity is constant over the area where and are correlated . for a given radius , that regionis defined by , where is slightly larger than the turbulence correlation length .since , variations in convection velocity are negligible over that region provided that , i.e. . using corco s equation , at high frequency , ( where is a constant of order one ) so the necessary condition becomes : the rotation speed should be much smaller than the source frequency .this corresponds to the assumption made by amiet ( see introduction of section [ sec : amiets - appr - rotat ] ) .substituting in  leads to , where  ^ 2},\\ \label{eq : in_init } & i_{n , n'}'(\omega , \varpi , r ) \equiv { { \int_{-\infty}^{+\infty } l_{n , n'}(k_r ' ) \phi_{qq}(k_x ' , k_r')\ , dk_r ' } } , \\ \label{eq : in1 }  & l_{n , n'}(k_r ' ) \equiv { \int_{r_h}^{r_t }     \frac{k_l c k_l ' c'}{u_c } \psi_l^ * \psi_l '    j_n\left(k_rr\right ) j_{n'}\left(k_r ' r ' \right ) e^{i k_r'(r - r ' ) }   \ , dr'}.\end{aligned}\ ] ] the integration over may be truncated at . in this narrow strip ,the flow variables can be considered constant in the amplitude terms of  ( , , , , and ) : they are equal to their values at .we seek to simplify  and put it in an exponential form to capture its phase .this is needed to analyse the spanwise behaviour of the solution .the main issue is with , which may be studied using a power series expansion .however , for simplicity , we will neglect the phase changes in and assume so that takes the form where , making the change of variable , the bessel function in  can be rewritten as an integral ( ) : substituting into , where . if is large relative to the wavelengths of interest in this problem , i.e. if , then the second integral in simplifies to the above simplification may not always be possible in which case the integral can be solved numerically . substituting  into  to get , and combining the result with and  yields   we can derive a different expression for , by approximating the integral over in equation ( [ eq : spp2 ] ) and by following the steps in equations ( [ eq : rpp ] ) to ( [ eq : q ] ) to get where where and the wavenumbers , , and are evaluated at radius .note that in deriving we have used the conjugate form of equation ( [ eq : jm_integral ] ) to express .we have also chosen this time instead of as the first argument of .this is justified since equals .if we change to and exchange and then turns into .this is sufficient to show that equals the conjugate of .we express as which satisfies the hermitian symmetry required for the instantaneous spectrum to be real valued .substituting equations ( [ eq : rpp ] ) and ( [ eq : rpp2 ] ) into ( [ eq : rpphermitian ] ) and then into , where we have made the change of variable and .note that the expression of equation ( [ eq : instant_spectrum_kg_final ] ) is always real but it can become negative due to a known defect of the wigner - ville spectrum , which does not define a true spectral density .   the time averaged psd is obtained by retaining the mode in , so and the prime and non - prime quantities are at the same frequency , and since , we can write where , from ( [ eq : in_old ] ) and , our numerical experiments using the above equations indicate that is always positive . for a small blade element of span ,. ]  consider a flat plate in a uniform flow of mach number at zero angle of attack ( figure [ fig : isolated_aerofoil_frame ] ) .the observer location is expressed using a cartesian coordinate system , with in the chordwise direction and pointing downstream and in the vertical direction . from , using the large span assumption ( equation ( [ eq : delta_simplification ] ) ) , the psd at frequency is given by where and from  the wavenumber spectrum and spanwise correlation length are given by  the term represents the exponential decay rate of the spanwise coherence function ; measured for a naca 0012 at mach 0.11 and zero angle of attack . for simplicity , this value will be used for all mach numbers in section [ sec : results ] . for a stationary flat plate in a uniform flow ( )    the acoustic lift , defined in equation ,can be expressed as \right\},\ ] ] where the overbar denotes normalisation by and the mach number of the turbulent eddies in the boundary layer close to the trailing edge .the square roots in equation ( [ eq : psi_l_expression ] ) have a branch cut along the negative real axis .the wavenumber is a function of the spanwise wavenumber .it is defined as so that the imaginary part of is always negative .this is required for the error functions in to converge in the far field .the square roots in equations  and  are classicaly defined with a branch cut along the negative real axis .figure [ fig : acoustic_lift_ky ] , shows how varies with spanwise wavenumber , for a constant frequency of 1 khz ( solid line ) .the transition between supercritical and subcritical gusts occurs at .it can be seen that a sharp increase occurs near .that increase is non - physical and occurs because the governing equation reduces from a helmholtz equation to a laplace equation .this was first pointed out by  .since is asymptotically constant at low and high values of we define as a piecewise function , such that it is constant for supercritical gusts ( ) , and for highly subcritical gusts ( ) . between those values, we use a linear interpolation of in terms of .the piecewise implementation of is shown as a dashed line in the figure [ fig : acoustic_lift_ky ] .   as a function of spanwise wavenumber , at mach and frequency .the radiating wavenumber is defined assuming an observer at .the transition between supercritical and subcritical gusts occurs at .the acoustic lift based on equation  becomes unphysical near the supercritical - subcritical transition ( solid line , see  ) , so we assume that is constant for and , and use a linear interpolation as a function of for ( dashed line ) ., height=188 ]   the psd of the incoming pressure fluctuations at the trailing edge , denoted , can be measured experimentally .if no experimental data is available , it can be estimated by using empirical low - order models .these low order models are expressed in terms of parameters characterising the boundary layer , such as the boundary layer thickness , displacement thickness , wall shear stress etc .a review of the different models is given by  .this paper uses the model of  which gives where is the boundary layer displacement thickness , the chordwise flow velocity and , where is the angle of attack and    a number of approximations were made in this section to simplify the comparison between the formulation of section [ cha : kim - george - approach ] and that of amiet ( section [ sec : amiets - appr - rotat ] ). these approximations may need to be replaced with more accurate models for practical applications .the first approximation is the equation , giving the correlation length , that was derived by corcos ( ) for a turbulent boundary layer over a flat plate with zero pressure gradient .a more general equation has been proposed by  .a second approximation is the spanwise dependence of the acoustic lift described in figure [ fig : acoustic_lift_ky ] .this approximation was made for convenience and does not affect the rest of the analysis .thirdly , the wall - pressure spectrum of chou and george ( ) in section [ sec : trailing - edge - noise]([sec : surf - press - power ] ) may be replaced by more complex models such as that of  .rozenberg s model takes into account the influence of an adverse pressure gradient and should therefore better predict the effect of aerofoil camber and thickness. however , more research is needed in this area and , ideally , the wall - pressure spectrum should be measured experimentally .this section is a detailed and corrected derivation of the theory presented by  . in particular ,the derivation of the present source position is both simpler and more general .this section explains how to apply a theory derived in the wind tunnel reference frame , where a stationary source and observer are immersed in a uniform flow ( see section [ sec : trailing - edge - noise ] ) , to the general case , where both the source and the fluid are moving relative to the observer . observed that the sound emitted by a rotating blade is approximately equal to that emitted by a translating blade at the same location .his observation was based on the expression of  for the pressure radiating from a rotating dipole : the difference between a rotating blade and a translating blade lies in the acceleration of the source in the direction of the observer .amiet argued that when the angular velocity is much larger than the source frequency , the acceleration term becomes negligible .this high frequency assumption will be validated in section [ sec : results ] against the new formulation of section [ cha : kim - george - approach ] . in the following, we will assume that the blade is in rectilinear motion .in the situation we are considering here , the observer is stationary relative to the hub and the blade moves through the fluid ( figure [ fig : rotor_reference_frame ] ) .however the formulae of section [ sec : trailing - edge - noise ] , for a blade in a uniform flow , assume that the observer is stationary relative to the blade .we therefore examine the current problem successively in these two reference frames . in the reference frame of the observer ( ) ,the observer is stationary and the blade moves rectilinearly at mach in a uniform flow of mach .consider a pulse of sound emitted at from the blade .when this pulse is received at the observer position , the blade has moved to the present source position , as illustrated in figure [ fig : reference_frames](a ) .   +  +  in the reference frame of the blade ( ) , the blade is stationary and the source is fixed to the present source position ( ) .the mach number of the flow is and the observer is moving at mach  ( figure [ fig : reference_frames](b ) ) . comparing figure [ fig : reference_frames](b ) with figure [ fig : isolated_aerofoil_frame ] ,this situation is equivalent to that of an isolated aerofoil in a uniform flow provided that the observer is stationary , i.e. . here, the observer is moving relative to the blade which leads to a doppler shift whose impact on the instantaneous psd will be discussed later .the above analysis indicates the steps required to apply the formulae derived for an aerofoil in a uniform flow to the present problem :  * move the origin of the coordinate system from to the present source position ; * rotate the axes of the coordinate system if necessary ; * account for the movement of the observer relative to the blade ( doppler shift ) .   from figure[ fig : reference_frames](a ) , where denotes the propagation time .the above result is more general than the equation ( 46 ) in  that is valid only if the chordwise mach number of the flow equals , i.e. if the angle of attack equals zero .the source position at emission time satisfies , where is the blade tip radius , while for propagation to the far - field , therefore ( provided  ) for an observer in the far field , we can hence assume so that the source is located at the hub .as demonstrated in the following , the propagation time is related to the convected source position .the convected source position is defined as the emission position in the reference frame of the fluid ( figure [ fig : reference_frames](c ) ) . in that reference frame , the fluid is stationary so the wavefronts are spherical and acoustic waves are propagating at the speed of sound .this allows us to relate the propagation time to and since where we have used the fact that is independent of the reference frame ( see figure [ fig : reference_frames ] ) .  in general the convected source position obtained by seeding the flow with a small particle that is released from the source position at emission time and convects at the mach number of the ambient fluid ; corresponds to the position of the particle at reception time . using the reference frame of the observer ( figure [ fig : reference_frames](a ) ) , we get   substituting  into  and taking the square of the result , the above equation leads to a second order polynomial equation in whose solution is given by where denotes the angle between and .   in the reference frame of the blade used in section [ sec : trailing - edge - noise ] , the -axis is pointing spanwise and the -axis is pointing chordwise .let denote the coordinate system centred on the present source position ( ) .for the -axis to point in the spanwise direction , the -coordinate system is rotated by around the -axis ( figure [ fig : x1_to_x2](a ) ) , i.e. where denotes the rotation matrix about the -axis ( see equation ) .similarly , for the -axis to point in the chordwise direction , the -coordinate system is rotated by around the -axis ( figure [ fig : x1_to_x2](b ) ) : where denotes the rotation matrix about the -axis ( see equation ) .  the transform from the observer reference frame ( figure [ fig : reference_frames](a ) ) to the blade reference frameis hence given by    the sound frequency at the observer is shifted compared to the emission frequency .the ratio is called the doppler shift and is a function of the source mach number and the flow mach number . as explained by , the doppler shift is given by where is the unit vector from the convected source position ( in figure [ fig : reference_frames ] ) to the observer position .[ sec : power - spectr - dens ]  it can be shown ( and appendix [ sec : inst - freq - doppl ] ) that in the far field the psd for a moving observer , , is related to the psd for a fixed observer , by , therefore where can be estimated using the isolated aerofoil expression of . furthermore , equation  is expressed in terms of the chordwise mach number of the flow relative to the blade .a derivation of , extending that of  to the case of a non - zero angle of attack , is given in appendix [ sec : chordw - mach - numb ] .the time averaged psd is obtained by averaging the instantaneous psd , i.e. equation , over one rotation of the rotor . since the instantaneous psd is a function of source time , the time increment must be expressed in terms of . if is the number of acoustic periods measured at the observer location during then .the time taken for the source to generate those periods is so .finally , it is convenient to express in terms of the blade azimuthal angle using the relation .the time averaged psd from a rotating blade element is hence given by    the above result agrees with  , provided that is replaced by in the left hand side of ( 54 ) in their report , and and are swapped in the left hand side of their equation ( 56 ) .note that there has been some discrepancy in the literature about the exponent in the doppler term of .for example , initially proposed a value of 1 for the exponent . the same is considered by .this ignores the additional weighting of the time increment and is equivalent to averaging over the angular position of a blade segment .we propose that the correct value of the exponent is 2 . in section[ sec : results ] , evidence for this is provided by comparison of predictions using the two methods presented in sections [ cha : kim - george - approach ] to [ sec : amiets - appr - rotat ] .in this section , both amiet s formulation ( equation ) and our new formulation ( equation ) are applied to several model blade elements .these blade elements have been chosen to cover the range of applications featured in the literature on trailing edge noise : cooling fans  , open propellers  and wind turbines .each blade element is described through its distance from the hub ( radius ) , its geometry ( chord and pitch angle ) , and the set of mach numbers ( blade mach , flow mach numbers relative to the observer and to the blade ) , whose values have been chosen to match a typical situation .for example , a typical wind turbine element has a large chord , low pitch angle and rotates at low mach number in a low wind speed .the specific values chosen here , presented in table [ tab : params ] , were the ones used by .the span is arbitrarily equal to the chord .note that for each blade element , the values are not independent ; the three mach number form a velocity triangle wherein the pitch angle is between and .furthermore , the values would usually change along a full blade ..typical parameters for the blade segments used in three different applications : open - propeller , cooling fan and wind turbine .these parameters are the ones proposed by  .the span is defined as a third of the radius .the wind mach numbers , relative to the observer ( ) or to the blade ( ) can be obtained from the pitch angle and blade speed but are given for completeness . [ cols="<,<,<,<",options="header " , ]   although this paper focuses on noise radiation from small blade elements , the noise spectrum of a full blade may be obtained using strip theory . in strip theory ,one divides the blade into small elements along its radius and sums the ( uncorrelated ) noise spectra generated from each element .limitations and extensions of strip theory are discussed in  .in addition to the approximations discussed in section [ sec : trailing - edge - noise]([sec : discussion ] ) , the thin - blade approximation has been used .this is not a good approximation for angles close to the propeller axis where a numerical approach taking the thickness of the blade into account may be preferable  .figure [ fig : exponents ] compares the sound pressure level directivity obtained using the new formulation and sclinker and amiet s approach , when the exponents of the doppler shift ( see equation ) takes the values 1 , 2 and -2 taken from the literature . a low mach number case ( wind turbine )is shown in figure [ fig : exponents](a ) , and a high mach number case ( open propeller ) in figure [ fig : exponents](b ) .differences of up to db and db can be seen when schlinker and amiet s approach are used with an exponent of ( square symbols ) and ( round symbols ) respectively . the best agreement between the new approach and schlinker and amiet s approachis obtained for an exponent of ( symbols ) , which validates equation .figure [ fig : results ] shows the sound pressure level directivity for the blade segments defined in table [ tab : params ] that model a cooling fan ( figure [ fig:3 ] ) , a wind turbine ( figure [ fig:4 ] ) , and an open propeller at take - off ( figure [ fig:1 ] ) and cruise ( figure [ fig:2 ] ) . in each case , the results are obtained using the new formulation ( dashed line ) and amiet s approach ( solid line ) . in figures [ fig : exponents ] and [ fig : results ] the levels are normalised to give an effective observer distance of and the observer is located at .the two approaches agree to within 1 db in all cases .these results suggest that schlinker and amiet s theory is valid when the acceleration effects are negligible , i.e. , and when amiet s blade response function is applicable , i.e. . since , this ratio is larger than for all the test cases in table [ tab : params ] .note also that schlinker and amiet s formulation is applicable even for high speed open rotors : figure [ fig:2 ] shows a good agreement for the open propeller model at cruise for which the chordwise mach number is very high ( ) .schlinker and amiet s ( ) theory of trailing noise from rotating blade has been validated by comparison with a new formulation that implicitly includes acceleration effects .the results presented in this paper show that the modified form of schlinker and amiet s predicts sound pressure levels within 1 db of our new formulation for subsonic chordwise mach numbers of up to 0.95 at high frequencies ( and ) .it is therefore applicable to both low speed applications ( cooling fans , wind turbines ) and high speed ones ( propellers ) .we propose that the correct exponent for the doppler weighting in schlinker and amiet s theory is .using , as in  and  gives results that agree with our formulation to within 5 db .however , all other things being equal , putting in schlinker and amiet s theory overestimates the result by up to 20 db at high mach numbers .the range of validity of amiet s theory for rotating blades is currently limited to high frequencies for two reasons : it uses a high frequency blade response function and neglects the effect of blade acceleration .more research is needed to identify the most important effect between the two .if the effect of acceleration can be neglected , i.e. at high frequency relative to the rotational speed , amiet s theory can be extended to lower frequencies by using a low frequency blade response function ( ) .taking the fourier transform of over , and using the far field approximation of  yields where and . using the jacobi - anger expansion which yields equation .the doppler shift indicates how the frequency of a pure tone varies for a moving observer ( relative to the source ) , compared to a fixed observer . following  ,an alternative physical interpretation can be obtained by observing that , for a pure tone , the pressure for a fixed observer and for a moving observer are given by where is the source frequency and the frequency received by the moving observer .the above equation implies that the measurement of the pressure time history is sped up ( or slowed down ) , by a factor , for a moving observer , compared to a fixed observer .   from equation , the autocorrelations and for a fixed observer and a moving observer ( relative to the source ) ,are such that where .since the power spectral density ( psd ) is defined as the fourier transform of the autocorrelation , the psd , for a moving observer , is related to the psd for a fixed observer by i.e.  [ fig:1_egamma_ez ] [ fig:2_ex_ey ]  we derive the chordwise mach number given the blade mach number and the flow mach number . as illustrated in figure [ fig : coordinates_isolated](b ) , can be decomposed into a component normal to the rotor plane ( inflow ) , and a component parallel to the rotor plane ( cross - flow ) : from figure [ fig : coordinates_isolated](a ) , the chordwise mach number is given by from figure [ fig : coordinates_isolated](b ) , the azimuthal mach number of the flow relative to the blade can be expressed in terms of and by substituting equation  into equation , ^ 2 + m_z^2 } . \ ] ]

Abstract: tthis paper presents a new formulation for trailing edge noise radiation from rotating blades based on an analytical solution of the convective wave equation . it accounts for distributed loading and the effect of mean flow and spanwise wavenumber . a commonly used theory due to schlinker and amiet ( ) predicts trailing edge noise radiation from rotating blades . however , different versions of the theory exist ; it is not known which version is the correct one , and what the range of validity of the theory is . this paper addresses both questions by deriving schlinker and amiet s theory in a simple way and by comparing it to the new formulation , using model blade elements representative of a wind turbine , a cooling fan and an aircraft propeller . the correct form of schlinker and amiet s theory ( ) is identified . it is valid at high enough frequency , i.e. for a helmholtz number relative to chord greater than one and a rotational frequency much smaller than the angular frequency of the noise sources .


one of the most challenging tasks when looking for a new phenomenon is the assessment of the background model and its uncertainties . in many cases the knowledge of the underlying background modelis limited , and the model is estimated and tested against a control measurement . in other cases ,a reliable functional description of the background model exists , but the relevant parameters fail to cover the true model .this can happen , for example , when the model is based on monte - carlo ( mc ) simulation with overly constrained parameters , or with too many parameters , lacking the ability to track the effect of each of their combinations .when using likelihood based statistical tests , it is assumed that the model is sufficiently flexible such that there exists a set of values of the parameters which can be regarded as `` true '' .however , if the model does not reflect the truth accurately enough it might lead to an enhanced false discovery rate , or to an artificially enhanced exclusion limit .there are several realistic cases in which it is challenging to provide a complete description of the background : ( i ) the background model depends on `` too many '' parameters and it becomes computationally challenging to perform the minimization .( ii ) the relevant parameters are hidden deep inside the mc unreachable by the user ( iii ) the shape is modeled based on a special dataset ( i.e. , calibration runs ) with no underlying physical model that reliably accounts for the systematic uncertainties .( iv ) the models , parameters or uncertainties used do not cover the true model .the excess events previously observed in some dark matter experiments have recently been demonstrated to be entirely attributable to underestimated backgrounds , adding once more the importance of proper treatment of background uncertainties in these experiments and others .a widely used procedure to establish discovery or exclusion in particle physics is based on a frequentist significance test using the profile likelihood ( pl ) ratio as the test statistic  .the pl method has been proven useful to incorporate model uncertainties into the likelihood test .in addition to the parameter of interest ( e.g. , the cross section or number of signal events ) , the signal and background models contain nuisance parameters whose values are not taken as known a - priori .the additional flexibility introduced to parameterize systematic effects through the nuisance parameters results in a loss in sensitivity , according to the uncertainty on the parameters and their influence on the interpretation of the data .  in this paperwe propose a procedure that provides a natural protection against mismodeling of the background , we use the unbinned pl ratio as a test statistic , and investigate its behavior .this paper is organized as follows : in sec .[ sec : likelihood ] we briefly discuss the pl method and motivate the use of unbinned likelihood over the binned likelihood . in sec .[ sec : theproblem ] we illustrate how mismodeling of the background causes loss of asymptoticness that enhances the false discovery rate ( in case of underestimated background ) or produces overly constrained exclusion limits ( in case of overestimated background ) . in sec .[ sec : thesolution ] we present a universal method to recover both the discovery potential and the exclusion sensitivity . in sec .[ sec : examples ] we present three realistic use cases , with user - defined parametric background models and empirical non - parametric models using kernel density estimates ( kde ) .in high energy physics and astroparticle physics , we wish to identify a small signal over a non - negligible background . in order to achievethat , the expected signal and background signatures in a detector have to show different behavior in some observable _ x_. this is modeled by the probability density functions ( pdfs ) , for signal and for background , where and represent one or more model parameters .if events were observed , the dataset is fully characterized be the vecotr . in a binned analysis ,the observed space is divided to bins , so each bin contains observed events , and .the probability for a signal ( or background ) event to be found in bin depends on the appropriate pdf , the bin location and its width   and the expected number of events per bin is = n_s \epsilon_s^j + n_b \epsilon_b^j,\ ] ] where and are the total number of signal and background events expected over all bins .the _ binned likelihood _function is the product of the poisson probabilities over all bins   the binned scheme is simple , well defined and widely used , see e.g. , .however , the projection of events into bins is associated with information loss .the choice of bins affects the sensitivity and the optimal choice varies for different hypotheses , detector performance and sample size ( see chapter 11 in  ) .an alternative is the unbinned extended likelihood function ( e.g. , )   eq .[ eq : ublikelihood ] is the infinitesimal version of eq .[ eq : blikelihood ] , and therefore contains all its information , and avoids the need of optimizing bins in order to enhance the sensitivity , as we show in appendix [ appendix : abinned ] .the unbinned scheme is invariant under coordinate transformation , as demonstrated in appendix [ appendix : atx ] .  throughout this paperwe will work with the unbinned likelihood function described above , with as the _ parameter of interest _ , and corresponding to the null , background only hypothesis . is treated as a nuisance parameter .though not explicitly shown here , additional nuisance parameters and their constraint terms can be added to the likelihood function in the usual way .the pl statistical inference approach is widely used for quantifying the level of agreement between a dataset and a null or signal hypothesis and its asymptotic formulae have been well studied  .  in this approach , given a likelihood function which is a function of a _parameter of interest _( in our case ) and a set of _ nuisance parameters _ , , the following test statistic is constructed   are the values of the nuisance parameters that maximize for a given , and and are the maximum likelihood estimators ( mle ) of the maximized ( unconditional ) likelihood function .if for a specific dataset , was observed when testing a specific , the level of disagreement between the data and the hypothesis can be quantified using the -value or the equivalent significance , ,   where is the distribution of under the assumption of signal strength , and is the quantile of the standard gaussian . according to wilks  , if has a gaussian distribution around its true value , and certain regularity conditions are met , asymptotically approaches a distribution with one degree of freedom .  for the statistical inference of discovery claim or exclusion limitswe follow the procedure described in  :  for a discovery claim the null hypothesis is to be rejected only when the data deviates from it with a positive signal . cases in which are not considered as an evidence against the null hypothesis    and can be asymptotically described by half a distribution plus half a delta function at zero .r0.5    for an exclusion limit we aim at rejecting a signal hypothesis , and do not consider data with as being less compatible with the hypothesis    and can be asymptotically described by half a distribution plus half a delta function at zero .improper modeling of background can cause devastating consequences such as a false discovery claim or overly constraint limit . to illustrate this , toy datasets were generated using a `` true '' background model , and unbinned analysis was performed as in eq .[ eq : ublikelihood ] alas using a `` slightly different '' background model .[ fig : fb_models ] shows its four variations .two of the variations overestimate the background in the signal region ( which can cause overly constraint exclusion limits ) , and two underestimate it ( which can lead to false discovery claims ) . while this modeling scenario is somewhat artificial , it aims at illustrating the challenges in a coherent way .more realistic examples will be presented in sec .[ sec : examples ] .  to test the false discovery rate eq .[ eq : discoveryteststat ] was used on background only datasets using the true background shape = , and each one of the four .[ fig : discovery_user](a ) shows for each cumulative distribution function ( cdf ) of ( ) .the deviation from the assumed ( cumulative ) distribution is evident . in case of underestimated background model the false discovery rateis enhanced , and the overestimated background model artificially strengthens the exclusion limit as will be discussed in more details along with the proposed solution in sec .[ sec : thesolution ] .   using the true background model ( solid ) and 4 variations as shown in fig .[ fig : fb_models ] . * ( a ) * with the regular procedure showing under ( over ) coverage when using background models that underestimate ( overestimate ) the background in the signal region . * ( b ) * with the correction the asymptotic behavior is regained .the expected asymptotic behavior is shown in gray.,title="fig:",scaledwidth=50.0% ]  using the true background model ( solid ) and 4 variations as shown in fig .[ fig : fb_models ] .* ( a ) * with the regular procedure showing under ( over ) coverage when using background models that underestimate ( overestimate ) the background in the signal region . *( b ) * with the correction the asymptotic behavior is regained .the expected asymptotic behavior is shown in gray.,title="fig:",scaledwidth=50.0% ]  in the next section we will show how to account for these unmodeled mismatches of the background , in a way that preserves the coordinate invariance and the optimal sensitivity .in this section we present a poka - yoke method to handle mismodeled or overlooked uncertainties in unbinned analyses , verify its asymptotic behavior , and discuss its implementation for discovery and exclusion .  starting with a benchmark background model , and a background only calibration measurementwe quantify the level of `` signal likeness '' that can be overlooked in the calibration sample by introducing a _ contamination parameter _ , .the benchmark background model will be added a signal like component quantified by which will serve as a nuisance parameter ,    this variation is motivated in appendix [ appendix : b ] using two approaches .this variation is universal and it does not depend on the way the background is modeled .in addition , it is unique with respect to other functional variations of the background , as it is the most conservative one with respect to the tested hypothesis .  as a result of this extra variation the likelihood function ( i.e. , eq .[ eq : ublikelihood ] ) is promoted to the signal - free calibration dataset provides additional constraints on , combined to   using this equation with eqs .[ eq : discoveryteststat ] and [ eq : exclusionteststat ] this method can be used for discovery or exclusion tests .the method can be made more conservative by limiting to be only positive ( for conservative discovery tests ) or only negative ( for conservative exclusion tests ) . however , as will be soon demonstrated this does not seem to be necessary , and the asymptotic behavior is recovered without any constraints on .while summarizing this paper , we became aware that a similar approach is used by the atlas collaboration . in , the `` spurious signal '' method was used to systematically compare background models , in a similar way to the method presented here . however , in this paper , we show that it is a valid and necessary addition to the likelihood in order to model shape uncertainties . in addition , we motivate the method ( appendix [ appendix : b ] ) , and investigate its test statistic asymptotic behavior in a systematic way .we also demonstrate how it can be used in different types of analyses , in order to restore the assumed asymptotic behavior ( sec .[ sec : examples ] ) , both for exclusion and discovery of a new signal .   to test the asymptotic behavior we repeat the procedure described in sec .[ sec : theproblem ] in conjunction with the procedure described here . as can be seen in fig .[ fig : discovery_user ] ( b ) , using the new procedure the test statistic distribution regains its expected asymptotic behavior , for both over- and underestimation of the background in the signal region . in both cases , the is an excellent approximation and therefore the significance ( or -value ) estimated using the pl will be reliable , and will not lead to false discovery .  to see how this procedure affects the discovery sensitivity we present it using fake data samples with 15 signal events over 100 background events .this is done using the true background model , and the 4 background model variations described in fig .[ fig : fb_models ] .  in fig .[ fig : discovery_significance ] on the upper panel ( in black ) the sensitivity using the true background is shown .we would like to have a distribution as similar as possible to this .if a distribution is shifted to the right , it means that we get an artificial enhancement of the sensitivity and we are exposed to false discoveries .alternatively , a left shifted distribution with respect the true one signifies a sensitivity loss .if the background is _ overestimated _ in the signal region the discovery potential is artificially reduced ( lower two panels , solid lines ) . however , using the new method ( dashed lines ) it is raised and becomes closer to the sensitivity with the true model in hand . if one wishes to be more conservative , one can force to be positive , and the original distribution will remain almost untouched . in case of _ underestimation_ , the sensitivity is artificially enhanced .looking at the second and third panels in fig .[ fig : discovery_significance ] , it can be seen that using the method of this paper ( dashed line ) the sensitivity is reduced to the level of the true model .this is due to the fact that the test statistic distribution is restored and can be described by the distribution ( fig . [ fig : discovery_user ] ) . as expected , when applying this correction a loss of sensitivity is apparent and in the examples shown here is less than .it can be seen that the proposed construction compensates for the unaccounted background shape without making any assumptions on its magnitude or functional shape .this is done naturally , without an inclusion of additional nuisance parameters .it is important to note that the distribution of the test statistic using the wrong model can not be estimated using mc as the true model out of which the data is really generated is unknown .   to illustrate the usage for an exclusion testwe compute the 90% confidence level ( cl ) upper limit distribution , using the true and false background models . in this casewe present the results for two background variations : the most overestimated and the most underestimated ones . for each modelwe compute the sensitivity with and without the correction , and compare it with the sensitivity using the true model .we use the same setup as before with 100 events in the physics dataset , and 1000 of background calibration events . as for discovery, the improper modeling will cause a breakdown of the distribution of the test statistic , leading to enhanced or reduced exclusion limits .[ fig : exclusion_user ] presents the tension between the limits using the wrong and the correct models .one can see that using the procedure defined in this paper cures the discrepancy . if the model is overestimated ( fig .[ fig : exclusion_user ] a ) , the sensitivity is artificially enhanced , and regions in the parameter space in which the experiment has no sensitivity might be excluded .the correction restores the true sensitivity of the experiment .alternatively , if the background model is underestimated ( fig .  [ fig : exclusion_user ] b ) , the limits will be higher than the true one , and can be restored using the correction .as before , to be more conservative , can be constrained to be negative throughout the procedure , and as a result the limits will remain practically unfixed in case of underestimation .for overestimation though , the limits will restore the true sensitivity .prevents too good limits ( left ) in case of overestimated background , and prevents sensitivity loss in case of underestimated background ( right).,title="fig:",scaledwidth=50.0% ]  prevents too good limits ( left ) in case of overestimated background , and prevents sensitivity loss in case of underestimated background ( right).,title="fig:",scaledwidth=50.0% ]in this section we present advanced examples in which the method defined in the previous section can be used .many times , the functional shape of the background is defined by either a mc or by some analytic expression , and the parameters of the background are extracted using a signal - free calibration sample . in the first example , we show how the method accounts for the uncertainties on the functional shape , and fix the asymptotic behavior of the test statistic .  in the second example , the background is estimated in a nonparametric way , this is done in order to dodge wrong assumptions on the model . however , in case the calibration size is not large enough the asymptotic behavior will be broken , and can be restored using the method suggested in this paper .  the third example is a 2-dimensional case study . we show that the method works in higher dimensions as well , and we present a breakdown due to nonparametric estimation with small a calibration sample .  as a general rule , all the applications depend on several aspects .a major aspect is the quality of the shape discrimination which can be quantified by ( see eq .[ eq : alphad ] in appendix [ appendix : atx ] ) .we give this measure for each of the examples as reference .in many experiments the background model constructed by fitting a function ( analytic  or mc based  ) to a control measurement ( calibration sample ) with one or more free parameters . this procedure assumes that the true background , , can be described by the chosen function and its parameters range . in case the functional space spanned by not cover , the results of the inference will be biased . in the following example we demonstrate how the method presented in this paper can resolve this issue .this is illustrated by constructing the background model using a fit to a calibration sample using different functional representations for  .[ fig : functional ] ( a ) shows an instance of a calibration dataset randomly generated from a tail of a gaussian ( ) , and its best fit to a second order polynomial ( ) .it is noted that very well the background . in this example .we generate 1000 trials of calibration dataset ( 100 events ) and physics dataset ( 100 events ) , where the physics dataset does not contain any signal . for each dataset pair, the likelihood function was calculated using eq .[ eq : discoveryteststat ] . to enhance the effect we did not include the fit uncertainties in the likelihood function as nuisance parameters .this represents the case in which the model s parameter do not cover the true model .[ fig : functional ] ( b ) shows the test statistic distribution and demonstrates the breakdown of the distribution .in addition , it shows how the method of this paper corrects the undercoverage introduced by the faulty background model . before applying the correctionthere is a clear undercoverage causing 4 times as many false discoveries than assumed , and around 2% of false discovery rate . after applying the correction the undercoverageis gone , and the distribution is reproduced .  for completeness , the discovery potential for 15 signal events over 100 background events is shown in fig .[ fig : functional ] ( c ) .it can be seen , by comparing the sensitivity to the one computed with , that the artificial enhancement of the sensitivity is relaxed when using the method of this paper at the price of a small sensitivity loss .( in black ) and a the best fit to a different function (in red ) .the signal model is shown in green . in this instancethe background is underestimated in the signal region . * ( b ) * cumulative distribution function ( cdf ) of discovery significance for a signal - less dataset , .the expected distribution is shown in gray , with the calculated distribution using (black ) .the red lines show the cdf using a best fit described in the text , before applying the correction ( red dashed ) and after ( blue solid ) .the undercoverage that causes enhanced false discovery rate is completely removed with the method . *( c ) * the discovery potential for 15 signal events .the distributions are presented for normal pl using the wrong model ( solid red ) and using the method presented in sec .[ sec : thesolution ] ( blue dashed ) .the distribution using presented for comparison ( gray dotted).,title="fig:",scaledwidth=33.0% ] ( in black ) and a the best fit to a different function (in red ) .the signal model is shown in green . in this instancethe background is underestimated in the signal region . * ( b ) * cumulative distribution function ( cdf ) of discovery significance for a signal - less dataset , .the expected distribution is shown in gray , with the calculated distribution using (black ) .the red lines show the cdf using a best fit described in the text , before applying the correction ( red dashed ) and after ( blue solid ) . the undercoverage that causes enhanced false discovery rateis completely removed with the method . *( c ) * the discovery potential for 15 signal events .the distributions are presented for normal pl using the wrong model ( solid red ) and using the method presented in sec .[ sec : thesolution ] ( blue dashed ) .the distribution using presented for comparison ( gray dotted).,title="fig:",scaledwidth=33.0% ] ( in black ) and a the best fit to a different function (in red ) .the signal model is shown in green . in this instancethe background is underestimated in the signal region . * ( b ) * cumulative distribution function ( cdf ) of discovery significance for a signal - less dataset , .the expected distribution is shown in gray , with the calculated distribution using (black ) .the red lines show the cdf using a best fit described in the text , before applying the correction ( red dashed ) and after ( blue solid ) .the undercoverage that causes enhanced false discovery rate is completely removed with the method .* ( c ) * the discovery potential for 15 signal events .the distributions are presented for normal pl using the wrong model ( solid red ) and using the method presented in sec . [ sec : thesolution ] ( blue dashed ) .the distribution using presented for comparison ( gray dotted).,title="fig : " ]   there are cases in which a calibration sample is available and one wants to construct an empirical model based on this sample without any prior assumptions .the problem is that in those cases , there are no natural nuisance parameters that can model the uncertainties . in this sectionwe show how the uncertainties can be modeled with the procedure described in this paper . in addition , we demonstrate how low calibration samples can lead to false discoveries if the procedure is not used .  in casea calibration sample is available we can use the kernel density estimation ( kde ) in order to produce a nonparametric estimation of the background pdf . let be a calibration sample , the kde is where is the _ kernel function _ ,a function that is normalized to one with a mean zero , and is a scale parameter called the _bandwidth_. there are many different ways to choose the bandwidth while throughout this paper we choose to work with a variable bandwidth , inversely proportional to the events density ( a thorough introduction of kde can be found e.g. , in  ) . the difficulty in using kde for inference , and nonparametric estimators in general , is the lack of natural nuisance parameters .then , any downward fluctuation of the calibration data in the signal region will result an enhanced probability for false discovery , due to the underestimated background model ( similarly to subsection [ subsec : usermodel ] ) .an example of such instance is shown in fig .[ fig : kde_example_pl ] ( a ) , with . where a small downward fluctuation in the calibration dataset causes an underestimation of the background in the signal region .this results in an overestimation of the profile likelihood function at as can be seen in fig .[ fig : kde_example_pl ] ( b ) .( red dashed ) , constructed using 1000 calibration events .it can be compared with (solid ) .the signal distribution is also shown ( blue dotted ) .in this instance is underestimated in the signal region . * ( b ) * the pl function using the the true background ( black solid ) , and using the underestimated before applying the correction ( red dashed ) .the dataset does not contain any signal events .the overestimation at is interpreted as a hint towards a signal like data , and is caused because of the underestimation of in the signal region .this can be avoided by the inclusion of the kde uncertainties as explained in this section and illustrated by the corrected curve ( dashed - dotted blue).,title="fig:",scaledwidth=50.0% ]  ( red dashed ) , constructed using 1000 calibration events .it can be compared with (solid ) .the signal distribution is also shown ( blue dotted ) . in this instance is underestimated in the signal region . *( b ) * the pl function using the the true background ( black solid ) , and using the underestimated before applying the correction ( red dashed ) .the dataset does not contain any signal events .the overestimation at is interpreted as a hint towards a signal like data , and is caused because of the underestimation of in the signal region .this can be avoided by the inclusion of the kde uncertainties as explained in this section and illustrated by the corrected curve ( dashed - dotted blue).,title="fig:",scaledwidth=50.0% ]  the false discovery presented in fig .[ fig : kde_example_pl ] is caused by the breakdown of the approximation .as for the previous example , this assumption on the test statistic distribution does not hold due to the improper model .this breakdown causes an enhanced false discovery rate .the 3 false discovery rate is presented in fig .[ fig : kde_discovery_comparison ] ( a ) as a function of the calibration size . as the calibration sizeis increased the kde approaches the false discovery rate is reduced . however , using the method described in this paper the false discovery can be corrected also for insufficient calibration sample size .discovery rate using kde background model , ( red line ) .the false discovery is removed using the method described in this paper as shown by the blue dashed line . * ( b ) * exclusion sensitivity using ( red circles ) , and using plus correction ( blue squares ) .the median 90% cl limit is shown along with its .the 1 sensitivity band using the true background model is shown for comparison ( gray band ) . *( c ) * discovery potential using pl with the true background ( gray band ) , using ( red circle ) , and using plus correction ( blue squares ) .it is computed for 15 expected signal events over 100 expected background events ., title="fig:",scaledwidth=45.0% ]  discovery rate using kde background model , ( red line ) .the false discovery is removed using the method described in this paper as shown by the blue dashed line .* ( b ) * exclusion sensitivity using ( red circles ) , and using plus correction ( blue squares ) .the median 90% cl limit is shown along with its .the 1 sensitivity band using the true background model is shown for comparison ( gray band ) . * ( c ) * discovery potential using pl with the true background ( gray band ) , using ( red circle ) , and using plus correction ( blue squares ) .it is computed for 15 expected signal events over 100 expected background events ., title="fig:",scaledwidth=45.0% ]  [ fig : kde_exclusion_comparison ]  another important property of the procedure is its discovery potential .ideally one would like to have a procedure that punishes lack of information by reduction in sensitivity . on the other hand, one does not wish to lose sensitivity in case there exists enough knowledge .our method is compared with the naive procedure , and with a pl using the in fig .  [ fig : kde_discovery_comparison ] ( b ) .the discovery potential is shown for various calibration sizes .it is computed by the injection of 15 signal events over 100 background events . for large calibration samplethe method approaches the sensitivity of the true model and of the kde .however , for small calibration samples the kde artificially enhances the sensitivity , while the correction reduces it .this is the penalty for lack of knowledge .last but not least , we examine the behavior of the new method for setting exclusion limits . in fig . [fig : kde_exclusion_comparison ] we compare the 90% cl exclusion limit of the method with pl using kde ( red squares ) , and with pl using the true model ( gray band ) .it can be seen that the sensitivity is comparable to kde for small calibration sizes , however it approaches the sensitivity of the true model for large calibration sizes .a third example is motivated by experiments trying to directly detect dark matter .these experiments typically expect signal in the order of a few events per year or less , and determining the background pdf lacks in general a complete physically motivated parametric model . therefore the pdf is , in many cases , estimated by a dedicated calibration dataset , see e.g. , .in other cases a model driven background model is used , as in , where a complementary 5 dimensional unbinned analysis is performed .  for this examplewe present the true pdfs of a two dimensional discrimination space in fig .[ fig:2d ] . in the casepresented here .the two dimensions can represent two measured channels ( e.g. , charge and light , or heat and charge ) or a combination of them .the natural invariance under transformations of the unbinned pl makes the specific choice of axes irrelevant .   and axes .the two dimensional discrimination between the two is based on the combination of them ., scaledwidth=50.0% ]  we use a two dimensional adaptive kde procedure to estimate the background model , similarly to the previous example . for each instance, we draw a number of calibration events and infer a background model , and then use this model for statistical inference of a 100 background events , with and without added signal of 15 events , both numbers subject to poisson fluctuations .[ fig:2dresults ] summarizes the results of this study . on the left panelone can see the case of no signal , and how often a false detection may occur with and without the correction presented in this paper , as a function of the calibration size .comparing this to the one dimensional example , it is pointing to a larger calibration sample needed when increasing dimensionality - even though the discrimination is somewhat better in the 2d case ( of 7 vs. 4.9 ) .the right panel shows the discovery potential of an injected signal , which for low calibration statistics exceeds the expected one by the true pdf .again , the discrepancy for two dimensions is larger than in one dimension for a given calibration size . in both caseswe see that the corrected pl is curing the false or overly confident discovery , but reduces accordingly the sensitivity - a natural result of less knowledge regarding the true nature of the background .discovery rate using kde background model , ( red line ) .the false discovery is removed using the method described in this paper as shown by the blue dashed line . * ( b ) * discovery potential using pl with the true background ( gray band ) , using ( red circles ) , and using plus correction ( blue squares ) .it is computed for 15 expected signal events over 100 expected background events ., title="fig:",scaledwidth=48.0% ]  discovery rate using kde background model , ( red line ) .the false discovery is removed using the method described in this paper as shown by the blue dashed line . * ( b ) * discovery potential using pl with the true background ( gray band ) , using ( red circles ) , and using plus correction ( blue squares ) .it is computed for 15 expected signal events over 100 expected background events ., title="fig:",scaledwidth=48.0% ]in this paper we have presented a new method that protects against biases created by mismodeling of the background .we show that statistical inference without this addition is prone to risks of false discovery and artificially enhanced sensitivity , which are cured when using eqs .[ eq : correctedlikelihood ] and [ eq : epsilonlikelihood ] .we work within the pl framework and present the asymptotic behavior restoration and explore a number of examples , using parametric models , non - parametric models and two dimensional parameter space .additional complexity and dimensionality of the data would increase the dangers of mismodeling , and make this addition an essential part of unbinned pl usage .application of the method to more complicated scenarios such as external data coming from other experiments can be done with proper care .one must bear in mind that uncertainties attached to detector performance or yet - unknown physics effects can not be eliminated by a model or simulation , however numerically accurate it is .the authors would like to thank jan conrad , eilam gross and daniel lellouch for fruitful discussions .we are also grateful to teresa marrodn undagoitia and manfred lindner for their scientific support .this work was supported by isf i - core grant 1937/12 `` the quantum universe '' and the dfg research training group `` particle physics beyond the standard model ''.  9  w. a. rolke , a. m. lopez and j. conrad , nucl .instrum .meth . a * 551 * , 493 ( 2005 ) doi:10.1016/j.nima.2005.05.068 [ physics/0403059 ] .  c. e. aalseth _[ cogent collaboration ] , phys .d * 88 * , 012002 ( 2013 ) doi:10.1103/physrevd.88.012002 [ arxiv:1208.5737 [ astro-ph.co ] ] .g. angloher _ et al ._ , eur .j. c * 72 * , 1971 ( 2012 ) doi:10.1140/epjc / s10052 - 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013 - 2501-z [ arxiv:1007.1727 [ physics.data-an ] ] .s. s. wilks , annals math . statist .  *9 * , no . 1 , 60 ( 1938 ) .doi:10.1214/aoms/1177732360  g. aad _ et al ._ [ atlas collaboration ] , phys .d * 90 * , no .11 , 112015 ( 2014 ) doi:10.1103/physrevd.90.112015 [ arxiv:1408.7084 [ hep - ex ] ] .d. s. akerib _ et al ._ , [ arxiv:1608.07648 [ astro-ph.co ] ].  k. s. cranmer , comput .commun .  * 136 * , 198 ( 2001 ) doi:10.1016/s0010 - 4655(00)00243 - 5 [ hep - ex/0011057 ] .marrodn undagoitia teresa and rauch ludwig , j. phys .g43 1 013001 ( 2016 ) .doi : 10.1088/0954 - 3899/43/1/013001  w. verkerke and d. p. kirkby , econf c * 0303241 * , molt007 ( 2003 ) [ physics/0306116 ] .here we will demonstrate the similarity between the binned and the unbinned likelihood function .  starting with the binned - likelihood function defined in eq .[ eq : blikelihood ] we can obtain the log - likelihood function  =\\ & \quad \sum_j^{n_{bins } } \left [ -\epsilon_s^j n_s - \epsilon_b^j n_b +  n_j \mathrm{log}(n_s \epsilon^j_{s } + n_b \epsilon^j_{b})\right]=\\ & \quad - n_s - n_b + \sum_j^{n_{bins } } n_j \mathrm{log}(n_s \epsilon^j_{s } + n_b \epsilon^j_{b}),\end{aligned}\ ] ]  where we use the fact that , and neglect constant terms .  alternatively , starting from eq . [ eq : ublikelihood ] , the unbinned log - likelihood function can be projected into bins .this is done by the replacement of with a binned probability density function , namely it can be replaced by , the probability to be found in the bin , divided by the bin width , . = \\ &\quad -n_s - n_b+\sum_{i=1}^n \mathrm{log}(n_s f_s(x_i ) + n_b f_b(x_i))= \\ &\quad -n_s -n_b + \sum_{i=1}^n \mathrm{log}(n_s \epsilon^{bin(x_i)}_{s}/w + n_b \epsilon^{bin(x_i)}_{b}/w ) = \\  & \quad - n_s - n_b + \sum_j^{bins } \sum_{i=1}^{n_j } \mathrm{log}(n_s\epsilon^j_{s } + n_b \epsilon^j_{b } ) + constant=\\ & \quad - n_s - n_b + \sum_j^{n_{bins } } n_j \mathrm{log}(n_s \epsilon^j_{s } + n_b \epsilon^j_{b } ) + constant,\end{aligned}\ ] ]  which yields an identical expression to the binned case , up to a constant .   in general, it is quite common to measure more than one variable for each event . out of those variablesone would usually like to get the maximum discrimination between signal and background .namely , to have the best sensitivity on the parameter of interest , .this question can be phrased in the following way : assuming that we have some data space in which the measurement has been done .we can apply transformation on the variables and work in other space .what is the space in which we get the smallest .working with eq .[ eq : ublikelihood ] , can be estimated by calculating the expectation values of the second derivatives of the log - likelihood function },\\  \beta \equiv -e\big{[}\frac{\partial^2 { { \mathrm{log}(\mathcal{l})}}}{\partial n_s \partial n_b}\big{]},\\  \gamma \equiv  -e\big{[}\frac{\partial^2 { { \mathrm{log}(\mathcal{l})}}}{\partial { n_b}^2}\big{]}.  \end{gathered}\ ] ]  the resolution of can be computed to be .for the unbinned likelihood it can be easily shown that , and are invariant under coordinate transformation - hence we can not gain or loose sensitivity by choice of coordinates   } = \int \frac{f_s^2(x ) } { n_s f_s(x ) + n_b f_b(x)}dx \equiv \alpha,\\  -e\big{[}\frac{\partial^2 { { \mathrm{log}(\mathcal{l})}}}{\partial n_s \partial n_b}\big { ] } = \int \frac{f_s(x ) f_b(x)}{n_s f_s(x ) + n_b f_b(x)}dx \equiv \beta,\\  -e\big{[}\frac{\partial^2 { { \mathrm{log}(\mathcal{l})}}}{\partial { n_b}^2}\big { ] } = \int \frac{f_b^2(x ) } { n_s f_s(x ) + n_b f_b(x)}dx \equiv \gamma .\end{gathered}\ ] ]  the resolution on can be computed to be , and we see that , and are invariant under transformation of .this means that for an unbinned analysis the sensitivity , , is optimal and invariant in any space we work in .this is another advantage of this procedure .  when rejecting the null hypothesis with eqs .[ eq : sensitivitynsnb2 ] , and we are left with the estimator for _ shape discrimination _ ranging from 1 to infinity , where if and only if . the higher , the better the shape discrimination between the background and the signal .this is again an invariant under transformations of the measured space . in the limit ,pl does not contribute to the sensitivity beyond simple counting , as the background leakage goes to zero .lets assume that we have a control measurement consists of background events . the unbinned likelihood ( eq . [ eq : ublikelihood ] ) can be written in order to constrain the background model .the calibration likelihood function is where and are the expected number of signal and background events .the component is introduced in order to quantify the level of signal like behavior of the background model , as we do nt expect to have any signal events in the calibration dataset . in the case of good background over signal discrimination as increases.  the log likelihood , after neglecting constant offsets , is with , and .we identify the term in the logarithm to be the background model .namely instead of keeping fixed we let it be contaminated with a signal like component , where is the contamination parameter , and the calibration dataset pose a constraint on it .  for a ` physics ' dataset , which might contain a signal we will use eq . [eq : ublikelihood ] , with one change .we will propagate the background model as follows , in order to reflect the uncertainty of the model .   where is the parameter of interest , and , , and are nuisance parameters . is constrained to be independently of the other variables , hence the last two terms can be neglected .lets take the unbinned likelihood function ( eq .[ eq : ublikelihood ] ) and treat the background model itself as a nuisance parameter .now , we have a semiparametric likelihood functional , with two parameters , and , and one nuisance function , , where the parameter of interest is .  for simplicitywe assume that is known , and check the first order variation with respect to and in order to identify functional variation on the background model that can be used .   for discoverywe take a negative variation of and we are scanning below .therefore the absolute value of the second term is greater than the value of the first term .then , if we allow variation of the background model , by setting we relax the difference between the two terms for any given instance of the data .

Abstract: we present a universal method to include residual un - modeled background shape uncertainties in likelihood based statistical tests for high energy physics and astroparticle physics . this approach provides a simple and natural protection against mismodeling , thus lowering the chances of a false discovery or of an over constrained confidence interval , and allows a natural transition to unbinned space . unbinned likelihood allows optimal usage of information for the data and the models , and enhances the sensitivity .   we show that the asymptotic behavior of the test statistic can be regained in cases where the model fails to describe the true background behavior , and present 1d and 2d case studies for model - driven and data - driven background models . the resulting penalty on sensitivities follows the actual discrepancy between the data and the models , and is asymptotically reduced to zero with increasing knowledge .


structural patterns in complex networks occurring in biological , social and technological contexts , have been a focus of study by physicists for a decade , since the groundbreaking discovery of small - world property and scale - free degree distribution for many networks .one of the common features seen in many networks is the occurrence of _ modules _ , namely , subnetworks whose members are highly inter - connected but have few links to nodes outside the module .many networks have also been seen to have _ hierarchical _ organization , i.e. , they are composed of successive interconnected layers or inter - nested communities . in the literature ,often the terms hierarchy and modularity have been used almost inter - changeably , although , as shown in fig .[ fig : hie_mod ] , they represent distinct properties of the network. however , it is interesting to note that these two properties have been found to coexist in many networks occurring in real life  , including the internet  and the network of cortical areas in the cat brain  .most of the complex systems seen in real life also have associated dynamics , and the structural properties of such networks have been sought to be linked with their dynamical behavior  . in this respect , one of the questions of obvious significance is whether there is a relation between the _ stability _ of the dynamics against small perturbations in the dynamical variables and the specific arrangement of the network s connections .if the perturbation decays quickly , so that it is unable to spread to the rest of the network , the network is said to be _stable_. such a property is necessary if networks are to survive the noisy environment that characterizes the real world .it has sometimes been argued that , networks with larger number of nodes , links and stronger inter - connections are more stable .such assertions are partly based on empirical observations , e.g. , in ecology , where it has been found that more diverse and strongly connected ecosystems are more robust than their smaller , weakly connected counterparts  . on the other hand , theoretical work on the stability of model networks have suggested the opposite conclusion . in particular , according to the may - wigner theorem  for random networks , increasing the complexity ( as measured by the number of nodes , density of connections and dispersion of interaction strengths ) always leads to decreased stability .however , this result is based on the study of networks whose connection topology shows none of the structures that are seen in real life networks , in particular , modularity and hierarchy .therefore , it is of interest to see whether introducing hierarchical organization and modular structures can result in refutation of the may - wigner theorem. early work on the stability of simple , structured model networks  seemed to indicate that such structures indeed promote stability , and this was also seen under certain conditions for hierarchically organized networks  .however , a later study of hierarchical , as well as , modular networks , concluded that these are less stable than corresponding random networks  .we revisit this problem in the present paper , by proposing a network model that exhibits both modular structure and hierarchical organization .in addition to looking at the stability of equilibria of the network dynamics , we also consider the stability of synchronization over the network . although these two stability phenomena are superficially similar , they involve looking at different properties of the network . the issue of network synchronization , in particular , has assumed importance in recent years , owing to its connection with , e.g. , brain dynamics  .  .]  an alternative model for hierarchical modular networks has been earlier proposed by ravasz and barabasi ( rb )  .this model generates a set of inter - nested modules in a hierarchical fashion using a deterministic procedure that has both high clustering ( because of the modular nature of the network at the most fundamental level ) and a scale - free degree - distribution .these two properties do not always co - occur in other network models that have been proposed in the literature .in particular , the barabasi - albert ( ba ) network model  allows generation of a network with scale - free degree distribution through the preferential attachment mechanism , but the average clustering coefficient of its nodes decays with system size .further , in the rb model , a scaling relation is observed between the clustering coefficient of a node and its number of connections ( i.e. , degree ) : similar relations were also observed in several real networks , such as the web of semantic connections between two english words which are synonyms  .this occurrence of the scaling relation between clustering and degree of the nodes in a network has often been taken as a signature for the existence of hierarchical modular structure in that network . recently, this scaling relation was shown to be actually an outcome of degree - correlation bias in the usual definition of clustering coefficient  .however , it can be easily seen that this scaling relation is not a necessary indicator for the existence of either modularity or hierarchy .for example , consider a modular network consisting of nodes and modules of equal size .let each node have degree , with the links initially occurring exclusively between nodes belonging to the same module ( i.e. , the modules are isolated from each other ) . to make the network connected we rewire a small fraction of the links keeping the degree of each node fixed .plotting clustering as a function of degree for this network will only show vertical spread of points at a single node degree value .let us consider another example , this time a hierarchical structure , viz . , the cayley tree with branches at each vertex .again , it is easy to see that the clustering versus degree curve will not show the characteristic scaling seen for the rb model .in fact , in the next section , we show that even for networks where both hierarchy and modularity are present , it is not necessary that this scaling relation between clustering and node degree will hold .the paper is organized as follows . in the next section we introduced a simple model of a modular network with hierarchical organization . in section [ sec : dynamics ] we introduce the formalism to analyze the stability of dynamical equilibria and synchronized states of a network .the proposed model allows a detailed study of the relation between dynamical stability and hierarchical modular organization of the network .we observe that both of these structural properties actually increase the instability compared to an equivalent random network .this may appear counter - intuitive as both modularity and hierarchy are observed in networks occurring in nature , which necessarily have to be robust to survive environmental fluctuations .however , the emergence of modular structures can be understood as a response to multiple ( and often conflicting ) constraints imposed on such networks  .we conclude with a discussion about how these observations can possibly be extended to explain the emergence of hierarchical organization .) indicated by broken lines , and the corresponding adjacency matrix ( right ) where indicates the density of connections within and , , between the different modules at each level . ]here we propose a general model for networks having modular as well as hierarchical structure .let us begin with a modular network consisting of modules , each containing nodes .the connectivity ( i.e. , the probability of a link between any pair of nodes ) within each module is , while the connectivity between modules is ( ) .we now introduce hierarchy by adding another set of modules ( each having nodes ) with the same and .the nodes belonging to these two different sets of modules are now connected , but with a probability ( ) .the resulting network has nodes and hierarchical levels ( fig .[ fig : model ] ) . to increase the number of hierarchical levels to , we add a similar network with nodes to the existing network and , as above , add links between these two networks with a probability ( ) .thus , to get a network with hierarchical levels , the above procedure is repeated times .the final network contains number of modules .note that , all connections between nodes are made randomly . to reduce the number of model parameters , we assume that the connectivities are related as : where, , the ratio of inter - modular connections between two successive hierarchical levels , is a control parameter . by varying , one can switch between isolated modular ( ) and homogeneous random ( ) networks , with intermediate values of giving hierarchical modular networks .we compare between networks having different number of hierarchical levels , keeping the total number of modules and average degree fixed .to consider the effect of hierarchy in isolation , while keeping modularity fixed ( e.g. , as measured by the newman modularity measure  ) , we use a variant of the above model , where ,  = constant , while other connectivities are still related by this implies that the average number of intra - modular ( ) and inter - modular ( ) connections per node are also constant . , and +  ] , where represent different modes of perturbation from the synchronized state .this is also referred to as the _ master stability equation _these equations have the same form but different effective couplings .the synchronized state is stable ,i.e. , the maximum lyapunov exponent is in general negative , only within a bounded interval $ ] . let the eigenvalues of the laplacian matrix be arranged as . then , requiring all effective couplings to lie within the interval , implies that a synchronized state is linearly stable , if and only if , .thus , a network having a smaller eigenratio , is more likely to show stable synchronized activity .here , we obtain the eigenvalues of the laplacian for a hierarchical modular network ( fig . [fig : synch ] , left ) and observe the eigenratio as a function of ratio of the inter - modular connections between two successive hierarchical levels , , and the total number of hierarchical levels , .first , keeping the number of hierarchical levels fixed , we vary the parameter .we find that with decreasing , i.e. , as the number of connections between two successive hierarchical levels decrease , the instability of the synchronized state increases .next , keeping the total number of modules fixed we increase the number of hierarchical levels ( ) in the network .[ fig : synch ] ( right ) shows that as the number of hierarchal levels of the network is increased , decreases , resulting in an increasing eigenratio .thus , arranging the modules of a network in a hierarchical fashion also makes a network difficult to synchronize .in previously published work  , we have shown that increased modularity in random networks leads to higher probability of instability for the equilibria of the network dynamics .thus , the work presented here is an extension and generalization of the above result , demonstrating that increased number of hierarchical levels also tend to destabilize these equilibria , and moreover , the same phenomena is observed for the stability of synchronized activity in a network with respect to increased modularity and hierarchy .this raises the question of how can systems with hierarchical modular structures be seen in nature at all , where they have to be robust enough to survive constant environmental fluctuations .an answer to this can be fashioned along the lines of our recent work showing that additional natural constraints operating on networks in real life , such as the minimization of ( a ) resource cost for maintaining each link and ( b ) the time required for communicating between nodes , in addition to linear stability of equilibria , will make modular networks the optimal configuration ( fig .[ fig : schematic_mod ] )  .we find such stable , modular networks to possess multiple hubs and a heterogeneous degree distribution .many types of networks , including scale - free networks  , can be seen as special cases of this general criterion .therefore , we can understand the large - scale occurrence of such networks in nature as a response to co - existing structural and dynamical constraints .one can ask , what will be the effect of introducing constraints other than the ones mentioned here .for example , replacing the criterion for linear stability by one demanding robustness with respect to removal of links ( selected by using a combination of random and targeted attack strategies ) does not qualitatively change our results .it turns out that this criterion is satisfied by networks with bimodal degree distribution , a property that our optimal modular networks possess .however , while this can explain the ubiquity of modularity , it does not answer the question of why hierarchical organization is so common in nature .the fact that tree - like networks with extensive ramifications occur so often in the context of resource transport ( e.g. , the circulatory system in plants and animals ) suggest that additional constraints related to flow maximization may be at work in this case .another possible candidate for such a constraint may be the need to minimize wiring cost , i.e. , the total link length  .this is applicable when the network is embedded on a geographic ( as opposed to topological ) space , so that the wiring cost can been defined as the sum of the euclidean distances between all connected pairs of nodes .as many of the networks showing hierarchical organization ( such as the internet and the network of cortical areas in the brain ) are indeed defined in metric space , this is a possibility that needs to be analysed in detail .

Abstract: several networks occurring in real life have modular structures that are arranged in an hierarchical fashion . in this paper , we have proposed a model for such networks , using a stochastic generation method . using this model we show that , the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks , as had previously been suggested on the basis of a deterministically constructed model . we also look at dynamics on such networks , in particular , the stability of equilibria of network dynamics and of synchronized activity in the network . for both of these , we find that , increasing modularity or the number of hierarchical levels tends to increase the probability of instability . as both hierarchy and modularity are seen in natural systems , which necessarily have to be robust against environmental fluctuations , we conclude that additional constraints are necessary for the emergence of hierarchical structure , similar to the occurrence of modularity through multi - constraint optimization as shown by us previously .