in their famous paper of 1935 , einstein , podolsky , and rosen ( epr ) questioned the completeness of quantum mechanics ( qm ) .their argument was based on the premise of no action at a distance ( locality ) and realism .several years later bell showed that the prediction of qm are incompatible with the premises of local realism ( or local hidden variable theories ) .experiments based on bell s result support qm , indicating the failure of local hidden variable theories .quantitative tests used mainly bohm s ( dichotomic ) version of the epr entangled states instead of the original epr states with continuous degrees of freedom .  in recent years , systems with continuousvariable has attracted much atttention in connection with the burgeoning field of quantum information . in such a field epr entanglement and quantum nonlocalityare of practical importance .nevertheless , the generalization of bell s inequalities to quantum systems with continuous variabbles still remains a challenging issue .  in continuous variable bipartite quantum systems ,epr aspects can arise when trying to infer quadratures of one subsystem from those of the other .these can be tested by exploiting the heisenberg uncertainty principle .such approach has been employed in refs. .few other theoretical proposals use the field quadrature measurements , while a phase space approach to quantum nonlocality has been developed and tested .  on the other hand quantum tomography a useful tool to reach a complete state knowledge .such knowledge could then be used for nonlocal tests .some aspects of epr problem have been considered in tomographic approach in .  here , we propose to use the tomographic data to do nonlocal tests . at first instance, we shall develop an inequality based on the tomograms .however , this will result quite difficult to violate .a state does not have to violate all possible bell s inequalities to be considered quantum nonlocal ; rather , a given state is nonlocal when it violates any bell s inequality .thus the degree of quantum nonlocality that we can uncover crucially depends not only on the given quantum state , but also on the bell operator .then , we shall introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state , thus giving the possibility to use standard bell s inequalities .we shall also present some illuminating examples .a quantum state is described by the density operator or by any its phase space representation like the wigner function , where , are canonical conjugate variables .quantum tomography is a technique which allows to recover the quantum state from a set of measured probability distributions .the latter , sometimes called tomograms , are line projections of the wigner function .such projections can be parametrized through a symplectic transform .practically , the tomograms can be expressed as where , are the parameters of the symplectic transformation and is a stochastic variable .it represents the random outcomes of the measurement of the observable , and results the probability distribution associated to such observable .it is worth noting that the above symplectic transformation , once thought as composition of rotations and squeezing , can be realized in optical systems .a remarkable property of the tomograms ( [ eq : ww ] ) is their homogeneity .  to get the phase space picture of the quantum state one has to invert the relation ( [ eq : ww ] ) , obtaining simply results an inverse fourier transform .neverthless , it becomes a more involved inverse transform , namely an inverse radon transform , when reducing the symplectic transformation to a mere rotation , i.e. , , .this is the case of optical homodyne tomography ( oht ) .  by expanding the density operator in terms of a complete set of operators, it is also be possible to directly relate the quantum state to the tomograms , that is where the kernel operator takes the form here , can be set equal to unity ; this freedom reflects the overcompleteness of the information obtainable by means of all possible tomograms ( [ eq : ww ] ) .  in ohtit is preferable to use eq .( [ eq : rhow ] ) instead of eq .( [ eq : ww ] ) since the kernel in the latter case is unbounded , thus requiring the use of an artificial cutoff to directly sample the wigner function . on the contrary , eq .( [ eq : rhow ] ) allows a direct sample of the density matrix elements on the basis where results bounded .the above arguments can be generalized to the case of a bipartite two - mode system .in such a case the tomograms will be given by where , , , are parameters .let us restrict to the case of oht , and suppose to perform homodyne detection in both subsystems and .then , , , , , where , are the rotation angles related to the local oscillators phase . as a consequence we have  .we now classify the results of the measurements to be if the quadrature result is greater than zero , and otherwise .then , we can construct the following probabilities from the tomograms then , the bell s inequalities can be written in terms of the above probabilities . in particular , the chsh inequality can be written as where now discussed in section 2 , measuring tomograms allows to statistically sample the density matrix elements at least in some basis , e.g. fock basis .this represents a complete knowledge of the state , also for a bipartite system .then , such information , say the density matrix elements in fock basis , can be used to derive the statistics of any hermitian operator , even if this is not directly measurable .the price one has to pay is the large amount of data needs to collect .let us now see how these arguments can be fruitful applied to the problem of quantum nonlocality .we introduce the local pseudo - spin operators  where and are fock states .the operators ( [ eq : s ] ) obey the commutation relation of spin- algebra , namely    = 2\,i\,\delta_{j , j'}\;\varepsilon_{xyz}\,s_{z}^{(j)}\,,\ ] ] with the totally antisymmetric tensor .spin tomography approach was considered e.g. in .  in terms of the operators ( [ eq : s ] )the chsh inequality reads in its standard form , that is with here , , , are unit vectors in , while and the dot indicates the ordinary scalar product in .furthermore , the angle brackets in eq .( [ eq : cale ] ) denote the average over the density matrix elements assumed available from tomographic reconstruction .let us now study the possible violations of inequality ( [ eq : calb ] ) for the two preceeding examples .for simplicity , we consider the directions , , , coplanar , in the plane , so that where ( ) is the relative angle betweeen ( ) and .let us going to study the possible violations of the inequalities ( [ eq : bdef ] ) and ( [ eq : cale ] ) in three paradigmatic cases .we first consider the two - mode squeezed vacuum state which can be generated in the non - degenerate optical parametric amplifier  where and is the squeezing parameter .such state can also be described by the following wigner function \bigr .\nonumber \\   & -&\bigl.e^{2s } \left[(q_1+q_2)^2+(p_1-p_2)^2\right]\bigr)\,.\end{aligned}\ ] ] notice that for the state ( [ eq : wepr ] ) becomes precisely the epr state .  inserting eq .( [ eq : wepr ] ) into eq .( [ eq : ww2 ] ) we obtain where in this case , the tomograms depend only on the sum of the parameters .  using the position of eqs .( [ eqs : wpm ] ) we obtain \ , ,    \label{eqs : wppepr } \\w_{+-}(\theta_1,\theta_2 ) & = & w_{-+}(\theta_1,\theta_2 )   = \frac1{4\pi}\left[\pi+2\arctan\left(\frac{b}{n}\right)\right]\ , ,    \label{eqs : wpmepr}\end{aligned}\ ] ] where the signs in front of refers to , and should be adjusted accordingly to the range of .  in fig .[ fig1a ] we show the behavior of the functions ( [ eqs : wppepr])-([eqs : wpmepr ] ) vs for different values of .as we can see they never exceed the value , thus the quantities , in eq .( [ eq : edef ] ) , will be always confined between and . in turns , the inequality ( [ eq : bdef ] )will be never violated .this results seems in agreement with bell s conclusions stating that the original epr state does not exhibit nonlocality because its wigner function is positive everywhere , and as such allows a hidden variable description .however , we shall see that is simply a matter of the type of measurements .  for the state ( [ eq : psiepr ] ) , eq .( [ eq : calexz ] ) becomes then , in fig .[ fig1b ] we show for different values of . in this case the epr state show its nonlocality .indeed a maximal violation of inequality ( [ eq : calb ] ) is achieved for ( ) .as second example we consider the entangled superposition since and are orthogonal for , the state ( [ eq : psin ] ) resembles a spin triplet state .the corresponding wigner function is where indicates the laguerre polynomial of order . by means of eqs .( [ eq : ww2 ] ) and ( [ eq : wn ] ) we get the tomogram of the state under consideration where indicates the hermite polynomial of order . using the position of eqs .( [ eqs : wpm ] ) we obtain probabilities   in fig . [ fig2a ] we show the behavior of the vs for different values of .also in this case they never exceed the value , thus the quantities , in eq .( [ eq : edef ] ) , will be always confined between and , and the bell s inequality ( [ eq : bdef ] ) will be satisfied .  for the state ( [ eq : psin ] ) , eq .( [ eq : calexz ] ) becomes then , in fig .[ fig2b ] we show for the case of .the nonclassical character of the entangled state ( [ eq : psin ] ) becomes now manifest .   as a third , and last , example we consider a pair - coherent state ,namely where norm is with the modified besssel function of order .the state ( [ eq : pc ] ) represents two coherent states , of the same amplitude , having a well defined phase relation although their own phase is random .the wigner function corresponding to the state ( [ eq : pc ] ) is \bigr)\,d\varphi\,d\varphi'.\end{aligned}\ ] ] the tomogram of the state ( [ eq : pc ] ) is where the integral reads \,d\varphi\,.\end{aligned}\ ] ] using the procedure shown in the appendix together with some algebra , we can obtain from eq .( [ eq : wpc ] ) the following expressions for the probabilities : \right)\right ) \times \\   & \times&\left(1\mp{\rm erf}\left(\frac{r}{\sqrt{2 } }   \left[e^{-i(\varphi_1+\varphi_0)}+   e^{-i(\varphi_2-\varphi_0)}\right]\right)\right )   \,d\varphi_1\,d\varphi_2\ , , \nonumber\end{aligned}\ ] ] where denotes the error function and .notice that the first ( second ) on the l.h.s. corresponds to the first ( second ) under the integral on the r.h.s.  the probabilities ( [ eq : wpmpc ] ) are no longer bounded between the values , thus , they can lead to violation of bell s inequalities . in particular , we show in fig .[ fig3a ] the dependence of from revealing violation of the chsh inequality in a small range of the amplitude .this result recall that of ref . where the state ( [ eq : pc ] ) was shown to violate another bell s inequality .for the state ( [ eq : pc ] ) , eq .( [ eq : calexz ] ) becomes where denotes the bessel function of order .then , in fig .[ fig3b ] we show for .the violation of chsh inequality becomes now more pronounced .in conclusions we have presented a tomographic approach to the quantum nonlocality of a bipartite continuous variable system . at first instancewe have proposed a rough use of tomograms for testing violation of local hidden variable theory .in such a case one does not really need of a complete set of tomograms. however , the presence of purely quantum correlations are not easily recognizable .as matter of fact , the epr state does not show the nonlocal character , while a pair - coherent state does .then , we have proposed to exploit the tomographically reconstructed state to get the statistics of pseudo - spin operators .this , practically , maps a continuous variable system into a discrete variable system making possible the use of standard bell s inequalities .this seeems a more powerful approach , but the price one ought to pay is the large amount of data ( a complete set of tomograms ) needed to reconstruct the state . in such a casethe epr state completely shows its nonlocal character .also the pair - coherent state evidenciates much larger violations .our result opens possibilities for testing qm against local hidden variable theories using very efficient detection methods like homodyne detection .finally , the present approach could be straightforward extended to multipartite systems , e.g. making possible test of grenberger - horne - zeilinger theorem in continuous variables version .moreover , it could be useful to characterize quantum states used in information processing .vim and evs thank the russian foundation for basic research for partial support under projects nos 00 - 02 - 16516 and 01 - 02 - 17745 .by refering to eq . ( [ eq : cali ] ) we note that the exponent of the imaginary argument is periodic , and since the integral is taken over a period of the exponent we can shift the angle argument as so the integral takes the form \,d\varphi\,,\end{aligned}\ ] ] where we have set then , the integral can be rewritten as \times \nonumber \\   & \times&\exp\left[-\frac{{(\alpha e^{-i\varphi})}^2}{2}+\sqrt{2}\alpha   e^{-i\varphi}x_2\right]\,d\varphi\,,\end{aligned}\ ] ] where we have set expanding each exponent in the above integral into a series of hermite polynomials , and taking into account the equality we obtain the following expression   let us now consider two generic series whose product yields how to express the sum of diagonal elements in terms of the ?the answer is found by taking into account the equality ( [ eq : exp ] ) , and it reads the sum in right - hand side of eq .( [ eq : i ] ) is the sum of diagonal terms of the product of two following series if we apply the formula ( [ eq : diag ] ) to this sum we obtain the representation ( [ eq : iint ] ) for it which we have started from . to obtain probabilities etc .we represent in the following way : to calculate probability we need to take the following integral : and analogously for other probabilities .swap integration order in these integrals we obtain the expression ( [ eq : wpmpc ] ) .a. aspect , p. grangier and g. roger , phys .lett . * 49 * , 91 ( 1982 ) ; a. aspect , j. dalibard and g. roger , phys .rev . lett . * 49 * , 1804 ( 1982 ) ; z. y. ou and l. mandel , phys .lett . * 61 * , 50 ( 1988 ) ; y. h. shih and c. o. alley , phys .. lett . * 61 * , 2921 ( 1988 ) ; j. g. rarity and p. r. tapster , phys . rev. lett . * 64 * , 2495 ( 1990 ) ; j. brendel , e. mohler and w. martienssen , europhys20 * , 575 ( 1992 ) ; p. g. kwiat , a. m. steinberg and r. y. chiao , phys .a * 47 * , 2472 ( 1993 ) ; t. e. kiess , y. h. shih , a. v. sergienko and c. o. alley , phys .lett . * 71 * , 3893 ( 1993 ) ; p. g. kwiat , k. mattle , h. weinfurter and a. zeilinger , phys .lett . * 75 * , 4337 ( 1995 ) ; d. v. strekalov , t. b. pittman , a. v. sergienko , y. shih and p. g. kwiat , phys .a * 54 * , 1 ( 1996 ) .

Abstract: we propose a tomographic approach to study quantum nonlocality in continuous variable quantum systems . on one hand we derive a bell - like inequality for measured tomograms . on the other hand , we introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state , thus giving the possibility to use standard bell s inequalities . illuminating examples are also discussed .  * keywords * : entanglement , nonlocality , quantum tomography , measurement theory .


ever since becker s ( 1974 ) theory of marriage , economists have been interested in the factors that lead to a stable marriage .it was becker who introduced the term `` assortative mating '' , which relates to the factors each party seeks in the other : men differing in physical capital , education or intelligence ( aside from their effects on wage rates ) , height , race , or many other traits will tend to marry women with like values of these traits " .the study of assortative mating is crucial to understand the formation of households , which is essential for the development of child human capital the basis for economic and social progress .there is a good deal of empirical evidence relating to becker s assortative mating hypothesis .positive spousal correlations have been found : ( a ) in anthropometric features , ( b ) in health an well - being , and ( c ) in smoking behaviour , see e.g. , . in this paper, we focus on a trait that is of particular interest to economists : risk attitude . also used the repeated data on married couples within the german socio - economic panel to investigate spousal correlation in risk attitude .they applied the bivariate panel ordered probit model to the self - reported risk attitude data from this source , and they found that the individual - specific effects ( in the risk attitude equation ) for the two members of a married couple , are positively correlated .this was interpreted in terms of positive assortative mating on risk attitude : individuals seek partners with similar risk attitude . as stressed by , it is not obvious that spousal correlation in risk attitude will be positive .risk - sharing arguments would lead to the hypothesis of negative correlation , or disassortative mating " . and , after all , according to a very well - known maxim , opposites attract " .  in this paperwe delve deeper into the investigation of assortative mating on risk attitude , by applying the copula approach .the main advantage of the copula approach is that it can allow a wide range of flexible tail dependence and asymmetry between the two variables ( here male s and female s risk attitude ) under investigation .the normality assumption used by is mathematically convenient , but one of our points is that this assumption is not reasonable if there is some skewness in the joint tails .for example , could it be that risk - lovers are particularly keen to match with other risk - lovers ( positive tail dependence ) ?could it be that risk - avoiders are particularly keen to match with risk - lovers ( negative tail dependence )? such features of the dependence structure may be captured when the copula approach is used .we are particularly interested in whether and how this dependence structure changes with years of marriage .a very interesting hypothesis is that of socialization : a process by which attitudes become more similar with time .this hypothesis can be investigated by estimating the copula model separately for different stages of marriage .we will use a general copula construction , based on a set of bivariate copulas to jointly model bivariate ordinal time - series responses with covariates .we call this the joint copula - based markov model " . for the proposed model ,we construct the joint distribution in terms of three bivariate copulas . for each ordinal time series we consider a copula - based markov model , where a parametric copula family is used for the joint distribution of subsequent observations andthen we relate these ordinal time - series responses using another copula to couple their conditional ( on the past ) distributions at each time point .much is known about properties of parametric bivariate copula families in terms of dependence and tail behavior . in this paperwe draw on this wealth of knowledge .the discretized bivariate normal ( bvn ) model is a special case of our general construction when all the bivariate linking copulas are bvn. other choices of copulas are better if ( a ) s have more probability in joint upper or lower tail than would be expected with a discretized bvn , or ( b ) s can be considered as discretized maxima / minima or mixtures of discretized means rather than discretized means .copulas that arise from extreme value theory have more probability in one joint tail ( upper or lower ) than expected with a discretized mvn distribution or an mvn copula with discrete margins .it is also possible that there can be more probability in both the joint upper and joint lower tail , compared with discretized mvn models .this happens if the respondents consist of a mixture " population ( e.g. , different genders ) .from the theory of elliptical distributions and copulas , it is known that some scale mixtures of mvn have more dependence in the tails .the theory and application of copulas have become important in finance , insurance and other areas , in order to deal with dependence in the joint tails . here ,we indicate that this can also be important into the concept of assortative mating . the remainder of the paper proceeds as follows .section [ overview ] has a brief overview of relevant copula theory .section [ themodel ] introduces the joint copula - based markov model for discrete ordinal responses with covariates and discusses its relationship with existing models .section [ sec - families ] discusses suitable parametric families of copulas for the joint copula - based markov model for discrete ordinal responses with covariates .estimation techniques and computational details are provided in section [ estimation ] . section [ sec - appl ]presents the applications of our methodology to assortative mating using data from the german socio - economic panel over the period 20042012 .it turns out that our model , with linking copulas selected according to the ordinal responses being plausibly based on mixtures of means , provides better fit than the discretized mvn model .we conclude with some discussion in section [ sec - discussion ] .a copula is a multivariate cumulative distribution function ( cdf ) with uniform margins .if is a bivariate cdf with univariate margins , then sklar s ( 1959 ) theorem implies that there is a copula such that the copula is unique if are continuous , but not if some of the have discrete components .if is continuous and , then the unique copula is the distribution of leading to where are inverse cdfs .in particular , if is the bivariate normal ( bvn ) cdf with correlation and standard normal margins , and is the univariate standard normal cdf , then the bvn copula is the major advantage of copulas for dependence modelling is that the dependence structure may be separated from the univariate margins ; see , for example , section 1.6 of .if is a parametric family of copulas and is a parametric model for the univariate margin , then is a bivariate parametric model with univariate margins . for copula models , the variables can be continuous or discrete .for ease of exposition , let be the dimension of a panel " and the number of clusters .the theory can be extended to varying cluster sizes .let be the number of covariates , that is , the dimension of a covariate vector .let be a latent variable , such that if where is the number of categories of ( without loss of generality , assume and ) , and is the -dimensional regression vector . from this definition, the response is assumed to have density where is a function of and the -dimensional regression vector , and is the -dimensional vector of the univariate cutpoints ( ) . note that normal leads to the ordinal probit model for ordinal response , logistic leads to the ordinal cumulative logit model .suppose that data are where is an index for individuals or clusters , is an index for the repeated measurements or within cluster measurements , and is an index for the gender : 1=male , 2=female .the univariate marginal model for is where and of dimension be the vector of univariate cutpoints .if for each , are serially independent conditional on , then the log - likelihood for each gender is   for estimation of when are dependent we consider copula - based markov models for ordinal time series with covariates , where a parametric copula family is used for the joint distribution of subsequent observations and then the corresponding transition probabilities are obtained .note in passing that studied copula - based markov models of continuous response data .the transition cdf of given is /f(y_{t-1,j}),\end{aligned}\ ] ] and the transition pmf is where .then the log - likelihood for each gender is   the bvn copula is a special case , and this is called autoregressive - to - anything " in as acknowledged by .other copulas would be useful for the transition probability if there is more clustering of consecutive large or small values than would be expected with bvn .so far we treat the ordinal response of the males and the ordinal response of the females separately as if they were independent . in the sequel ,we novel propose to relate these responses using a copula to couple their conditional ( on the past ) distributions at each time point . from , there is a bivariate copula such that .then it follows that the joint pmf is   for the joint copula - based markov model , we let and be parametric bivariate copulas , say with parameters and , respectively . for the set of all parameters , let .we model the joint distribution in terms of three bivariate copulas .there is much known about properties of parametric bivariate copula families in terms of dependence and tail behavior .note that the copula models the time - series for the response and the copula links the ordinal response for males to the ordinal response for females .our general statistical model allows for selection of and independently among a variety of parametric copula families , i.e. , there are no constraints in the choices of parametric copulas .in our candidate set , families that have different strengths of tail behaviour ( see e.g. , ) are included . in the descriptions below , a bivariate copula is _ reflection symmetric _ if its density satisfies for all .otherwise , it is reflection asymmetric often with more probability in the joint upper tail or joint lower tail . _upper tail dependence _ means that as and _ lower tail dependence_ means that as . if for a bivariate copula , then , where is the survival ( or rotated by 180 degrees ) copula of ; this reflection " of each uniform random variable about changes the direction of tail asymmetry .* reflection symmetric copulas with tail independence satisfying and as , such as the frank copula with cdf * reflection asymmetric copula family with upper tail dependence , such as the gumbel extreme value copula with cdf ,\quad \th\geq 1.\ ] ] the resulting model in this case has more probability in the joint upper tail compared to the bvn copula .that is there is more dependence of large ordinal values that would be expected with bvn .* reflection asymmetric copula family with lower tail dependence , such as the survival gumbel ( s.gumbel ) copula with cdf ,\quad \th\geq 1.\ ] ] the resulting model in this case has more probability in the joint lower tail compared to the bvn copula .that is there is more dependence of small ordinal values that would be expected with bvn . *[ t ] copulas with reflection symmetric upper and lower tail dependence , such as the bivariate student copula with cdf where is the univariate student t cdf with ( non - integer ) degrees of freedom , and is the cdf of a bivariate student t distribution with degrees of freedom and correlation parameter .a small value of , such as , leads to a model with more probabilities in the joint upper and joint lower tails compared to the bvn copula .that is there is more dependence of large and small ordinal values than would be expected with bvn .  for this paper ,the above copula families are sufficient for the applications in section [ sec - appl ] , since tail dependence is a property to consider when choosing amongst different families of copulas and the concept of upper / lower tail dependence is one way to differentiate families .have shown that it is hard to choose a copula with similar properties from real data , since copulas with similar ( tail ) dependence properties provide similar fit . to depict the concept of reflection symmetric ( asymmetric ) tail dependence ( independence ) , we plot contour plots of the corresponding copula densities with standard normal margins and dependence parameters corresponding to kendall s value of in figure [ contours ] .cc contour plots of bvn , , frank , gumbel and s.gumbel copulas with standard normal margins and dependence parameters corresponding to kendall s value of ., title="fig:",scaledwidth=30.0% ] &  contour plots of bvn , , frank , gumbel and s.gumbel copulas with standard normal margins and dependence parameters corresponding to kendall s value of ., title="fig:",scaledwidth=30.0% ] +  + contour plots of bvn , , frank , gumbel and s.gumbel copulas with standard normal margins and dependence parameters corresponding to kendall s value of ., title="fig:",scaledwidth=30.0% ] & contour plots of bvn , , frank , gumbel and s.gumbel copulas with standard normal margins and dependence parameters corresponding to kendall s value of . , title="fig:",scaledwidth=30.0% ] +  note also also that these copulas satisfy the conditions under which a copula function generates a stationary markov chain that satisfies mixing conditions at a geometric rate .the log - likelihood of the joint copula - based markov model is where is given in ( [ jointserpmf ] ) and   the maximum likelihood ( ml ) estimates can be derived using the steps below :  1 .  for each :1 .  at the first step the in ( [ indlik ] )is maximized over the univariate marginal parameters assuming time independence .2 .  at the second step the in ( [ serlik ] )is maximized over the copula parameter with univariate parameters fixed as estimated at the first step .3 .  at the third step the in ( [ serlik ] )is maximized over both the univariate parameters and copula parameter with initial parameters the estimates for the preceding steps .2 .  at the fourth step the in ( [ jointserlik ] )is maximized over the copula parameter with all the other parameters fixed as estimated at the preceding steps .3 .  at the final step the in ( [ jointserlik ] )is maximized over with initial parameters the estimates for the preceding steps .  in the steps above the inference function of margins method used to get initial estimates .each of the estimated parameters can be obtained by using a quasi - newton method applied to the log - likelihood .this numerical method requires only the objective function , i.e. , the joint log - likelihood , while the gradients are computed numerically and the hessian matrix of the second order derivatives is updated in each iteration .the ses of the ml estimates can be also obtained via the gradients and the hessian computed numerically during the maximization process .assuming that the usual regularity conditions for asymptotic maximum likelihood theory hold for the bivariate model as well as for its margins we have that ml estimates are asymptotically normal .therefore one can build wald tests to statistically judge the effect of any covariate .couples were followed during 20042012 .interviews were administered in 2004 , every 2 years until 2008 , and every year thereafter , resulting in a maximum of 7 measurements per couple .we use the 2201 couples observed at all seven time points , although our methodology does not depend on a constant cluster " size .the outcome variable in our analysis is the risk attitude question which is asked in each of the seven survey years .the question is reproduced as follows :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ how do you see yourself ?are you generally a person who is fully prepared to take risks or do you try to avoid taking risks? please tick a box on the scale , where 0 means risk averse " and 10 means fully prepared to take risks " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ]   [ joint - copula-2009 - 12 ]  standard errors are shown in parentheses for the best fit ; age and age are scaled by factor and respectively .tables [ joint - copula-2004 - 8 ] and [ joint - copula-2009 - 12 ] show results from the joint models estimated with data from 20048 and 200912 respectively , and assuming the univariate models found to be best - fitting in tables [ time - series-2004 - 8 ] and [ time - series-2009 - 12 ] . comparing tables [ joint - copula-2004 - 8 ] and [ joint - copula-2009 - 12 ], we note two key differences .firstly , the estimate of kendall s rises from 0.146 in 20048 to 0.191 in 200912 , and moreover , it may be verified from the associated standard errors that the two 95% confidence intervals do not overlap .hence we have evidence that kendall s rises with years of marriage .the second key difference is that the best - fitting joint model for the 20048 data is , while the best - fitting joint model for 200912 is .there is a reduction of one in the degrees of freedom of the best - fitting copula . both of these differences between the results in tables [ joint - copula-2004 - 8 ] and [ joint - copula-2009 - 12 ]provide evidence that dependence increases with years of marriage ; both in the middle of the distribution , and in the joint tails .at this point it is useful to recall that the sample used in this study is restricted to couples observed in every year .hence attrition ( poorly - matched couples dropping out of the sample ) is not an issue .this means that the increase in dependence seen in the comparison of tables 4 and 6 may be interpreted as clear evidence of `` socialization '' : the two members of the couple become more similar in terms of risk - attitude as their marriage progresses .assortative mating is a concept of great interest to economists , as evidenced by the extensive theoretical and empirical literatures devoted to it , dating back to .the econometric modelling of assortative mating is clearly a setting in which the precise nature of the dependence between two variables representing the relevant outcomes for the two members of the couple , becomes the central focus of the analysis .the copula approach provides the ideal framework for this sort of modelling , since it brings richness to the dependence structure .for example , it allows the modelling of both joint tails and moreover allows a variety of types of asymmetry in the dependence structure .importantly , this richness is introduced without the requirement of estimation of additional parameters .much is known about copulas .they have been used to great benefit in many areas ( e.g. , insurance , finance and hydrology ) , but have not been used much in economics .this paper illustrates their usefulness in a mainstream economics application : assortative mating on risk attitude . in this application , we have found evidence of both centre dependence and tail dependence .that is , over the entire range of of risk attitudes , there is a tendency for individuals to match with other individuals with similar risk attitude , and this positive dependence is particularly marked in the tails .this conclusion has been arrived at using a novel model which we have labelled the joint copula - based markov model " .the model consists of two stages . in the first stage ,males and females are considered separately , and a copula is used to model the joint distribution of neighbouring ( in time ) outcomes for a given individual .the best fitting copulas thus found are then combined using a third copula that couples the two conditional distributions at each time point .in all three cases , the best - fitting copula is found to be a with a small , which leads to a model with more probabilities in the joint upper and joint lower tails compared to the bvn copula . in this economics application , it is highlighted that a joint model with a copula is plausible for a population that is a mixture of subpopulations , while a bvn model might be adequate for smaller homogeneous subgroups .this fact leads to the conclusion of positive dependence in the tails , and we have found strong statistical evidence to confirm this , in the form of the vuong s statistic .  having reached this conclusion , we then extended the analysis by applying the same two - stage procedure separately for the 2004 - 8 and 2009 - 12 data .the key results here were that both middle and tail dependence were stronger in the second time range .we interpreted this in terms of the phenomenon of socialization " : the two members of the couple become more similar in risk attitude with the accumulation of years of marriage .we acknowledge access to the german socio - economic panel for use in this research under licence number : 2596 .thanks to phil bacon for outstanding data management .nikoloulopoulos , a. k. ( 2013 ) .copula - based models for multivariate discrete response data . in durante , f. , hrdle , w. , and jaworski , p. , editors , _copulae in mathematical and quantitative finance _ ,pages 231249 . springer .

Abstract: a novel copula - based bivariate panel ordinal model is applied to self - reported ( ordinal ) data on risk attitudes for males and females within married couples , from the german socio - economic panel over the period 2004 - 2012 . in the first step , a copula - based markov model is used to relate the marginal distribution of the response in different time periods , separately for males and females . in the second step , another copula is used to couple the males and females conditional ( on the past ) distributions . this coupling copula accounts for the dependence between males and females , and is therefore the focal point of the paper . we find positive dependence , both in the middle of the distribution , and in the joint tails , and we interpret this as positive assortative mating on risk attitude . by estimating the model separately for different time ranges , we then find evidence that middle- and tail - dependence both increase with years of marriage , and this is interpreted as evidence of `` socialization '' . +  + _ keywords : _ copula models ; joint tail probabilities ; markov models ; panel ordinal data ; risk - lovers / avoiders .   u # 1 # 1


the motion of spin bearing particles can be studied by the judicial use of pulsed magnetic field gradients in nmr experiments  . pulsed gradient spin echo ( pgse ) nmr is best known for diffusion measurements in porous media as well as for velocimetry studies  .the potential of pgse nmr for nmr microscopy has been demonstrated with the appearance of diffraction patterns when self - diffusion was studied in a bead pack sample  .magnetic resonance pore imaging ( mrpi ) is a recent advancement in pgse nmr which dramatically increases the resolution as compared to mri for suitable closed pore systems  .mrpi relies on a weak gradient pulse which can be several hundreds of milliseconds long . during this gradient pulsethe nuclear spins have to be kept in the transverse plane  .we suggested therefore a cpmg based design in order to refocus the internal gradients and other experimental imperfections  .however , the introduction of many rf - pulses in the presence of magnetic field gradients leads to a complicated coherence pathway spectrum  .this may not always be a problem in cpmg relaxation studies , since the decay due to transverse relaxation is still obtained  . in mrpihowever , it is essential to preserve the spin phase throughout the rf - pulse train , while the unwanted pathways do interfere adversely  . since traditional phase cycling techniques proved impractical for our cpmg pulse sequence design , it was necessary to explore alternative ways to filter for the correctly encoded signalthis study focuses on the application of a novel phase cycling technique called phase incremented echo train acquisition ( pieta )  .  in pulsed nmr the spectrum of coherence pathways ( cps )is multiplied with each additional rf - pulse  .this creates a very rich spectrum of cps , where each cp contains specific information which is influenced by its history since excitation .since each coherence may experience different physical interactions , it is paramount to select the right cps of the spin system under study .the most popular strategy to achieve this selection involves the nested cycling of the rf - pulse phases and the receiver phase from scan to scan according to a master equation  .however , the number of cps grows exponentially with the number of rf - pulses and finding the right phase cycling scheme may become complicated . throughout the article we consider a spin- system without coupling between the individual spins and where the desired cp does not involve any -storage . in this case , an nmr pulse sequence with rf - pulses would need phase steps following the nested phase cycling scheme  .several algorithms were designed to solve this problem with the help of computer programs  .two other widely applied strategies are improved perfection of the spin rotations induced by specially designed rf - pulses  and the application of pulsed field gradients . these two latter strategies can reduce the complexity of the problem , but they are often harder to implement or face limitations due to the available hardware .especially the carr - purcell - meiboom - gill ( cpmg ) nmr pulse sequence  and its derivatives can contain several hundreds or even thousands of rf - pulses and the number of phase steps with traditional phase cycling would be forbiddingly high .a first solution emerged with the development of cogwheel phase cycling which when applied to cpmg like pulse trains promised significant reductions in the number of phase steps needed  . when cogwheel is applied to cpmg the receiver phases have to be cycled such that they span a angle range after acquisitions , where is the number of rf - pulses  .this renders this method appealing for experiments with a fixed number of rf - pulses .however , in the case of cpmg echo trains with a variable number of rf - pulses it would require as many independent measurements ( including the cogwheel phase cycle ) as different numbers of rf - pulses have been chosen .a recent approach called phase incremented echo train acquisition ( pieta ) offers the extraction of all echoes in a given rf - pulse train and a de - convolution of the contributing coherence pathways with the same number of scans as cogwheel  .  in this work, pieta is utilized to investigate the coherence pathway spectrum contributing to the spin echo signal of mrpi .furthermore , it is shown how the artifact free coherence pathway can be selected .in nmr experiments the desired information is usually contained in one specific coherence pathway ( cp ) or a small subset of all cps  . the values denote the coherence order of the spins after the -th rf - pulse of the nmr pulse sequence .the elements can assume integer values which are determined by the spin quantum number and the coupling between multiple spins . for a system of uncoupled spin- nuclei the values of can be and . it is assumed that every cp starts from a well defined equilibrium with ( associated with magnetization in -direction ) before the first rf - pulse . for a cp to be measured with quadrature nmr detection it has to end with  .the magnetization of all three levels is mixed with each additional rf - pulse giving rise to a large number of cps growing exponentially with , where is the number of rf - pulses  .the complex nmr signal for a single scan with scan counter can be expressed as where is the signal arising from a specific coherence pathway after the last rf - pulse . denotes the final phase with which the cp is recorded in the spectrometer memory and it is given by  here , are the phases of the -th rf - pulse and is the coherence transfer difference induced with the -th rf - pulse .the signal after scans can be expressed as the complex sum given by    where we inserted eq .[ eq : signal ] in the first step and in the second step we used the fact that does not depend on .the last term in eq .[ eq : signal_all_scans ] shows that each individual cp signal is multiplied by the sum of phase factors added over all scans .thus , in order to filter any desired pathway after scans , the following condition has to be full - filled the most popular strategies to achieve this condition are the nested phase cycling approach  and cogwheel phase cycling scheme  . both methods involve the design of a set of for each scan and to cycle the receiver phase such that the wanted cp is always recorded with the same phase , while all other cps cycle in phase and cancel upon completion of the full set of scans . in the case of many rf - pulses , such as they are common in cpmg type experiments ,these methods are hard to apply because of the shear number of cps and they do not provide any extra information about the importance of the unwanted cps .  a different concept was proposed with multiplex phase cycling  where each scan is recorded separately in the spectrometer memory and the wanted cps are selected in a post - processing step .it will be shown in the next section how pieta is similar to both multiplexing and cogwheel phase cycling .baltisberger et al . introduced a new phase cycling scheme which allows one to extract the direct cp from echo train acquisition ( eta ) experiments .the authors use eta as a general expression for a train of spin echos due to a train of rf - pulses , which can be appended to an nmr experiment or may represent a stand - alone experiment like cpmg . this novel approach was called phase incremented echo train acquisition ( pieta ) , because the phase of every other refocusing pulse is incremented as a single variable from scan to scan.[pp : one_echo ] note that the application of pieta in this contribution slightly deviates from the use cases suggested by baltisberger _ . instead of recording many spin echo amplitudes for each scan , here only one spin echois recorded at the end of the rf - pulse train for each scan as will be explained below .( 1200,700)(-110,-350 ) ( 0,50 ) ( 0,50 ) ( 0,0)(0,1)200 ( 3,13)(0 , ) ( 0,0 ) ( 0,0)(0,1)50 ( 200,0)(0,1)50 ( 200,50 ) ( 0,0)(0,1)200 ( 5,15)(0 , ) ( 200,0 )  ( 400,50 )  ( - , ) ( - , ) ( -,0 ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( 0 , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , 0 ) ( 0,0)(,)( 600,50 ) ( 0,0)(0,1)200 ( 5,15)(0 , )  ( 800,50 )  ( - , ) ( - , ) ( -,0 ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( - , ) ( 0 , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , 0 ) ( 0,0)(,)   ( 600,0 ) ( 600,0)(0,1)50 ( 180,-350)(820,600 ) [ ] ( 920,-380)(30,-5)[l]loop ( -90,-225)(30,-5)[l] ( -45,-225)(30,-5)[r] ( -45,-150)(30,-5)[r] ( -45,-300)(30,-5)[r]  ( 0,-225)(0,-1)75 ( 0,-300)(1,0)185 ( 185,-300)(1,6)25 ( 210,-150)(1,0)375 ( 585,-150)(1,-6)25 ( 610,-300)(1,0)390 ( 0,-225)(0,15)5 ( 0,1)10 ( 0,-150)(15,0)13 ( 1,0)10 ( 187,-160)(2,-10)15 ( 0,1)7 ( 215,-300)(15,0)25 ( 1,0)10 ( 587,-290)(2,10)15 ( 0,-1)7 ( 615,-150)(15,0)26 ( 1,0)10  pieta applied to the cpmg pulse sequence is shown in fig .[ fig : pieta_sequence ] . the part enclosed by the dashed lineis repeated multiple times to create a train of refocusing rf - pulses .the rf - pulse phase is incremented from scan to scan , while the phases of the leading rf - pulse , of every other rf - pulse and of the receiver phase stay fixed for all scans .the level diagram below the rf - pulses in fig .[ fig : pieta_sequence ] shows the three coherence orders . in this level diagram the direct cp which is measured for even echos indicated as a solid line , while its conjugate for the odd echos is shown as a dashed line .with pieta the vector of spectrometer phases for the -th scan is given by , where is a winding number given by thus , the phase is rotated by when all scans are completed .the final signal phase for the pieta experiment as shown in fig .[ fig : pieta_sequence ] is given by where the phases which remain fixed for all scans have been neglected .the fixed phases lead to a constant phase for all scans and will not be needed in the following analysis .one may write eq .[ eq : phase_pieta ] as a sum leading to where is defined as the cumulative coherence transfer difference induced by the even rf - pulses [ pp : delta_p_def ] which are phase cycled with .therefore , the nmr signal as given by eq .[ eq : signal ] can be rewritten according to it can be seen that assumes the role of a modulation frequency when the signal is acquired as a function of the phase .similar to the multiplex phase cycling scheme  , all the spin echos in the cpmg train are stored separately for each scan in the computer memory , while is incremented from scan to scan .let the number of spin echos recorded be denoted by then the stored data is the signal matrix . in this matrix , represents the discrete time steps which start all over again with each spin echo .one may recall that is connected with as the underlying modulation frequency ( see eq .[ eq : deltap ] ) which implies that and are a fourier pair similar to the relation found in  .thus , after all scans are completed one can perform a discrete fourier transform in the direction of the data matrix to obtain the signal in the dimension according to the crucial observation of baltisberger et al. is that is unique for the direct cp and can be extracted from without interference of any other cps .[ pp : comment_9 ] for the isolated spin- system considered here , the direct cp difference vector for even is given by and for odd is given by .the cp difference for each rf - pulse cycled with is equal to , while for odd spin echos it is equal to since the conjugate cp is measured .all other cp vectors have one or more elements with and thus they have a smaller cumulative coherence pathway difference . for the direct coherence pathway in dependence of the spin echo count is given by  where is the integer floor function .note that for cpmg the spin echo count as a function of the number of rf - pulses is given by in order to account for the excitation pulse .there is a minimum number of scans needed in order to completely de - aliase the direct cp , since the number of phase increments has to be chosen such that the nyquist - shannon criterion is full - filled . in the case of pieta the sampling frequency is given by .this frequency has to be at least double the frequency which shall be detected .the highest frequency to be detected is is given by for the direct cp as given by eq .[ eq : direct_selection ] .thus , if the spin echo count is even then and for odd spin echoes it is .therefore , the number of phase increments in dependence on the echo count is given by   [ pp : one_echo_ii ] in this work , the spin echo count will not be increased as compared to conventional cpmg relaxation experiments .instead , stays fixed and the gradient amplitude is stepped as will be explained in the next section .   magnetic resonance pore imaging ( mrpi )can be regarded as a pulsed gradient spin echo ( pgse ) nmr technique .pgse nmr techniques measure the nmr signal as a function of the wave vector , where denotes the gradient amplitude , is the gyro - magnetic ratio of the observed nucleus and is the length of the gradient pulses .the pgse nmr signal is usually normalized with the nmr signal obtained with zero gradient pulse amplitude .recently , laun _ et al . _ suggested the gradient pulse pattern used for mrpi , which created a hybrid between mri and pgse nmr .with mrpi the spin echo attenuation is resembling the averaged form factor of the pores as measured in the direction of .the form factor is the fourier transform of the pore shape .therefore , one may obtain an average image of the pores by performing an inverse fourier transformation of the mrpi spin echo attenuation if certain requirements are met , i.e. .note that the pore shape equals the spin density distribution if the spin density is homogeneous throughout the pore space . in this contributionwe discuss the experimental results for a cylindrical capillary sample for which the form factor is given by where denotes the cylindrical bessel function of first order and is the radius of the cylindrical domain . for obtaining eq .[ eq : cyl_formfactor ] the wave - vector has to be applied perpendicular to the cylinder axis . in the following discussion it will be assumed that the gradient pulses will always be applied in this direction and therefore the vector notation is dropped .mrpi relies on the application of a long gradient pulse with duration and amplitude as shown in fig .[ fig : mrpi_vs_pgse ] ( a ) .the long gradient pulse allows for diffusion encoding , while the intense gradient with amplitude and duration acts similar to a phase imaging gradient in mri  .these two gradient pulses have to full - fill the spin echo condition which requires .[ pp : def_of_t ] here , is defined as the time from the onset of the first gradient pulse until the the end of the last gradient pulse .  ( 900,250)(-100,-170 ) ( 0,0 ) ( 0,-10 ) ( 30,0 ) ( 10,0 ) ( 610,0 ) ( 10,0 ) ( 610,0 ) ( 10,0 ) ( 530,20)(0,-1)20 ( 530,-40)(0,1)20 ( 535,20) ( 735,0 ) ( -80,-10 )   +  ( 900,250)(-100,-110 ) ( 0,0 ) ( 0,-10 ) ( 0,10)(1,0)850 ( 30,0 ) ( 10,0)(120,0)5 ( 0,0 ) ( 70,0)(120,0)5 ( 0,0 ) ( 610,0 ) ( 670,0 ) ( 60,0)(60,0)12 ( 0,0 ) ( 0,0)(0,1)100 ( 5,15)(0 , ) ( 30,0 ) ( 0,0)(0,1)100 ( 5,10)(0 , ) ( 690,0 ) ( 0,0)(0,1)100 ( 5,12)(0 , )  ( 770,30)(0,-1)20 ( 770,-20)(0,1)20 ( 780,35) ( -80,-10 )  ( 900,150)(-100,-70 ) ( 0,0 ) ( 0,-10 ) ( 30,0 ) ( 10,0)(60,0)10 ( 0,0 ) ( 610,0)(60,0)2 ( 0,0 ) ( 60,0)(120,0)6 ( 0,0 ) ( 120,0)(120,0)5 ( 0,0 ) ( 30,0 ) ( 750,10)(0,-1)10 ( 750,10)(1,0)100   ( 460,25 ) ( 475,-30)(1,0)25 ( 545,-30)(-1,0)25 ( 500,-35)(0,1)10 ( 520,-35)(0,1)10 ( 500,-70) ( -80,-10 )  fig .[ fig : mrpi_vs_pgse ] ( b ) shows schematically the replacement of the gradient pulse pattern as initially suggested in  with the one used in this work .the gradient pulses were applied with alternating polarity in the laboratory reference frame while they were interspersed with rf - pulses .the replacement was chosen to minimize the dephasing due to the internal gradients by refocusing their influence on the spin phases  .[ fig : mrpi_vs_pgse ] ( c ) shows the effective gradient pattern as experienced by the spin system when the effect of the pulses is accounted for .the pulsed gradients are effectively adding up to yield a pattern similar to fig .[ fig : mrpi_vs_pgse ] ( a ) , while the internal gradients are compensated .note that many more gradient pulses were used in the actual implementation , which will be shown in sec .[ sec : exp ] along with the rf - pulse phases .  for illustration purposes ,consider the gradient pattern as shown in fig .[ fig : mrpi_vs_pgse ] ( a ) . in the ideal case that the weak gradient pulse is infinitely long and the intense pulse is applied instantaneously the spin echo attenuation is given by  which yields the form factor averaged over the pores in the sample . under experimental conditions the requirements and can not be full - filled .fortunately , the need for infinitely long gradient pulses can be relaxed to the condition that the molecules traverse the pore space multiple times such that[pp : inequality ] .however , this less stringent requirement has been linked to blurring of the obtained average pore image  .secondly , the diffusion encoding during has to be accounted for as well , since the intense gradient pulse can not be applied instantaneously . on averagethe molecules push away from the pore walls during , which leads to edge enhancement similar to the phenomenon in mri  . these two effects together with experimental adaptations of mrpi may pose challenges for the formulation of fitting functions especially when the gradient pattern involves many pulses as shown in fig .[ fig : mrpi_vs_pgse ] ( b ) and ( c ) .therefore , the multiple correlation function ( mcf ) technique was employed for the simulation of the spin echo decay .this simulation technique is flexible enough for most gradient pulse patterns and it can adequately account for both blurring and edge enhancement  .  in mcfthe pgse nmr signal is found by evaluating the ordered matrix product \right]_{\text{0},\text{0}}\,,\ ] ] where {0,0}$ ] denotes the first diagonal element of the resulting matrix .the piece - wise constant function is the dimensionless temporal profile of the normalized spatial magnetic field profile with maximum amplitude , such that the magnetic field at any time is given by  is the correlation matrix of the spatial gradient profile with the eigenfunctions of the laplace operator and is the diagonal eigenvalue matrix of the laplace operator .furthermore , is the reduced self - diffusion coefficient and denotes the generalized gradient intensity .more detailed descriptions of the mcf technique and its application to mrpi can be found in  .  for calibration purposesthe spin echo decay has been measured previously in an isotropic liquid  . when mrpi is applied to nuclear spins in an isotropic unrestrictedliquid one obtains a gaussian attenuation similar to other pgse nmr techniques given by where is the self - diffusion coefficient of the spin bearing molecules and is the effective observation time , which is a function of the pulsed gradient pattern . in the nmr self - diffusion literature ,the term is usually captured as one variable in the so called -factor  .however , in this contribution the case of unrestricted self - diffusion will be compared to the case of -space imaging with mrpi .therefore , we chose to express the spin echo attenuation as a function of the -value .the effective diffusion time can be calculated in analogy to the well known double integral for the calculation of the -factor in the pgse nmr literature  .it is given by ^ 2\,,\ ] ] where the factor is appearing , because it has been factored into the -value which is given by for our implementation .[ pp : delta_eff ] for this work , only two intense gradient pulses were applied and gradient pulses are adding up to yield the weak long gradient .[ pp : comment_16 ] evaluation of eq .[ eq : double_int ] for the cpmg based mrpi pulse sequence as shown in fig .[ fig : mrpi_vs_pgse ] ( c ) yields  \,,\ ] ] where is the gradient pulse duration and where is the separation between the gradient pulses , i.e. the time from the end of one gradient pulse until the leading edge of the next gradient pulse .figure [ fig : mrpi_seq ] a ) shows the rf - pulse sequence of the cpmg based mrpi experiment  .( 1200,450)(-110,-100 ) ( -80,120 ) ( 0,50 ) ( 0,50 ) ( 0,0)(0,1)170 ( 3,13)(0 , ) ( 0,0 ) ( 100,50 ) ( 0,0)(0,1)170 ( 5,15)(0 , ) ( 100,0 ) ( 300,50 ) ( 0,0)(0,1)170 ( 5,15)(0 , ) ( 300,0 ) ( 500,50 ) ( 0,0)(0,1)170 ( 5,15)(0 , ) ( 500,0 ) ( 700,50 ) ( 0,0)(0,1)170 ( 5,15)(0 , ) ( 700,0 ) ( 900,50 ) ( 0,0)(0,1)170 ( 5,15)(0 , ) ( 900,0 ) ( 1000,50 ) ( -,0)(-,0)(- , ) ( -,)(0 , ) ( , ) ( , ) ( , 0)(,0 ) ( 0,0 ) ( , ) ( 0,0)(0,- ) ( 1000,50 ) ( -,0)(-,0)(- , ) ( -,)(0 , ) ( , ) ( , ) ( , 0)(,0 ) ( 10,30)(,) ( 0,0)(0,- ) ( 1000,50 ) ( -,0)(-,0)(- , ) ( -,)(0 , ) ( , ) ( , ) ( , 0)(,0 ) ( 0,0 ) ( , ) ( 0,0)(0,- ) ( 1000,40)(0,1)20 ( 0,0)(0,1)50 ( 100,0)(0,1)50 ( 300,0)(0,1)50 ( 500,0)(0,1)50 ( 700,0)(0,1)50 ( 900,0)(0,1)50 ( 1000,0)(0,1)50  ( 1200,350)(-110,-100 ) ( 0,60 ) ( 0,20 )  ( -80,130 ) ( 0,100 ) ( 150,100 ) ( 350,100 )   ( 150,100 ) ( 350,100 )   ( 150,100 ) ( 350,100 )   ( 150,125 ) ( 10,100 ) ( 950,100 )   ( 550,100 ) ( 750,100 )   ( 550,100 ) ( 750,100 )   ( 550,100 ) ( 750,100 )   ( 750,125 )  ( 93,40)(400,500 ) [ ] ( 250,-5)(30,-10)[l]loop ( 855,-49 ) ( 985,150)(0,-1)40 ( 985,60)(0,1)40 ( 1005,125) ( 360,170)(0,-1)40 ( 360,60)(0,1)40 ( 375,150)  the rf - pulse phases were implemented according to the pieta phase scheme , where the odd rf - pulses are phase cycled with .the phase in units of radians of the rf - pulse was fixed at , the phases of the even numbered rf - pulse were set to and the receiver phase was kept at .the gradient pulse scheme is depicted in fig .[ fig : mrpi_seq ] b ) . here, shows the relative amplitude of the gradient pulses as they are applied in the laboratory reference frame . there is a read gradient applied in the beginning of the pulse sequence and during the detection period for the detection of gradient pulse mismatches . for this study ,the number of long gradient pulses with amplitude ranged from to .note that no spin echo is formed until after the last rf - pulse .thus , the spin echo count is fixed and for the implementation shown in fig .[ fig : mrpi_seq ] it is given by .[pp : eq_delta_p ] considering eq .[ eq : direct_selection ] and the fact that is an odd number it follows that the desired signal is contained in the matrix elements of where .previous mrpi results as discussed in and  have been obtained with the same pulse sequence , but with a -step phase cycle .table [ tab : phase_cycle ] shows the phase cycle , which has been designed in analogy to a cyclops phase cycle adapted to the special case of the mrpi pulse sequence .ccccc & + & & & & +  + & & & & + & & & & + & & & & + & & & & +  the rf - pulse phases are not shown in fig .[ fig : mrpi_seq ] , but have been named for the odd rf - pulses , for the even rf - pulses and for the last rf - pulse before detection . the rf - pulse phase has been introduced to filter the pathway , which does not experience the stepped gradient pulses and can therefore not be dephased .the experiments were carried out on a bruker avance ii spectrometer with a proton resonance frequency of .the gradient pulses were generated by a bruker great 60 gradient amplifier in conjunction with a micro 2.5 imaging probe . at maximum current output of the gradient system provided gradient strengths up to .the utilized rf - coil had a birdcage coil design with an inner diameter of .typical rf - pulse durations were and when utilizing as the nmr active nuclei .a spherical sample tube ( ) filled with doped was utilized for testing of the mrpi pulse sequence and to measure the spin echo attenuation for unrestricted self - diffusion . for the mrpi experiments in a model porous system a capillary samplewas prepared , which contained ca . polyimide coated glass capillaries with inner diameter of .these capillaries were filled with distilled according to the procedure outlined in  .it has been shown previously that the mrpi pulse sequence can be calibrated by measuring the free self - diffusion coefficient of a suitable isotropic liquid  .thus , the cpmg based mrpi pulse sequence was applied to the spherical sample described in sec .[ sec : hardware ] and the pieta phase cycle was applied to investigate the coherence pathway spectrum .the number of long gradient pulses was , which were refocused by narrow gradient pulses .the measured spin echo corresponds to the spin echo count and therefore the number of phase steps was set to , respecting the condition of eq .[ eq : nscan_pieta ] .the resulting data matrix had dimensions of , where was incremented in steps , points were acquired in the time domain for each scan and phase steps were acquired . a subsequent fourier transform analogous to eq .[ eq : ft ] in the direction yielded the signal matrix .[ fig:3dcp_freediff ] shows the signal matrix with dimension , where the instance in time is chosen to show only the spin echo centers .  .the desired pathway is the direct coherence pathway with .the inset shows the cp spectrum for zero pulsed field gradient amplitude ( ) . ]the coherence pathway spectrum shows several peaks in the dimension for the first row in the spectrum where .since the contribution of unwanted pathways is the highest for zero gradient amplitude the corresponding spectrum is shown separately in the inset .the direct coherence pathway can be selected at according to eq .[ eq : direct_selection ] .it is apparent that the attenuation of the pathway with follows a gaussian shape as expected by eq .[ eq : gaussian ] , while all other pathways experience a sharper drop in amplitude when is increased .this is a consequence of their dephasing with increasing gradient pulse amplitudes .one exception being the cp at , with .this cp is not dephased , since it is only influenced by the constant read gradient in the beginning and the end of the pulse sequence for which the spin echo condition is full - filled regardless of the amplitude of the other pulsed gradients .this pathway was accounted for when designing the -step phase cycling scheme discussed in sec .[ sec : design ] .furthermore , the relative amplitude of all signal contributions at low values is high enough to significantly impact on the spin echo amplitude if they are not filtered in an experiment without pieta .the most important pathways are the ones close to , which are stimulated echo pathways with many intervals where the magnetization is stored in the -direction ( i.e. many elements with ) , while coherence pathways close to the direct pathway with also appear with significant intensity .the latter pathways are characterized by a small number of time intervals where the magnetization is stored in -direction . during the -storage timeintervals the magnetization associated with these cps is subject to relaxation instead of relaxation .thus , their relative amplitude with respect to the direct cp may be very high even in the case of nearly perfect rotations of the magnetization .[ fig : pieta_vs_4step ] shows the normalized spin - echo attenuation as obtained by selecting the direct coherence pathway with ( squares ) in comparison to an mrpi experiment performed with the -step phase cycle ( triangles ) .-step phase cycling scheme ( triangles ) and the pieta phase cycling scheme with subsequent selection of the direct coherence pathway ( squares ) .the insert shows a zoom of the signal amplitudes at low values . ]all parameters were kept the same for both experiments except for the phase cycling scheme . for both experiments the gradient duration was and the inter rf - pulse spacing was .the solid line in fig . [ fig : pieta_vs_4step ] represents the fit of the experimental data against eq .[ eq : gaussian ] .the fitted self - diffusion coefficient was , which is higher than the literature value for of  .however , it has been shown that the right self - diffusion coefficient can be obtained with mrpi when less gradient pulses are utilized  .this suggests that the gradient ramp times which are not accounted for in eq .[ eq : delta_eff ] may play an increasingly important role with increasing numbers of gradient pulses .this is consistent with findings for pulse sequences specifically designed for the measurement of self - diffusion coefficients  .nonetheless , the expected gaussian attenuation is obtained and the effect of the coherence pathways on the spin echo amplitude for low values can be studied . clearly , the amplitudes of the first four data points are lower than expected for the -step phase cycle ( triangles ) . especially the fist point measured with zero gradient amplitude is below the expected amplitude .this signal suppression is caused by spurious coherence pathways which interfere with the direct coherence pathway destructively .it may appear surprising that the spin echo amplitude first increases before it asymptotically follows the expected gaussian attenuation according to eq .[ eq : gaussian ] .this surprising increase in amplitude is caused by the fact that the recorded amplitude is obtained by the sum of the individual cps .each cp is associated with a corresponding complex magnetization . when the magnetization of the cps is recorded with a different magnetization phase during detection, they do interfere destructively . on the other hand ,the spin echo attenuation extracted from the mrpi pieta experiment does not suffer from destructive interference and thus the correct amplitude for low -values was recovered .following the bulk experiments , mrpi pieta was applied to the capillary sample with the gradient pulses applied perpendicular to the cylinder axis .the number of long gradient pulses was and therefore .the number of phase steps was set to , which was phase steps less than required by eq .[ eq : nscan_pieta ] .this was due to the requirement of fft algorithms to operate on sampling points of multiples of power of two , else the necessary number of phase steps would have been .this would have extended the experimental time beyond practical limits .in hindsight , a bluestein algorithm  accepting an arbitrary number of samples would be favorable in the future .analysis of the results , however , can still be carried out with the under sampled data set , since fold - back signals do not overlap in the resulting coherence pathway spectrum .[ fig:3dcp_cyl ] shows the real part of the matrix with the dimension , which was obtained from the matrix by selecting the signal at the time instance of the spin echo center .   for the center of the spin echo .the desired pathway is the direct coherence pathway with which appears at due to fold - back . ]further parameters were the gradient duration of and the echo time of . due tothe under sampling of the spectrum in the dimension , one obtains a cp spectrum with fold backs of the desired signal which appears at .similar to the experiment shown for the free self - diffusion case there are several coherence pathways close to and in the centre of the spectrum at for . here, however also cps close to are contributing significantly .these pathways are initially following the pathway which is conjugate to the desired pathway and they are transferred with the last few rf - pulses into coherence pathways which are detectable .these cps appear in the cylindrical capillary spectrum , while they do not contribute appreciably to the bulk spectrum shown in fig .[ fig:3dcp_freediff ] .this may be explained by higher internal gradients in the capillaries which render the rf - pulse rotations less ideal than in the bulk sample .thus , the contributions of coherence pathways depends on the sample as well as the careful adjustment of the rf - pulse durations .regardless , their relative importance can be measured directly with pieta and the desired coherence pathway can be extracted .figure [ fig : pieta_vs_4step_cylinder ] ( squares ) shows the real part of the signal as obtained by selecting the direct coherence pathway appearing at .-space profile of the capillary sample as obtained by the mrpi pulse sequence .the spin echo amplitude for the experiment utilizing the -step phase cycle ( triangles ) is distorted at low gradient strengths due to destructive coherence pathways .the solid line was obtained by mcf simulation  .the insert shows a zoom of the signal amplitudes at low values . ]  the spin echo amplitude for the -step phase cycle ( triangles ) is suppressed by below the expected amplitude at zero gradient strength ( ) .the -values in this experiment are over a factor of larger than in the experiments with the bulk sample shown in fig .[ fig : pieta_vs_4step ] .therefore , only the first and second data points are affected by the destructive coherence pathways , while for all the following -steps these cps have been dephased .furthermore , the number of rf - pulses is only versus in the bulk experiments shown in fig .[ fig : pieta_vs_4step ] .thus , the relative importance of undesired coherence pathways to the desired coherence pathway is higher in experiments with more rf - pulses .the solid line in fig . [ fig : pieta_vs_4step_cylinder ] is the simulated signal as obtained with the multiple correlation function technique  .the signal extracted from the pieta experiment recovers the signal amplitudes for low values of .furthermore , the signal phase is preserved with this phase cycling technique as can be seen by the positive and negative lobes in fig .[ fig : pieta_vs_4step_cylinder ] .thus , pieta may not only prove useful for relaxation studies in cpmg like experiments , but it can also help with the development of imaging techniques where phase preservation through a sequence of rf - pulses is a major concern .such techniques are for example rapid imaging with refocused echoes ( rare ) experiments  and spatially resolved experiments  .it has been shown that pieta can recover the desired nmr signal in cases where a broad coherence pathway spectrum complicates the design of complete phase cycles with the nested phase cycling approach .the results with mrpi pieta show that the nmr signal phase due to gradient pulses can be preserved with this phase cycling technique , which may prove beneficial for applications involving other nmr imaging techniques .the only two steps required for implementing pieta are to cycle every second rf - pulse phase and to respect the nyquist - shannon criterion for the number of phase increments recorded .this standardized scheme of implementation may shorten the time required for pulse sequence design by allowing one to extract the desired signal without a deep analysis of the cp spectrum .however , pieta requires an extra fourier transform processing step as compared to cogwheel phase cycling .thus , cogwheel phase cycling may be more efficient during routine application of pulse sequences where the extra information about contributing coherence pathways is not needed .prime candidates for applications of pieta are cpmg like pulse sequences .such pulse sequence designs are increasingly utilized for a broad range of nmr experiments .examples involving spin- nuclei include the spectral characterization of the self - diffusion of molecules  , compartment sizing in mri  and the enhancement of the signal in low field nmr .it can be assumed that the pieta approach will prove beneficial for these applications in the future .we gratefully acknowledge financial support by the ministry of business , innovation and employment of new zealand .s.h . thanks victoria university of wellington for receiving support through a ph.d . scholarship .+  38 natexlab#1#1[2]#2 , , , ( ) ., , , , . , , , , , , ( ) . , , , , , ( ) . , , , , , , ( ) . , , , , , ( ) . , , , , ( ) ., , , , , , , ( ) . , , , ( ) ., , , , , , , ( ) . , , , ., , , , ( ) . , , , , ( ) . , , , ( ) ., , , ( ) . , , ( ) ., , , ( ) . , , , , , ( ) . , , , ( ) ., , , ( ) ., , , , ( ) . , , , , , , ., , , , ( ) . , , , , , , ( ) . , , , , , , ( ) ., , , in : , volume , , , pp . ., , , , ( ) . , , , ( ) ., , , , ( ) . , , , , ( ) ., , , ( ) ., , , , ( ) .

Abstract: efficient phase cycling schemes remain a challenge for nmr techniques if the pulse sequences involve a large number of rf - pulses . especially complex is the carr purcell meiboom gill ( cpmg ) pulse sequence where the number of rf - pulses can range from hundreds to several thousands . our recent implementation of magnetic resonance pore imaging ( mrpi ) is based on a cpmg rf - pulse sequence in order to refocus the effect of internal gradients inherent in porous media . while the spin dynamics for spin- systems in cpmg like experiments are well understood it is still not straight forward to separate the desired pathway from the spectrum of unwanted coherence pathways . in this contribution we apply phase incremented echo train acquisition ( pieta ) to mrpi . we show how pieta offers a convenient way to implement a working phase cycling scheme and how it allows one to gain deeper insights into the amplitudes of undesired pathways .


we are interested in the problem of quickest detection of the onset of a signal in a system of sensors .we consider the situation in which , although the observations in one sensor can affect the observations in another , the onset of a signal can occur at different times ( i.e. , change points ) in each of the sensors ; that is , the change points differ from sensor to sensor . as an example in which this situation arisesconsider a system of sensors monitoring the health of a physical structure in which fault conditions are manifested by vibrations in the structure . before a change affects a given sensor, we have only noise in that sensor .then , after a change , the system is vibrating and thus the signal received in any location reflects a vibrating system .thus , observations at any given sensor are coupled with those received in other locations .the change points observed at different sensors can occur at different times because the source of the vibrations ( i.e. , the excitation ) may arrive at different structural elements at different times .relevant literature related to such models includes , for example , .we assume that the probability law of the observations is the same across sensors .this assumption although seemingly restrictive , is realistic in view of the fact the system of sensors is coupled .we model the signals through continuous - time it processes .the advantage of such models is the fact that they can capture complex dependencies in the observations .for example , an autoregressive process is a special case of the discrete - time equivalent of an ornstein - uhlenbeck process , which in turn , is a special case of an it process .other special cases of this model include markovian models , and linear state - space systems commonly used in vibration - based structural analysis and health monitoring problems .it is important to stress that the fact that the system of sensors is coupled makes the probabilistic treatment of the problem equivalent to the one in which all observations become available in one location .the reason is that one integrated information flow is sufficient for describing such a system .our objective is to detect the first onset of a signal in such a system .so far in the literature of this type of problem ( see ) it has been assumed that the change points are the same across sensors .recently the case was also considered of change points that propagate in a sensor array . however , in this configuration the propagation of the change points depends on the identity of the first sensor affected and considers a restricted markovian mechanism of propagation of the change .in this paper we consider the case in which the change points can be different and do not propagate in any specific configuration .the objective is to detect the minimum ( i.e. , the first ) of the change points .we demonstrate that , in the situation described above , at least asymptotically , the minimum of cusums is asymptotically optimal in detecting the minimum of the different change points , as the mean time between false alarms tends to , with respect to an appropriately extended kullback - leibler divergence criterion criterion that incorporates the possibility of different change points .  in the next sectionwe formulate the problem , discuss special cases of our it models and demonstrate asymptotic optimality ( as the mean time between false alarms tends to ) , in an extended min - max kullback - leibler sense , of the minimum of cusum stopping times .we finally discuss extensions of these results to the case of different structures of observations in each sensor .we sequentially observe the processes for all . in order to formalize this problemwe consider the measurable space , where ^n ] with reflecting boundary at and absorbing boundary at .the same holds for with respect to the variable .  in order to solve ( [ pde_g1 ] ), we apply a laplace transform in and obtain for the ordinary differential equation making use of the fact that is small , we find as leading order approximation to the solution of ( [ bvping ] ) : for this approximation it is simple to find the inverse laplace transform to obtain using this formula for both and we obtain immediately for in ( [ pdet ] ) the asymptotic formula from which it follows that . setting , and using , we further obtain that as , .  for the asymptotic formula of of ( [ pdeg ] ), we also let and use the same change of variable as in the previous case .the only difference is that we have to solve for the different problem in this case , the approximate solution takes the form from here we obtain after inverse laplace transform where denotes the heaviside function and denotes the dirac delta distribution . combining the formulas ( [ gtplus ] ) for and ( [ gtminus ] ) for we find as approximation of for the problem ( [ pdeg ] ) from which we obtain , from which it follows that as .  using the same derivational steps it is possible to generalize to sensors .in particular , in this case the integrand for in ( [ integral ] ) becomes the product ( see ( [ product ] ) ) of functions , each of which satisfies equation ( [ pde_g1 ] ) with the same boundary conditions with respect to their respective variables .their respective laplace transforms satisfy ( [ bvping ] ) .this leads to   similarly , takes the form ( [ integrals ] ) , with integrand consisting of the product of functions , the laplace transform of the first of which satisfies ( [ pde+ ] ) and the laplace transforms of the others satisfy ( [ bvping ] ) . following the same steps as before, this leads to the asymptotic formula using ( [ asapn ] ) and ( [ asapns ] ) , we derive as .this completes the proof of lemma [ asymptotics ] .

Abstract: this work considers the problem of quickest detection of signals in a coupled system of sensors , which receive continuous sequential observations from the environment . it is assumed that the signals , which are modeled a general it processes , are coupled across sensors , but that their onset times may differ from sensor to sensor . the objective is the optimal detection of the first time at which any sensor in the system receives a signal . the problem is formulated as a stochastic optimization problem in which an extended average kullback - leibler divergence criterion is used as a measure of detection delay , with a constraint on the mean time between false alarms . the case in which the sensors employ cumulative sum ( cusum ) strategies is considered , and it is proved that the minimum of cusums is asymptotically optimal as the mean time between false alarms increases without bound .  * keywords : kullback - leibler divergence , cusum , quickest detection *