delayed - choice , quantum eraser walborn-2002 and delayed - choice quantum eraser experiments have led to suggestions that interference patterns formed by particles impacting on a screen may be influenced in some way by decisions made long after those particles had landed on that screen .our objective in this paper is to show by a detailed analysis of various experiments that quantum principles do not support those suggestions .  in our analysis, we shall make out a case for the adoption of a perception of time in quantum observation different to that used in classical mechanics . to understand what we mean, it is important to keep a clear distinction between the concepts of _ systems under observation _ ( suos ) , such as photons , and _apparatus_. our formalism will reflect this difference consistently .standard unitary schrdinger evolution can be maintained for states of suos in between preparation and outcome , but apparatus appears to follows different rules .recent quantum experiments walborn-2002,jacques+al-2006,kim+al-2000,kim-2003 are consistent with and support the view that the passage of time is synonymous with quantum information acquisition occurring in a sequence of _ stages _ .stages have rules which are not precisely those of classical information acquisition , and it appears to be this which accounts for much of the well - known difficulty we have in explaining on a classical level various quantum mechanical experiments involving quantum interference .these rules conform with known physics .for example , quantum information acquisition never violates the light - cone constraints of relativity : classical information can not be acquired between spacelike intervals .quantum correlations which appear to violate the principle of einstein locality actually always require observations to be completed before those correlations can be defined , and this completion always takes place in a classically consistent matter .another rule is that quantum information in the form of suo states _ can _ be shielded against the effects of decoherence and preserved in a state of stasis for arbitrarily long periods of laboratory time .this is most evident in the heisenberg picture and is confirmed by the observation of light from distant stars and galaxies .it is also one way to understand particle decay experiments .yet another rule is that the observation of different components of entangled states is best discussed in terms of stages rather than linear laboratory time .this rule is responsible for the apparent acausality in the delayed - choice experiments we are interested in : observations involving separate detectors can be taken in apparently random order relative to laboratory time without affecting correlations .this was confirmed in the case of the double - slit quantum eraser experiment by walborn et al .walborn-2002 , who specifically looked at this issue .our approach uses a formalism that we have developed for the analysis of time - dependent quantum apparatus networks .we have recently applied it to the franson - bell experiment franson-1989,kwiat-1993,j2008e , an experiment that appears to involve acausal quantum interference .in contrast to standard approaches which tend to focus on the quantum mechanics of systems under observation ( suos ) such as photons , our approach focuses on the detecting apparatus as well . this permits a stage - by - stage analysis of the processes involved in typical quantum optics experiments , starting from state preparation , through the various modules making up the apparatus and ending up with the final state detectors .the formalism is particularly good at giving coincidence rates , which are crucial to many recent quantum optics experiments such as the delayed - choice quantum eraser .our notation serves two purposes .first , it provides an efficient method for dealing with quantum networks of great complexity and can be readily encoded into computer algebra packages .second , it distinguishes between the quantum states of apparatus detectors and those conventionally associated with suos such as photons .the formalism is completely consistent with all standard quantum principles .we make a number of standard assumptions throughout our analysis .first , complete efficiency is assumed , but of course , no real experiment is like that .however , many experiments show precisely those important quantum features such as interference bands which idealized discussions such as ours predict . because of that, there is no need to introduce environmental factors such as decoherence into the discussion .there seems to be no need either to use full - scale quantum field theory in order to draw out the important features of the processes we discuss .  in our analysiswe shall for economy frequently refer to _ photons _ as if they existed in some physical sense as particles. a better interpretation consistent with our modelling would be in terms of detector signals .photon spin is then most naturally interpreted in terms of the specific physical properties of photon detectors .in this section we discuss the double - slit ( ds ) experiment .this experiment features prominently as a component of the delayed - choice quantum eraser experiment and wheeler s delayed - choice experiment discussed subsequently .figure shows a schematic diagram representing the main features of the ds experiment .our notation is as follows .in such a diagram , represents the labstate of the elementary signal detector located at the indicated place in the apparatus network at stage . here is the associated signal creation operator and is the void state of the apparatus at stage .a stage can often be identified with a particular moment or period of laboratory time ( i.e. , something approximating an instant of simultaneity in the laboratory rest frame ) , but this need not be the case at all. in particular , a given stage during a delayed - choice experiment may involve an enormous interval of laboratory time , far beyond any notion of simultaneity .what is crucial in the definition of a stage is that all the detectors associated with a given stage are _ effectively and mutually spacelike _ relative to each other .in other words , no information in any form is transmitted between them within that stage .successive stages are defined in terms of either actual or potential transmission of information between them in either classical and quantum forms .exactly what this means will be made clearer during our discussion of delayed - choice experiments .  in the ds experiment ,the source of the photon beam impinging on the double - slit is denoted by in figure , the subscript denoting that it occurs at stage and the superscript denoting that there is a single photon source .the initial labstate is taken to be of the form  is the normalized spin state of the photon concerned and is a complex valued normalization factor related to the initial beam characteristics . in our notation , .the initial labstate is not normalized to unity because the formalism actually determines _ relative probability rates _ for photon signal production during relevant photon coherence times and related times connected with the passage of wave - trains through the apparatus .related to this is the requirement for specific contextual information about the apparatus to be taken into account .for instance , given a very large detecting screen , some of its detectors would signal photon detection much earlier on in a given run than other detectors further way from the slits .wheeler s delayed - choice experiment is an example where such contextual information is crucial to the discussion .  in the basic ds experiment , there are no issues with photon spin , so actually the term in ( [ eraser-01 ] ) is redundant here .however , photon pair spin is a factor in the double - slit quantum eraser experiment discussed later on , so we include a photon spin term here to show how we deal with it in our formalism .  the next step is to compute the effective evolution operator which takes the initial labstate from to in the transition from stage to . referring to figure , we write and are complex coefficients satisfying the rule .we do not require at this point to have a symmetrical double - slit device , so and are not assumed equal in magnitude .moreover , each slit could induce a separate phase change in that part of the beam passing through it , so this is left open as well .the transition from stage to stage involves a change in hilbert space dimension from to , according to the counting rules of our formalism .however , the effective dimension involved with stage is just and we find and are the respective duals of and and denotes .the next step is to calculate , the effective evolution operator from to , at which point we have completed our dynamical account of the ds experiment . referring again to figure , we write we have modeled the detecting screen as consisting of a large number of photon detectors .this accords with actual experiments , which never actually involve a continuum of detectors . if necessary, we are free to take as large as required in order to model an observed probability pattern to the accuracy required .note that the detectors in this collection need not be assumed to be coplanar : they could be distributed throughout a three - dimensional region of physical space .our discussion is perfectly general .what keeps a track of total probabilities are the semi - unitarity relations between the complex coefficients , which represent the transition amplitudes from emitters based in to detectors in .these relations take the form denotes the complex conjugate of . again , this is perfectly general .there is no need for our purposes to specify any particular form for the coefficients , provided the basic semi - unitarity conditions ( [ eraser-02 ] ) are satisfied .it is traditional in standard discussions of the double - slit experiment to solve the schrdinger equation for a monochromatic beam impinging on a screen from two point sources , but this is not necessary in a discussion such as ours : a more general formulation reveals the important features of the experiment more clearly and precisely .  from ( [ eraser-04 ] )we find the effective transition operator from to to be given by total effective evolution operator is given by the product and is found to be  the next step is to calculate the generalized kraus operators , defined by in principle , there are such operators , counting multiple coincidence cases , but in the case of the ds experiment , we know we are dealing with single photon interference .this means that only the one - signal kraus operators are non - zero , as given by ( [ eraser-03 ] ) and there is one of these for each of the screen detecting sites .next , we construct the generalized povm elements from the generalized kraus operators . by definition , , where there is no sum over .we find construction these are all positive operators . as a check on the physical correctness of our formalism , we can use the semi - unitary relations ( eraser-02 ) to show that  is the effective identity operator for stage .the outcome signal rates are defined by and are found to be rates contain the quantum interference contributions expected from quantum principles . using the semi - unitarity relations ( [ eraser-02 ] ) , these rates add up as expected to the beam production rate , as required from probability conservation .this completes our analysis of the double slit experiment .we turn now to the delayed - choice quantum eraser .the basic experiment is shown in figure  .a photon source sends a beam onto a crystal which is arranged to produce a superposition of coherent photon pairs associated with sites and as shown .one component of each pair is collimated and passed onto a screen containing detectors , exactly as for the ds experiment discussed in the previous section . in effect , positions and act as a pair of slits for a ds experiment .  in this case, however , the situation is more complicated . for each pair, one component is passed into the ds screen whilst the other passes into a prism which deflects it onto a beam - splitter .component is passed onto beam - splitter whilst component is passed onto beam - splitter . each of these beam - splitters acts on its own incident beam and splits it further into two . in each case , the reflected component is passed onto a photon detector , either at or at , whilst the transmitted component is passed onto a third beam - splitter , where interference takes place , with subsequent detection at and   various discussions of this arrangement suggest that choices made by the experimentalist at can influence the interference patterns seen on the screen containing , even though the signals in that screen may have been captured much earlier .we proceed with our stage analysis as follows . as with the ds experiment, we represent our initial source at stage by the labstate .the next stage defined by the production of two correlated photon pairs .these pairs are each assumed spinless. the formalism can readily deal with any situation where this is not the case .the first pair is generated at point on the crystal whilst the other pair is generated at point .the transformation from is given by{1}^{1}a_{1}^{2}+\notag \\ & & \beta \frac{1}{\sqrt{2}}[s_{1}^{1,l}s_{1}^{2,r}+s_{1}^{1,r}s_{1}^{2,l}]a_{1}^{3}a_{1}^{4}\end{aligned}\]]where and represents the spin state of a left - handed circularly polarized photon moving along direction , and so on . hereterms such as represent the two - signal labstate j2007c .hence the effective transition operator is\bar{a}_{0}^{1},\]]where\]]represents an entangled two photon state of total angular momentum zero at stage , .although individual photon wave components get channeled into four possible directions as shown , the internal total angular moment state remains unaffected during this particular experiment .the next stage change is from to and given by , one component beam from each pair is focused on the detecting screen whilst the other component is channeled onto either beam - splitter or , as shown .the represents the amplitudes for landing on the screen from the pair sourced from point , and similarly for the pair sourced from point .the coefficients are characteristic transmission and reflection parameters associated with .it is very useful not to set these parameters to the conventional value but to keep them open and available to be changed .it is in these parameters that we shall encode the observer s freedom of choice in this particular experiment .  from the above we find the effective transition operator.\end{aligned}\]]the final transition is from stage to and involves four terms: gives .\end{aligned}\]]the complete evolution operators is given by . using the above results we find{3}^{s+2 } \\ + [ t_{3}v^{i , a}t_{1}\alpha + ir_{3}v^{i , b}t_{2}\beta ] a_{3}^{s+3}\}\end{array}\right\ } \bar{a}_{0}^{1}\text{.}\ ] ]there are four kraus operators associated with each detector on the screen , each of the form find\bar{a}_{0}^{1 } ,  \notag \\ m_{3,0}^{i , s+3 } & = s_{3}^{1}\bar{s}_{0}^{1}[t_{3}v^{i , a}t_{1}\alpha + ir_{3}v^{i , b}t_{2}\beta ] \bar{a}_{0}^{1 } , \\m_{3,0}^{i , s+4 } & = s_{3}^{1}\bar{s}_{0}^{1}\beta v^{i , b}ir_{2}\bar{a}_{0}^{1}. \notag\end{aligned}\]]these give four povms associated with each detector on the screen : is straightforward to check that effective identity operator for the initial stage hilbert space .there are four coincidence rates associated with each detector on the screen , involving one of the detectors , .these rates are defined by findlatexmath:[\ ] ]  from this we find the kraus operators then the povms the two coincidence transition rate patterns are given by now show interference , with one showing what would normally be described as a _pattern whilst the other showing an _ antifringe _ pattern . essentially , the insertion of the polarizer in front of the photon detector erases the which - path information which previously gave a non - interference pattern on the screen .most significantly , walborn et al repeated the experiment with the screen and photon detection order reversed with significant time differences and found no change in the results .this is strong evidence for the validity of the stages concept in such quantum process .our analysis supports the notion that quantum mechanics never actually involves acausality .we should be worried if it did , for then our entire view of what probability and information represent would need drastic revision .however , several factors would appear to conspire to make it look otherwise .it is the case that some quantum interference experiments do suggest acausality to the unwary .we have in mind here not only the delayed - choice scenarios discussed here but also the franson - bell experiment franson-1989,kwiat-1993,j2008e . in that experiment , the lack of which - path information involves non - locality in time as well as non - locality in space in a most spectacular fashion .however , detailed analysis always reveals the basic fact that interference phenomena arise from a lack of information about quantum _states _ and have nothing specifically to do with the properties of particles per se. it may be reasonable to talk about self - interference when we know we are dealing with one - signal experiments , but as the franson - bell experiment and more recent ones demonstrate , two - photon state interference occurs under circumstances when individual photons simply do not .conceptual problems arise when our classical conditioning is relied on too much .we would like to believe in photons as particles and we would like to believe that time runs continuously .both concepts have their uses , but quantum mechanics requires a generalization of both . in the case of the former ,experiments tell us that we have to deal with interference of states not particles . in the case of the latter, we can not expect quantum processes to evolve strictly according to an integrable timetable , such as coordinate time , or even the physical time in a laboratory .what is important is whether or not quantum information has been extracted .if it has been placed hold , as can be seen in our analysis of the delayed - choice eraser and the double - slit eraser , then it can remain in a stage which could in principle persist until the end of the universe .this is one way of understanding unstable particles : an undecayed unstable particle is one trapped in an information bubble .99 j. a. wheeler , _ the _ the _ _ - choice__ _ _ double - slit experiment _ _ in mathematical foundations of quantum theory , a. r. marlow , editor . academic press ( new york ) , 1978 .j. eakins and g. jaroszkiewicz , _ a quantum computational approach to the quantum universe ._ in new developments in quantum cosmology research , a. reimer , editor , volume 247 of _ horizons in world physics _ , chapter 1 , pages 151 , nova science publishers , inc .new york , 2005 .

Abstract: delayed choice and quantum eraser experiments have attracted much interest recently , both theoretically and experimentally . in particular , they have prompted suggestions that quantum mechanics involves acausal effects . using a recently developed approach which takes apparatus into account , we present a detailed analysis of various double - slit experiments to show that this is never the case . instead , quantum experiments can be described in terms of a novel concept of time called stages . these can cut across the conventional linear time parameter as experienced in the laboratory and appear to violate causality .


linear canonical correlation analysis ( lcca ) is a technique for multivariate data analysis and dimensionality reduction , which quantifies the linear associations between a pair of random vectors . in particular , lcca generates a sequence of pairwise unit variance linear combinations of the considered random vectors , such that the pearson correlation coefficient between the elements of each pair is maximal , and each pair is uncorrelated with its predecessors .the coefficients of these linear combinations , called the linear canonical directions , give insight into the underlying relationships between the random vectors .they are easily obtained by solving a simple generalized eigenvalue decomposition ( gevd ) problem , which only involves the covariance and cross - covariance matrices of the considered random vectors .lcca has been applied to blind source separation , image set matching , direction - of - arrival estimation , , data fusion and group inference in medical imaging data , localization of visual events associated with sound sources , audio - video synchronization , undersea target classification among others .the pearson correlation coefficient is only sensitive to linear associations between random variables .therefore , in cases where the considered random vectors are statistically dependent yet uncorrelated , lcca is not an informative tool .  in order to overcome the linear dependence limitation several generalizations of lccahave been proposed in the literature . in information - theoretic approach to canonical correlation analysis , called icca , was proposed .this method generates a sequence pairwise unit variance linear combinations of the considered random vectors , such that the mutual - information ( mi ) between the elements of each pair is maximal , and each pair is uncorrelated with its predecessors . since the mi is a general measure of statistical dependence , which is sensitive to non - linear relationships , the icca is capable of capturing pairs of linear combinations exhibiting non - linear dependence .however , in contrast to lcca , the icca does not reduce to a simple gevd problem .indeed , in each pair of linear combinations must be obtained separately via an iterative newton - raphson algorithm , which may converge to undesired local maxima .moreover each step of the newton - raphson algorithm involves re - estimation of the mi in a non - parametric manner at a potentially high computational cost .another approach to non - linear generalization of lcca is kernel canonical correlation analysis ( kcca ) - .kcca applies lcca to high - dimensional non - linear transformations of the considered random vectors that map them into some reproducing kernel hilbert spaces .although the kcca approach can be successful in extracting non - linear relations , - , it suffers from the following drawbacks .first , the high - dimensional mappings may have high computational complexity .second , the method is highly prone to over - fitting errors , and requires regularization of the covariance matrices of the transformed random vectors to increase numerical stability . finally , the non - linear mappings of the random vectors may mask the dependencies between their original coordinates .  in this paperwe propose a different non - linear generalization of lcca called measure transformed canonical correlation analysis ( mtcca ) .we apply a structured transform to the joint probability distribution of the considered pair of random vectors , i.e. , a transformation of the joint probability measure defined on their joint observation space .the proposed transform is structured by a pair of non - negative functions called the mt - functions .it preserves statistical independence and maps the joint probability distribution into a set of probability measures on the joint observation space . by modifying the mt - functions classes of measure transformationscan be obtained that have different properties .two types of mt - functions , the exponential and the gaussian , are developed in this paper .the former has a translation invariance property while the latter has a localization property .mtcca applies lcca to the considered pair of random vectors under the proposed probability measure transform . by modifying the mt - functions the correlation coefficient under the transformed probability measure , called the mt - correlation coefficient ,is modified , resulting in a new general framework for canonical correlation analysis . in mtcca, the mt - correlation coefficients between the elements of each generated pair of linear combinations are called the mt - canonical correlation coefficients .the mt - functions are selected from exponential and gaussian families of functions parameterized by scale and location parameters . under these function classesit is shown that pairs of linear combinations with non - linear dependence can be detected by mtcca .the parameters of the mt - functions are selected via maximization of a lower bound on the largest mt - canonical correlation coefficient .we show that , for these selected parameters , the corresponding largest mt - canonical correlation coefficient constitutes a measure for statistical independence under the original probability distribution . in this caseit is also shown that the considered random vectors are statistically independent under both transformed and original probability distributions if and only if they are uncorrelated under the transformed probability distribution .  in the paper an empirical implementation of mtccais proposed that uses strongly consistent estimators of the measure transformed covariance and cross - covariance matrices of the considered random vectors .the mtcca approach has the following advantages over lcca , icca , and the kcca discussed above :  in contrast to lcca , mtcca is capable of detecting non - linear dependencies . moreover , under appropriate selection of the mt - functions , the largest mt - canonical correlation coefficient is a measure of statistical independence between the considered random vectors .  in comparison to the icca , mtcca is easier to implement from the following reasons .first , it reduces to a simple gevd problem , which only involves the measure transformed covariance and cross - covariance matrices of the considered random vectors .second , while mtcca with exponential and gaussian mt - functions involves a _maximization for choosing the mt - functions parameters , the icca involves a _ sequence _ of maximization problems , each having the same dimensionality as in mtcca .  in the paperwe show that unlike the empirical icca and kcca , the computational complexity of the empirical mtcca is linear in the sample size which makes it favorable in large sample size scenarios .unlike kcca , mtcca does not expand the dimensions of the random vectors , nor does it require regularization of their measure transformed covariance matrices .  finally , unlike kcca , in mtccathe original coordinates of the observation vectors are retained after the probability measure transform .therefore , mtcca can be easily applied to variable selection by discarding a subset of the variables for which the corresponding entries of the measure transformed canonical directions are practically zero .the proposed approach is illustrated for two applications .the first is a simulation of graphical models with known dependency structure . in this simulated examplewe show that in similar to icca , the mtcca outperforms the lcca in selecting valid linear / non - linear graphical model topology .the second application is construction of networks that analyze long - term associations between companies traded in the nasdaq and nyse stock markets .we show that mtcca and kcca better associate companies in the same sector ( technology , pharmaceutical , financial ) than does lcca and icca . furthermore , mtcca is able to achieve this by finding strong non - linear dependencies between the daily log - returns of these companies .the paper is organized as follows . in section [ cca_review ], lcca is reviewed . in section [ pmt_cca ] ,lcca is generalized by applying a transform to the joint probability distribution .selection of the mt - functions associated with the transform is discussed in section [ choiceofkernels ] . in section[ estimation ] , empirical implementation of mtcca is obtained . in section [ examp ] ,the proposed approach is illustrated via simulation experiment . in section [ disc ] , the main points of this contribution are summarized .the propositions and theorems stated throughout the paper are proved in the appendix .let and denote two random vectors , whose observation spaces are given by and , respectively .we define the measure space , where is a -algebra over , and is the joint probability measure on .the marginal probability measures of on and are denoted by and , where and are the -algebras over and , respectively .let denote an integrable scalar function on .the expectation of under is defined as \triangleq\int\limits_{\xcalsc\times\ycalsc}g\left({{\bf{x}}},{{\bf{y}}}\right)d\pxy\left({{\bf{x}}},{{\bf{y}}}\right),\ ] ] where and .the random vectors and will be said to be statistically independent under if ={\rm{e}}\left[g_{1}\left({{\bf{x}}}\right);\px\right ] { \rm{e}}\left[g_{2}\left({{\bf{y}}}\right);\py\right]\ ] ] for all integrable scalar functions , on and , respectively .the random vectors and will be said to be uncorrelated under if ={\rm{e}}\left[{{\bf{x}}};\px\right]{\rm{e}}\left[{{\bf{y}}}^{t};\py\right],\ ] ] where denotes the transpose operator .lcca generates a sequence of pairwise unit - variance linear combinations , in the following manner .the first pair is determined by maximizing the pearson correlation coefficient between and over and with the constraint that both and have unit variance .similarly , the -th pair  is determined by maximizing the pearson correlation coefficient between and over and with the constraints that both and have unit variance and are uncorrelated with all the previously obtained pairs , .the pairs and are called the -th order _ linear canonical directions _ and the -th order _ linear canonical variates _ , respectively .the pearson correlation coefficient between and is called the -th order _ linear canonical correlation coefficient_.  the pearson correlation coefficient between and under is given by \triangleq\frac{{\rm{cov}}\left[{{\bf{a}}}^{t}{{\bf{x}}},{{\bf{b}}}^{t}{{\bf{y}}};{\pxy}\right]}{\sqrt{{\rm{var}}\left[{{\bf{a}}}^{t}{{\bf{x}}};{\px}\right]}\sqrt{{\rm{var}}\left[{{\bf{b}}}^{t}{{\bf{y}}};{\py}\right]}}=\frac{{{\bf{a}}}^{t}\bsigma_{{{\mbox{\boldmath \tiny }}}{{\mbox{\boldmath \tiny }}}}{{\bf{b}}}}{\sqrt{{{\bf{a}}}^{t}\bsigma_{{{\mbox{\boldmath \tiny }}}}{{\bf{a}}}}\sqrt{{{\bf{b}}}^{t}\bsigma_{{{\mbox{\boldmath \tiny }}}}{{\bf{b}}}}},\end{aligned}\ ] ] where  ] denote the variance and covariance under and , respectively . the last equality in ( [ rhodef ] ) can be easily verified using the basic definitions of variance and covariance , where , and denote the covariance matrix of under , the covariance matrix of under , and their cross - covariance matrix under , respectively , and it is assumed that and are non - singular .hence , lcca solves the following constraint maximization sequentially over . where denotes the -th order linear canonical correlation coefficient .since the number of constraints in ( [ ccaoptprob ] ) increases with , it is implied that the linear canonical correlation coefficients satisfy the following order relation .it is well known that the constrained maximization problem in ( [ ccaoptprob ] ) reduces to the set of distinct solutions of the following generalized eigenvalue problem \left[\begin{array}{c}{{{\bf{a } } } } \\{{{\bf{b}}}}\end{array}\right]=\rho\left[\begin{array}{cc}{\bsigma_{{{\mbox{\boldmath \tiny } } } } } & { { { \bf{0 } } } } \\ { { { \bf{0 } } } } & { \bsigma_{{{\mbox{\boldmath \tiny }}}}}\end{array}\right]\left[\begin{array}{c}{{{\bf{a } } } } \\{{{\bf{b}}}}\end{array}\right],\ ] ] where is the -th largest generalized eigenvalue of the pencil in ( [ gevd ] ) , and ^{t}=\left[{{\bf{a}}}^{t}_{k},{{\bf{b}}}^{t}_{k}\right]^{t} ] , are called the mt - functions .  in the following proposition ,some properties of the measure transform ( [ measuretransform ] ) are given .[ prop1 ] let be defined by relation ( [ measuretransform ] ). then  1 .[ p1 ] is a probability measure on .[ p2 ] is absolutely continuous w.r.t . , with radon - nikodym derivative given by 3.  [ p3 ] if and are statistically independent under , then they are statistically independent under .[ p4 ] assume that the mt - functions and are strictly positive .if and are statistically independent under , then they are statistically independent under .[ a proof is given in appendix [ prop1proof ] ]  by modifying the mt - functions and , such that the conditions in definition [ def1 ] are satisfied , an infinite set of joint probability measures on can be obtained .mtcca generates a sequence of pairwise linear combinations , that have the following properties under the transformed probability measure : and have unit variance , the correlation coefficient between and is maximal , and are uncorrelated with for all . in mtcca ,the pairs and are called the -th order _ mt - canonical directions _ and the -th order _ mt - canonical variates _ , respectively . the correlation coefficient between and under called the -th order _ mt - canonical correlation coefficient_.  the correlation coefficient between and under is given by \triangleq\frac{{\rm{cov}}\left[{{\bf{a}}}^{t}{{\bf{x}}},{{\bf{b}}}^{t}{{\bf{y}}};{\qxy}\right]}{\sqrt{{\rm{var}}\left[{{\bf{a}}}^{t}{{\bf{x}}};{\qx}\right]}\sqrt{{\rm{var}}\left[{{\bf{b}}}^{t}{{\bf{y}}};{\qy}\right]}}=\frac{{{\bf{a}}}^{t}\bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}{{\mbox{\boldmath \tiny }}}}{{\bf{b}}}}{\sqrt{{{\bf{a}}}^{t } \bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}{{\bf{a}}}}\sqrt{{{\bf{b}}}^{t}\bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}{{\bf{b}}}}},\ ] ] where  ] , where is some arbitrary matrix function of and .therefore , one can easily verify that -{\rm{e}}\left[{{\bf{x}}}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right]{\rm{e}}\left[{{\bf{x}}}^{t}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right],\ ] ] -{\rm{e}}\left[{{\bf{y}}}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right]{\rm{e}}\left[{{\bf{y}}}^{t}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right],\ ] ] and -{\rm{e}}\left[{{\bf{x}}}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right]{\rm{e}}\left[{{\bf{y}}}^{t}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right].\ ] ] equations ( [ rxmod_pxy])-([rxymod_pxy ] ) imply that , and are weighted covariance and cross - covariance matrices of and under , with weighting function .mtcca solves the following constrained maximization sequentially over . where denotes the -th order mt - canonical correlation coefficient .since the number of constraints in ( [ ccaoptprobmod ] ) increases with , the mt - canonical correlation coefficients satisfy the following order relation   similarly to ( [ ccaoptprob ] ) the constrained maximization problem in ( [ ccaoptprobmod ] ) reduces to the following generalized eigenvalue problem \left[\begin{array}{c}{{{\bf{a } } } } \\{{{\bf{b}}}}\end{array}\right]=\rho\left[\begin{array}{cc}{\bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny } } } } } & { { { \bf{0 } } } } \\ { { { \bf{0 } } } } & { \bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}}\end{array}\right]\left[\begin{array}{c}{{{\bf{a } } } } \\{{{\bf{b}}}}\end{array}\right],\ ] ] where is the -th largest generalized eigenvalue of the pencil in ( [ gevd_pmtcca ] ) , and ^{t}=\left[{{\bf{a}}}^{t}_{k},{{\bf{b}}}^{t}_{k}\right]^{t} ] , and the lcca is obtained .other choices of and are discussed below .in this section we parameterize the mt - functions and with parameters and under the exponential and gaussian families of functions . this will result in the corresponding cross - covariance matrix gaining sensitivity to non - linear relationships between the entries of and .optimal choice of the parameters and is also discussed .let and be defined as the parameterized functions where and . using ( [ varphidef ] ) ,( [ rxymod_pxy ] ) and ( [ expkernel ] ) one can easily verify that the cross - covariance matrix of and under takes the form where \ ] ] is the joint moment generating function of and , and it is assumed that is finite in some open region in containing the origin . note that the cross - covariance matrix in ( [ offhessxy ] ) involves higher - order statistics of and .additionally , observe that reduces to the standard cross - covariance matrix for and .finally , note that the quantity in ( [ offhessxy ] ) has been proposed in - for blind source separation , blind channel estimation , blind channel equalization , and auto - regression parameter estimation . to the best of our knowledgethis paper is the first to propose this quantity for generalizing lcca .  in the following theorem , which follows directly from ( [ offhessxy ] ) and the properties of  , , one sees that preserves statistical independence and can capture non - linear dependencies when they exist .[ offoriginindepprop ] let denote an arbitrary open region in containing the origin , and assume that is finite on .the random vectors and are statistically independent under the joint probability measure if and only if [ a proof is given in appendix [ expkerth ] ] .the `` if '' is the interesting part of the theorem since the `` only if '' part follows directly from property [ p3 ] of proposition [ prop1 ] .in particular , if and are statistically dependent under , then there exist , , and , such that .thus , ( [ mtrhodef ] ) implies that if and are statistically dependent under then there exist linear combinations of the form and whose mt - correlation coefficient under is non - zero .finally , we show that mtcca with the exponential mt - functions in ( [ expkernel ] ) is translation - invariant .let and , where and are deterministic vectors in and , respectively . according to ( [ varphidef ] ) and ( [ expkernel ] ) , by ( [ rxmod_pxy])-([rxymod_pxy ] ) : , , and .thus , by ( [ ccaoptprobmod ] ) , the mt - canonical correlation coefficients are invariant to translation , i.e. for .next we define the mt - functions and by where , , , , and denotes the -norm .since and are strictly positive and bounded , one can easily verify that the condition in ( [ assumption2 ] ) is satisfied .relations ( [ varphidef ] ) and ( [ rxymod_pxy ] ) imply that the mt - functions ( [ gausskernel ] ) produce a weighted cross - covariance matrix , for which the observations are weighted in an inverse proportion to the distances and . hence , the resulting mt - correlation coefficient is a measure of local linear dependence in the vicinity of .we note that local linear dependence exists whenever there are global non - linear dependencies .sensitivity of to non - linear relationships between and is shown via the following theorem .[ gaussindepprop ] let , be fixed and positive . additionally , let denote an arbitrary open region in containing the origin .the random vectors and are statistically independent under the joint probability measure if and only if [ a proof is given in appendix [ gausskerth ] ] .hence , if and are statistically dependent under , then there exist , , and , such that .therefore , again , non - linear dependencies can be detected using mtcca .unlike mtcca with gaussian mt - fucntions ( [ gausskernel ] ) , mtcca with exponential mt - functions ( [ expkernel ] ) is translation invariant .moreover , in mtcca with gaussian mt - functions , in addition to the location parameters , , which share the same dimensionality of the scaling parameters of the exponential mt - functions , one has to set two width parameters and . on the other hand , unlike the exponential mt - functions , the gaussian mt - functions are bounded in the joint observation space .hence , mtcca with gaussian mt - functions is more robust to outliers .additionally , the gaussian mt - functions has the property that they localize linear dependence over the observation space .this property is illustrated in subsection [ simexamp1 ] .additional common properties of the exponential and gaussian mt - functions are given in the following remarks :  since the exponential and gaussian mt - functions are strictly positive , by property [ p4 ] of proposition [ prop1 ] we conclude that and preserve statistical dependence under .the exponential and gaussian mt - functions preserve gaussianity in the sense that if and are jointly gaussian under , then they are jointly gaussian under and .   a natural choice of the parameters and , would be those that maximize the first - order mt - canonical correlation coefficient in ( [ ccaoptprobmod ] ) .however , this maximization is analytically cumbersome . therefore , as an alternative , we propose maximizing a lower bound on . we show that the resultant first - order mt - canonical correlation coefficient will be sensitive to dependence between and .  [ lowerboundth ] define the following element - by - element average : ^{2}_{i , j } } { \left[\bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}\left({{\bf{s}}},\tvec\right)\right]_{i , i}\left[\bsigma^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}\left({{\bf{s}}},\tvec\right)\right]_{j , j}}\right)^{1/2},\ ] ] where {i , j} ] via the formula which produce the empirical informational canonical correlation coefficients .the empirical kcca was performed using gaussian radial basis function kernels .since kcca masks the original coordinates of and , it is not illustrated for the graphical model selection tasks in simulation examples 1 and 2 , which involve variable selection . in simulation examples 1 and 2 , the canonical correlation coefficients and canonical directions were estimated using i.i.d .samples of and .the statistical significance of the empirical canonical correlation coefficients was tested using empirical estimates of -values associated with rejecting the null - hypothesis of no statistical dependence between and ( see appendix [ pval ] ) .   in this example, we consider the random vectors ^{t} ] , where and , , , and are mutually independent standard normal random variables .for this example , the pair of linear combinations of the form having maximal dependency is obtained for the vector pair ^{t},{{\bf{b}}}_{1}=\left[1,0\right]^{t}) ] , ^{t} ] and ^{t} ] and finding the graph with minimum edit distance .the symmetric difference graphs are shown in the third column .the red lines in the symmetric difference graphs indicate edges found by mtcca and not by the other compared method , and vice - versa for the black lines .note that for all of the threshold parameters investigated , the mttca graph shows equal or larger number of dependencies than the closest lcca , icca and kcca graphs .this result suggests that mtcca has captured more dependencies than lcca , icca and kcca . while there is no ground truth validation , the fact that mtcca clusters together companies in similar sectors ( banking , pharmaceuticals , and technology ) provides anecdotal support for the power and applicability of mtcca .[ fig3 ] depicts the distribution of the empirical mt , linear , and informational first - order canonical directions .let ^{t} ] on the unit circle .observe that in mtcca ( first and second columns ) and are relatively small in comparison to and .one can conclude that , unlike lcca and icca , mtcca is zeroing in on the strong non - linear dependencies between the daily log - returns of these companies and is de - emphasizing the daily log - volume ratios .this analysis is not performed for kcca since the empirical canonical directions obtained by kcca do not correspond to the original coordinates of and .we note that in this example the difference between mtcca and icca may possibly arise from the sensitivity of fixed kernel density estimation , preformed in icca , to the heavy - tailed financial data .in this paper , lcca was generalized by applying a structured transform to the joint probability distribution of and . by modifying the functions associated with the transform , this generalization , called mtcca , preserves independence and captures non - linear dependencies .two classes of mtcca were proposed based on specification of mt - functions in the exponential and gaussian families , respectively .the proposed mtcca approach was compared to lcca , icca and kcca for graphical model selection in simulated data having non - linear dependencies , and for measuring long - term associations between pairs of companies traded on the nasdaq and nyse stock markets .it is likely that there exist other classes of mt - functions that have a similar capability to accurately detect non - linear dependencies .  in the paperwe have shown that the hessian of the joint cumulant generating function ( [ offhessxy ] ) is a special case of measure transformed covariance matrix with exponential mt - functions .therefore , in similar to the generalization proposed in this paper , the techniques in - , which are based on hessians of the cumulant generating function , may also be generalized by the measure - transformation framework .this research was supported in part by aro grant w911nf-11 - 1 - 0391 .   1 .  * property [ p1 ] : * + since is nonnegative , then by corollary 2.3.6 in  is a measure on . furthermore , so that is a probability measure on .* property [ p2 ] : * + follows from definitions 4.1.1 and 4.1.3 in .* property [ p3 ] * : + let and denote the marginal probability measures of , defined on and , respectively .additionally , let and denote arbitrary sets in the -algebras and , respectively . using ( [ measuretransform ] ) and ( [ varphidef ] ) , the assumed statistical independence of and under , and tonelli s theorem : }d\pxy\left({{\bf{x}}},{{\bf{y}}}\right ) \\\nonumber & = & \int\limits_{a_{x}}\frac{u\left({{\bf{x}}}\right)}{{\rm{e}}\left[u\left({{\bf{x}}}\right);\px\right]}d\px\left({{\bf{x}}}\right ) \int\limits_{\ycalsc}\frac{v\left({{\bf{y}}}\right)}{{\rm{e}}\left[v\left({{\bf{y}}}\right);\py\right]}d\py\left({{\bf{y}}}\right ) = \int\limits_{a_{x}}\frac{u\left({{\bf{x}}}\right)}{{\rm{e}}\left[u\left({{\bf{x}}}\right);\px\right]}d\px\left({{\bf{x}}}\right).\end{aligned}\ ] ] similarly , it can be shown that }d\py\left({{\bf{y}}}\right) ] , and denotes the kronecker delta function .it is easily verified that hence , by ( [ lowerboundproofeq1 ] ) where the last equality stems from the invariance of to normalization of and . therefore , according to ( [ ccaoptprobmod ] ) , ( [ psidef ] ) and ( [ lowerboundproofeq2 ] ) , the relation in ( [ lowerboundeq ] ) is verified .if , then by ( [ lowerboundeq ] ) and the positivity of  therefore , since by ( [ maxprob ] ) are the maximizers of over , which is a closed region in containing the origin , we have that hence , by the definition ( [ psidef ] ) of , on the interior of , which is an open region in containing the origin . thus , since the mt - functions and are chosen according to ( [ expkernel ] ) or ( [ gausskernel ] ) , by theorems [ offoriginindepprop ] and [ gaussindepprop ] and must be statistically independent under .conversely , if and are statistically independent under , then by property [ p3 ] of proposition [ prop1 ] we have that for all , and in particular for . therefore , by ( [ ccaoptprobmod ] ) , .it suffices to show that if the conditions in ( [ cond1 ] ) and ( [ cond2 ] ) are satisfied , then almost surely as .convergence proofs for and are very similar and therefore omitted .   in the following , the limits of the series in the r.h.s . of ( [ xy_lim])-([muy_lim ] )are obtained .additionally , in remark [ remark3 ] below , we show that the assumption in ( [ nonzeroassumption ] ) is satisfied .since , is a sequence of i.i.d .samples of , then the random matrices , , in the r.h.s . of ( [ xy_lim ] ) , define a sequence of i.i.d .samples of . moreover , if <\infty ] , for any , <\infty ] , then & \leq & \left({\rm{e}}\left[\left(x_{k}y_{l}\right)^{2};\pxy\right]{\rm{e}}\left[\left(u\left({{\bf{x}}}\right)v\left({{\bf{y}}}\right)\right)^{2};\pxy\right]\right)^{\frac{1}{2 } } \\\nonumber & \leq & \left({\rm{e}}\left[x^{4}_{k};\px\right]{\rm{e}}\left[y^{4}_{l};\py\right]{\rm{e}}\left[u^{4}\left({{\bf{x}}}\right);\px\right]{\rm{e}}\left[v^{4}\left({{\bf{y}}}\right);\py\right]\right)^{\frac{1}{4}}<\infty,\end{aligned}\ ] ] for any and any , where the second and third semi - inequalities stem from the hlder inequality for random variables .therefore , by khinchine s strong law of large numbers ( kslln ) \hspace{0.2cm}{\rm{a.s.}}\ ] ] similarly , it can be shown that if the conditions in ( [ cond1 ] ) and ( [ cond2 ] ) are satisfied , then by the kslln \hspace{0.2cm}{\rm{a.s.}},\ ] ] \hspace{0.2cm}{\rm{a.s.}},\ ] ] and \hspace{0.2cm}{\rm{a.s.}}\ ] ]   therefore , since the sequences in the l.h.s .of ( [ xy_lim])-([muy_lim ] ) are obtained by continuous mappings of the elements of the sequences in their r.h.s ., then by ( [ nomxylim])-([denlim ] ) , and the mann - wald theorem }{{\rm{e}}\left[u\left({{\bf{x}}}\right)v\left({{\bf{y}}}\right);\pxy\right ] } = { \rm{e}}\left[{{\bf{x}}}{{\bf{y}}}^{t}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right]\hspace{0.2cm}{\rm{a.s.}}\ ] ] }{{\rm{e}}\left[u\left({{\bf{x}}}\right)v\left({{\bf{y}}}\right);\pxy\right ] } = { \rm{e}}\left[{{\bf{x}}}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right]\hspace{0.2cm}{\rm{a.s.}}\ ] ] and }{{\rm{e}}\left[u\left({{\bf{x}}}\right)v\left({{\bf{y}}}\right);\pxy\right ] } = { \rm{e}}\left[{{\bf{y}}}\varphi_{u , v}\left({{\bf{x}}},{{\bf{y}}}\right);\pxy\right]\hspace{0.2cm}{\rm{a.s.}},\ ] ] where the last equalities in ( [ xy_lim_closed])-([muy_lim_closed ] ) follow from the definition of in ( [ varphidef ] ) .  thus , since the sequence in the l.h.s . of ( [ rxy_uv_est_lim ] ) is obtained by continuous mappings of the elements of the sequences in its r.h.s ., then by ( [ xy_lim_closed])-([muy_lim_closed ] ) , the mann - wald theorem , and ( [ rxymod_pxy ] ) it is concluded that a.s . as .    1 .  estimate the optimal mt - functions parameters in ( [ expkernel ] ) and ( [ gausskernel ] ) according to where is defined in ( [ psidef ] ) , and , , and are the estimates in ( [ rx_uv_est])-([rxy_uv_est ] ) of the covariance matrices , , and , respectively . the maximization in ( [ maxprob2 ] ) was carried out numerically using gradient ascent over the search region , which was selected as follows : 1 .  for the exponential mt - functions , we chose where , and is a quadratic empirical approximation of the joint moment generating function in ( [ mongenfunc ] ) .the vectors and denote the sample expectations of and , respectively .the matrices , , and denote sample auto - correlation matrix of , the sample auto - correlation matrix of , and their cross - correlation matrix , respectively .since and is quadratic and takes a unit value at the origin , then defines a closed region in containing the origin .2 .  for the gaussian mt - functions ,the search region was set to where and are the -th and -th entries of and , respectively , and is the empirical -th percentile of the random variable .one can notice that defines a closed rectangle in . in the considered examples it was verified that contains the origin .we note that in case where does not contain the origin , one can always subtract the expectations of and and perform mtcca on  ] .2 .  obtain estimates of the mt - canonical correlation coefficients , and estimates of the mt - canonical directions , by solving the following gevd equation \left[\begin{array}{c}{{{\bf{a } } } } \\{{{\bf{b}}}}\end{array}\right]=\rho\left[\begin{array}{cc}{\hat{\bsigma}^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}\left(\hat{{{\bf{s}}}}^{*},\hat{\tvec}^{*}\right ) } & { { { \bf{0 } } } } \\ { { { \bf{0 } } } } & { \hat{\bsigma}^{\left(u , v\right)}_{{{\mbox{\boldmath \tiny }}}}\left(\hat{{{\bf{s}}}}^{*},\hat{\tvec}^{*}\right)}\end{array}\right]\left[\begin{array}{c}{{{\bf{a } } } } \\{{{\bf{b}}}}\end{array}\right],\ ] ] where is the -th largest generalized eigenvalue of the pencil in ( [ gevd_pmtcca_2 ] ) , and ^{t}=\left[\hat{{{\bf{a}}}}^{t}_{k},\hat{{{\bf{b}}}}^{t}_{k}\right]^{t}$ ] is its corresponding generalized eigenvector .  in all considered examples the width parameters and of the gaussian mt - functions ( [ gausskernel ] )were set to and , where denotes the empirical standard deviation the random variable .let and denote sequences of i.i.d .samples of and , respectively . additionally , let denote the empirical -th order canonical correlation coefficient based on and . a bootstrap based procedure for testing the statistical significance of the empirical -th order canonical correlation coefficientis specified below :  1 .repeat the following procedure for times ( with index ) : 1 .generate a randomly permuted version of the sequence , denoted by .2 .  compute the statistic .2 .  construct an empirical cumulative distribution function from the sample statistics , , as where is an indicator random variable on its argument .3 .  compute the -value where is the true detection statistic .4 .  if , then we have that is significant at level , leading to rejection of the null - hypothesis of no dependence between and .       t.k .kim , j. klitter , and r. cipolla , `` discriminative learning and recognition of image set classes using canonical correlations , '' _ ieee trans .pattern analysis and machine intelligence , _ vol .1005 - 1018 , 2007 .x. wang , h. ge , and i. p. kirsteins , `` direction - of - arrival estimation using distributed arrays : a canonical coordinates perspective with limited array size and sample support , '' _ proc . of the icassp 2010 , _ pp .2622 - 2625 , 2010 .y. yamanishi , j. p. vert , a. nakaya , and m. kanehisa , `` extraction of correlated gene clusters from multiple genomic data by generalized kernel canonical correlation analysis , '' _ bioinformatics , _ vol .19 , pp . 323i-330i , 2003 .h. suetani , y. iba , and k. aihara , `` detecting hidden synchronization of chaotic dynamical systems : a kernel - based approach , '' _ journal of physics a : mathematical and general , _ vol .10723 - 10742 , 2006 .

Abstract: in this paper linear canonical correlation analysis ( lcca ) is generalized by applying a structured transform to the joint probability distribution of the considered pair of random vectors , i.e. , a transformation of the joint probability measure defined on their joint observation space . this framework , called measure transformed canonical correlation analysis ( mtcca ) , applies lcca to the data after transformation of the joint probability measure . we show that judicious choice of the transform leads to a modified canonical correlation analysis , which , in contrast to lcca , is capable of detecting non - linear relationships between the considered pair of random vectors . unlike kernel canonical correlation analysis , where the transformation is applied to the random vectors , in mtcca the transformation is applied to their joint probability distribution . this results in performance advantages and reduced implementation complexity . the proposed approach is illustrated for graphical model selection in simulated data having non - linear dependencies , and for measuring long - term associations between companies traded in the nasdaq and nyse stock markets .   * *  * *  * *  * s *  * *  association analysis , canonical correlation analysis , graphical model selection , multivariate data analysis , probability measure transform .


due to advances in technology , it is now possible to have low - cost , stand - alone sensor and actuator devices that can communicate through the wireless medium .such devices have applications in areas like epidemic detection , biological attack detection , intruder detection . in some applications ,these sensor nodes are deployed in hostile environments , and therefore security of communication becomes critical . to provide security , well - developed public key cryptographic methodshave been considered , but these are compute - intensive and too demanding for resource - constrained devices .so , symmetric key based encryption is the only way for secure communication between nodes .however , to do that , two nodes should agree upon a common key first . for this, various key distribution schemes have been proposed in the literature .eschenauer and gligor proposed a random key predistribution scheme , referred to as the basic scheme or eg scheme .based on this scheme , various improvements have been proposed in the literature .  recently , there has been research on key distribution schemes which make use of predeployment knowledge of expected locations of the nodes , and these schemes are shown to perform better than the basic schemes .but in some cases predeployment knowledge is not available , so in liu et.al .proposed a scheme which uses group - based deployment model and showed that their scheme is better than the basic schemes .although in , anjum proposed a scheme which does not use predeployment knowledge of the nodes , and group - based deployment model , the scheme requires some nodes which can transmit at different power levels .our scheme also uses the same concept of groups as used by but we have dropped the assumptions of predeployment knowledge of expected locations of nodes and group - based deployment .moreover , our scheme does not require nodes which can transmit messages at different power levels .our contributions in this paper are as follows .  * we propose a scheme , in which nodes form groups _ after _deployment on the basis of their physical locations , and generate keys which depend on the group they are part of .each group is assigned a key pool and no two key pools have a common key .if a node is compromised , then it can affect communication in its group only ; so the proposed scheme is more resilient to node capture than the basic schemes .* we show using simulations that our scheme performs better than the basic scheme and the schemes which assume pre - deployment knowledge of node locations .our scheme assumes that there exists some constant time before which the adversary is unable to extract keys from the nodes , as assumed in .we show that this time can be considerably less for our scheme than that in , with certain tradeoff .* we also look at the problem of connectivity of the key graph formed when our scheme is followed .this problem is studied in the framework of the `` ab random geometric graph '' . using results in , the number of tagged nodesis calculated such that the whole key graph is connected with high probability .the rest of this paper is organized as follows . in section [ s :related_work ] , we discuss related work .section [ s : proposal ] describes our proposal .section [ s : abrandom ] gives the expressions for the parameters ensuring connectivity of the key graph .section [ s : sim_results ] gives the comparison with the scheme proposed by du.et.al .section [ s : otherapp ] describes other applications where our scheme could be useful .section [ s : conclusion ] concludes the paper .various key distribution schemes have been proposed in the literature for wireless sensor networks , keeping in view the resource - constrained devices used in these networks .eschenauer and gligor proposed a scheme in which for every node , keys are picked randomly ( with replacement ) from a key pool and assigned to it before deployment ; this scheme is known as the basic or eg scheme .after key discovery , two neighbor nodes that have a common key use that as the key for secure communication . based on this basic scheme, several schemes with enhanced security features have been suggested in .there is another class of schemes called `` threshold schemes . '' in these schemes , all nodes can communicate with one another , and no communication is compromised until some fixed number of nodes is compromised .blundo et.al. and blom proposed such threshold schemes .blundo s scheme uses symmetric bivariate polynomials to obtain pairwise keys , while blom s scheme also uses a similar idea , in which symmetric matrices are used instead of symmetric polynomials . improve upon blom s scheme by combining it with the random key distribution scheme .similarly , liu and ning improve upon blundo s scheme by combining it with random key distribution scheme .both these schemes perform better than the eg scheme in terms of connectivity and resilience against node capture .but threshold schemes do not scale with the number of nodes in the network . for a fixed resilience against nodecapture , if the number of nodes is increased , then they require large memory .  in any sensor network , generally nodes need to talk to their neighbor nodes only .so it is quite intuitive that nodes which are near should share the same key pool .this will lead to more efficient use of memory , and will give better connectivity and better resilience against node capture . because of this reason , various location - based key distribution schemes have been proposed . and liu and ning independently proposed schemes which assume predeployment knowledge of expected locations of the nodes .nodes are assumed to be deployed in groups ( group - based deployment model ) and nodes in the same group have the same expected location , so that after deployment , they lie close to one another . further in , nodes in the same group are allocated keys from the same key pool , while the groups which lie far from each other are allocated disjoint key pools . therefore , compromise of any node jeopardises transmissions of nearby nodes only .due to this reason , performance is better than that of the eg scheme .all the location - based schemes which depend on the knowledge of expected locations of nodes perform well , but they are all prone to estimation errors in the expected positions of the nodes . so other schemes which do not assume predeployment knowledge of the expected locations of the nodes have been proposed . in ,liu et.al . proposed a scheme which does not use expected locations of the nodes but still uses group - based deployment .this scheme proposes a framework , and any basic scheme like random key distribution or polynomial - based scheme can be used with this framework .the authors showed that basic schemes used with their proposed framework perform better than when used alone .further , anjum removed the assumption of group - based deployment model and also removed the assumption of knowledge of expected locations of nodes .he showed that the scheme performs better than the basic scheme ; but the scheme requires nodes which can transmit at different power levels . in this schemethere are some special nodes which generate different random numbers ( nonces ) and transmit them at different power levels .nodes receiving the same nonce can communicate , provided they are neighbors .our scheme is different from this scheme , since we do not require the presence of nodes which can transmit at different levels . instead of using different power levels ,our scheme uses ttl scoping . in ttlscoping , after the deployment phase , some nodes transmit a broadcast packet containing the ttl ( time to live ) field , similar to that of ip packets in data networks .  in addition , our scheme is also different in the way nodes choose their key rings . in , on receiving the nonce , nodes map it to some different value . in contrast , in our scheme , some nodes transmit their i d , and corresponding to every i d there is an associated key pool . nodes sample keys from the key pool corresponding to the received i d . the main advantage of doing this is the improved resilience against node capture . in  ,all nodes receiving the same nonce use the same key for secure communication ; so if any node is compromised , all the secure links formed by the nonce received by this node will be compromised . on the other hand , in our scheme , communication with other nodesis compromised _ with some probability only _ , because nodes receiving the same i d sample keys from the key pool instead of using the same key .in this paper , we consider static sensor networks .nodes are uniformly distributed across the deployment region .we use the following cryptographic primitives :  * pseudo random number generator ( prng ) this is a deterministic function , which takes an bit number as input and produces output of pseudorandom bits : * hash function this is a deterministic function which takes an input of any length and returns a number of fixed bit - size . given the output of the hash function , one can not find the input and it is highly unlikely that for two different inputs , the output is same : where is variable , and is fixed .   in our scheme ,all nodes are divided into two sets : the `` tagged node '' set and the `` normal node '' set .tagged nodes are similar to normal nodes in terms of memory , storage , and transmission range .they are deployed in the same way as normal nodes .the difference is that tagged nodes are programmed to broadcast a packet after the deployment phase is over .subsequently , tagged nodes behave like normal nodes .once deployment is over , a tagged node broadcasts a packet with ttl value .nodes within distance from the tagged node receive this packet , and all these nodes associate themselves in one group .different groups are associated with _key pools , and nodes in a group `` sample '' keys from the same key pool .since we are using disjoint key pools , so compromise of any node results in compromise of communications in _ its _ group only ; in this way _ localization _ of the effects of node compromise is achieved .our scheme relies on the assumption that the adversary will not be able to extract keying material from a captured node before a small time interval has elapsed .this is a reasonable assumption because breaking into a node and extracting keying material will take some time .the same assumption has been made in and .our scheme consists of the following four phases :  * predeployment phase * broadcast phase * key generation phase * shared key discovery phase   in this phase , two keys are stored in the nodes .* global key ( ) : this is common to all the nodes and is used for authentication and encryption of packets during the broadcast phase . *root key ( ) : this key is a single key stored in all the nodes .it is used to derive the other keys during the key allocation phase ; this procedure is explained subsequently .after nodes are deployed , all tagged nodes broadcast a packet up to hops , containing two fields : tagged node i d field and hop count field .each node ( both normal and tagged node ) receiving this packet fetches the tagged node i d from the packet , and compares it with previously stored tag ids .if there is no match with any of the previously stored tag ids , then its value is stored . then, the hop count value is fetched from the packet and its value is decreased by . if , after decreasing , the value is , then the packet is discarded ; otherwise , the packet is broadcast again with the new value of the hop count .this broadcast packet is encrypted and authenticated using the global key ( ) .all nodes are able to decrypt and authenticate this packet since this key is stored in all the nodes . after the end of this phase , all nodes which are within distance from the tagged node receive the packet . in this way , all the nodes are divided into groups , on the basis of their physical locations .consider a tagged node .it will broadcast a packet with tagged node i d field set to .all nodes within the radius of of this tagged node will receive this packet and associate themselves with tagged node i d .all these nodes will consider themselves as a part of group .so there will be a group corresponding to each tagged node .we note that , since a node can receive broadcasts from more than one tagged node , a node can be part of multiple groups .there is a key pool corresponding to each group and each node samples keys from the key pool of each group to which it belongs . since a node can be part of multiple groups , so different nodes can choose different key ring sizes . to bound the number of keys chosen by any node, a limit is put on the number of key pools from which a node samples keys .let be the maximum number of groups to which a node can belong .we define two sets for any node , and . contains all the distinct tagged node ids received during the broadcast phase . is a subset of .a selects tag ids out of the received tag ids , and the set contains these selected values .  randomly selecting the tagged node ids from not the best thing to do .consider an example with .if two neighbor nodes receive broadcasts from the same nodes , then on randomly selecting the tag i d , the probability that both choose the same tagged node i d is .but if both the nodes plan to choose the least tag node i d , then with probability they will choose the same tag node i d . andintuitively one can say that two neighbor nodes are more likely to receive broadcasts from the same set of tagged nodes .so we set the selection criterion as : node selects the smallest tagged node ids from the set .   once the broadcast phase is over , nodes select smallest tag ids from the received tag ids . if we consider node , then it is a part of groups in the set .after the node has associated itself with the groups , it has to sample keys from the key pools corresponding to the selected groups .one way to do this is to store all the key pools in all the nodes before deployment , but this is not feasible because of memory constraints .so we propose a way in which nodes can _ compute _ the keys such that it is _ equivalent _ to first selecting the key pools and then sampling keys from them .let be the key pool associated with tag i d or group i d . the pool is generated by using [ eqn : key_pool ] .  [ eqn : key_pool ]  here , represents the concatenation operator , is the key pool size per group , is the root key as defined earlier and represents hash operations on ; for example .however , instead of deriving the key pool and then sampling the keys , one can first select numbers uniformly distributed in the range from to , ( called key indices ) , and then applying the function to them .each key can be identified by the tuple ( group number , key index ) .keys are stored along with this tuple to identify them during the subsequent key discovery phase .  to understand the procedure followed by nodes to derive their key rings, we consider a node , and examine what it does .\1 ) node generates the set , and arranges it in ascending order of its index values , , with .let us call these values `` group keys . ''if there are large number of tagged nodes then it will be expensive to compute the group key corresponding to tag nodes with large tag i d s .so to minimize the computation , some fraction of group keys , uniformly distributed across the full range , can be computed offline and stored in the node before deployment .for example , if the number of tagged node is , then group keys ( ) could be pre - stored , so that the average number of hash computations done by any node will be .\2 ) in this step , node generates sets each containing values in the range of to .these sets are generated using the prng , with as the seed values .let us call the elements of these sets `` key indices . ''since a node can generate a maximum of sets , so it will use seed values in the range to .further , nodes and will use different seed values since node will use seed values in the range to , while node will use values in the range to . in this way ,all sets of key indices are generated independently , and hence the key rings are also generated independently .\3 ) in this step , mapping of key indices to actual keys is done .node has group keys and same number of key index sets .node will pair each key index set with a single group key .pairs are formed by first arranging the group keys in ascending order of their indices , and the group key with the smallest index is paired up with the key index set generated using as the seed value . from each pair , tuples are formed , where the first element of the tuple is the group key and the second element is the key index of the set .for example , suppose . if the key index set produced using is and that using is , then following set of tuples is produced : + = \{ , , , , , }  note that each tuple can be identified by ( group number( ) , key index ) .so even after the key is deleted from the memory of the node , tuples can be identified using the group number ; for the above example , tuples can be identified by  .this is important because during the key discovery phase , node will send it s node i d ( ) and the set , and from this information , other nodes should be able to identify the common keys .final keys are obtained by concatenating the elements of the tuple and then hashing the resultant value .this procedure is also illustrated in fig .[ figure : keygen1 ] .our scheme requires nodes that are close to each other to be in the same group , nodes in the same group to sample keys from the same key pool and key pools selected by distinct groups to be disjoint .the procedure described above satisfies all our requirements .as soon as the key allocation phase is over , and the set should be deleted from memory , because given this information , an attacker may be able to generate all the keys in the network , and that will lead to compromise of all communication .   in this phase ,node broadcasts its node i d and the set .if node is a neighbor of node , then node will receive the broadcast by node .on receiving this broadcast packet , node will fetch the set and then compare its elements with .if there is no matching element between the two sets , then there is no common key .if there are common elements between the two sets , then call the set of matching elements as , .let us call an element of as .now node will find the index of the elements present in the set , in sets and .what is done with the index is explained using the example below .for example , let and , so the set , and the index of these elements in set is and , while in set it is and .now node will generate numbers for seed values , and , the prng . then it will compare the values produced from seed values and , and also compare the values produced from the seed values and .if any value matches , then the node and share a common key , and as mentioned earlier , the key is identified by the tuple ( group number , key index ) .if multiple keys are found to be shared , then xor of all the keys will be used as the common key . on following a similar procedure ,node can also find common keys with node .after keys have been generated and discovered , the natural question which arises is : can any two nodes exchange information securely ?this question is addressed by considering the notion of the key graph and connectivity of the key graph .a node is represented by a vertex in the graph .there exists an edge between two vertices if the corresponding nodes share at least one common key , and lie in the coverage radius of each other . a graph formed in this wayis called a `` key graph . ''there are two properties of connectivity : local connectivity and global connectivity .local connectivity of any node is the probability of sharing at least one key with the neighbor nodes , while global connectivity is the percentage of the nodes in the key graph that is reachable from any node .  in this section ,we are concerned with the problem of how to find the number of tagged and normal nodes such that the resulting key graph is connected .the problem of connectivity of the key graph can be broken down into followingthe three sub problems :  * inter group connectivity ( ieg ) since key pools are disjoint , so nodes belonging to two different groups will have zero probability of sharing a key .so , two groups can only be connected if there exist nodes ( ) , which belong to both the groups . as these common nodes sample keys from the key pools of both the groups , so these nodes are reachable from the nodes of both the groups ; thus , these nodes act as `` gateways . '' + now consider a graph in which a _ group _ is represented by a vertex , and there exists an edge between two vertices , if there exists at least one common node between two groups . in_ this _ graph , if all the vertices are reachable from any vertex , then the ieg property holds .we will use the results in on ab random geometric graphs to find the minimum number of groups or tagged nodes required such that all the groups are connected .+ in , two kinds of nodes are considered : a type and b type .two a type nodes can communicate via a b type node only .let the graph formed in this way _ among a type nodes _ , be denoted as , where is the number of a type nodes and is the number of b type nodes .we apply this framework by taking type a nodes as tagged nodes , and type b nodes as normal nodes .+ define as the largest nearest neighbor radius of the ab random geometric graph , i.e. , the radius below which there exists at least one node with degree equal to zero .then , shows that for transmission radius ( ) equal to .+ also , the thereshold transmission radius of the nodes for which the graph is connected with high probability as is given by eqn .[ eqn : iyer_radius ] : we will fix , which will translate to a target small probability of `` graph isolated '' groups . then , we find the value of number of tagged nodes ( ) required , for the given value of and total number of nodes to be deployed ( ) , by substituting and in eqn .[ eqn : iyer_radius ] .if is the solution obtained and if , then the number of tagged nodes greater than will satisfy ieg property . *tagged node - covered nodes which do not receive broadcasts will not be part of any group ; so these nodes are isolated from rest of the network .the number of tagged nodes should be such that all the normal nodes are covered by broadcast from at least one tagged node .we will use the result from to calculate the expected number of normal nodes not covered by any broadcast .it is given by eqn .[ eqn : kcovered0 ] . = ( n - n ) e^{-\frac{n}{a } \pi ( hr)^2 } \label{eqn : kcovered0}\ ] ] where , is the deployment area . to satisfy this property ,  ] , and if , then the number of tagged nodes greater than will satisfy node - covered property .+ so , to satisfy both the node - covered property and and inter group connectivity , the number of tagged nodes is given by eqn .[ eqn : calct ] . * intra group connectivity ( iag ) all the nodes within a group should be reachable from any node in the key graph .we will ensure this by using expressions to calculate the keyring size in the eg scheme .since the number of nodes in a group has decreased , so less number of nodes will share the key pool .so , key pool size could be reduced , which , in turn , will reduce the requisite keyring size .  if all the above mentioned requirementsare met , then our objective of connectivity of the key graph is achieved , because all the nodes are reachable within a group , all the groups are reachable from any group and all the nodes are part of at least one group .we compare our proposed scheme with the scheme  which makes the stronger assumption of availability of expected knowledge of positions of nodes , which our scheme does not .however our scheme makes another assumption : that there exists an interval ( vulnerable time ) , after the deployment , during which attacker should not be able to extract the keys .it may be noted that in our scheme , keys are generated by nodes _after _ they are deployed , while in , keys are loaded into nodes before deployment . moreover, our scheme also has the features of random key distribution .so it is appropriate to compare the connectivity and resilience of our scheme with that of schemes which use random key pre distribution ( rkd ) schemes ( for example , ) and a scheme like in , which uses rkd with the assumption of knowledge of expected locations of the nodes .we will also compare our scheme with leap in , which is also a post - deployment key generation scheme and considers the notion of vulnerable time .we will argue that our scheme leads to a smaller vulnerable time than that required by in leap , and discuss a related trade - off .  for the evaluation of our scheme , we use the following metrics.  * connectivity of the nodes in the key graph : local and global connectivity . * resilience against node capture this is the ratio of the number of links compromised to the total number of links formed .links which are formed among the compromised nodes and the links which are between the compromised nodes and the noncompromised nodes are not taken into account while calculating the number of compromised links and the total links .* memory requirement as the sensor nodes are resource - constrained , so there is always the requirement of attaining high connectivity using minimum memory . * vulnerable time it is the time interval after the deployment during which attacker should not be able to extract key material from the nodes , and before which all nodes should delete the key material from their memory .it is desirable to keep it as less as possible .our simulations are done for the following values of parameters .the number of tagged nodes is calculated using the analysis in the previous section .* deployment region : * total number of nodes is taken as * transmission radius of the node is taken as * hop count ( ) is taken as * , keys per key pool * tagged nodes : ,   also , simulations for the scheme are done using the same set of values .figure [ fig_all_1 ] shows the plot of probability of sharing at least one key between two neighbor nodes versus the key ring size of our scheme , scheme and eg scheme .we have simulated our scheme for . from the graph it is clear that our scheme performs better than or as well as the scheme . for lower values of key ring size , should be chosen which gives better connectivity than that of the scheme , while for larger value of key ring sizes should be chosen , which gives same connectivity as that of the scheme .we have derived analytical expressions for the probability of sharing at least one key between two neighbor nodes , and the results match well with the simulated values as shown in fig .[ fig_sim_anal_h2 ] .the analytical expressions are not reported here due to lack of space .for , table [ tab : locvsglob ] gives the comparison of local connectivity with global connectivity of the nodes . as we can see from the last row of the table, there are small percentages ( and ) of nodes which are isolated even at large value of key ring size .this is because they do not receive broadcasts from any of the tagged nodes , and hence remain isolated from the network .however , an algorithm could be developed to tackle this problem of a small number of nodes not receiving broadcast from any tagged node .the simplest algorithm is as follows : all nodes which do not receive a broadcast from any tagged node send a packet containing their node i d to the sink node , and broadcast it up to one hop . nodes which receive _ this _ broadcast packet fetch the node i d from the packet , and send a message to the sink node ( authenticated and encrypted with their own key ) containing the node i d they received from the broadcast packet and their group i d . the sink node samples keys from the group key pool of the neighbor nodes and communicates it to the uncovered node ..local vs global connectivity [ cols="^,^,^,^,^",options="header " , ]   the basic threat model that we have considered is as follows : if a node is captured , then all the keys contained in it are revealed to the adversary . to evaluate the resilience of our scheme, we consider that nodes are compromised and they are distributed uniformly across the deployment region .key ring size of any node is , where is the key ring size per group .total number of keys in the key pool is given by , where is key pool size per group , and is number of tagged nodes . if one node is captured , then the probability that a link between two non - captured nodes is compromised is at most .( this is a worst case value since links between non - captured nodes could be secured by multiple keys ) . . for compromised nodes , the probability that a link is not compromised is at least .then , the probability that a link is compromised is at most . while comparing this metric with other schemes , local connectivity ( ) needs to be kept the same .figure [ fig_security33 ] , [ fig_security50 ] show a comparison of the resilience metric of our scheme with that of other schemes , at local connectivity and .it shows that our scheme performs better than other schemes .we see that the fraction of links compromised due to node capture is very small compared to that in .this is attributed to the fact that our scheme has small group size , which is equal to the total number of nodes falling in the transmission radius of the node .so when a node is compromised , it affects communication links of the groups it is part of , and these are very small in size ; so the fraction of total links compromised is also very small . in ,the group size is bigger , so the number of affected nodes is larger , and so is the fraction of compromised links .however , it could be argued that in this scheme also the number of groups can be increased , simultaneously decreasing the number of nodes a group ; but this requires expected locations to be known with more precision , with a corresponding increase in the complexity of deployment .   since in our scheme keysare computed after the nodes are deployed , so it is important to consider another threat , which is the vulnerable time during which the compromise of a single node can lead to compromise of whole network .we will compare our scheme with leap , which also needs to address the issue of vulnerable time . since we have shown that our scheme for ( ) performs better than the schemes using deployment knowledge , so we will compare the vulnerable time for only .vulnerable time required by our scheme for ( ) is just the time taken by all the tagged nodes to broadcast up to one hop or time taken by all the nodes to hear broadcast from all the neighbor tagged nodes ; the latter constitute a fraction ( ) of the total number of nodes . in leap , it is the time taken by all the nodes to hear broadcast from all the neighbor nodes . since the number retransmission attempts required by the node to transmit in any mac protocol depends on the number of active neighbor nodes , so our scheme , which requires that nodes should hear from only neighbor tagged nodes which are only the fraction of total number nodes , requires less time , than what leap+ does .however , there is a trade off between performance achieved and the requirement of the vulnerable time , since leap+ acheives perfect resilience and connectivity while our scheme does not . nevertheless , still our scheme is able to achieve better performance than the schemes which assumes predeployment knowledge of the nodes .   to add new nodes after the initial deployment , the base station or sink informs tagged nodes to broadcast their tag i d s for the new nodes .root key is stored in the new nodes before their deployment , using which they generate the keys .key generation and shared key discovery procedure is same as described earlier .after the generation of the key ring they delete the root key .apart from the main objective of key distribution , our scheme could be used for other applications as well .tagged nodes could be considered as the virtual base stations ( vbs ) distributed all across the deployment region , and other nodes around them associated with them .if the main base station wants to convey a broadcast message only to some nodes lying in a certain region , then this could be accomplished by a single unicast message to the tagged node ( assuming tagged nodes can identify their location ) lying near that region and then that tagged node can broadcast that message with a flag indicating a `` regional broadcast , '' so that receiving nodes check the hop count value before rebroadcasting the message .all the tagged nodes can maintain a group key which is known only to its group members ; this will help secure delivery of the message .this scheme can also be used for group key management . in a distributed environment where all nodes can not communicate directly with base station, tagged nodes can act as the virtual base stations .such a scheme involving decentralization is proposed in ; however , in this scheme , cluster heads ( tagged nodes ) have larger transmission range than the cluster members ( normal nodes ) , so that they can communicate directly with sink nodes , and are less energy - constrained than the normal nodes .another application could be the collection of data , where the nodes in the group sends their data to the vbs of their respective group , and then vbs creates a single packet and send it to bs .we have proposed a key distribution scheme , which does not assume node predeployment knowledge and also does not require nodes to be deployed in groups , still our scheme achieves better performance in terms of connectivity and security than the scheme  which take these assumptions . however our scheme assumes existence of vulnerable time , which is less than that of the leap+  , with tradeoff of connectivity and resilience .our future work is to propose a way to allocate polynomial based keys  , to make our scheme more robust against node capture attack .another future work is to evaluate and compare the performance of our scheme for non - uniform deployments .this work was supported by drdo under project 571 : a research and implementation project on sensor networks .

Abstract: many basic key distribution schemes specifically tuned to wireless sensor networks have been proposed in the literature . recently , several researchers have proposed schemes in which they have used group - based deployment models and assumed predeployment knowledge of the expected locations of nodes . they have shown that these schemes achieve better performance than the basic schemes , in terms of connectivity , resilience against node capture and storage requirements . but in many situations expected locations of nodes are not available . in this paper we propose a solution which uses the basic scheme , but does _ not _ use group - based deployment model and predeployment knowledge of the locations of nodes , and yet performs better than schemes which make the aforementioned assumptions .  in our scheme , groups are formed _ after _ deployment of sensor nodes , on the basis of their physical locations , and the nodes sample keys from disjoint key pools . compromise of a node affects secure links with other nodes that are part of its group only . because of this reason , our scheme performs better than the basic schemes and the schemes using predeployment knowledge , in terms of connectivity , storage requirement , and security . moreover , the post - deployment key generation process completes sooner than in schemes like leap+  .   security ; key distribution ; sensor networks