we consider the model reduction problem for stable , discrete - time periodic systems , using an approximation of balanced trunctation .in particular , we combine the lifting approach developed in  with a snapshot - based approximation described in  , which is tractable even for very high - dimensional systems , on the order of millions of states .several different algorithms are available for extending balanced truncation from linear time - invariant systems to time - periodic systems ( see , e.g. , ) . here , our interest is in systems with very large state dimension , on the order of tens of thousands or millions , for which the algorithms used in these previous approaches become intractable .such systems arise in the control of systems governed by partial differential equations , for instance in fluid mechanics , where often one approximates the full infinite - dimensional dynamics by a high - dimensional discretization .periodic systems may arise in these settings either as linearizations around a periodic orbit ( e.g. , vortex shedding  ) , or as linearizations of a system with periodic open - loop forcing ( e.g. , periodic pulsed blowing ) .the goal of this paper is to describe a model reduction procedure that closely approximates balanced truncation , yet is computationally tractable even for these high - dimensional systems .the resulting reduced - order models may be used for control synthesis , or other studies where the full high - dimensional system is unwieldy .we suppose that while the number of states and outputs are both very large , the number of control inputs is small ( by duality , we may alternatively assume that the number of outputs is small ) .this case is also typical in practice , in which one often has a small number of actuators , or sources of disturbances , that one wants to model .we also assume that the system is asymptotically stable .key ideas in the method presented here involve the computation of the balancing transformation directly from snapshots , without computing the controllability and observability gramians , which become prohibitively large for these systems ; and two different output projection methods for tractable computation with systems with high - dimensional output spaces .the paper is organized as follows : in section [ background_sec ] we summarize an existing approach to balanced truncation for time - periodic systems , and the corresponding lifted time - invariant reformulation . in section [ bpod_sec ]we present the main results , an approximate method based on snapshots , and an output projection based on the lifting approach that makes the snapshot method feasible for large numbers of outputs . in section [ example_sec ] ,we demonstrate the method on a numerical example .consider a discrete linear time - periodic system with state variable , input and output , and , where the matrices , and are -periodic ( e.g. , for all  ) .we assume that this system is asymptotically stable , in the following sense : defining for , with , the identity matrix , we consider the case where the spectral radius  is smaller than  .note that periodicity implies that the non - zero eigenvalues of are independent of time  , whereby asymptotic stability is equivalent to uniform geometric decay , as in the lti case .many different approaches are available for model reduction , including singular perturbation  , hankel norm reduction , and balanced truncation .balanced truncation  has become a widely used method for linear systems , since it has a priori error bounds comparable to other methods , and is computationally tractable , at least for systems of moderate dimension , say .however , for very large systems , exact balanced truncation becomes computationally intractable , as the procedure involves finding coordinate transformations that simultaneously diagonalize non - sparse matrices ( controllability and observability gramians ) . to overcome this difficulty , a snapshot - based method has been proposed for approximate balanced truncation  , sometimes referred to as _ balanced proper orthogonal decomposition _( balanced pod ) . here , the idea is to use `` empirical gramians '' defined using snapshots of a simulation of the system , as in  , and to compute the balancing transformation directly from the snapshots , without ever computing the gramians .the method of balanced truncation has been applied to linear time - periodic systems using several different approaches .one approach , used in  , is to perform separate balanced truncations along a period .an alternative , suggested by  , is a lifting approach , in which balanced truncation is applied to a time - invariant reformulation .however , both of these approaches require the solution of lyapunov equations or inequalities and are not computationally tractable for very large systems . thus , the objective of our work is to obtain more computationally efficient algorithms for computing approximate balanced truncations for these large systems .  before reviewing the theory for periodic systems , first recall the main idea of balanced truncation for linear time - invariant ( lti ) stable systems ( for more detail , see standard textbooks , such as  ) .one begins by defining controllability and observability gramians and to measure to what degree each state is excited by an input , and each state excites future outputs , respectively .one then seeks a coordinate transformation , called balancing transformation , so that the transformed gramians and are equal and diagonal ( hence balanced ) : , where are the hankel singular values .finally , in these new coordinates , one truncates the states that are least controllable and observable , corresponding to the smallest hankel singular values .   for time - periodic systems, one may also define controllability and observability gramians , as in  .the _ controllability gramian at time _ , , and _observability gramian at time _ , , for the complex stable periodic system ( [ lps ] ) are defined similarly by the following two positive - semidefinite matrices where the superscript denotes the adjoint of a linear operator .see the appendix for the specific properties these gramians satisfy .both and are -periodic .also , for each , and satisfy the following respective discrete lyapunov equations : as discussed in the appendix , when convergent , the definitions ( [ grammian_def ] ) ( equivalently , ( [ lyapunoveqn ] ) ) extend to general linear time - varying systems of the form ( [ lps ] ) , with the same interpretations in terms of input - to - state and state - to - output mappings .if the dimension  of the system is not too large , it is possible to solve the above pairs of lyapunov equations for gramians , for each , .one can then do balanced truncation _ at each time . refer to , , for detailed discussions .note that farhood et al . also compute the gramians / generalized gramians by solving corresponding lyapunov equations / inequalities at each time step ; however , the balanced truncation is then realized in a lifted time - invariant setting , not in the original periodic one . in this paperwe will also use the lifting approach , in conjunction with a method of snapshots as in .   as discussed in  , the linear periodic time - variant system ( [ lps ] )can be equivalently rewritten as a linear time invariant ( lti ) system : where is fixed and and \\ \tilde{c}_j&=\begin{bmatrix } c(j)\\c(j+1)f(j+1,j)\\\vdots\\c(j+t-1)f(j+t-1,j ) \end{bmatrix}\\ \tilde{d}_j&=\begin{bmatrix } 0 & & & 0\\f_{j,2,1}&0 & & \\\vdots & \vdots & \ddots & \\f_{j , t,1 } & f_{j , t,2 } & \cdots & 0 \end{bmatrix},\end{aligned}\ ] ] where .we refer to the time invariant system ( [ lti_sys ] ) as a _ lifted system at time _ , in which , , and are constant matrices .the lifted system may be viewed as a poincar map of the original periodic system in the state space at times , while the input and output information for an entire period is kept at each iteration of the poincar map .the controllability gramian and observability gramian for this lti lifted system are conventionally defined by the two positive - semidefinite matrices they respectively satisfy the following discrete lyapunov equations : the above setting allows one to study the lifted lti system instead of the original periodic one such that the well - developed balanced truncation techniques for lti systems are available .we emphasize that in the lifted lti system , only one time of balanced truncation is needed .however , the drawbacks of working on the lifted system are that it has even higher dimensional inputs and outputs , and that we usually do not have the explicit form of the lifted system in computations .the approach taken in this paper is as follows : _ by using the equivalence between the lti ( lifted ) and time - periodic systems , we apply the balanced pod procedure in the lti setting but do computations in the original periodic setting_. to realize this plan , first we list the following statement establishing the equivalent relations between the gramians of the lifted system and the original periodic system .proofs are by direct calculations , using periodicity .  consider an arbitrary time .the controllability and observability gramians at time of the periodic system ( [ lps ] ) , i.e. , and given by ( [ grammian_def ] ) are respectively equal to the controllability and observability gramians and of the lifted system ( [ lti_sys ] ) at time given by ( [ grammian_def_lti ] ) .[ prop gramian contrl observ ]  an immediate corollary from the above statement is that , as stated in  ,  , solutions of the time - varying lyapunov equations ( [ lyapunoveqn ] ) are equivalent to the solution of the time - invariant lyapunov equations ( [ lyapunoveqn_lti ] ) in the lifted setting .this result plays an important role in traditional lifting approaches of balanced truncations for periodic systems , where the gramians are found by solving lyapunov equations . in the following section ,we show how proposition [ prop gramian contrl observ ] allows us to apply a snapshot - based algorithm for computing balancing transformations for periodic systems , thereby avoiding the need to solve lyapunov equations .the method of snapshots provides a numerical approximation of gramians directly based on their definitions .refer to and the references therein for the application of this method to linear time - invariant systems .  for time - varying discrete systems , first, we define the _ empirical controllability gramian _ and _ empirical observability gramian _ at time for the periodic system ( [ lps ] ) as where is a nonnegative integer . clearly , ; .further , the following result gives upper error bounds of and .consider an arbitrary time .let , where is a nonnegative integer . for the linear periodic system ( [ lps ] ) , where an induced norm is used .[ lemma empirical gramian error bound lps ]  the proof is based on the following result for linear time - invariant systems .consider a linear stable time - invariant system where , and are constant matrices and whose controllability and observability gramians are and . define the empirical controllability and observability gramians for this lti system as where is a nonnegative integer .thus , where an induced norm is used .[ lemma empirical gramian error bound lti ]  under any induced norm , similarly , we have .  by proposition [ prop gramian contrl observ ] , . also , it is straightforward to show , where is the empirical controllability gramian of the lifted lti system at time , defined in form of ( [ gramian_controllability_empirical lti ] ). thus , and the result follows by applying lemma [ lemma empirical gramian error bound lti ] to the lifted lti system at time .the result for the observability gramian follows from a similar argument .indeed , for any , there is an induced norm such that ( by lemma 5.6.10 in  ) .thus , with an satisfying , lemma [ lemma empirical gramian error bound lps ] implies that under a certain matrix norm , where .thus , a large enough guarantees that the error between empirical and exact gramians is small .this result encourages one to use empirical gramians instead of exact ones to realize approximate balanced truncations , whenever it is difficult to compute exact gramians .the following subsections introduce the method of snapshots by which we can calculate those empirical gramians for linear periodic systems .also note that it is clear that the condition is not crucial to obtain a result similar to that in the lemma [ lemma empirical gramian error bound lps ] .   without loss of generality ,let .the empirical controllability gramian can be obtained by running a series of impulse - response simulations for system ( [ lps ] ) as follows .consider the -th column of , denoted by ^d ] ,one starts the first simulation with the initial condition and a unit impulse ^{\top} ] , and by running steps we get ^i ] , where the will be used to construct the finite sum . repeat the above process until the -th simulation , with initial condition and unit impulse ^{\top} ] . by running the simulations with steps in all , we can thus construct the sum ^d\left([b(k)]^d\right)^*f(j , i+1)^* ] is obtained by running the ` -th ' simulation mentioned above for steps , with initial condition and corresponding unit impulse . however , to run this simulation is not necessary , since ^d = f(j - t , j - m+1)[b(j - m)]^d ] , starting from run a simulation for steps . then , start from , running for steps . repeat the process until the simulation starting from and running for steps .the data set obtained is sufficient for construction of .   to calculate for a fixed , , by the method of snapshots, one needs to construct an _ adjoint system _ where , the state , the control input , and as above , here we run a series of impulse - response simulations for the adjoint system .consider the -th column of , ^d=\left[c(2j+m - k-1)^*\right]^d ] whose -th entry is  . by running the simulation for steps ,we obtain ^d;\\ z(j+2)&=f(j+m-1,j+m-2)^*\left[c(j+m-1)^*\right]^d;\\ & \;\vdots\\ z(j+m)&=f(j+m-1,j)^*\left[c(j+m-1)^*\right]^d,\end{aligned}\ ] ] where is a snapshot we need for computing the .the second simulation starts with initial condition and unit impulse ^{\top} ] , another snapshot we need .repeat the above process until the -th simulation , with initial condition and unit impulse ^{\top} ] .since has columns , we need to run simulations , with steps total , to obtain snapshots , with which we have the following statement :  * ( observability gramian from snapshots ) * define an dimensional matrix ^ 1,\cdots,\left[c(j)^*\right]^1,\cdots,\\ f(j+m-1,j)^*\left[c(j+m-1)^*\right]^q,\cdots,\left[c(j)^*\right]^q\big ] , \end{gathered}\ ] ] whose columns are the snapshots obtained through the impulse - response simulations described above .then [ prop empirical observb gramian ]  again , the above procedure is valid for general stable time - varying systems . if the system is -periodic , then as for the empirical controllability gramian case , one only needs to run simulations to obtain the snapshots .suppose we have obtained the factors and mentioned above , where may be different .then for the _ lifted lti system at time _ , by proposition [ prop gramian contrl observ ] , [ prop empirical contrl gramian ] and [ prop empirical observb gramian ] , its gramians are approximated by for lti systems , the method of snapshots presented in  gives an algorithm for computing the transformation that exactly balances the empirical gramians and , directly from the factors and , without computing the gramians themselves :  * ( balanced truncation using the method of snapshots  ) * let be a real diagonal matrix including the non - zero hankel singular values , obtained by singular value decomposition ( svd ) of the matrix  in which is the rank of , and , satisfy .the balancing transformation is then found by computing matrices if , then is the balancing transformation and is its inverse . if , then the columns of form the first columns of the balancing transformation , and the rows of form the first rows of the inverse transformation .we emphasize that computing the balancing transformation by the method of snapshots _ exactly _ balances the empirical gramians .the only approximation here is to use and to approximately construct the gramians as in ( [ approximation of gramians ] ) .balanced truncation of order is then done as follows .let denote the first columns of , and the first rows of .the reduced state variable satisfies . the reduced model , in the lifted setting ,is then given by in simulations , the reduced output equation ( [ reduced system part2 ] ) shall be un - lifted to the original periodic setting : for each , , where denotes the entry of at -th row and -th column .when the number of outputs is very large , direct construction of as described in section [ sec : obsv_gram ] will be computationally intractable , because adjoint simulations are required . to overcome this , an _ output projection method _ may be used , by which one can substantially reduce the number of adjoint simulations . the starting point of this methodis to define an optimization problem that minimizes the error between the input - output behavior of the original system and that of a projected system with a smaller - dimensional output space . for a time - periodic system ,this optimization problem is cumbersome to define directly . we instead consider the lifted lti system , for which we design a version of output projection , and then relate the method back to the periodic system .fix a time  .the lifted system at time  , though with an even higher dimension of output  , is a standard lti system , and its input - output behavior can be measured by a sequence of dimensional impulse - response matrices , the -th column of each representing the output response corresponding to a unit impulse input ^{\top} ] in which the column vectors are the orthonormal eigenvectors of .it is a typical eigenvalue problem in pod reduction and can be numerically solved by the method of snapshots , where the snapshots are provided by the data sets obtained through simulations .the above output projection generally produces a full matrix .thus the components of the projected no longer cleanly correspond to the outputs of the periodic system at different time steps respectively , but the intermixed combinations of them , which is not desirable both for physical understanding of the system , and for numerical simulation purposes ( recall that in simulations we do not really compute with the lifted system , so the projected lifted system should be ` unlifted ' back to a periodic system for computation ) . to deal with this difficulty , we propose a modified version of output projection , in which we seek a sub - optimal solution that solves ( [ error_outputprojection_lifted ] ) with a constraint imposing that the orthogonal projection with rank takes a diagonal form \label{p_rtilde constrained}\ ] ] where each block is an orthogonal projection on with rank in form of where and .note that in this case .\label{phi_rtilde constrained}\ ] ] here we need .the diagonal makes it easy to unlift the projected lifted system ( [ lti part1 projected ] ) & ( [ lti part2 projected ] ) to a _ projected time - periodic system _ for , where the -periodic orthogonal projection of rank on is defined by we write where the -periodic is defined by .  by the above unlifting , though solving for the sub - optimal is no longer a standard pod reduction problem in the lifted setting , we can attack it in the periodic setting such that standard pod reduction techniques are available .  to realize that , first we rewrite as where each block is a matrix .let , .the -th column of represents the output response of the original periodic system at time to the unit impulse ^{\top} ] .this definition matches that proposed in bamieh and pearson 1992 .the following statement links the constrained optimization problem given above for the lifted system to an equivalent optimization problem defined in the periodic system .  under the frobenius norm ,the optimization problem ( [ error_outputprojection_lifted ] ) , with its solution constrained in form of ( [ p_rtilde constrained ] ) , is equivalent to the optimization problem which implies , for each , is the solution of the optimization problem [ prop optimization ]  by direct calculation , using ( [ p_rtilde constrained ] ) , ( [ gtilde_blocks ] ) , ( [ p def ] ) , ( [ g matrix def ] ) and the linearity of the operation .this statement allows us to obtain the in form of ( [ p_rtilde constrained ] ) by solving unconstrained optimization problems for , , in the periodic setting , each of which is a typical eigenvalue problem in pod reduction with solutions satisfying that the columns of each , , are orthonormal eigenvectors of , i.e. , they satisfy   numerically , the eigenvectors can be solved by the method of snapshots , in which svds are needed to obtain , , .see details of this method in , for example , and .the snapshots are the columns of impulse - response matrices for .a convenient computational feature of the output projection method is that by periodicity we see all the snapshots have already been obtained during the computation of described in section [ sec : ctrb_gram ] , as long as , except that the data obtained there needs to be left - multiplied with a corresponding matrix .for instance , the matrix includes all the columns of matrices for .suppose we have found the in form of ( [ p_rtilde constrained ] ) . by definition , the observability gramian of the projected lifted system at time is and the observability gramian at time for the projected periodic system is   the following statement for the projected systems is an analog to proposition [ prop gramian contrl observ ] for the full case .its proof is by direct calculation .the observability gramian for the projected lifted system at time , , is equal to the observability gramian at time for the projected periodic system , .[ prop gramian observ projected ]  therefore , , where is the empirical observability gramian at time of the projected periodic system obtained by the method of snapshots introduced in section 3.1 .now the input is only dimensional , , in the corresponding _ projected adjoint time - periodic system _ where , the state , the control input , and thus , only adjoint simulations in total is needed for computing .  instead of seeking sub - optimal solutions for ( [ error_outputprojection_lifted ] ) in form of ( [ p_rtilde constrained ] ) , ( [ phi_rtilde constrained ] ) , where each , , are different and given by the solutions of ( [ eigenvalue problem pod ] ), one can also alternatively impose a stronger constraint that all the projection matrices at each time step are the same , which means , and .\label{phi_rtilde constrained_one phi}\ ] ] it is straightforward to show that , similar to ( [ eigenvalue problem pod ] ) , the single is obtained by solving the eigenvalue problem where are the columns of , and .the obtained by this setting reflects the overall impulse - responses of the periodic system , mixing those at different time steps along each time period .the stronger constraint given here let us expect that the solution is even less optimal .however , a numerical advantage is that only one svd is needed in the method of snapshots for solving the single .[ remark single projection ]   following the terminology in , the approximate balanced truncation based on the method of snapshots given in section [ snapshot_subsec ] & [ sec : balpod ] , plus the output projection introduced in section [ sec : outputproj ] , is named _balanced pod_. we summarize the computational procedures of the balanced pod method for linear asymptotically stable time - periodic systems :  * step 0 : pick a time , , as the `` base point , '' based on which the balanced truncation will be done . *step 1 : run impulse - response simulations to obtain snapshots and form dimensional as described in section [ snapshot_subsec ]. * step 2 : for each , ( corresponding to one whole period ) , left - multiply the state responses stored during computing by corresponding matrices , and then use the method of snapshots to solve the eigenvalue problems defined by ( [ eigenvalue problem pod ] ) for the output projection matrices along one time period. * step 3 : construct the `` projected adjoint system '' ( [ adjoint system_projected ] ) and run impulse - response simulations for the adjoint system to form dimensional , as described in section [ snapshot_subsec ] * step 4 : compute the svd of defined in ( [ svd ] ) and compute the balancing pod modes for the lifted system given by ( [ t1 s1 ] ) . * step 5 : obtain the reduced lifted system in form of ( [ reduced system part1 ] ) & ( [ reduced system part2 ] ) . for computational purposes ,rewrite the output equation ( [ reduced system part2 ] ) as ( [ reduced system part2_periodic setting ] ) .if the dimension of output is small , then we can skip step 2 for output projection , and directly run impulse - response simulations for the adjoint system ( [ adjoint system ] ) as in section 3.1 to form the dimensional .on the other hand , in step 2 , instead of solving for the time varying -periodic output projection matrices , one can alternatively seek a single time - invariant projection matrix by solving the eigenvalue problem ( [ eigenvalue problem pod one phi ] ) as we mentioned in remark [ remark single projection ] .note that to pick up an optimal base time in the sense that the upper error bound of the reduced system of a fixed order is minimized , one needs to solve for the hankel singular value matrices corresponding to the lifted systems at different times during one whole period ( see farhood et al .this method is therefore not attractive or even computationally feasible for large systems .[ remark optimal base time ]   a `` dual '' version of the above algorithm is readily available for balanced truncations of linear periodic systems with high - dimensional states and inputs , but only few outputs . in that case, one can start with the construction of by running a series of impulse - response simulations for the adjoint system , since the dimension of outputs is small .then an _ input projection _ based on pod reduction shall be done in the same spirit as that underlies the output projection to obtain a projected system whose dimension of inputs is reasonably small .one then runs a corresponding number of impulse - response simulations for the projected system to construct .[ remark input projection ]to validate and demonstrate the balanced pod algorithm , we consider the following example . consider a linear periodic system ( [ lps ] ) with period , state dimension , output dimension , control input dimension , and randomly generated diagonal matrices with diagonal entries bounded in  ] .the setting is similar to that used in the example in farhood , et .al .it is not a high - dimensional problem , such that exact balanced realizations for the corresponding lifted system can be done by solving lyapunov equations for gramians of the lifted system , and the result can thus be compared with that by the balanced pod approach .we pick the `` base time '' .choose , and test different cases of balanced truncation , including balanced pods with the order of output projection for the periodic system at each time step and respectively .recall that , to the lifted system , the order of output projection is .figure [ figure hankel singular values ] shows the hankel singular values in the lifted setting , obtained by the exact balanced realization , the balanced truncation based on the method of snapshots given in section [ sec : balpod ] but without output projection , and the balanced pod ( with different output projection matrices along one period for the periodic system ) .figure [ figure g inf norm ] shows the error plots of the infinity norm , versus , for those different cases . here is the impulse - response matrix of the reduced lifted system of order .we do not show the balanced pod with case since the result is almost identical to that by balanced truncation based on the method of snapshots but without output projection .we see that the balanced truncation based on the method of snapshots gives a good approximation of the exact balanced truncation , and further , the balanced pod , even with low orders of output projection , generates satisfying results .figure [ figure g inf norm diff output projections ] shows comparisons between balanced pod results with the same order of output projection , one set based on different projection matrices along one period in the periodic setting , and the other only having one time - invariant projection matrix ( see section [ sec : outputproj ] ) . for the cases where are low ,these two approaches give almost identical results , or even the latter one gives better results .however , when the order of output projection increases , such as , the results based on different projection matrices at each time step are better than those by a single projection matrix , as we expect .: exact balanced truncation ( ) ; balanced truncation by the method of snapshots but without output projection ( ) ; balanced pod with ( ) ; balanced pod with ( ) ; balanced pod with ( ) ; balanced pod with ( ) . ]  , for exact balanced truncation ( ) , balanced truncation by the method of snapshots but without output projection ( ) , balanced pod with ( ) , balanced pod with ( ) , balanced pod with ( ) , and the lower bound for any model reduction scheme ( ) . ]-periodic output projections versus time - invariant output projections : error , for balanced pod with ( ) , balanced pod with ( ) and balanced pod with ( ) .solid lines correspond to cases using different projection matrices along one period for the periodic system , and dashed lines using one single projection matrix .the black solid line is the lower bound for any model reduction scheme ( ) . ]we have proposed a version of the balanced pod method to realize approximate balanced truncation for linear asymptotically stable periodic systems , especially with very high - dimensional states and outputs but a small number of inputs .it is a generalization of the balanced pod method for linear time invariant system developed in  .the development of this balanced pod is based on a lifting approach , and key parts include the method of snapshots for computing empirical gramians , and a version of output projection that gives a periodic projected system based on pod reduction , with which the number of adjoint simulations needed for computing the empirical observability gramian is substantially decreased .simulation results given in the previous section validate the approach .this snapshot - based approach is also readily applicable to high - dimensional systems with high - dimensional inputs and few outputs , where an input projection is needed .a future direction of this work is to apply the balanced pod method to construct reduced - order models of high - dimensional ( linearized ) periodic systems arising in engineering applications and then , based on the low - dimensional models , to design closed - loop control laws based on these models . for instance , such periodic orbits may arise as periodic shedding in the wake of a bluff body  , or from open - loop forcing at a prescribed frequency , as in the recent results of  , which show that a periodic blowing and suction of flow added at the walls of a channel flow may reduce drag .the dimension of states of such systems , including the three components of velocity and pressure at each grid point in the channel , can be on the order of , and thus reduced - order models of such systems are quite valuable for designing model - based control laws .the observability gramian provides a measure of the influence of an initial state on future outputs with zero control inputs . to see that ,first , for a fixed time , we define a linear operator for system ( [ lps ] ) to describe the state - output behavior with zero inputs and an initial state .more precisely , , . to measure to what degreethe state excites the output , it is natural to compute the square of the induced norm denotes an inner product , with subscripts specifying the vector space where the inner product is defined .the observability gramian given in ( [ grammian_def ] ) has the following property :    first , it is clear that , we consider , for an arbitrary , where matrices are considered as linear operators and denotes the corresponding adjoint .so , and .controllability gramian for linear time - varying systems similarly , the controllability gramian provides a measure of the influence of a sequence of input history on the current state . in other words , to reach a given current state ( if possible ) , the controllability gramian measures how much effort from control inputs is needed . for a fixed current time , define a linear operator \rightarrow \mathbb{c}^n ] is the adjoint of . and, is the inverse of .[ prop controllability gramian property ]   similar to the observability gramian case , by calculations under standard inner products of and $ ] , one obtains , , .it follows from definition that .

Abstract: we introduce an algorithm based on a method of snapshots for computing approximate balanced truncations for discrete - time , stable , linear time - periodic systems . by construction , this algorithm is applicable to very high - dimensional systems , even with very high - dimensional outputs ( or , alternatively , very high - dimensional inputs ) . an example is shown to validate the method .


dss , experiment scripts and data are available at http://homepages.inf.ed.ac.uk/gsanguin/dss.zipcyclic behaviour is ubiquitous in biology .the importance of oscillatory systems stems both from the necessity to adapt to the many environmental cycles ( circadian , annual , etc . ) , as well as to maintain intrinsically periodic processes such as the cell cycle .both of these type of oscillations are essential to many physiological processes , and malfunctions in the cellular time keeping mechanisms are frequently associated with disease , further motivating the study of these systems .genetic regulatory networks are at the core of many of these biological oscillators. these networks can sustain oscillatory behaviour in protein levels through specific architectures involving multiple feedback loops of transcriptional regulation .for example , a transcriptional oscillator is thought to drive the _ arabidopsis thaliana _circadian clock through mutual repression of three transcriptional regulators .the cell cycle is another oscillatory process , which controls cell division and duplication . in the case of _ saccharomyces cerevisiae _ , experiments and dynamical models suggest that the cell cycle is the result of a transition between two self maintaining steady states , driven by two antagonistic classes of proteins .evidence suggests that a transcriptional network is an important part of this mechanism .these oscillators have been the subject of study for many years , but uncovering the exact mechanism is a challenge that involve many complex chemical , genetic and physiological components .it is therefore important to devise computational statistical methods which may guide experimental analyses by inferring potential regulatory interactions directly from time series gene expression data , which is usually easier to obtain .network inference is a well established and rich domain of research in systems biology .state of the art methods for regulatory network inference include a wide variety of techniques from statistics and machine learning .for example , mutual information between gene expression levels under different experimental conditions is used by aracne and clr , two of the most widely used methods for network reconstruction .genie3 , another method which was a top performer at the dream network inference challenges , uses random forests to produce a weighted ranking over the network edges .other methods recently used include regularized regression , anova and hierarchical gaussian models most of these methods focus on steady state data , which is by definition not available for oscillatory networks .regularisation - based and bayesian methods can also be adapted to time series data .dynamic bayesian networks have long been a popular choice in network inference ( e.g. ) .such methods present considerable advantages in being able to quantify uncertainty and to incorporate prior knowledge , but are often severely limited by computational constraints .optimisation - based methods based on regularised regression present often a scalable alternative at the cost however of some modelling flexibility .  here, we use a first order model of the system dynamics to constrain the network inference , but we explicitly take advantage of the oscillatory behaviour of the system by pursuing frequency - based estimation . we build a hierarchical bayesian model over the network dynamics which can set and infer structural constraints and account for the inevitable uncertainty that experimental settings convey . furthermore , our method can easily integrate non - trivial side information , for example in the form of sequence similarity between promoter sequence of genes .experimental results on real and simulated data highlight that the method offers an effective and flexible platform for statistical inference in oscillatory systems , and can uncover non - trivial biological information .the rest of the paper is organised as follows : the next section describes the methodology we use , reviewing the linear time - invariant approximation we use as well as introducing the bayesian hierarchical framework for network inference .we then present an experimental evaluation on three data sets : a synthetic data set from the dream network inference challenge , a simulated data set obtained from a state of the art model of the _ a. thaliana _ circadian clock , and a real data set from the yeast _s. cerevisiae _ cell cycle .we then conclude the paper by discussing our method in the light of these experimental results and the existing literature on network inference .our approach is centred on the assumption that the oscillatory dynamics of the regulatory network can be reasonably approximated , in fourier space , by a linear time invariant system .this is of course a simplification , but it is not an unreasonable one , and has been previously proposed as a formalism to model oscillatory genetic circuits with considerable success , see for a recent review . from the inferential point of view , adopting a frequency domain perspective is convenient , as it enables us to transform the network reconstruction problem in a regression problem , for which many advanced estimation tools exist .we choose a bayesian regression approach , as it provides an effective methodology to integrate diverse information in the inferential machine . as a proof of principle of how non - trivial information can be incorporated ,we discuss how sequence similarity between promoter regions could be used within a hierarchical model framework .   the starting point for our modellingis the approximation of the system s dynamics as a linear time invariant ( lti ) model :   here the expression level of gene , denoted as , depends on the expression levels of the other genes ( potential regulators ) through activating or repressing intensity .gene expression levels decay linearly with rates .additionally , gene expression depends on a set of inputs which can be either external signals ( light for example ) or any other gene signal that is not modelled explicitly in the network .finally , each gene has a basal transcription rate .having a set of samples from an experiment ( e.g. mrna levels from a microarray experiment ) , let the vector denote the set of expression level measurements for gene .we can further construct the matrix , which contains the sample points for the set of genes .let be the derivative of , so equation ( [ eq : aproxi ] ) in matrix form for this set of gene expression levels is given by :    where is the matrix with diagonal elements and off - diagonal elements , the input signals are contained in matrix . to complete the notation, we denote with vector of basal expression levels , which multiplies the matrix of ones to add a constant term to the equation .we proceed to compute the derivative by first projecting the gene expression levels into a set of orthogonal basis functions .the chosen set of basis functions is the one given by the discrete fourier transform of the gene expression levels .we emphasize that the choice of basis function is dictated by the nature of the problem : while in the limit of a continuously sampled signal this choice would be irrelevant ( any complete basis would yield perfect reconstruction ) , for discretely sampled signals the quality of the approximation to the signal ( and its derivative ) will depend on the expressiveness of the chosen finite set of basis functions .our choice of basis functions is motivated by the prior knowledge that the signals of interest should be oscillatory , making the choice to work in the frequency domain particularly appealing .we denote , for brevity , as the frequency representation of , with each column containing the frequency spectrum of the expression of a gene over the time points .the frequency domain derivative can be computed analytically by , so the frequency domain representation of the system is given by :    basal rates are included in the zero frequency component of .the frequency representation of the inputs is given by .  to account for any discrepancies between the linearised model and the true system dynamics ,we assume normally distributed error with variance .the likelihood function for equation ( [ eq : matrix form-1 ] ) is :  \left[\mathbf{\begin{array}{c } \mathbf{a_{i}^{t}}\\ \mathbf{c}_{i}^{t}\end{array}}\right]\right)^{\mathbf{t}}\mathbf{\left(\mathrm{\dot{\mathbf{x_{i}}}-}[\begin{array}{cc } \mathbf{x } & \mathbf{u}\end{array}]\left[\mathbf{\begin{array}{c } \mathbf{a_{i}^{t}}\\ \mathbf{c}_{i}^{t}\end{array}}\right]\right)}\end{gathered } .\label{eq : likelihood}\ ] ]  in general , multiple replicate time series may be available . denoting with the number of replicate time series , the overall likelihood , under an assumption of normal i.i.d error between series , can be generalized as :    which is a product of gaussian densities .notice that the form of equation is identical to a regression problem where the output variables ( fourier coefficients of the derivatives of the signals ) are regressed onto the fourier coefficients of the signals .the inference problem of estimating the interaction and input response matrices ^{\mathrm{t}} ] between gene promoters , derived from a multiple alignment method like or an alignment - free method ,can be related to the structural adjacency matrix at the top of the hierarchical model .assuming for simplicity a gaussian observation model , we can then incorporate sequence similarity by positing the following relationship between promoter similarity scores and the structural adjacency matrix    here the parameter is the similarity `` induced '' by the transcription factor ( a proportionality constant ) , and the product equals 1 if and only if genes and are both regulated by .this model is a form of additive clustering . by conditioning on , we can derive the distribution , which is a gaussian with non - negative constraints , ( see appendix eq .4 ) . this distribution can be used for sampling posterior values of ; in our applications , however , we preferred to fix the value of to its non - negative maximum likelihood solution , effectively approximating this conditional posterior with a function .the similarity score variance is given a weakly informative inverse gamma prior . by completing the square we can derive a gaussian distribution for the parameters , for its derivation and estimation see appendix section 1 .the overall structure of the model is depicted graphically in fig .[ fig : model ] .   and sparsity parameter . in chequered circles the frequency - domain gene expression model and its parameters . in yellow the stripes sequence similarity and its parameters.,scaledwidth=50.0% ]inference of parameters is done through a simple gibbs sampling scheme .given conjugacy among distributions , sampling of these parameters is straightforward for all distributions except .this distribution is not conjugate , so a metropolis within gibbs would be necessary for exact inference . in order to improve performance and given the fact that retrieving the distribution over is not an objective ; we use the non - negative least square estimate for the vector .convergence was tested by applying geweke diagnostic over the last 1000 samples of matrix .mathematical derivations of the required conditional posteriors and the general sampling algorithm are described in the appendix .in this section we assess the performance of our method on two realistic simulated data sets and a real data set , comparing its performance to two other state of the art methods .we call our method dss , for dft - based spike and slab model . the first simulated dataset was generated from a well known model for the _ a. thaliana _ circadian clock network .this model is a non - linear ode - based model which exhibits regular oscillations ( for suitable parametrisations ) , thus matching one of our main modelling assumptions .however , it is a non - linear model , hence introducing an element of model mismatch . as a second synthetic benchmark data set we used one of the data sets provided by the dream 4 challenge is again a non - linear model , which exhibits damped oscillatory dynamics in some of the nodes ; thus , this data set presents considerably more elements of model mismatch .the last experiment tested the method on a real data set of gene expression levels obtained in a micro - array experiment for the _ s. cerevisae _ cell cycle transcriptional network .results were assessed in terms of area under the precision - recall ( aupr ) curve ; pr curves plot the fraction of correctly called instances versus the ratio of true positives over true positives plus false negatives .an ideal classifier would give a aupr of 1 , while a random baseline would return the ratio of positives negatives .inference of the models parameters was conducted by gibbs sampling from the model presented in ( fig .[ fig : model ] ) . in total , 5000 samples were obtained .the last 1000 samples were selected and averaged to compute the conditional probability of a link given the model and the expression data , see appendix sections 1.1.1 and 1.1.2 for details into the inputs and outputs of the program .   as a first comparison , in order to establish the validity of our claim that frequency domain analysis is beneficial for oscillatory networks , we sought to compare our results with a complete analogue in time domain . to do this , we implemented a spline - based alternative to the dft , using cubic splines interpolation as means of computing the time domain derivative , while the rest of the hierarchical model was left unchanged .as competing methods to assess the performance of dss we selected genie3 , which is based on random forests , and the ode - regression based inferelator .  in a network of n genes , genie3 solves n regression problems by predicting , using random forests , the expression level of each gene as a function of the other n-1 genes ( putative regulators ) .then the relative importance of each gene expression is evaluated and the putative gene interactions are ranked .genie3 was designed for steady state data , but time - series adaptation can be readily derived and was provided to us by one of the authors .the inferelator estimates the parameters of an ode system using regression with l1-regularization over a finite element approximation of the derivative .the method has been extended , with new functionalities to incorporate prior information over the network links , and to use alternative optimisation methods for model selection , including the elastic - net ( regularization over l1 and l2 norms ) and bayesian regression with best subset selection .finally , as a simple baselines , we implemented a l1 regularised version of the regression problem in equation , using the lasso implementation .as a first example we used data generated from a well known oscillatory network model , the _a. thaliana _ circadian clock .the data consists of simulated mrna measurements from the model found in .this non - linear model has 7 transcription factors and 2 post transcriptional elements ztl and lhymod . in order to replicate experimental conditions ,we assume that only mrna data is available , so protein concentrations for the transcriptional and post - transcriptional elements are assumed unobserved .the transcription factors used for network inference are lhy , toc1 ,prr5 , hypothetical gene y , gi , prr9 and prr7 , the post - transcriptional elements are not considered. a graphical representation of the model can be observed in ( appendix fig .this model was simulated for 3 cycles obtaining 28 samples .the procedure was performed with a light / dark photo period of 12/12 , 6/18 , 8/16 , 18/6 and 20/4 hours which are represented in our model by binary input signals .this design of our study is created to mimic a realistic experimental setting as in ; the biological rationale for such design is that stimulating the system with these different inputs may tease out the contribution of the main drivers of the clock at different times of day .we also simulated knock - out mutants , , and by the same procedure as presented in with photo periods of 12/12 hours . these experiments amount to 14 time series ; as these data are directly the outputs of an ode model ( without any additional noise ) we define this idealised data set as the _ noiseless _ data set .an additional dataset of 14 time series was produced by adding gaussian white noise with a signal - to - noise ( snr ) of 10 .an example of the simulated expression levels is plotted in the upper left panel of fig [ fig : circ ] .  using the model specification as ground truth, we proceeded to draw the pr - curves for the different methods and computed the area under the pr - curve for all the resulting networks .these areas are plotted for the noiseless ( simulated data without added noise ) and noisy data in the upper right panel of fig .[ fig : circ ] . for the case of noisy data, we generated 20 data sets by adding 10snr - white - gaussian noise to the noiseless data set .the simulated data has a very well defined oscillatory behaviour , as such the dss method proved superior over the alternatives , as can be appreciated in the chart .the dss method achieved an aupr of 0.57 and mean 0.55 with std of 0.1 in the noisy case , being over the mean genie3 and inferelator estimates with these later yielding less variance in their scores ( 0 variance in case of the inferelator ) .the lasso solution to the quadratic form also shows an improvement over both methods , even though dss still has the best results .interestingly , the spline solution achieves comparably good performance with dss for the noiseless case , but under performs in the noisy case .we speculate that this may be due to the sensitivity of the spline estimate of the derivative to noise , confirming our intuition that adopting a set of basis functions well adapted to the problem may convey an advantage .it is intriguing that the method actually obtains a better performance on noisy data sets ; we speculate that this may be due to the fact that adding noise alleviated the effects of model mismatch ( resulting from the lti approximation ) .intuitively , in the absence of noise the attempts to fit non - linear data with a linear model could become more problematic .  to test the effect of including side information , we simulated a between - gene similarity matrix by drawing from a uniform distribution and using equation [ eq : additive ] .in this case we notice an important improvement by observing an increment in the aupr to 0.68 in the noiseless case and a mean of 0.57 with std of 0.1 for the noisy data .the principal objective of using this simple simulated similarity matrix was to confirm that structural information can be retrieved and used as aid for inference . by clustering the co - regulated elements we added additional structural constraints into the inference scheme .finally we included a graphical representation of the true network ( fig .[ fig : circ ] bottom left ) and a network resulting from setting a threshold of 0.5 over the inferred matrix ( dss - with - similarity , noiseless data ) ( fig .[ fig : circ ] bottom right ) .we notice that the 0.5 threshold , while reasonable , is still arbitrary and is used here only for the purposes of graphical visualisation .the full output from the method is a probability over the existence of edges , and can be better visualised as a heatmap , see appendix sections 1.2 and 2 . directed edges go from blue ( regulators ) to red ( targets ) , black edges mean bidirectional regulation . as can be appreciated important features such as the bidirectional regulation between lhy-prr7 and lhy-prr9 are recovered .errors are related to the roles of prr7 and prr9 regulating gi instead of toc1. this may be due to the method confounding the effects of toc1 over these former elements as being closer to the expression patterns of gi .this difficulty discriminating between the roles of the prr genes is also expressed by inferring the spurious bidirectional edge between prr7-prr9.  , scaledwidth=100.0% ]   as a second example , we considered a data set from the fourth edition of the dream competition .this data set is obtained from simulating a 10-node network , of which three nodes are input nodes ; 15 regulatory links are present .three simulations were present , one with an ode - based system , another one with a stochastic differential equation ( sde ) system and a third one with sde - based system and added experimental noise .five time series are provided for each system , a time series contains 21 samples .the network is subjected to a single node perturbation , which mathematically corresponds to a change in the basal expression parameter , so the mean expression level of the node changes for half of the time points .the expression profiles for the set of ten genes in one time series is presented in ( fig .[ fig : dream ] top left ) .this data set does not comply with the main assumption of the model ( it shows irregular damped oscillations ) ; we therefore expect performance not to be optimal , but it is still useful to evaluate comparatively the model under such a model mismatch scenario .  figure ( fig .[ fig : dream ] ) shows a comparison of the area under the p - curve for the three simulated systems . of these , dss achieves better performance in the ode - based simulation , by having an aupr of 0.31 , higher than the nearest best method ( genie3 ) .inferelator could not be executed on this data set due to numerical issues ( some expression levels are exactly zero in this example ) .the performance improved for the sde based simulation , by achieving an aupr of 0.35 , above inferelator s 0.27 .slightly worse results were achieved for the sde model with experimental noise , achieving an aupr of 0.3 . by simulating a sequence similarity matrix performancewas improved for both ode and sde solutions . in the case of sdethe solution improved dramatically to 0.42 .as in the previous experiment , the network and its inferred counterpart are presented in fig [ fig : dream ] bottom left and bottom right respectively .the inferred network is obtained by setting the threshold to 0.5 over the inferred adjacency matrix for the sde data with added similarity matrix . as can be observed in the true network ,nodes `` g1 '' and `` g10 '' are constant inputs .node `` g9 '' is subjected to perturbation for half the time points , thus its effect is propagated through the network by node `` g5 '' .  in the inferred networkwe can observe some interesting characteristics .first , nodes `` g1 '' and `` g9 '' are identified as input nodes , node `` g10 '' is incorrectly identified as an output only node .node `` g2 '' maintains its out - degree of 4 even though it s regulators are not correctly identified .nodes `` g9 '' and `` g5 '' are shown with increased in and out - degree , this may also be due to the confounding effects of the their `` parent - son '' relationship , specially considering that the perturbed `` g9 '' node has the biggest amplitude of the gene expression profiles , as appreciated by the red curve in the top left plot in fig .[ fig : dream ] .  , scaledwidth=100.0% ]   for the last experiment we used a real time series data set collected during the _s. cerevisiae _ cell cycle .our evaluation is based on the genes identified by and some of their interactions on the dynamical model found in .the main transcriptional elements selected were swi5 , yhp1 , swi4 , fkh1 , sic1 , ace2 , yox1 , stb1 , nrm1 , whi5 , fkh2 , mcm1 , swi6 , hcm1 , ndd1 and mbp1. their putative regulations were extracted from literature \{see appendix } for the putative network used as ground truth .the source for the gene expression data is , it contains 2 wild type replicates and two mutant replicates ( ) each one containing 14 samples for each gene during approximately 2 cell cycles .additionally , we downloaded promoter sequence information from for all the network elements .we then proceeded to use the multiple alignment software clustal omega to obtain an alignment - based similarity matrix between sequences .as an alternative way of encoding sequence information , an alignment - free similarity matrix was built using the method described in .we tested three subsets of data , one containing only the wild type expression profiles , other containing only the mutants expression levels , the last data set was the normalized concatenation of both .as an example of the observed gene expression levels , fig .[ fig : yeast ] top panel shows the gene expression levels for the wild type conditions .the aupr from applying the various methods to this data are presented in fig .[ fig : yeast ] bottom left panel . in this case dss identifies the putative network well above the random baseline of 2.1 and above the competing methods . in the case of wildtypesthe aupr of dss was of 0.24 . in the mutant datasets the performance of dss improves by including sequence similarity achieving an aupr of 0.2607 and 0.2608 for s and s2 respectively .the best overall performance was achieved by using the combined data set with sequence similarity matrix s2 , resulting in an aupr of 0.267 .  the network in ( fig .[ fig : yeast ] ) bottom right is obtained by setting the threshold of 0.9 to the inferred network from the combined wild type and mutant dataset with added similarity matrix . in this case fkh1has a central role in the inferred network , being fully connected to the other elements . even though this fully connected position is biologically implausible, it does reinforce the important role of fkh1 in the cell cycle , e.g. its role in regulating the m - phase response .another noticeable inferred link concerns the post transcriptional regulation of swi6 by whi5p ; this regulation was also considered as part of the ground truth network , as in the case of the yeast cell cycle transcriptional and post transcriptional regulations are intertwined .also worth noticing the regulation of swi6 by yox1 ( member of the sbf complex ) even though evidence suggests causality may be in the opposite direction .swi4 and swi6 form part of transcription factor complexes sbf and mbf , as such , their regulations may be confounded .this can be appreciated in the regulation of nrm1 by swi4 in the inferred network , when in fact nrm1 appears to be regulated by swi6 .the transcriptional activator ndd1 is essential during the s - phase , ndd1p along mcm1p bind to fkh2p , this effect may be observable in the inferred network by directed edges from ndd1 to yox1 and from yox1 to fkh2 .  by observing the aupr plotwe see that mutant data appears to be more informative in this case than wild type , being only marginally inferior to the combined data set with similarity matrix .part of the experimental design in selecting mutations in was aiming at attenuating the effects of the post - transcriptional elements of the cell cycle ; the stronger performance of our method on the mutant data sets may be explained by this experimental design .generally , the dss solution will find the most relevant edges in the network to explain the observed dynamics , while the dss with similarity method will find the most relevant solution that includes a grouping of the proposed edges according to the similarity matrix .so both results can be analysed separately and may offer additional insight over the whole network behaviour . with this purpose the six inferred networks and the putative ground truth are included in ( appendix fig .3 ) for analysis .   , scaledwidth=100.0% ]inference of gene regulatory networks from expression data is one of the best studied problems in systems biology . despite this considerable collective effort , the general problem remains ill - posed and , in the absence of extensive data sets and strong domain expertise , a solution to this problem remains elusive . in this light , it is of interest to consider more delimited problems which may be amenable to specialised but more effective solutions .oscillatory systems present a prime example of such a problem : while they obviously constitute a specialised subset of regulatory networks , in our opinion they are sufficiently widespread to warrant tailor - made solutions .dss couples a simplified mechanistic approach ( lti ) with frequency domain information to provide such a method .lti methods in the time domain for _ a. thaliana _ with experimental data have been studied in .our results on the circadian clock simulation suggest that this frequency domain approach can indeed be fruitful when the model assumptions are reasonably met . as results over the dream and _ s. cerevisae _data sets suggest that the method can perform competitively with state of the art methods also when the model assumptions are not precisely met ( damped oscillatory behaviour ) ; however , in these cases the methods competitive advantage is smaller or inexistent .the use of derivative and ode information in a network inference framework has some precedents .a method that is in spirit similar to our approach is inferelator .it casts the network inference problem as a a parameter inference problem over a first order differential equation system , then estimates the system parameters via regularized regression over a finite differences solution to the system .recently bayesian approaches that make use of the derivative information have also been proposed . in a probabilistic model for integrating a linearised version of network dynamics in a regression frameworkis presented . attacked the problem of parameter inference of an ode system jointly with a bayesian regression over the gene expression levels .the basis of this model is a gaussian process with product of experts likelihood , not dissimilar from our model in equation .however , the authors in did not attempt a joint parameter estimation and variable selection problem , stopping short of formulating the problem in terms of network inference .basis functions in time domain ( splines ) have already been applied to network inference problems in systems biology , primarily to model unknown non - linear transition functions .the distinctive part of our work is the proposal of a frequency domain approach for oscillatory systems , and in particular the embedding of our method within a hierarchical framework where integration of additional information is natural .we expect that non - linearities encoded as basis functions as in would be a valuable extension of our work and likely result in an improvement in performance .while we believe that the dss method provides promising results , there are several inherent limitations in our approach .importantly , the lti approximation implies that self regulation is confounded with decay , so such types of interactions can not be identified .empirical results also seem to suggest that post transcriptional interactions may be confounded with transcriptional interactions ; this is to be expected , as post - transcriptional interactions are not modelled in our framework . for such reasons , direct application to models that include complex post - transcriptional interactions , such as ,is not advised .furthermore , as all bayesian network inference methods , dss also suffers from multi - modal posterior distributions . the use of auxiliary information , such as sequence similarity , can be beneficial to ameliorate this problemmany different types of auxiliary information can be considered , and indeed alternative models for incorporating sequence similarity could also be used .a major strength of a bayesian hierarchical model is that different models for auxiliary information could be easily incorporated within the dss framework .we thank botond cseke , vn - anh huynh - thu and daniel seaton for useful discussions .the genie3 software adapted for time series data was kindly provided to us by dr vn - anh huynh - thu .dtb is funded by a microsoft research studentship .gs acknowledges support from the european research council under grant mlcs30699 .synthsys was founded as a centre for integrative biology by bbsrc / epsrc award d19621 to ajm and others .bell _ et al_. 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( 1999 ) , scpd : a promoter database of the yeast saccharomyces cerevisiae._bioinformatics _ , * 15(7 ) * , 607 .likelihood funcion for the lti system .\left[\begin{array}{c } \mathbf{a}^{\mathrm{t}}\\ \mathbf{c}^{\mathrm{t}}\end{array}\right]\right)\end{aligned}\ ] ]  conditional distribution of  ] and  ] representing the off - diagonal elements of the upper triangular matrix of , vector representing the row vector of and representing the elementwise product ( hadamard product ) . then the distribution of the upper diagonal elements is :  {i < j}\boldsymbol{\beta}\right)^{\mathrm{t}}\left(\bar{\mathbf{s}}-\left[\mathbf{\bar{h}}_{i}\circ\mathbf{\bar{h}}_{j}\right]_{i < j}\boldsymbol{\beta}\right)\right)\end{aligned}\ ] ]  conditional distribution for   {i < j}\boldsymbol{\beta}\right)^{\mathrm{t}}\left(\bar{\mathbf{s}}-\left[\mathbf{\bar{h}}_{i}\circ\mathbf{\bar{h}}_{j}\right]_{i < j}\boldsymbol{\beta}\right)}{2}\right)\nonumber \end{aligned}\ ] ]  sampling   instead of sampling from we use the non - negative least square estimate of , by solving the quadratic form  {i < j}\boldsymbol{\beta}\right)^{\mathrm{t}}\left(\bar{\mathbf{s}}-\left[\mathbf{\bar{h}}_{i}\circ\mathbf{\bar{h}}_{j}\right]_{i < j}\boldsymbol{\beta}\right);\ ; s.t.\beta_{l}\geq0\label{eq : betas}\ ] ]  conditional distribution for   we define matrices{i < j} ] such that .       the algorithm [ alg : algorithm - for - the ] presents the gibbs sampler for parameter inference in the hierarchical model . a brief overview of the inputs and outputs is also presented .due that the dft computed by the fft algorithm yields complex numbers , a real representation of the coefficients is needed .we work with the so - called real discrete fourier transform .it consists of stacking the real over the imaginary part of the first fft coefficients .we denote this matrix . figures ( [ fig : spectra - for - the ] ) and ( [ fig : dspectra - for - the ] ) show plots for the dft real and imaginary parts of the _ a. thaliana _ data - set , with a photoperiod of 12 hours .the magnitude and phase spectra is also shown , finally the rdft is presented at the bottom of the picture ., scaledwidth=100.0% ]  , scaledwidth=100.0% ]  hyperparameters and are set to 1 so is uniformly distributed in the interval $ ] .parameters and are set so the hypervariance has a continuos bi - modal distribution , according to the recommendations of , in which they set them to 5 and 50 respectively .alternative parametrization of 50 and 500 was also tested yielding better results in some cases .the hyperparameters and are set to 0.001 and 0.001 , this is a weak prior reflecting uncertainty about the linearity of the system .hyperparamters and are set to 10 and 0.001 , this parametrization required a manual tunning , as the scale parameter having a weak prior resulted in the effects of the sequence similarity model to banish . by modifying this priorwe can give more `` weight '' to the sequence similarity clustering thus , the flexibility of the hierarchical model .the gibbs sampler presented in the previous section allows us to draw samples from the joint conditional distribution .the marginal distribution for each of the models parameters can be drawn from this joint distribution , and the expected value for each parameter equals to the average of the samples .for example , figure illustrates the expected value for matrix obtained from averaging over 1000 samples drawn from the marginal distribution .this figure shows in dark red those elements with higher probability of a regulatory interaction under the model assumptions , except the diagonal elements , which represent the decay rates of the equation model .the aupr were computed by thresholding the off - diagonal elements of this matrix for each data - set .inputs : time series of time - points for gene expression levels , encoded in matrices .prior hyper - parameters . optional similarity matrix .  outputs : joint conditional posterior distribution   1 .obtain the dft of and the corresponding rdft coefficient matrices 2 .compute the derivatives 3 .sample from the conditional distribution over the lti coefficients , given in eq .( [ eq : bcond ] ) 4 .  sample from the conditional distribution over given by eq .( [ eq : ptau ] ) 5 .sample from eq .( [ eq : ph ] ) , to account for the decay rates we set diagonal elements to 1 , and set the diagonal elements of matrix to negative . 6 .sample from eq .( [ eq : tw ] ) 7 .sample from eq .( [ eq : sigmad ] ) 8 .optional sample from eq .( [ eq : psigma ] ) and from the nonnegative least squares solution to equation ( [ eq : betas ] ) . 9 .  return to step 3  note : a burn in period of 4000 samples is considered in the general purpose implementation of the model .   obtained by averaging the last 1000 samples .rows represent targets and columns regulators .the diagonal indicates the decay parameters .[fig : heat - map - representing - the],scaledwidth=60.0% ], scaledwidth=100.0% ]  , scaledwidth=100.0% ]  , scaledwidth=100.0% ]  , scaledwidth=100.0% ]  , scaledwidth=100.0% ]  , scaledwidth=100.0% ]the arabidopsis thaliana circadian clock model as presented in , is shown in fig . [fig : pokhilkho ] .the 10-node oscillatory network that was part of the dream4 challenge supplementary information data set is presented in fig . [fig : dream4-challenge - network ] .in fig . [ yeast nets ]the inferred networks after thresholding the value of are presented , the putative ground truth matrix is presented on fig .[ yeastputative ]     * regulation of sic1 by swi5 as in .* regulation of swi4 by yhp1 as in .* regulation of yhp1 , swi4 , yox1 and hcm1 by swi4 as in .* regulation of swi5 and ace2 ; yhp1 ; sic1 , yox1 and hcm1 ; ndd1 ; by fkh1 .* regulation of swi6 and mbp1 by nrm1 as in .* regulation of swi6 , swi4 and mbp1 by whi5 as in .* regulation of swi5 , yhp1 and fkh1 ; ace2 and ndd1 by fkh2 .* regulation of swi4 and swi5 by mcm1 as in .* regulation of swi4 , fkh1 , yox1 , nrm1 , hcm1 and ndd1 as in .* regulation of yhp1 , fkh1 , fkh2 , whi5 and ndd1 by hcm1 as in .* regulation of swi5 and ace2 ; yhp1 and hcm1 as in .* regulation of yhp1 , fkh1 , yox1 , nrm1 and hcm1 by mbp1 as in .* regulation of ndd1 ; swi6 and mbp1 by the interaction of transcription factor mbf with stb1 .* regulation of swi4 and swi6 by its cobinding with mcm1p as in .bahler ( 2005 ) , cell - cycle control of gene expression in budding and fission yeast ._ annu rev genet _ , * 39 * 6994 .et al _ ( 2014),topology and control of the cell - cycle - regulated transcriptional circuitry , _ genetics _ , * 196(1 ) * , 6590 .et al_. 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( 2010 ) , data assimilation constrains new connections and components in a complex , eukaryotic circadian clock model , _ molecular systems biology _ , 6 .( 2006),the forkhead transcription factor hcm1 regulates chromosome segregation genes and fills the s - phase gap in the transcriptional circuitry of the cell cycle , _ genes dev _ ,* 20(16 ) * , 226678 .stillman ( 2013),dancing the cell cycle two - step : regulation of yeast g1-cell - cycle genes by chromatin structure , _ trends biochem sci _ , *38(9 ) * , 46775 .et al _ ( 2011 ) , a comprehensive genomic binding map of gene and chromatin regulatory proteins in _ _s__accharomyces , _ mol cell _ ,* 41(4 ) * , 48092 . weiss ( 2012 ) , yeastbook - cell signaling and development : mitotic exit and separation of mother and daughter cells , _ genetics _ * 192 * , 11651202 ;

Abstract: oscillations lie at the core of many biological processes , from the cell cycle , to circadian oscillations and developmental processes . time - keeping mechanisms are essential to enable organisms to adapt to varying conditions in environmental cycles , from day / night to seasonal . transcriptional regulatory networks are one of the mechanisms behind these biological oscillations . however , while identifying cyclically expressed genes from time series measurements is relatively easy , determining the structure of the interaction network underpinning the oscillation is a far more challenging problem . here , we explicitly leverage the oscillatory nature of the transcriptional signals and present a method for reconstructing network interactions tailored to this special but important class of genetic circuits . our method is based on projecting the signal onto a set of oscillatory basis functions using a discrete fourier transform . we build a bayesian hierarchical model within a frequency domain linear model in order to enforce sparsity and incorporate prior knowledge about the network structure . experiments on real and simulated data show that the method can lead to substantial improvements over competing approaches if the oscillatory assumption is met , and remains competitive also in cases it is not .