cells are able to sense gradients of chemical concentration with extremely high sensitivity and accuracy .this is done either directly , by measuring spatial gradients across the cell diameter , or indirectly , by temporally sensing gradients while moving . in temporal sensing ,a cell modifies its swimming behavior according to whether a chemical concentration is rising or falling in time .this mode of sensing is typical of small , fast moving bacteria such as _ escherichia coli _ , which can respond to changes in concentration as low as 3.2 nm of the attractant aspartate .in contrast , direct spatial sensing is prevalent among larger , single - celled eukaryotic organisms such as the slime mold _ dictyostelium discoideum _ ( dicty ) and the budding yeast _ saccharomyces cerevisiae _ .dicty cells are able to sense a concentration difference of only 1 - 5% across the cell , corresponding to a difference in receptor occupancy between front and back of only 5 receptors .spatial sensing is also performed by cells of the immune system including neutrophils and lymphocytes , as well as by growing synaptic cells and tumor cells .interestingly , a direct spatial mode of sensing has also been demonstrated for the large oxygen - sensing bacterium _ thiovulum majus _ , indicating that direct gradient sensing is widespread among the different kingdoms of life .there has been great progress in understanding the limits of concentration sensing in bacteria such as _ e. coli _ , pioneered by berg , and in understanding the origins of sensitivity in the underlying signaling network , pioneered by bray , and followed by others . by contrast , very little is known about what determines the accuracy of direct gradient sensing by eukaryotic cells .recently , we derived the fundamental physical limits of direct gradient sensing , where the accuracy is limited by the random arrival of particles at the cell surface due to diffusion .we used as models a perfectly absorbing sphere and a perfectly monitoring sphere ( la berg and purcell ) . in these two models ,gradients are inferred from the positions of particles absorbed on the surface of a sphere or the positions of freely diffusing particles inside a spherical volume , respectively .the latter case simulates rebinding of particles , as particles can enter and exit the spherical volume freely . in comparison , for the perfectly absorbing sphere ,previously observed particles are never remeasured . as a result , we found that the perfectly absorbing sphere is superior to the perfectly monitoring sphere , both for concentration and gradient sensing ( table i ) .the superiority of the absorbing sphere may help explain the presence at the surfaces of cells of signal degrading enzymes , such as pde for camp in dicty and bar1 for mating factor in_ s. cerevisiae_. those surface enzymes could reduce or eliminate rebinding ( and therefore remeasurement ) of the same signal molecule .quantitatively , our theory compares favorably to recent measurements of dicty moving up shallow camp gradients , suggesting that these cells operate near the physical limits of gradient detection .  while our recent models of the absorbing and monitoring spheres allowed us to derive the fundamental limit of gradient sensing , the models neglect the details of biochemical reactions , such as particle - receptor binding and downstream signaling events , which might further increase measurement uncertainty . to study the effects of particle - receptor binding , we here extend a formalism for the uncertainty of concentration sensing recently developed by bialek and setayeshgar , to the case of gradient sensing .this formalism uses the fluctuation - dissipation theorem to infer the fluctuations of the receptor occupancy ( and hence the accuracy of concentration sensing ) from the linear response of the average receptor occupancy to changes in receptor binding free energies .the effect of particle rebinding is included by coupling particle - receptor binding to the diffusion equation , leading to correlations in time among the receptors .we report analytical results for two receptors ( fig .1 ) , as well as two coaxial rings of receptors , _ e.g. _ one at each cell pole ( fig .2 ) . by assuming diffusion - limited particle binding to the receptors , we are able to directly compare to our previous model for the fundamental limits of gradient sensing . for realistic receptors , we find that particle rebinding lowers the accuracy of gradient sensing in line with our previous results for the absorbing and monitoring spheres ( table i ) ..[tab : formulas ] uncertainties in measured concentration and concentration gradient for two idealized cell models : a perfectly absorbing sphere ( second column ) and a perfectly monitoring sphere ( third column ) . also provided is the ratio of the uncertainties of the absorber and monitor .parameters : diffusion constant , radius of sphere , averaging time , and average chemical concentration .table reproduced from ref .[ cols="^,^,^,^",options="header " , ]   .,width=283 ]bialek and setayeshgar previously presented a method based on the fluctuation - dissipation theorem ( fdt ) to calculate the accuracy of measurement of chemical concentration by receptors .we extend their method to calculate the accuracy of measurement of concentration gradients , and derive analytical results for ( 1 ) two receptors and ( 2 ) two coaxial rings of receptors , _ e.g. _ one at each cell pole .we start our derivation by considering an arbitrary number of receptors .the kinetics of the ensemble - average occupancy of receptor due to binding and unbinding of chemical ligands at the local concentration is given by -k_-n_j(t).\ ] ] linearization about the mean steady - state occupancy at concentration gives by thermodynamics , the ratio of binding and unbinding rates is related to the free - energy difference between the unbound and bound states of the receptor according to variations of the rate constants in eq .[ eq : dn ] are equivalent to a variation or external perturbation of this receptor free - energy difference combining eqs .[ eq : dn ] and [ eq : dk ] , one obtains after fourier transforming in time to obtain a frequency representation where , _e.g. _ , describes the variations in the receptor occupancy as a function of frequency .the dependence on can be eliminated by using the ( linearized ) diffusion equation in eq .[ eq : ftn ] using the fourier transforms where and where we have introduced a convergence factor in eq .[ eq : df ] to regulate the function ( effectively assigning a size scale to the receptor ) , eq .[ eq : diff ] yields inverting the spatial fourier transform back into real space , one obtains the concentration fluctuations at the locations of the receptors in terms of the occupancy fluctuations \\ & = & \frac{i\omega}{2\pi^2}\left[\delta \hat n_j(\omega)\int_0^{\lambda}\frac{k^2\,dk}{dk^2-i\omega}+ \sum_{l\neq j}^m\frac{\delta \hat n_l(\omega)}{|\vec{x}_j-\vec{x}_l|}\int_0^{\infty}\frac{k\sin(k|\vec{x}_j-\vec{x}_l|)}{dk^2-i\omega}dk\right].\label{eq : dcx}\end{aligned}\ ] ] the cut - off , which accounts for the physical dimensions of the receptor , is used in eq .[ eq : dcx ] to set an upper limit of integration .following bialek and setayeshgar , we imagine that the mechanism that reads out the receptor occupancy averages over a time long compared to the correlation time between binding and unbinding events of a receptor ( see below ) . in this case, we can apply the low - frequency limit ( ) . using the expression where is real and is the principal value of the associated integral , eq .[ eq : dcx ] becomes inserting eq .[ eq : dcx2 ] for in eq .[ eq : ftn ] yields \delta\hat n_j\nonumber\\ & & + k_+(1-\bar n_j)\frac{i\omega}{4\pi d}\sum_{l\neq j}^m\frac{\delta\hat n_l}{|\vec x_j-\vec x_l| } + \frac{k_+(1-\bar n_j)\bar c_j}{k_bt}\delta\hat f_j,\label{eq : dni}\end{aligned}\ ] ] which depends only on the and their conjugate variables .these equations describe how deterministic frequency - dependent changes in the free - energy differences affect the frequency - dependent occupancies of all the receptors .+  . also shownis receptor dimension .,width=283 ]  * two receptors * + we first consider two identical receptors ( fig .1 ) for which eq .[ eq : dni ] simplifies to two coupled linear equations \delta\hat n_1\nonumber\\ & & + k_+(1-\bar n_1)\frac{i\omega}{4\pi d}\frac{\delta\hat n_2}{|\vec x_1-\vec x_2|}+\frac{k_+(1-\bar n_1)\bar c_1}{k_bt}\delta\hat f_1\\ -i\omega\delta\hat n_2&=&\left[k_+(1-\bar n_2)\frac{i\omega\lambda}{2\pi^2d}-(k_+\bar c_2+k_-)\right]\delta\hat n_2\nonumber\\ & & + k_+(1-\bar n_2)\frac{i\omega}{4\pi d}\frac{\delta\hat n_1}{|\vec x_1-\vec x_2|}+\frac{k_+(1-\bar n_2)\bar c_2}{k_bt}\delta\hat f_2,\end{aligned}\ ] ] in matrix form these equations can be written in terms of a complex susceptibility as where and .the inverse susceptibility relates changes in to changes in the occupancies , _  according to the fluctuation - dissipation theorem , the noise power spectra of the occupancies can be calculated in the low - frequency limit from the above deterministic response functions , obtained by inverting the matrix in eq .[ eq : ex ] , where is the total variance in each occupancy and is the correlation time for receptor occupancy in the absence of rebinding .the correlation time is modified when rebinding is included via coupling to particle diffusion so that the effective correlation time for receptor occupancy increases to as shown by the above equations . for an equilibrium system with time - reversal symmetry , the power spectra are necessarily symmetric in the zero - frequency limit , _i.e. _ .however , ( eq . [ s12 ] ) and ( eq . [ s21 ] ) are different depending on and , respectively .hence , our approach based on the fluctuation - dissipation theorem is only valid for shallow gradients with .  to determine the uncertainty of gradient sensing , we need to know the uncertainty in measuring a concentration difference .this uncertainty can be obtained from the noise power spectra for ligand binding and unbinding ( eqs .[ s11]-[s21 ] ) .starting with the expression for the average occupancy in terms of the average concentration , where , we obtain the following relation which expresses how a fluctuation in the occupancy of receptor affects the cell s best estimate of the local concentration . to obtain the uncertainty of the gradient measurement, we require the variance of the estimated concentration difference ^ 2\rangle$ ] , which we calculate using eq .[ eq : dc_n ] and substituting the previously calculated noise power spectra . for an averaging time , we obtain for the variance of the inferred concentration difference between two receptors separated by a distance : \tau^2\rangle&= & \frac{(\bar c_1+k_d)^4}{k_d^2\tau}s_{n_1n_1}(0)+\frac{(\bar c_2+k_d)^4}{k_d^2\tau}s_{n_2n_2}(0)\\ & -&\frac{(\bar c_1+k_d)^2(\bar c_2+k_d)^2}{k_d^2\tau}[s_{n_1n_2}(0)+s_{n_2n_1}(0)]\nonumber\\ & = & \frac{2\bar c_1}{k_+(1-n_1)\tau}+\frac{2\bar c_2}{k_+(1-n_2)\tau}+\frac{\bar c_1+\bar c_2}{\pi d\tau}\left(\frac{1}{s}-\frac{1}{2r}\right)\label{eq : var_diff}.\end{aligned}\ ] ] assuming that the true gradient is shallow , , to estimate the variance in eq .[ eq : var_diff ] we set and for identical receptors in equilibrium , and obtain \tau^2\rangle = \frac{4c_0}{k_+(1-n)\tau}+\frac{2c_0}{\pi d\tau}\left(\frac{1}{s}-\frac{1}{2r}\right).\label{eq : var_diff2}\ ] ] * two rings of receptors * + following bialek and setayeshgar , we next consider the case where a cell is equipped with two rings of receptors , one at each pole .we further assume that the cell length is large enough so each receptor in ring 1 is at nearly the same distance from each receptor in ring 2 ( fig .the equation governing receptor in ring 1 is \delta\hat n_j^{(1)}(\omega)\\ & & + k_+(1-\bar n_1)\frac{i\omega}{4\pi d}\left[\sum_{l\neq j}^{\text{ring}\,1}\frac{\delta\hat n_l^{(1)}(\omega)}{|\vec x_j-\vec x_l|}+\sum_{l=1}^{\text{ring}\,2}\frac{\delta\hat n_l^{(2)}(\omega)}{r}\right]\\ & & + \frac{k_+(1-\bar n_1)\bar c_1}{k_bt}\delta\hat f_j^{(1)}(\omega).\end{aligned}\ ] ] summing over all receptors of ring 1 , and using the notation and , we obtain for ring 1 \right\}\delta n^{(1 ) } -\frac{i\omega mk_bt}{4\pi dr}\delta n^{(2)}=\delta f^{(1)},\ ] ] where , with a similar expression for ring 2 . these equations can be solved for the uncertainty in measuring the concentration difference between the rings in a similar manner to the two - receptor case . for an averaging time and , one obtains for the uncertainty in measuring a gradient using two rings of receptors each \tau^2\rangle=\frac{2\bar c_1}{mk_+(1-\bar n_1)\tau}+\frac{2\bar c_2}{mk_+(1-\bar n_2)\tau}+ \frac{\bar c_1+\bar c_2}{m\pi d\tau}\left(\frac{1}{s}+\frac{\phi}{2}-\frac{1}{2r}\right).\label{eq : var_diffring}\ ] ] assuming again that the true gradient is shallow , , to estimate the variance in eq .[ eq : var_diffring ] we set and for identical receptors in equilibrium , which results in \tau^2\rangle=\frac{4c_0}{mk_+(1-n)\tau}+ \frac{2c_0}{m\pi d\tau}\left(\frac{1}{s}+\frac{\phi}{2}-\frac{1}{2r}\right).\label{eq : var_diffring2}\ ] ]many types of cells are known to measure spatial chemical gradients directly with high accuracy . in particular , _ dictyostelium discoideum _ is known to measure extremely shallow gradients of camp important for fruiting body formation and _ saccharomyces cerevisiae _ ( budding yeast ) detects shallow gradients of mating pheromone .direct spatial sensing of gradients is also performed by cells of the immune system including neutrophils and lymphocytes , as well as by the large marine bacterium _thiovulum majus _ .the question arises what are the limits of the accuracy of gradient sensing set by chemical diffusion ?recently , we derived fundamental physical limits for gradient sensing using as model cells a perfectly absorbing sphere and a perfectly monitoring sphere .we found that a perfectly absorbing sphere is superior to a perfectly monitoring sphere for both concentration and gradient sensing since the perfectly absorbing sphere avoids the noise due to remeasuring previously detected particles ( table i ) .consequently , our results for the perfectly absorbing sphere represent the true fundamental limits of both concentration and gradient sensing by cells .our models of the absorbing and the monitoring spheres neglect all biochemical reactions , such as particle - receptor binding and downstream signaling events , which might further increase measurement uncertainty . to study the effects of particle - receptor binding , we extended a formalism for the uncertainty of concentration sensing , recently developed by bialek and setayeshgar , to gradient sensing .this formalism uses the fluctuation - dissipation theorem to infer the fluctuations of the receptor occupancy ( and hence the accuracy of concentration sensing ) from the linear response of the average receptor occupancy to changes in receptor binding free energies .the effect of particle rebinding is included by coupling particle - receptor binding to the diffusion equation , leading to correlations in time among the receptors .+ * single receptor for concentration sensing * + it is instructive to first review the result for a single receptor without and with coupling to particle diffusion ( corresponding , respectively , to preventing and allowing rebinding of already measured particles ) .the uncertainty of sensing concentration without rebinding ( _ i.e. _ assuming that upon unbinding the particle is removed from the system ) is given by where is the rate constant for binding , is the average receptor occupancy , and is the averaging time . the right hand side of eq .[ eq : no_rebind ] is obtained for diffusion - limited binding , in which case with the diffusion constant and the receptor dimension .this is the well - known fundamental limit derived by berg and purcell .in contrast , the uncertainty of concentration sensing including particle diffusion and possible rebinding is given by bialek and setayeshgar as  where .comparison of eq .[ eq : no_rebind ] and eq .[ eq : with_rebind ] shows that the uncertainty of concentration sensing by a single receptor is larger by the term when allowing for rebinding of already measured particles . for the minimum uncertainty case set by diffusion - limited binding , and given by the right hand side of eq .[ eq : with_rebind ] , this additional term simply leads to a factor of 3 increase to the fundamental limit derived by berg and purcell ( eq .[ eq : no_rebind ] ) . +* two receptors for gradient sensing * + as derived in methods ( see eq .[ eq : var_diff2 ] ) , we find that the uncertainty of gradient measurement is given by \tau^2\rangle / r^2 = \frac{4c_0}{k_+(1-n)r^2\tau}+\frac{2c_0}{\pi dr^2\tau}\left(\frac{1}{s}-\frac{1}{2r}\right)\label{eq : resultdc1}.\ ] ] as expected , the larger the receptor - receptor separation , the smaller the uncertainty in the gradient , because of the larger `` lever arm '' between receptors .note that the result in eq .[ eq : resultdc1 ] for the uncertainty in the gradient is independent of the magnitude of the actual gradient , including the case when there is no real gradient present . for comparison ,the uncertainty of mean concentration measurement is \tau^2\rangle = \frac{c_0}{k_+(1-n)\tau}+\frac{c_0}{2\pi d\tau}\left(\frac{1}{s}+\frac{1}{2r}\right)\label{eq : resultdc2}.\ ] ] analogous to the single receptor case , eq .[ eq : with_rebind ] , the first term in eqs .[ eq : resultdc1 ] and [ eq : resultdc2 ] arises from particle - receptor binding kinetics , whereas the second term is due to diffusion and includes the effects of possible rebinding of already measured particles . due to the proximity of the receptors , separated by distance , a particle can unbind one receptor and subsequently rebind the other receptor because of diffusion ( see fig .1 , trajectory 2 ) .hence , there is an additional noise component , which actually improves the accuracy of gradient measurement ( term ) due to cancellation with the noise due to rebinding to the same receptor , but degrades the accuracy of mean concentration measurement ( term ) since rebinding to the other receptor can only increase noise in the estimate of the mean concentration .this correlated noise was recently also investigated with monte carlo simulations . + * two rings of receptors for gradient sensing * + we next consider two rings of receptors , parallel to one another at opposite cell ends a distance apart ( fig .2 ) . as derived in methods ( see eq .[ eq : var_diffring2 ] ) , we find that the uncertainty of gradient measurement is given by \tau^2\rangle / r^2 = \frac{4c_0}{mk_+(1-n)r^2\tau}+\frac{2c_0}{m\pi dr^2\tau}\left(\frac{1}{s } + \frac{\phi}{2}-\frac{1}{2r}\right)\label{eq : resultdc3},\ ] ] where is the number of receptors per ring and is a geometric factor close to unity . for comparison , the uncertainty of meanconcentration measurement is \tau^2\rangle = \frac{c_0}{mk_+(1-n)\tau}+\frac{c_0}{2m\pi d\tau}\left(\frac{1}{s } + \frac{\phi}{2}+\frac{1}{2r}\right)\label{eq : resultdc4}.\ ] ] the factor in eqs .[ eq : resultdc3 ] and [ eq : resultdc4 ] reflects signal averaging by multiple receptors , which reduces the measurement uncertainty with respect to the case of two receptors .the possibility of rebinding to other receptors within the same ring leads to correlations among the signals , which are reflected in the extra term in the rebinding noise . +* comparison with the perfect monitor and the perfect absorber models * + to make comparison to our results for the perfectly absorbing and monitoring spheres ( table i ) , which do not include particle - receptor kinetics , we replace by for the minimum uncertainty case set by diffusion - limited binding .to specifically compare with the perfectly absorbing sphere , we neglect the second term in eqs .[ eq : resultdc3 ] and [ eq : resultdc4 ] ( thereby neglecting rebinding of particles ) and obtain for gradient and concentration sensing \tau^2\rangle / r^2}{(c_0/r)^2}&=&\frac{1}{\pi da'c_0\tau}\label{eq : diff_limited1}\\ \frac{\langle[\delta(c_1 + c_2)/2]_\tau^2\rangle}{c_0 ^ 2}&=&\frac{1}{4\pi da'c_0\tau}\label{eq : diff_limited2}\end{aligned}\ ] ] respectively .to specifically compare with the perfectly monitoring sphere we keep both terms in eqs .[ eq : resultdc3 ] and [ eq : resultdc4 ] and obtain for gradient and concentration sensing \tau^2\rangle / r^2}{(c_0/r)^2 } & = & \frac{1}{\pi da'c_0\tau}\left[3+s\left(\phi-\frac{1}{r}\right)\right]\label{eq : diff_limited3}\\ \frac{\langle[\delta(c_1+c_2)/2]_\tau^2\rangle}{c_0 ^ 2 } & = & \frac{1}{4\pi da'c_0\tau}\left[3+s\left(\phi+\frac{1}{r}\right)\right]\label{eq : diff_limited4}.\end{aligned}\ ] ] the parameter is the combined receptor dimension , ultimately limited by the cell dimension .note that in eqs .[ eq : diff_limited1 ] and [ eq : diff_limited3 ] for gradient sensing we normalized by , and in eqs .[ eq : diff_limited2 ] and [ eq : diff_limited4 ] for concentration sensing we normalized by in order to use the same notation as table i and ref . .  as a result , for , _i.e. _ receptor separation larger than receptor size , the measurement uncertainty with rebinding ( eqs .[ eq : diff_limited3 ] and [ eq : diff_limited4 ] ) is always larger than the measurement uncertainty without rebinding ( eqs .[ eq : diff_limited1 ] and [ eq : diff_limited2 ] ) for both gradient and concentration sensing .hence , the absorber is superior to the monitor even when receptor binding kinetics are explicitly included in line with our previous finding ( table i ) .specifically , for diffusion - limited binding , the dominant effect of particle rebinding ( eqs .[ eq : diff_limited3 ] and [ eq : diff_limited4 ] ) is simply an increased numerical prefactor , also in line with our results for the perfect absorber and perfect monitor models .+ in conclusion , we found that the accuracy of concentration and gradient measurement without ligand rebinding is higher than the accuracy with rebinding , confirming the superiority of the absorber over the monitor .our model of two coaxial rings qualitatively resembles the polar clusters found abundantly in bacteria and archaea .hence , our model may be directly suitable for describing the concentration sensing by these organisms and possibly also for oxygen - gradient sensing by the bacterium _ thiovulum majus _ .furthermore , a number of mechanistic models for gradient sensing and chemotaxis by eukaryotic cells have addressed the important questions of cell polarization , signal amplification , and adaptation , cell movement of individual cells , cell aggregation , as well as sensing of fluctuating concentrations .our results on the accuracy of gradient sensing complement these models , and may ultimately help lead to a comprehensive description of eukaryotic chemotaxis .rge acknowledges funding from the biotechnology and biological sciences research council grant bb / g000131/1 and the centre for integrated systems biology at imperial college ( cisbic ) .nsw acknowledges funding from the human frontier science program ( hfsp ) and the national science foundation grant phy-0650617 .gestwicki , j.e . ,lamanna , a.c . ,harshey , r.m . ,mccarter , l.l .kiessling , l.l . , adler , a. , 2000 .evolutionary conservation of methyl - accepting chemotaxis protein location in bacteria and archaea .j. bacteriol .182 , 6499 - 6502 .dawes , a.t ., edelstein - keshet , l. , 2007 .phosphoinositides and rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a one - dimensional model of a motile cell .j. 92 , 744 - 768 .

Abstract: chemotactic cells of eukaryotic organisms are able to accurately sense shallow chemical concentration gradients using cell - surface receptors . this sensing ability is remarkable as cells must be able to spatially resolve small fractional differences in the numbers of particles randomly arriving at cell - surface receptors by diffusion . an additional challenge and source of uncertainty is that particles , once bound and released , may rebind the same or a different receptor , which adds to noise without providing any new information about the environment . we recently derived the fundamental physical limits of gradient sensing using a simple spherical - cell model , but not including explicit particle - receptor kinetics . here , we use a method based on the fluctuation - dissipation theorem ( fdt ) to calculate the accuracy of gradient sensing by realistic receptors . we derive analytical results for two receptors , as well as two coaxial rings of receptors , _ e.g. _ one at each cell pole . for realistic receptors , we find that particle rebinding lowers the accuracy of gradient sensing , in line with our previous results .


precision measurements of the cmb and the lss have opened a window to the physics of the early universe ( ( * ? ? ?* wmap 2012 ) , ( * ? ? ?* planck 2013 ) ) .in particular it has become possible to measure the exact statistics of the primordial curvature perturbations on uniform density hypersurfaces , , which turned out to be almost gaussian ( ( * ? ? ?* planck 2013(2 ) ) ) .these small deviations from gaussianity are commonly represented by the parameters and and allow to write where denotes the gaussian curvature perturbations .  on the other side ,the exact statistics of are predicted by inflation models .this suggests to relate current observations directly to inflation models , parametrized by inflationary parameters , , by higher order statistics of the curvature perturbations ( see fig .[ cosmo ] ) . how to set up such an inference approach in the framework of information field theory ( ( * ? ? ?* enlin et al . 2011 ) ) was originally developed in ( ( * ? ? ?* dorn et al . 2013 ) , ( * ? ? ?* dorn et al . 2014 ) ) and is addressed in the work at hand .  to do inference of inflationary parameter we consider the observation ( cmb , lss ) to be a discrete data set , , given by where denotes some gaussian noise , , and the so - called response operator .if we consider the cmb , the latter is a linear operator that transfers the curvature perturbations into temperature anisotropies , i.e. the radiation transfer function , and contains all measurement and instrumental effects .  ]since the non - gaussianity parameters depend on a particular inflation model , the calculation of their posterior , ,\ ] ] is done first ( ( * ? ? ?* dorn et al . 2014 ) ) . denotes the information hamiltonian , .a straightforward calculation of this hamiltonian shows that it contains terms up to .therefore the integral of eq .( [ inte ] ) can not be performed analytically . however , this obstacle can be circumvented by conducting a saddle - point approximation in around $ ] up to the second order in to be still able to perform the path - integration analytically .this means , we replace by the gaussian , with including the saddle - point approximation and performing the path integral in eq .( [ inte ] ) yields the final expression of the posterior ( ( * ? ? ?* dorn et al . 2014 ) ) , \mathcal{p}(f_\text{nl},g_\text{nl}).\ ] ] eq . ( [ end ] ) enables to calculate the posterior of fully analytic without expensive monte carlo sampling techniques .this analyticity has been conserved by conducting a saddle - point approximation , whose sufficiency has been validated by the dip test ( ( * ? ? ?* dorn et al . 2013(2 ) ) ) .note that as well as denpends on the two point correlation function of and thus requires some a priori knowledge on the primordial power spectrum .one might use the currently measured and therefore well motivated primordial power spectrum ( pure power law ) with best fit parameters from _ planck _ ( ( * ? ? ?* planck 2013(3 ) ) ) .  to obtain the posterior of inflationary parameters we replace bytheir parameter dependent expressions , predicted by inflation models , e.g. , for the simplest curvaton model with potential  ( ( * ? ? ?* bartolo et al .2002 ) , ( * ? ? ?* sasaki et al .2006 ) ) one obtains the posterior for the curvaton parameter for a toy example in two dimensions as well as possible values of are illustrated in fig .[ figgf ] .   and within current _ planck _ constraints parametrized by the curvaton parameter .( right ) normalized likelihood distributions for in a two - dimensional test case in the sachs - wolfe limit with data generated from .figures taken from ( ( * ? ? ?* dorn et al . 2014)).[figgf],title="fig : " ] and within current _ planck _ constraints parametrized by the curvaton parameter .( right ) normalized likelihood distributions for in a two - dimensional test case in the sachs - wolfe limit with data generated from .figures taken from ( ( * ? ? ?* dorn et al . 2014)).[figgf],title="fig : " ]we presented a novel and generic method to infer inflationary parameters from observations ( cmb , lss ) by local non - gaussianity .the method is fully analytic and thereby avoids expensive sampling techniques .the introduced approximation , necessary to conserve the analyticity , has been validated successfully .2013 , _ apjs _ 208 20 , arxiv : astro - ph/12125225 2013 , arxiv : astro - ph/13035076 2013 , arxiv : astro - ph/13035084 2011 , _ phys . rev .d _ 80 105005 , arxiv : astro - ph/08063474 2013 , _ phys .d _ 88 103516 ,arxiv : astro - ph/13073884 2014 , arxiv : astro - ph/14035067 2013 , _ phys .e _ 88 053303 ,arxiv : astro - ph/13073889

Abstract: the presence of multiple fields during inflation might seed a detectable amount of non - gaussianity in the curvature perturbations , which in turn becomes observable in present data sets like the cosmic microwave background ( cmb ) or the large scale structure ( lss ) . within this proceeding we present a fully analytic method to infer inflationary parameters from observations by exploiting higher - order statistics of the curvature perturbations . to keep this analyticity , and thereby to dispense with numerically expensive sampling techniques , a saddle - point approximation is introduced whose precision has been validated for a numerical toy example . applied to real data , this approach might enable to discriminate among the still viable models of inflation .


sleep is an important part of life , and quality of sleep has a significant impact on individual well - being and performance .this calls for methods to analyze sleep patterns in large populations , preferably without laborious or invasive consequences , as people typically disapprove of the use of intrusive technologies  .large scale studies of human sleep patterns are typically carried out using questionnaires , a method that is known to be unreliable .it is possible to perform more accurate studies , but these are currently carried out within small controlled environments , such as sleep labs . in order to perform accurate measurements of sleep in large populations consisting of thousands of individuals without dramatically increasing costs ,alternative methods are needed .smartphones have become excellent proxies for studies of human behavior , as they are able to automatically log data from built - in sensors ( gps , bluetooth , wifi ) and on usage patterns ( phone calls , sms and screen interaction ) , from which underlying user behavioral patterns can be derived .smartphone data has been used to infer facets of human behavior such as social interactions , communication , mobility , depression and also sleep patterns . either paired with additional sensors or on their own , mobile app solutionsare able sometimes very ingeniously to track individual sleep patterns and visualize them .we cite as examples _ smart alarm clock _ , _ sleep cycle _  , _ sleepbot _  , and _ sleep as android _  .using mobile phone data to derive sleep patterns has thus already been demonstrated and verified , and offers advantages ( i.e. reduced cost ) as an alternative to dedicated sleep monitoring devices . in this paperwe suggest extending previous approaches , using a bayesian model to infer rest and wake periods based on smartphone screen activity information .the advantages of our proposed bayesian approach _ sensiblesleep _ ,as compared to previous work , are that it :  * is less sensitive to `` noisy '' data , for instance infrequent phone usage during sleep interruptions ( such as checking the phone at night ) * is able to quantify not only specific rest and wake times but also characterize their distributions and thus uncertainty * can encode specific prior beliefs , for instance on expected rest periods ( when desirable ) * can capture complex dependencies between model variables , and possibly even detect and relate patterns that are common to a group of people with diverging individual patterns ( when using one of the proposed hierarchical models ) , such as detecting how available daylight may modulate sleep patterns across an otherwise heterogeneous group of users  our method , moreover , only needs screen on / off events and is thus _ non - intrusive _ , _ privacy - preserving _ , and has _lower battery cost _ than microphone or accelerometer based ones .we start by providing an overview of the related work .we then describe the collected data , and introduce the bayesian model .we compare the model results with ground truth obtained by sleep trackers , and show how the model is able to infer the sleep patterns with high accuracy .finally we describe the individual and collective sleep patterns inferred from the data .a key finding by zhang et al . shows a global prevalence of sleep deprivation in a group of students , partly linked to heavy media usage . in this study sleep patternsare largely deduced from the teachers perception or based on individual self - reports , lacking more direct measurements .  corroborating this finding , orzech et al . report that digital media usage before bedtime is common among university students , and negatively impacts sleep .the findings are based on studies involving self - reports through ( online ) sleep diaries and digital media surveys , and also lacks more direct measurements of sleep patterns .additionally , this would make it possible to increase the scale of the experiment and enable the study of larger populations .abdullah et al . have previously demonstrated using 9 subjects how a simple rule - based algorithm is able to infer sleep onset , duration and midpoint based on a ( filtered ) list of screen on - off patterns with the help of previously learned individual corrective terms , and further analyzed behavioral traits of the inferred _ circadian rhythm _the algorithm uses an initial two weeks of data with journal self - reported sleep for learning key corrective terms in order to improve the accuracy and compensate for differences between _ actual _ sleep and _ inferred _ nightly rest period .the method has been verified against a daily online sleep journal and results in differences less than 45 minutes of average sleep duration over the entire analysed period .while our proposed bayesian model , which has been applied to more than 400 users , may be more complex , it increases the robustness and allows us to better quantify the uncertainties of the inferred resting periods as well as offer the possibility of building more advanced models across heterogeneous groups of users .in particular , our model may better be able to handle short midnight interruptions , which appear to be not uncommon , without any additional filtering .in contrast to abdullah et al . using ( only ) screen on -off events , a fine - grained sleep monitoring by `` hearing '' and analyzing breathing through the earphone of a smartphone is suggested by ren et al . . heresix users tested the system over a period of 6 months , demonstrating the feasibility of using smartphones for the purpose of analysing breathing patterns , using a respiration monitor logger as ground truth .sleep estimates are not directly inferred in this paper , however .this technology is also non - invasive , although it does requires capturing and analyzing large samples of audio data ._ isleep _ proposes detecting sleep patterns by means of a decision tree model , also based on audio features .the system was evaluated with 7 users for a total of 51 days , and shows high accuracy in detecting snoring and coughing as well as sleep periods , but report drops in performance due to ambient noise .  increasing the number of features , the _ best effort sleep model _ is based on a linear combination of phone usage , accelerometer , audio , light , and time features using a self - reporting sleep journal , and subsequently achieved a 42 minutes mean error on 8 subjects in a test period of 7 days .other work also tries to estimate sleep quality , for example _ intelligent sleep stage mining service with smartphones _ , which uses conditional random fields on a similar set of features trained on 45 subjects over 2 nights , and reports over 65% accuracy of detection of sleep phases , compared to eeg ground truth on 15 test subjects over 2 nights .  _ candy crushing your sleep _ uses the longest period of phone usage inactivity as heuristic for sleep , with some ad - hoc rules for merging multiple periods , and proceeds to quantify the sleep quality and to identify aspects of daily life that may affect sleep .the inferred sleep period was however not validated against any ground truth .the _ sleep well _framework  deploys a bayesian probabilistic change - point detection , in parallel with an unsupervised classification , of features extracted from accelerometer data , in order to identify fine - grained sleep state transitions .it then uses an active learning process to allow users to incrementally label sleep states , improving accuracy over time .it was evaluated both on existing datasets with clinical ground truth , and on 17 users for 8 - 10 days with user diary data as ground truth , reaching an average sleep stage classification accuracy approaching 79% .  in comparison , even though sleep quality is not estimated , our non - intrusive model only needs screen on / off events and has been tested on a large user - base , and can suitable for very large - scale deployment .we have analyzed two datasets in this work .the first dataset ( a ) was provided by sony mobile , and contains smartphone app launches coupled with sleep tracking data from the swr10 and swr30 fitness tracking armbands  . for each userwe have a set of records containing an anonymized unique user identifier , a timestamp and the unique app package name .note that the model only uses the app launch timestamp and completely ignores the app identifier , therefore no privacy risks related to app names are present .the sleep tracking data indicates when each user is detected asleep or awake with a granularity of one minute , serving as ground truth that we will compare our results against . from this datasetwe select 126 users that have at least 3 hours of tracked sleep per day , and have between 2 and 4 weeks of contiguously tracked sleep .the second dataset ( b ) originates from the _ sensibledtu _ project , which collected smartphone sensor data for more than 800 students at the technical university of denmark . in this datasetwe focus on the screen interaction sensor that records whenever the smartphone screen is turned on or off , either by user interaction or by notifications .each record contains a unique user identifier , a timestamp , and the event type ( on or off ) . from this datasetwe select users in november 2013 that have at least 10 events per day , thus filtering out users with gaps in the collected data or with very sparse data .there is on average screen - on activations pr .user in this period .data collection for the sensibledtu dataset was approved by the danish data protection agency , and informed consent has been obtained for all study all participants .data collection for the sony dataset has been approved by the sony mobile logging board and informed consent has been obtained for all study participants according to the sony mobile application terms of service and the sony mobile privacy policy .the underlying assumptions of the model are ( 1 ) that the user is in one of two modes : being _ awake _ or _ sleeping _ , and ( 2 ) that mobile phone usage differs between the two modes .in particular a user will have many screen interactions when awake , and very few or even no interactions when sleeping .sleeping is here considered as an extended _ resting _ period that typically takes place once every 24 hours at roughly similar times , as governed by the users circadian rhythm and influenced by socio - dynamic structures , during which the owner physically rests and/or sleeps .resting periods , however , might be interrupted by short periods of activity , such as checking the time on the phone or responding to urgent messages .this behavior leads to two different activity levels , which we label and , one for each mode .  if we can deduce when the switchpoint between the two distributions occur during each hour period, we can also infer the time during which the owner is _ resting _ for the night , and thereby also the period within which sleeping takes place .short of using the more invasive eeg or polysomnographic methods , properly differentiating the resting period and actual sleep is difficult ; even sleep diaries may easily contain reporting bias or be somewhat inaccurate . to remove self - reporting bias and to study a larger population we have therefore decided on using a motion - based detector ( sony fitness tracking armbands ) as ground truth .if higher accuracy would be required , applying individual corrective terms ( i.e. average sleep / rest time differences ) learned from an initial period by more accurate means ( polysomnography , external observer or possibly a careful user diary ) might be possible , similar to what as demonstrated by abdullah et al .  .each user is considered independently .we divide time into periods starting at 16:00 and ending at 15:59 on the next calendar day , so that the night period and the expected sleep midpoint is in the middle , for convenience .each day is divided into time bins of size minutes .we count the number of events that start within each time bin , where an event is an app launch for dataset a and a screen - on for dataset b. information about the duration of the events is purposely discarded , as phone usage typically takes place in short bursts .this is supported by the median duration of screen events in dataset b , which is seconds .it is reasonable to assume that the count of events in each time bin follows a poisson distribution :   + with or , depending on the mode of the user .it is , furthermore , assumed that the user mode , and consequently the value for , is determined by two switchpoint variables and , both assuming values from to :    for simplicity , all models assume that is identical for all days of a given user .it can be expected that users have a very low number of screen events during sleep mode , which is encoded in this prior belief :    here represents the exponential distribution :    the rate parameter is set to a very large value to encode our prior belief that almost no events should happen during the sleep time .  fig .[ fig : model_illustration ] shows an illustration of the model idea .  ) and one for awake periods ( rate ) .furthermore we assume that two switchpoints and determine the rate ( i.e. the poisson distribution ) that generates the events . ]we now propose four different models , which differ in the assumptions made on the relation of the rate and sleep / awake time parameters for different days .the simplest model assumes that for a given user there is a single ; i.e. the user has very similar phone interaction patterns each day .also and are each identical for all days , that is : the user goes to sleep , and wakes up , at the same times each day :    here represents a uniform probability to choose a timebin between 0 and .no additional prior knowledge of and is assumed ; there is equal probability of any bin value . in other words ,sleep and awake time are equally probable at any time of the day . the prior for chosen to represent our prior belief of a reasonable rate of events , specifically with both mean and variance = 2.5 ( events / bin ) and a longer tail than a normal distribution .a somewhat more realistic model would assume that each day has independent and times , while still sharing rates .therefore in this model there are and , with , one for the each of the considered days :    the rest of the model remains as above .it may further be assumed that each day could have its own specific activity rate .we modeled this as separate for each of the days , in addition to and for each of the days :     the assumption that each day s interaction rate is completely independent may not be correct .it may not be unreasonable to imagine that the daily rate(s ) arise from an underlying user - specific rate ; i.e. the user may have certain habits that varies from day to day but share some similarities specific to that user .this is modeled by adding and hyperparameters to the gamma priors for :    we do not have strong prior beliefs for and , so we set their prior distributions to generic exponential distribution with rate parameter = 1 , .finally we could assume that each day s sleep and awake times derive from an underlying circadian rhythm that is specific to the user , but still modulated by events that take place during the week .this can be modeled by changing the and priors to a normal distribution , with hyperparameters , and as follows :    the are here chosen to be centered at the bin corresponding to 23:00 , while the are centered at the bin corresponding to 07:00 .also in this case we have no strong prior knowledge of the , and parameters , so we set their prior distribution to a non - informative exponential and gamma respectively .the models are fitted using markov chain monte carlo ( mcmc ) sampling  , where the parameter values are estimated by a random walk in the parameter space guided by the log likelihood .we use the _ pymc3 _ python library for running the sampling , but any mcmc framework could be used to implement our model .the result of the bayesian inference is a trace that captures the most probable values of the parameters , and also gives an indication of the uncertainty of the estimation .it is important to note that the models are unsupervised , which means that they are fitted only to the number of events without having access to the ground truth of the actual sleep patterns .this allows the model to be fit to other datasets where we do not have ground truth of sleep patterns , which is desirable if the sleep inference has to be deployed on a large scale . for dataseta we verify the fit by comparing with the sleep patterns from sleep trackers , while for dataset b we evaluate the fit by inspecting the inferred sleep patterns .  in order to find the model that provides the best overall fit for the intended purpose without introducing too many degrees of freedom , we compare the log posterior from the traces of the models , , and see how they converge .one example of a plot of traces for the five models is shown in fig .[ fig : logp ] , which shows that the hyper - hyper model ( blue ) has the highest ( least negative ) , followed by the independent - hyper model for dataset b. the three other models appear with lower . in of the analyzed cases of dataseta ( for dataset b ) , the hyper - hyper model has the highest score , followed by the independent - hyper model with the highest in ( ) of the cases .traces ( a top , b bottom ) ]  the estimation does not , however , take into account the added complexity of the more advanced models .an attempt to do so is the deviance information criterion ( )  , which penalizes the increased degrees of freedom ( more model parameters ) that usually result in a model that is easier to fit to the data .[ fig : dicrel ] shows the relative dic score ( vs. the simplest model , pooled - pooled ) .the order is identical for both datasets .further , table [ table : dic ] compares the 5 models by ranking the calculated dic for all and users .the _ median _rank shows that the hyper - hyper model is the `` best '' model ; it has a probability of being the best ranked model ( _ p(best ) _ ) in of the cases for dataset a ( for dataset b ) .the independent - hyper model follows as a somewhat distant 2nd best , ranking highest in ( ) of the cases .it should be noted that , in addition to their different abilities to reflect the underlying assumptions and provide varying levels of fit to the actual data , the models also differ in their runtime ; the most complex model typically takes times longer to execute than the simplest . in particular , the hyper - hyper model on average had a runtime that is longer than the independent - hyper model , so there may be cases where the latter would be a better model to use despite the slightly worse dic ranking .all five models have been run on both datasets , producing an estimation of the times of sleep and wake up for each day , as well as estimates for the other hyperparameters , for each user .moreover , we calculated logp and dic as discussed in the previous section .we firstly verify the accuracy our method using the ground truth from the sleep trackers .we then provide a qualitative analysis of some key examples of individual sleep patterns , and a description of the aggregated sleep patterns for both datasets . for the remainder of the paperwe restrict our analysis to the model with the best fit , the hyper - hyper model .   to assess the results , we compare the sleep periods inferred by our model and those inferred by a previously suggested rule - based method to the ground truth collected by the sony sleep trackers .  for each daywe calculate the time of sleep and time of awake inferred by our model as the mean of the and respectively , and we consider the user asleep ( z = 1 ) for all time bins between and , and awake ( z = 0 ) for the remaining bins .  for a representative and comparable method , we chose to implement a rule - based algorithm similar to what is proposed by abdullah et . to derive sleep data for dataset a. this rule - based method essentially works by finding the longest contiguous sleep period , with a prior assumption that sleep must start after 10 pm and before 7 am next morning . note that the original algorithm is based on screen on - off events and furthermore discards events of short duration during the night ; in our case we use app launches with no available duration , and thus can not discard events of short duration .  for the sleep trackers we can directly mark each time bin as sleep ( z = 1 ) if the trackers have detected at least one sleep status in that bin , and awake ( z = 0 ) otherwise .we again consider one user at a time .for each user we now have three binary matrices : two inferred sleep status values per time bin from either model , and one measured sleep status value per time bin ( ground truth ) .we evaluate this as two binary classification problems , and calculate accuracy , precision , recall and f1 for each model and for each user according to the definitions :          fig .[ fig : mcmc_acc ] shows the resulting distribution of accuracy , precision , recall and f1 scores for the proposed method .the sensiblesleep method achieves a mean accuracy of 0.89 , and a mean f1 score of 0.83 .the below - average scores for some users are expected , since it is likely that among the large population under study there will be people having irregular sleep schedule or noisy sleep ground truth .[ fig : model_compare_acc ] shows the corresponding _ complementary cumulative distributions _ of the accuracy , precision , recall and f1 scores of the proposed sensiblesleep model vs that of the rule - based model  .the results are generally comparable between the two models , on this particular dataset .our model has slightly better accuracy and precision whereas the previously suggested rule - based model has a slightly better recall . the f1 scores , which weights precision and recall equally , are comparable .this particular dataset has only very limited sleep interruptions during the night . for populations with more interrupted sleep, we expect our model to maintain a high score .we now analyze individual sleep patterns to show the results of the model in details .for each user we create a visualization of sleep schedules .we call this the _ sleep matrix_. each row represents one day , and each column represents one time bin .the blue color shows the probability that sleep takes place within the interval ; the darker the color the higher the probability .the red dots show activity count per bin ; the larger the radius the more events are registered within that particular bin .this compact representation is able to capture at a glance the sleep patterns of individuals over time .we have created one such sleep matrix for each of the users , which allows us to inspect hundreds of sleep patterns quickly .large individual variability both in sleep schedules ( regular , irregular ) and in phone activity ( low , high , during day or night ) are noticeable .still , in most cases it is evident that the model is able to capture a reasonable sleep period , even if it may have been somewhat interrupted .let us consider the inferred sleep patterns for two example users in fig .[ fig : grid ] .the top user has a pretty regular schedule , waking up around 5:30 except every few days , when he / she wakes up later presumably due to vacation or weekends .notice the light blue sections that indicate how the model is less confident about the probability of sleep due to events that do not follow the usual patterns .the bottom user instead has a much more unstable app usage , therefore the model infers a correspondingly more unstable sleep schedule .the bottom user has also some events in the middle of the night throughout many days ( which is presumably checking the phone at night ) yet the model is still able to correctly infer this being a sleep phase .finally notice how the two users have significantly different intensity of app usage ( the bottom one uses the phone much more than the top one ) , yet this is not a problem since the model learns individual activity rates .     in this sectionwe also quantify the aggregated sleep patterns . from the posterior probability distribution functions ( ) , and , the probability that the user is sleeping can be estimated as follows :    this is equivalent to stating that a user is currently sleeping if he has passed the time of falling asleep but has not yet passed the time waking up .the derived values of sleep - length and mid - sleep time can be calculated directly from the values of and for each sample of the trace , and the posterior density can be estimated for these derived values in a similar way as for the model parameters .[ fig : tpdf ] shows the aggregate posterior probability density functions for and for the users of dataset a over days , and for the users of dataset b over a selected period of days ( just after semester start ) . it may not be entirely meaningful to average the sleep patterns from all users , but it serves to illustrate the distribution of and for a larger population .table [ table : sleeptimes ] summarizes the sleep and wake times .( blue ) and ( green ) ( a top , b bottom ) , showing what the probability is for the specific population to go to sleep or wake up at the specified time . ]  .aggregated sleep and wake times [ cols="<,^,^,^,^ " , ]   across the 30 ( 14 - 28 ) analyzed days for the ( ) users of the study , the distribution of sleep durations are as shown in fig [ fig : sleepdur ] .the model allows us to easily compute such metrics . the mean value is around 8:02 ( 36 m ) for dataset a and 7:20 ( 28 m ) for dataset b. notice how the distributions are not completely similar ; this is likely due to the fact that the larger dataset b captures the sleeping behavior of students as opposed to dataset a that may have a more diverse demographic distribution .[ fig : weekdays ] shows the probability density functions for the and times for all users of dataset b , grouped according to weekday .mondays to thursdays appear quite similar , but friday shows a much wider distribution ; users typically go to bed much later on friday and sleep in on saturday .the distributions start to narrow down saturday and sunday but are more `` week - like '' only from tuesday morning again .( blue ) and ( green ) over weekdays for dataset b ]the main contribution of this work is to show how simple counts of smartphone interactions can be used to infer sleep patterns with reasonably high accuracy .we have demonstrated how the seemingly weak signal of screen events carry significant information of the user status .our method has several advantages :  * the method requires only a smartphone and can therefore be deployed without the need for special equipment or methods , such as fitness or sleep tracking bands , or sleep diaries . * the data collectionis completely automated , as no action is required from the user in setting up the tracking or remembering to log his / her activity .* since the model requires only screen interactions , it is absolutely non - intrusive and privacy - preserving .although in this work we stored the data on a central server for analysis purposes , the data could remain on the phones and the sleep analysis could in principle be run directly on the phones as well . *compared to accelerometer or microphone - based methods , using only screen events is much more battery - efficient .although solutions using screen events have been proposed before  , our model provides a number of key improvements :  * it is more robust to noise such as screen events generated by checking the phone at night . *using a bayesian formulation allows us to provide confidence intervals for the sleep and awake times , instead of point estimates only .* it does not depend on ad - hoc rules , but it is based on a well - defined statistical formulation . * it is fitted and verified on a much larger userbase of over 400 users , and a longer time duration ( between 2 and 4 weeks ) .  demonstrating the feasibility of inferring reasonable sleep patterns from simple event counts opens the way for new exciting research directions .in particular we believe that similar methods can be applied to large datasets of user activity .for example on social network ( such as twitter , facebook , meetup , gowalla ) users leave a trace of their activity in the form of messages , posts , likes , etc .another great example is call detail records , the logging information kept by telecom providers about user calls and sms .these events could be treated again as a proxy for sleep and wake cycles .the main drawback of the proposed method is that it requires that users periodically interact with their phones during their wake time . in line with other recent polls ( see for example ) , we show that in most cases this does happen , as the population of users analyzed here tend to check their phone from the early morning to the late night when awake .different populations , however , such as elderly people less accustomed to smartphone usage , may not show similar usage patterns .there is therefore a need for additional work in order to understand how increased sparsity would affect sleep pattern reconstruction .we have presented a bayesian model to infer sleep patterns from smartphone interactions , which we have applied to two datasets of more than 400 users in total .we have compared the model output with ground truth from sleep trackers , and we have shown how the model is able to recover the sleep state with a mean accuracy of 0.89 and a mean f1 score of 0.83 .furthermore , we have shown how the model is capable of producing very reasonable individual and aggregated sleep patterns .our method represents a cost - effective , non - intrusive and automatic alternative for inferring sleep patterns , and can pave the way for large - scale studies of sleep rhythms .ac is funded in part by the high resolution networks project ( the villum foundation ) , as well as social fabric ( university of copenhagen ) .pb is supported in part by the innovation fund denmark through the project eye tracking for mobile devices . vs and hj received funding from sony mobile .the funders had no role in study design , data collection and analysis , decision to publish , or preparation of the manuscript .choe ek , consolvo s , watson nf , kientz ja .opportunities for computing technologies to support healthy sleep behaviors . in : proceedings of the sigchi conference on human factors in computing systems .new york , ny , usa : acm ; 2011 .p. 30533062 .available from : http://doi.acm.org/10.1145/1978942.1979395 .onnela jp , saramki j , hyvnen j , szab g , lazer d , kaski k , et al .proceedings of the national academy of sciences of the united states of america .2007;104(18):73327336 . doi:10.1073/pnas.0610245104 .chen z , lin m , chen f , lane nd , cardone g , wang r , et al . .in : 7th international conference on pervasive computing technologies for healthcare ( pervasivehealth ) .ieee ; 2013 .p. 145152 .available from : http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6563918 .gu w , yang z , shangguan l , sun w , jin k , liu y. intelligent sleep stage mining service with smartphones . in : proceedings of the 2014 acm international joint conference on pervasive and ubiquitous computing .ubicomp 14 .new york , ny , usa : acm ; 2014 .p. 649660 .available from : http://doi.acm.org/10.1145/2632048.2632084 .jayarajah k , radhakrishnan m , hoi s , misra a. candy crushing your sleep . in : adjunct proceedings of the 2015 acm international joint conference on pervasive and ubiquitous computing and proceedings of the 2015 acm international symposium on wearable computers .ubicomp / iswc15 adjunct .new york , ny , usa : acm ; 2015 .p. 753762 .available from : http://doi.acm.org/10.1145/2800835.2804393 .hossain hms , roy n , khan maah .sleep well : a sound sleep monitoring framework for community scaling . in : proceedings of the 2015 16th ieee international conference on mobile data management - volume 01 .washington , dc , usa : ieee computer society ; 2015 .available from : http://dx.doi.org/10.1109/mdm.2015.42 .

Abstract: we propose a bayesian model for extracting sleep patterns from smartphone events . our method is able to identify individuals daily sleep periods and their evolution over time , and provides an estimation of the probability of sleep and wake transitions . the model is fitted to more than participants from two different datasets , and we verify the results against ground truth from dedicated armband sleep trackers . we show that the model is able to produce reliable sleep estimates with an accuracy of 0.89 , both at the individual and at the collective level . moreover the bayesian model is able to quantify uncertainty and encode prior knowledge about sleep patterns . compared with existing smartphone - based systems , our method requires only screen on / off events , and is therefore much less intrusive in terms of privacy and more battery - efficient .