pose estimation from still images has wide applications in image and video indexing , video surveillance and human computer interaction .for example , online solutions of this problem can be applied for single frame initialization in tracking human poses .yet pose estimation from still images , that is , 2d body part localization is a difficult problem , due to the fact that human body is highly flexible resulting human poses with high degrees of freedom even in 2d images .a state - of - art and currently widely used solution for 2d body part detection is the mixture - of - parts ( mop ) method  , in which a human body is modeled as a tree structure and body parts are encoded as nodes in the tree .maximum responses from detection are passed from the leaf nodes to the root .one problem with this solution is the double - counting problem , that is , one detected body part is counted twice for both sides of the human body . in this paper , we tackle the double - counting problem in mop model with multiple feature inputs .additional input features are incorporated so we are able to verify body part localization from more feature responses .compared with the greedy solution in the original mop method , we are able to solve double - counting problem with a global optimization .with detected 2d body part locations from the proposed method , we are able to predict 3d poses by feeding 2d body part detections to a 2d to 3d pose estimator . for 2d to 3d pose estimator , we choose gaussian process regression , which has been proved to be effective in modeling non - linear regression problems .we further validate the whole pipeline on a public available dataset for pose estimation : humaneva dataset .we visualized two types of results : enhanced 2d body part detections and 3d poses estimated from enhanced 2d body part detections .figure [ fig : pipelineoverall ] shows main steps as a pipeline for the whole algorithm .the pipeline includes several major steps : feature extractions ( mop and background subtraction , in our case ) , global optimization , and 2d to 3d pose estimation . from the input ,the original mixtures of part model is trained and applied to detect body part positions . meanwhile , human blobs extracted with background subtractions are used as another cue in our method . then in the third step , this two cues are combined with the proposed algorithm .the generalized framework in our algorithm are able to incorporate multiple features other than human blobs .we extract multiple cues from input images and combine them under the proposed framework . with augmented inputs from multiple features ,we are able improve 2d body part localization and solve double counting problem . as in mop method ,a human body is modeled by a tree structure where kinematic constraints between connecting parts are kept .first , feature models of different features are trained separately and the optimization target is modified to reach a global optimization target by incorporating multiple feature cues .the inference of the optimal pose in a test image is also carried out with multiple cues .the bottom up message passing procedure combines multiple feature cuesso as to reduce false positive body part detections and the top down back tracing procedure are optimized globally so as to tackle double counting problem .the contributions of this papers are as followings : by combing multiple cues , we boost 2d body part localizations under a general framework ; enhanced 2d body part detectors are validated on a public available dataset : humaneva dataset ; 3d pose estimation is shown as an examplar application of detected 2d poses and this application is also validated on humaneva dataset .the rest of the paper is organized as following : in section [ sec : rw ] , we introduce related works on 2d pose estimation and related works on solving double counting problems ; in section [ sec : themethod ] , we introduce details of the proposed method including training models from different features and the global optimization taget ; section [ sec : results ] shows boosted 2d body part localizations by combining multiple feature cues on a standard public available dataset and the examplar application of 3d pose estimation from localized 2d body parts ; in section [ sec : conclusions ] we conclude the work and discuss about possible future works .as mentioned in the previous section , human bodies are highly flexible , thus results in a huge amount of possible guesses in the solution space .but human body joints are not completely under no constraints .models like pictorial structure model  and tree models  are exploited and successfully applied to represent human body models in 2d .these models keep kinematic constraints between connecting body parts .that is , the body parts that are connected physically are also connected in the tree structure . using tree structures has the advantage of tractable inference of the optimal pose .however , spatial constraints between body parts without direct connections are not incorporated in the tree structure .due to this reason , the original tree structure can not deal with occlusion and has the problem of double counting , where an image evidence is counted twice for different body parts .one important example of tree models is mixture of parts model  .mop defines body parts as an area surrounding body joints and has the advantage in dealing with the foreshortening problem of body limbs caused by viewpoint changes . while the traditional pictorial structure ( ps ) model  , which defines body parts as body limbs , needs to deal with foreshortening problem by explicitly training on body limbs of different lengths .also in mop model , the orientation of a limb is naturally represented by the connection of detected body joints . while in a traditional ps model ,limb orientations need to be learned and detected explicitly .so we choose mop model as the human body model .a body part in the mop model is represented as a mixture of several templates , each of which is trained with one subset of samples of this body part . in this way , the trained body part is able to deal with different limb layouts from different poses .  as mentioned in the first paragraph , although the tree structured human model is efficient in training and referencing , it has the double counting problem due to occlusions and lack of constraints between body parts denoted as a node in the tree structure . to deal with these problems ,authors in  propose multiple tree models .the models contain a tree structure to account for kinematic constraints between connected body parts , tree structures for spatial constraints among body parts without direct connections , and tree structures for occluded body parts .different tree structures are combined with a boosting procedure .other research also explore the possibility of imposing constraints in the optimization target .for example , authors in  modify the optimization target and incorporate spatial constraints to deal with double counting problem . in referencing , those poses who violate the spatial constraintswill get a comparatively lower score .given training images with only one human in each image , we train 2d body part detectors with image patches cropped within bounding boxes surrounding the body parts . for a test image , we localize 2d body part positions with trained detector and optimize the detection with multiple feature cues .we name the detector enhanced mop model since it is based on the mop model proposed in  . in the following subsection , we are going to split the method into modules and explain in details .the idea of mixture of parts detector in  is to represent a body part with a mixture of several ( or ) templates , each represent a different appearance of the corresponding body part .so the body part which has more variances in appearance , for example , elbows and knees , are apt to contain more templates . after cropping the image surrounding the bounding box with a proper size ,all samples of a body parts are clustered into several clusters , whose total number is predefined according to the variance of the body part .training templates are formulated as optimizing parameters in a support vector machine , which is carried out with eq optimization .note that the size of the bounding box is a crucial factor in adapting the method to custom data .considering the different notation in each dataset , body joints might correspond to different position and if the size of the bounding box is defined too big , it might contain information from other body part and if the size is too small , it might be lack of information for identifying the body part or joint .after training templates for each body part , given a test image , we compute the response of the image with respect to all trained templates by convolution. then a distance transform  is performed so that the maximum response of the image to the test template is highlighted .later on , we start from the leaves of the human tree structure ( rooted at the head ) , and pass maximum responses of all mixtures from the child body part to its parent .thus , when we come to the root node , all the body part nodes contribute by passing messages .the score of the root is considered the final score of the human detection .this tree structure is very effective in referencing but it has problem dealing with double counting problem . in the following subsections , we are going to explain how we are going to enhance the algorithm based on the original mop model .   instead of imposing spatial constraints or modifying tree structure model, we explore the possibility of combining multiple cues from input images .we argue that multiple feature cues provides richer information so that effectively combining multiple cues reduce false positives and ease double counting problem . in our experiments , we consider histogram of gradient ( hog )  and human blobs extracted from background subtraction  .let us write for an image , for the pixel location of part and for the mixture component of part .we write , and .we call the `` type '' of part . for notational convenience ,we define the lack of subscript to indicate a set spanned by that subscript ( e.g. , ) . the kinematic constraints of human body between connected body parts are modeled as following :    the parameter favors particular type assignments for part , while the pairwise parameter favors particular co - occurrences of part types .we write for a k - node relational graph whose edges specify which pairs of parts are constrained to have consistent relations .we can now write the full score associated with a configuration of part types and positions :    where is a hog vector extracted from pixel location in image . ^t$ ] , where and , the relative location of part with respect to .  until now , this is the original mop model from  . sincemultiple features are extracted separately , we can train models from each candidate features separately . for example , when we use extracted human blobs as another feature cue .we get the human blob model from background subtraction as following :    where is the background model and can be updated with new added frames in the following way ,    and is the learning rate .we denote this model as human blob ( hb ) model .  after training mop model and hb model separately , given a test image , we need to find the optimal human pose with respect to certain criterion .this criterion should take into account both of the trained models . since we suppose each image features are extracted separately .we can get the joint probability of matching two models as :    where represents the mop model and represent the hb model .this probabilities formulation can be easily extended to other image feature cues , given the definition of the model probabilities and conditional probabilities .  in our method, we consider hog and detected human blobs .we define for each pixel as following :    since the hog feature and the human blob feature are extracted separately and thereafter mop model and hb model are trained separately , equals which is defined in equation ( [ eq : pmop ] ) . in implementation, we calculate the probability of a certain pixel belonging to a certain body part by convoluting with image evidence of this pixel with trained body part template .   after training mop model and hb model separately from hog and human blob features .we can detect human pose from an unseen image by find the optimal human pose . in  , with all trained mixtures of parts models , the test image is convoluted with each trained templates .then starting from the leaves of the tree structure , responses of all body parts are passed to their parent parts .after one pass , all the body parts contribute their score to the root part of the tree structure .  in our combined model , before passing the score from the child node to its parent node, we check if this pixel also confirms with the evidence from human blob detection .if the current pixel belongs to the detected human blob , we keep the current score , otherwise the score is set to a very small value .this procedure guarantee that the final probability is the joint probability of two candidate feature models .the advantage of this procedure is obvious , we can remove some false positives by verifying that the current pixels confirm with both models trained from different image feature cues .so the detected root position is more accurate . after we find the root position of the human , we go through the whole tree to fix each body parts with global optimization .   from the detected root position of the tree structure ,authors in  employ a backtracking algorithm to fix all the body part positions .it starts from the root of the tree structure and fix its child node by picking the maximum response from all the child nodes .this method causes the double counting problem .since each body part is fixed only considering the response of the test image with the trained templates , when sibling body parts ( the same body part , but on different sides of the human body , like a left hand and a right hand ) resemble each other , the same image patch might be picked repetitively . in this case , the estimated pose is occluded while in the real case it is not .  to solve this problem , we use global optimization to fix each body part positions . in the original mop model , where there is only hog feature , optimizing body part position is very time consuming .for example , if the model uses body parts , and each body parts use or mixtures , the minimum number of possible combination for all body parts is .this is a huge amount of possible guesses . in our case , where we consider human blob as another feature cues .the possible number of mixtures for each body parts is great reduced due to the constraint .so we can optimize the tree structure globally .the body part positions are optimized to maximize the score of the proposed model for combining multiple cues :    where is the ratio of overlap between the body part and the foreground model define in equation([eq : foregroundhb ] ) .the gaussian process regressor is one of the most widely used regression model for learning 2d to 3d mapping in the pose estimation since it has been proved to be an effective approach for the nonlinear 2d to 3d pose mapping problem  .gaussian process regression ( gpr ) is considered a model - free framework .gpr defined as a distribution over functions , extends statistics from data points to functions . with kernel trick, we can even get rid of the function definition , and only concentrate on kernel matrix instead . once we normalize the training input to have a zero mean , we only need to define a covariance matrix ,that is , the kernel matrix , for gpr . frequently used covariance matrices include squared exponential covariance function , matrn covariance function and so on . in the following subsections , we will explain detailed representations and settings for the gaussian process regressor used here .   according to  ,gaussian process is defined as : _ a collection of random __ variables , any finite number of which have ( consistent ) joint _ _gaussian distribution_. a gaussian process is completely specified by its mean function and a covariance function . integrating with our problem , we denote the mean function as and the covariance function as , so a gaussian process is represented as :    where   , \nonumber\\ k(\mathbf{s},\mathbf{s}^\prime ) & = & e[(\mathbf{\zeta}(\mathbf{s } ) -  m(\mathbf{s } ) )  ( \mathbf{\zeta}(\mathbf{s}^\prime ) -m(\mathbf{s}^\prime))],\label{eq : mk_gp}\end{aligned}\ ] ]   we assume prediction noise as a gaussian distribution and formulate finding the optimal hyperparameters as an optimization problem .we seek the optimal solution of hyperparameters by maximizing the log marginal likelihood ( see  for details ) :    where is the calculated covariance matrix of the target vector .  with the optimal hyperparameters ,the prediction distribution is represented as :  ^{-1}\varpsi^{\prime } ,  \nonumber \\k(s^{\ast},s^{\ast})+\sigma_{noise}^{2}-\mathbf{k}(s^{\ast},\mathbf{s})^{t}[k  + \sigma_{noise}^{2}i]^{-1}\mathbf{k}(s^{\ast},\mathbf{s } ) ) , \label{eq : gprreference}\end{aligned}\ ] ]  where is the calculated covariance matrix from training 2d image features and is the covariance of gaussian noise .equation[eq : gprreference ] for referencing test data is deducted from marginal and conditional properties of gaussian distributions .the following is the marginal property of gaussian distributions : the marginal of a joint gaussian is again a gaussian , that is ,  ,\left [ \begin{array}{cc } a & b  \\ b^{t } & c  \end{array } \right])\quad  \nonumber\\ & \rightarrow & \quad  p(\mathbf{x } ) = \mathcal{n}(\mathbf{a},a ) .\end{aligned}\ ] ]  and the conditional property of gaussian distributions are : the conditionals of a joint gaussian are again gaussian , that is ,  ,\left [ \begin{array}{cc } a & b  \\ b^{t } & c  \end{array } \right])\quad  \nonumber\\ \rightarrow \quad p(\mathbf{x}|\mathbf{y } ) & = & \mathcal{n}(\mathbf{a}+bc^{-1}(\mathbf{y}-\mathbf{b } ) , a - bc^{-1}b^{t } ) .\end{aligned}\ ] ]  thus we are able to predict the distribution of given the distribution .  in most cases ,we assume that gaussian process priors have zero means , that is ,    this leads to a gaussian process posterior    where  ^{-1}\mathbf{y}.\ ] ]  with this posterior , we only need to define covariance matrices , known as kernel in machine learning community .the most frequently used covariance matrices ( kernels ) include : squared exponential ( se ) , rational quadratic ( rq ) , matrn and periodic , smooth covariance functions .the function of covariance function is to define the distance measure in a newly transformed space where the original data samples have one to one correspondences with their mapped points and due to the transformation , data samples of different attribute classes in the new spaces are easier to classify or identify . with the kernel trick , we can get rid of directly defining the mapping model and only define the kernel matrix , the covariance matrix here .gaussian processes yield a method for specifying a probability distribution over functions by specifying a mean and a covariance function for the function values . by training a gaussian process with sample data the variance of the gaussian process becomes small for function values at supporting points included in the training data , which corresponds to an increased certainty about the function values at these points , while at other points the variance of the gaussian process remains high which corresponds to a high uncertainty about the function values at such points .  in our algorithm , we select the most commonly used covariance matrix : squared exponential covariance matrix . given a 2d pose estimate which is represented as the 26 dimensional vector ( , where is the number of body joints in mop and is the dimension size ) , we train one gaussian process to predict each of the 60 dimensions of the 3d pose vector ( , where is the number of body joints in humaneva motion capture data and is the dimension size ) separately . then given features from test samples, we predict 3d poses with trained gpr .to demonstrate the effectiveness of the proposed method , we first show improved 2d body part localization results with the proposed feature fusion method ; then , the trained 2d to 3d pose estimator is carried out on detected 2d body part locations and 3d poses are estimated and shown.   from humaneva - i data set , we select two different actions ( `` walking '' and `` box '' ) performed by three different actors ( `` s1 '' , `` s2 '' and `` s3 '' ) .all three performers perform the actions within a fixed area ( confined with a carpet ) .`` walking '' is performed in a cyclic way , while in `` box '' , performers are moving in a very small area positions notwithstanding different performing style .  as a result, we have six different experiments in total . training and testare carried out within each experiment .that is , we train a detector on a single experiment setting and validate the trained models on the test frames of same experiment setting .this experiment setting is designed to compare the influence of different action type and different performer to the body part localization and pose estimation results .the detailed splits between training and test is shown in the following table .   & walking & & & + & walking & & & + & walking & & & + & box & & & + & box & & & + & box & & & +  for each action performed by a specific actor , training data are composed of consecutive frames , which is close to the number of frames in a cyclic walking sequence .test data are sampled with a equal step among the whole motion sequence excluding the training frames , so that the test poses covers all possible poses for an action .the proposed 2d part localization method aims to solve double counting problem , that is , a pixel location ( a body part location in our case ) might be designated to two body limb positions even there is no occlusions between these body limbs .the reason for the double counting problem in mop detection is :  1 .the responses of a pixel location ( or a body part observation ) to all trained body part templates are calculated separately , 2 .then , from leaf nodes to root nodes , a best response is selected for a each node among all candidate mixtures and this response is passed as a message to its parent node , that is , a locally optimal solution .the limitation of this solution is that usually , the chained body part position calculated from local optimums are not a global optimum and the essence of the solution gives no globally target .  cccc mop detection & our detection & mop detection & our detection +  +  &  &  &  +  &  &  &  +  &  &  &  +  &  &  &  +  &  &  &  +  &  &  &  +  +  in our solution , we introduce another feature cue ( extracted human blobs in current experiments ) . in this way , not only the localizationcould be verified by two features , but also we are able to optimize the target in a global way .this is due to the newly introduced feature cue gives global description .the following are main module that are incorporated :  * we keep response scores from all mixtures of the current body part for later use . instead of selecting the mixture with the maximum response in , we select pre - defined set of body part candidates whose overlap with another feature are over a certain ratio and put them in a candidate list , and calculate the best configuration whose overlap of two features are the maximum .the reason that we select a _pre - defined _ set of body part candidates is that if we consider all body limbs , the calculation number might be too much ( in our method , where we use fourteen body parts , with five or six mixture for each body part ) and redundant because not all body limbs are possible to have double counting problem .the body parts that are possible to have double - counting problems are : two elbows ( left and right ) , two hands , two knees and two legs .so we _ pre - define _ a set of body parts that could be optimized .* we add a mixture - selection module , where any mixture that has a overlap ratio of over a certain threshold ( in our experiment , due to the noisy extracted silhouettes ) are considered as a candidate mixture that can pass messages to it parent .* we optimize the position among all kept candidate positions for a body part by fixing all other body part positions . if the overlap of two feature cues at a pixel location is within a interval ( in our experiment ) , for all pair of body part that might encounter double counting problem ( that is , two elbows , including the left elbow and the right elbow , two hands , two knees and two legs ) , we check if they overlap . if they do , there is a possibility that they are double - counted , then they are added to the candidate list for local optimization .note that , in our experiments parameters are set by experience .it is also straightforward to acquire them from training data .for example , we can calculate all overlap ratios between bounding boxes of training body parts and extracted human blobs , fit gaussian distribution , and take the mean of the fitted gaussian as .body part localization results are shown in figure [ fig : qualitative_results ] . from the figure, we can see that double counted body parts are correctly localized after optimization .   [ cols="^ " , ]   to have a qualitative comparison , we show in figure [ fig : leftelbowwalkings1 ] 3d joint positions of the ground truth pose and the estimated 3d pose .both figures shows values from the first dimension of 3d joints .the figure in the first row is from the left elbow of the actor `` s2 '' performing `` walking '' and the figure in the second row is from the left hand of the actor `` s2 '' performing `` walking '' .one direct application of the proposed method is for controlling 3d poses of avatar .motion capture systems usually requires invasive body markers . while in our pipeline , performers are able to get rid of invasive body markers once training 3d poses are attained .what s more , we only need image sequences from one single view point .in this paper , we design an algorithm to enhance the performance of 2d body part localization based on mixture of parts models which recently achieved good performances in 2d body part localization .later on , we take the estimated poses as an input to estimate 3d poses . we validate our method in two ways : 2d body part localization visualized results and 3d pose estimation errors .one interesting further work is to incorporate physical constraints into 3d human model so we can optimize 2d body parts accordingly .we are also interested into validate this method on other public data set , like youtube data set where 3d pose ground truth are not provided .1 y. yang and d. ramanan , _ articulated pose estimation with flexible mixtures - of - parts _ , in proceedings of computer vision and pattern recognition , 2011 , pp .1385 - 1392 .p. f. felzenszwalb and d. p. huttenlocher , _ pictorial structures for object recognition _ , int .j. comput . vision , vol .1 , pp . 55 - 79 ,y. xiao , h. lu , and s. li , _ posterior constraints for double - counting problem in clustered pose estimation _ , in ieee international conference on image processing , 2012 , pp . 5 - 8 .y. wang and g. mori , _ multiple tree models for occlusion and spatial constraints in human pose estimation _ , in proceedings of european conference on computer vision , 2008 , pp .710 - 724 .p. f. felzenszwalb and d. p. huttenlocher , _ efficient graph - based image segmentation _ , int .j. comput .vision , vol .167 - 181 , sep 2004 .n. dalal and b. triggs ._ histograms of oriented gradi- ents for human detection_. in cvpr , pages 886893 , 2005 .a. amato , m. mozerov , a. bagdanov , and j. gonzlez , _ accurate moving cast shadow suppression based on local color constancy detection _ , ieee transactions on image processing , vol .2954 - 2966 , 2011 .gregor gregori and gordon lightbody ._ gaussian process approach for modelling of nonlinear systems_. engineering applications of artificial intelligence , 22(4 - 5):522533 , 2009 .b . sofiane and a. bermak . _ gaussian process for nonstationary time series prediction_. computational statistics and data analysis , 47(4):705712 , 2004 . j. m. wang , d. j. fleet , and a. hertzmann ._ gaussian process dynamical models for human motion_. pami , 30(2):283298 , 2008 .rasmussen , c.k.i . williams ._ gaussian processes for machine learning_. mit press , 2006 .

Abstract: mixture of parts model has been successfully applied to 2d human pose estimation problem either as explicitly trained body part model or as latent variables for the whole human body model . mixture of parts model usually utilize tree structure for representing relations between body parts . tree structures facilitate training and referencing of the model but could not deal with double counting problems , which hinder its applications in 3d pose estimation . while most of work targeted to solve these problems tend to modify the tree models or the optimization target . we incorporate other cues from input features . for example , in surveillance environments , human silhouettes can be extracted relative easily although not flawlessly . in this condition , we can combine extracted human blobs with histogram of gradient feature , which is commonly used in mixture of parts model for training body part templates . the method can be easily extend to other candidate features under our generalized framework . we show 2d body part detection results on a public available dataset : humaneva dataset . furthermore , a 2d to 3d pose estimator is trained with gaussian process regression model and 2d body part detections from the proposed method is fed to the estimator , thus 3d poses are predictable given new 2d body part detections . we also show results of 3d pose estimation on humaneva dataset .   gong : an effective solution to double counting problem in human pose estimation  pose estimation , double counting problem , mixture of parts model


in astronomy today , the standard technique for acquiring deep astronomical images is through a series of short , often dithered , exposures .applying this procedure enables the removal of cosmic rays , the filtering of bad pixels , and the masking of device defects .it also increases the dynamic range of the resulting image ( as bright sources in single , long exposures saturate ) and provides better control of the underlying point spread function ( psf ) as long as the psf is fully sampled .  to achieve the depth of a long exposure from a series of exposuresrequires that we combine the individual images , accounting for the variation in seeing , sky transparency , and background variability .this process is typically referred to as `` image coaddition '' and is at the heart of many image processing systems .the criteria that we optimize in the image coaddition depends on the science in question at hand .for example , in faint galaxy surveys ( e.g. the hubble ultra deep field , ) photometric accuracy might be paramount , but for weak lensing surveys ( e.g. the canada france hawaii telescope legacy survey , ) , the preservation of the underlying image resolution may be the primary concern . given these differing objectives two open questions remain ; how do we optimize the coaddition of astronomical images and is there is an optimal criterion that is relevant to all scientific goals ?  with a new generation of deep imaging surveys coming on - line throughout this decade , including the panoramic survey telescope & rapid response system ( pan - starrs ) , the dark energy survey ( des ) , and the large synoptic sky survey ( lsst ) , each producing petabytes of data , image coaddition can not just be assed by the statistical analysis ; it is also a question of the computational efficiency of the different approaches . in this paper , we compare several different approaches for image coaddition , taking into account their statistical properties and their computational cost .we consider in detail the fourier - domain methods proposed in along with several straightforward methods , such as running means or medians .we compare these methods using two performance criteria : flux conservation ( i.e. , pixelwise photometric accuracy ) and image resolution ( using a metric we call image quality ) . using these comparisons we can make statements about whether the additional complexity outweighs the extra costs they incur .our goal is to provide a framework for assessing the magnitude of the practical advantages more complex methods might provide .  in section 2, we set up the statistical model underlying our analysis and outline the properties of the various techniques . in section 3 ,we describe the simulations and define the performance criteria , and , in section 4 , we present the results of our simulations .consider a single image of a particular patch of sky .the object of interest is the true scene , which we represent as a function that takes a two - dimensional argument corresponding to position in the image and returns an intensity . for ground - based viewing ,however , we observe only a blurred , discretized , and noisy version of .we represent the blurring ( caused by atmospheric seeing and instrument artifacts ) by an operator . in this case , is associated with a psf such that the action of on is to integrate against .  now , suppose instead of getting one image we get a stack of images , all of the same patch of sky with science .we assume for this discussion that the images have been registered . then , due to atmospheric conditions changing over time , for each image is a possibly different operator with associated psf .then , if we define a regular grid of pixel centers , the observation on the image at the pixel is ) for points in the sky / image .note that integrals over such two dimensional arguments are actually double integrals .] where the s are noise random variables with suitable distributions .note that equation ( [ eq : fredholm ] ) also defines the operator as integrating against the psf .the statistical problem is to estimate given observation of .what makes the problem challenging is that the operators are _ ill conditioned _ , meaning that large changes in can make relatively small changes in , making recovery of from unstable .this is so because blurring dampens high frequency structure , so the singular values of this integral operator decay quickly to zero .thus , even small levels of noise can critically obscure structure in .  to see this in a simple example ,assume that the psf does not vary in shape across the image , which makes a convolution kernel .it follows that by using the fourier transform , we can recover exactly in the absence of noise by because .while the inverse exists in the absence of noise , the decay of at high frequencies can produce catastrophically high variances in the presence of noise , which produces nontrivial and random contribution at high frequencies ( so the numerator is non - zero but the denominator goes to zero ) .figure [ fig : fourierinvoperator ] demonstrates this effect in a 1-dimensional analog of the problem , comparing noise free reconstruction via equation ( [ eq : fourierinvoperator ] ) with reconstruction under noise with a 10e9:1 signal - to - noise ratio ( snr ) .the estimate in the latter case is wholly untenable .+   we assume that each observation is drawn from a distribution with mean because the observation process in the telescope involves counting photons , the distribution of is well modeled by a poisson distribution .thus , we assume that the s are independent poisson random variables .if the mean counts are large enough , the gaussian approximation to the poisson distribution is accurate , and we can model the s as independent normal random variables .  the statistical problem is to form an estimator of given observations . in the present context , is the template or coadd we are attempting to create . for each need ways to measure its misfit relative to the true scene . in this paperwe consider two .the first criteria corresponds to flux conservation , which we measure through mean integrated squared error ( mise ) , and the second criteria we refer to as image quality .we give a brief description of these here .see section [ sec : methods ] for a more thorough discussion .flux conservation gives an indication about how well a method maintains the spatial location of a given amount flux .mean integrated squared error ( mise ) , which corresponds to the expected value of the integrated squared difference between any recovered image and the true image is the measure of flux conservation we use . specifically , suppose we form a template .then see for a thorough overview of mise .  while mise measures the degree to which maintains flux in the correct location , it does nt distinguish between different scatterings of the remaining misspecified flux .we define a second comparison , which we call image quality , that is designed to measure how spread out the source is , as this is nt captured by mise . for image quality, we do a very rough psf estimation on a source in the coadded image .the relative size of these psfs give us an indication about how spread out an object gets by the coaddition .[ fig : misesharpnesscomparison ] for an example of proceedures with combinations of good and bad mise and image quality .+  one significant complication in practice is that the psfs are unknown and must be estimated as part of the procedure . while it is ideal to simultaneously estimate the psf and , it is common , and typically effective , to first estimate the psf and then use the estimate as a known psf in the coaddition procedure . because our focus is on the relative performance of techniques for estimating , we will follow standard practices and assume that the psfs are known .this produces a comparison of the techniques at their best but does not account for how well the procedures tolerate mis - estimation of the kernels .note that the straightforward comparison methods do not require psf estimates .so , any method that does require knowledge of the psf would need to be substantially better , given it has access to much more information .in this section , we describe a variety of methods that have been applied to the coaddition problem , with particular focus on the fourier domain approach we are analyzing .note that we restrict our attention to non - iterative methods ._ lucky imaging_. in lucky imaging , a large number of images are observed and only the best, according to some criterion such as seeing , are retained . and describe such implementations in detail .an advantage of this approach is that the reconstruction of the true scene is based entirely on high quality data .a disadvantage is that the method requires storing many images to determine which are best .moreover , the images that are discarded can contain useful information about the scene that is , in effect , wasted ._ pixelwise statistics_. if a stack of images is aligned , the values at a particular pixel in each image represent a random sample from a common distribution .the aligned pixel stacks can thus be used to estimate the parameters of these distributions and in turn the true scene .basic choices would be a mean , median , or perhaps a trimmed mean to account for heavy - tailed noise , but more sophisticated estimators tailored to particular distributional assumptions can be constructed .the advantages of these approaches are computational and conceptual simplicity .moreover , the mean can be computed sequentially with a fixed amount of storage , although this is not true of the median or trimmed mean , for which the entire image stack needs to be maintained .a big disadvantage is that pixelwise methods tend to create rough , discontinuous images as no information sharing is permitted between pixels ._ fourier deconvolution_. in spatially constant seeing case is that equation ( [ eq : fredholm ] ) has a convolutional structure .the advantage of fourier based methods is that the operator decomposes nicely in the fourier domain .an inherent disadvantage to the fourier approach is that great care must be used to avoid outcomes such as fig .[ fig : fourierinvoperator ] .the approach outlined in is to fourier transform each image and estimate the fourier coefficient of by a weighted average of the fourier coefficient of each image .the weighting is accomplished so that the images with better seeing are weighted more heavily in the average .this method has some associated optimality properties .however , in practice , these properties turn out to not be useful .see below for a more thorough description .note that , as presented , this method makes three nontrivial assumptions .first , it assumes that each , that is , spatially constant seeing . as we describe in section[ sec : results ] , while this assumption could have significant consequences when seeing varies spatially , in practice , it does not appear to cause much problem .the second assumption is that the s are large enough that the gaussian approximation to the poisson is accurate across the image .the third assumption is that the variance of the gaussian is a _ known _value that depends only on the image .these last two assumptions make the analysis easier , and while they are often reasonable , they need not hold in practice .to define the fourier deconvolution estimator , first expand into the fourier basis , which we write as , in general , we use the notation for the fourier transform of the function .  using the above assumptions , and the resulting form of the s, proposes two estimators of .see table 1 for these estimators and some of their cumulants .the first estimator in the table , , has the smallest variance of all unbiased estimators of . also , the resulting estimator of , defined to be is the best linear unbiased estimator of .  . , which is the result of a variance stabilizing transformation .[ cols="<,^,<,<,^ , < " , ]   however , as alluded to previously , these optimality properties are not useful . to see this ,observe that the variance of is the sum of the variances of .also , since the are smoothing operators , is small for large .in fact , the worse the seeing , the smaller becomes and can be smaller than machine error in many cases .hence , as  ]   provides a correction that prevents the variance from exploding for large by multiplying by a term that is proportional to the reciprocal of its standard deviation .see the second row of table 1 .this multiplication biases the estimator but bounds the variance .we define and we spend the balance of the paper investigating its properties .we refer to as the fourier deconvolution ( fd ) method . as an aside ,since the correction corresponds to multiplication in fourier space , we see that the bias of is given by , where the fourier transform of is defined in the caption of table 1 and is the convolution of and .the computational complexity of the fd method is dominated by the need to fourier transform the data and kernel .if we suppose the images are by and , then we can perform a fourier transform in flops via the fast fourier transform .this is the dominating computation in the fd method .however , it also has several order computations .the mean method requires flops and the median method requires comparisons to update for each new image .  the fd method must maintain two pixel images and the mean method requires one pixel image , each of which get updated after each new image and kernel is recorded .the median method requires the entire stack must be maintained at all times , though not necessary in ram .hence , after images have been observed , all images must be kept for possible updating of the median .for our evaluation metrics , we need to know the true sky before any distortions . hence , we simulate an idealized view of the sky ( above the atmosphere ) using a catalog of point and extend sources .extended sources are represented by sersic profiles and the density of point and extended sources is designed to match observations to a depth of r .figure [ fig : simulation ] shows a representation of one such image . from these true scenes , we apply a blurring operator and noise with variance to represent the effect of the atmosphere , telescope , and instrument .  under the assumptions of the fd method , we assume the psfs are convolutional ( by that we mean spatially constant seeing ) , but this leaves open the specific functional form for the kernel .we choose to set to be the two dimensional gaussian probability density function .our results are , however , robust against more complicated parametrization of ( e.g. , a mixture of gaussians ) .    for evaluating the effectiveness of the fd method , we compare it to the pixel - wise mean and pixel - wise median .our goal is to assess the value added by the fd method and evaluate whether the improved performance overcomes the added cost and complexity of the method .the mean and median procedures are intended as extreme comparisons rather than serious proposals and we are nt endorsing their use in this paper .however , if the value added of more sophisticated methods relative to even these simplistic methods is not great , then it suggests that simple methods may suffice in practice .note that the computational complexity of the methods is not a purely academic concern .in large imaging surveys such as the lsst , a near constant data stream requires efficient , near real - time processing , and the cost of generating template images is an important factor .  for this comparison, we use the two general criteria introduced in section [ sec : model ] corresponding to mise and image quality . in principle, at least two different errors can be made when comparing an estimated template to the true scene .these errors can most easily be thought of in the context of a point source such as a star .our estimated image can remove some of the flux from the point source and/or it can smooth the image ( degrading the photometric accuracy and the resolution of the resulting template ) .figure [ fig : misesharpnesscomparison ] shows examples of a pixelated point source that has been reconstructed by four made - up methods .  in the mise comparisonwe compute the expected value of the integrated squared difference between the true image and our estimate , assuming zero background . under the non - gaussian distributions , a larger variance results in more total flux in the image .this , in turn , results in a large mise . to compensate for this increase , and to make all the cases more directly comparable , we rescale the mise by the total signal of each image to compensate for this additional flux .  for the evaluation of image qualitywe fit a two dimensional , spherical gaussian density to a source using least squares .the covariance matrix of this fitted gaussian is then used as a metric for the width of the point source .in this section , we present the results of our simulations .the section is divided into two parts , corresponding to the two evaluation criteria outlined above .when looking at equation ( [ eq : fredholm ] ) we see two major influences on the quality of the observation .one is the type and severity of the seeing , determined by the psf .the other is the distribution of the noise . in this section ,we look at several different scenarios for comparing the mean or median to the fd method by changing these two factors .overall , we wish to understand the impact of seeing and noise on the performance of the methods . to do this ,we simulate a stack of images with each image having a fwhm drawn from a distribution with mean , .we do this procedure for ranging from 2 pixels to 7 pixels ( 0.4 to 1.4 ) .this interval is chosen so that it contains the expected seeing width for the lsst .this procedure creates a sequence of stacks of images .we apply this increasing fwhm to six different noise parameterizations . for both high variance ( 5:1 snr ) and low variance ( 20:1 snr )we have the noise term be distributed gaussian , heavy - tailed shot , and inhomogeneous poisson noise . in all six combinations, a lower mise indicates that the method conserved the flux better , on average. see figure [ fig : results ] for the results .  in all cases , the mean severity of seeing , , does not impact the ranking of the methods . in the gaussian case ,the methods are ranked , from best to worst , as fd , mean , and then median . in both the low and high noise cases ,absolute difference between the mean and the median is constant for all levels of due to the well - known efficiency gains of the mean over the median in the gaussian case .however , in the high noise case , the difference in mise between the fd and mean methods increases 11% as ranges from 2 to 7 pixels .as the fd method uses the additional information of a known seeing kernel , this non - constant difference is expected .  in the inhomogeneous poisson or heavy - tailedshot noise cases , the median outperforms both other methods .this owes to the particular and well known feature of the median to be more robust against heavier - tailed distributions . in the shot - noise case, the median ignores the random noise spikes with overwhelming probability and we get a noise free version of the true scene with the median amount of seeing . the very small advantage of the fd over the mean method owes to the fd s slightly narrower impulse response function , which we discuss in section [ sec : imagequality ] and figure [ fig : expectedirf ] . in the inhomogeneous poisson noise case ,the observed image is created by simulating independently from a poisson from equation ( [ eq : theta ] ) .this creates a very noisy image , especially near sources , and hence the median s ability to ignore large , transient spikes results in the large mise advantage .  under all three noise distributions, the fd method displays a small but increasing benefit in flux conservation over the mean as goes from 2 to 7 .it should be kept in mind that the fd method uses the known kernel assumption .hence , as the seeing becomes more severe this informational advantage should become more pronounced . in reality , the kernel would need to be estimated and this advantage would most likely decay .   +  +  +   to do this comparison we use an image where the function is a -function ( i.e. a point source ) .we use the expected distribution of seeing at the lsst stack to draw 10 independent and identically distributed gaussian psfs and apply them to the point source .this mimics the forward process of an image getting blurred by the atmosphere and corresponds to operating on with for . for this stack of imageswe apply the three methods described earlier , fd , mean , and median , to derive a template image .we then fit a spherical gaussian kernel to the template via least squares and use the diagonal element of the covariance matrix as a measure of the width of the method s psf .we make 1000 draws from the seeing distribution and compute this statistic for each method and each draw .the results are summarized in figure [ fig : sharpnessboxplot ] with boxplots of the fwhm of the fitted gaussian kernel .boxplots are a graphical tool for quickly conveying the distribution of observed data and are composed of a ` box ' and ` whiskers . 'the ` box ' is the main rectangle , which has horizontal lines , increasing with fwhm , at the 25th , 50th , and 75th percentiles of the data .the ` whiskers ' correspond to the dotted vertical lines with horizontal lines at the end and the plus signs .these horizontal lines are at the 1st and 99th percentile .the pluses are at extreme data points .the boxplots are very similar and show little difference in the fwhm of the fitted kernels .the fd method has longer ` whiskers ' than the other methods , indicating a wider range of fitted fwhm that can be expected .it also has a very slightly lower ` box ' than the other two methods , indicating that the majority of draws result in a slightly better fitted fwhm .for instance , the 50th percentile of the fd method s fitted fwhm is 1.2% lower then the mean method s fitted fwhm .interestingly , one of the draws resulted in six very good seeing images and four poor images .this resulted in the median method having the smallest fitted fwhm recorded for any method on any draw .this is part of a more general property of the median to behave non - continuously with the composition of the stack .for instance , if there were instead 4 very good seeing images and 6 poor seeing images , the result would be a large fitted fwhm .+   the mean and fd methods are very similar in that they are both based around summing the images and hence are both linear filters on the observed images .let s denote the estimated coadds of the two methods as and , respectively .we can understand the filtering that we are applying to the true image by considering  ] .note that these are = \left ( \frac{1}{l } \sum_{i=1}^l k_i\right)g = \int k_{mean}(x - y ) g(y ) dy \qquad \textrm{and } \qquad \mathbb{e}[\hat{g } _ { * } ] = \int k^{*}(x - y)g(y)dy \label{eq : maybe}\ ] ] where the fourier transform of the function is defined in the caption under table 1 .  by examining the impulse response functions ( irfs ) of these filters we can find the average width of the psfs that will result from these methods .note that in this case this corresponds to the bias of each method .see figure [ fig : expectedirf ] for a one dimensional representation of the irfs of the expected filters .the mean irf and the fourier deconvolution irf are virtually identical and would create no visible difference in the image quality .   has more mass at lag = 0 , bear in mind that it uses a weighting scheme that depends on knowledge of both the true psfs and true variances .taking this into consideration , the difference is slight.,title="fig:",width=384 ] +a variety of template generation / image coaddition techniques have been applied in current imaging surveys , including the cfhtls ( ; mean technique ) , panstarrs ( , mean technique ) , mopex for sirtf images ( , mean and median technique ) , and the virmos survey ( , median technique ) . in this paper , we address the broad question of whether sophisticated methods of template generation and image coaddition are worth the added cost and complexity , in the context of modern image surveys with large and nearly constant data streams . to this end , we compare fourier domain reconstruction techniques which have a variety of nice theoretical properties to straightforward pixelwise statistics .we consider two cost metrics to evaluate the image reconstructions ; image quality by way of the resulting gaussian fwhm of an image and the conservation of flux , measured by mise .  under the assumptions that motivate the fd method , namely gaussian noise and spatially constant seeing, the fd and mean methods both have lower mise than the median .however , in the heavy - tailed shot and inhomogeneous poisson noise cases , the median outperforms the fd method .  in situations when there is a small amount of seeing in an image stack , corresponding to pixels , we see the mean and fd methods have nearly the same mise properties ( less than 0.7% difference across all considered noise distributions ) .however , as the severity of the seeing increases , the disparity between the methods increases as well . as the fd method incorporates knowledge of the seeing kernel , this property is to be expectedhowever , the mise for the high noise gaussian , shot , and inhomogeneous poisson cases are only 5.4% , 1.2% , and 0.13% lower , respectively , at the most severe levels of seeing .these benefits are rather small , particularly in the more realistic heavy - tailed and poisson noise cases .this is especially so considering the extra computational complexities involved in the fd method over the mean and median methods .  for comparing image quality of the templates, we find that the resolution of the resulting images is very similar as the 50th percentile of the fitted fwhm of the fd method is less than 2% smaller than the 50th percentile of the fitted fwhm of the mean method .we find that the value added by the fd method over the mean or median procedures does not overcome its added cost and complexity .this leads us to the conclusion that there is room for improvement over all three methods .we are currently exploring various possible improvements and theoretical justifications for some of the approaches outlined in this paper .beckwith , steven et al ., 2006 , , 132 , 5 , pg .1729 fried , d.l ., 1978 , j. optical soc .amer , 86 , pgs .1651 - 1657 gwyn , s. d. j. , 2008 , , feb ., 120 , pgs .212 - 223 hu , yuchng et al . , 2007 , physics letters a , 367 , pgs .173 - 176 kaiser , nick , 2004 , private communication mokovoz , d. et al ., 2005 , astronomical data analysis software and systems xiv , 347 , dec , pg. 81 parker , laura et al ., 2007 , , 669 , 1 , pg .21 price , p.a and pan - starrs team , bulletin of the american astronomical society , 38 , may , pg .183 radovich m. et al . , 2004 , ,apr . , 417 , pgs . 51 - 60 scully , marlan et al . , 1969 ,physics review .179 , pgs .368 - 374 silverman , brian , 1985 , chapman & hall / crc , monographs on statistics and applied probability , vol 26 tubbs , r.n . , 2003 , phd thesis , cambridge univ wasserman , larry , 2006 , chapter 7 , springer science

Abstract: large , multi - frequency imaging surveys , such as the large synaptic survey telescope ( lsst ) , need to do near - real time analysis of very large datasets . this raises a host of statistical and computational problems where standard methods do not work . in this paper , we study a proposed method for combining stacks of images into a single summary image , sometimes referred to as a template . this task is commonly referred to as image coaddition . in part , we focus on a method proposed in , which outlines a procedure for combining stacks of images in an online fashion in the fourier domain . we evaluate this method by comparing it to two straightforward methods through the use of various criteria and simulations . note that the goal is not to propose these comparison methods for use in their own right , but to ensure that additional complexity also provides substantially improved performance .


bitcoin , the best - known cryptographic currency , draws as many harbingers of imminent failure as heralds of long - term success as a mainstream currency . throughout its 5-year existence, it has been the subject of growing attention , due in part to its rapidly increasing and very volatile exchange rate to other currencies . amidst the hype surrounding the cryptocurrency , it is difficult to recognise which factors participate in its growth , and influence its value .bitcoin s decentralised structure , based on the contribution of its users rather than a central authority , implies that the dynamics of its economy may be strongly driven by social factors , which are composed of interactions between the actors of the market .this paper reveals the interdependence between social signals and price in the bitcoin economy , namely a social feedback cycle based on word - of - mouth effect and a user - driven adoption cycle .the bitcoin economy is indeed growing at a staggering speed : the market value of all bitcoins in circulation went from about usd 14b in december of 2013 , hence a fifty - thousand - fold increase in that period .this came along with a stark increase in public interest , as shown by internet search data : during the same period , the volume of bitcoin - related searches on the google search engine grew by over 10000% .our hypothesis is that this growth is driven by the online actions and interactions of individual users , which leave traces of their activity .how can we use these digital traces to capture the link between market dynamics and public interest ? in this paper , we provide evidence that this feedback can be understood by incorporating two types of signals : price and social information .the influence of the first was investigated through the foundational work of fama and later grossman , who demonstrated that economic agents rapidly integrate common sources of information to assign a price to a good , including price itself .the role of purely social information for price formation was first studied by bikhchandani , who showed that imitation is a rational strategy in periods of large volatility or in the absence of other sources of information .  in the bitcoin economy ,the fixed supply and predictable scarcity , both independent of the user base , create a strong link between public interest , user adoption , and price ( illustrated in the time series of fig .[ fig : timeseries]a ) . following from our hypothesis on the role of social interactions ,a key issue is characterising the effect of social influence in the price variations .we quantify these socio - economic signals to provide an analytical perspective on the relationship between the bitcoin exchange rates and the social aspects of its economy . by adopting this perspective, we reveal multiple temporal dependencies leading to the formation of bitcoin price bubbles .  .]  [ [ four - tiered - data ] ] four - tiered data + + + + + + + + + + + + + + + +  we use a four - tiered data set ( table [ tab : samplesizes ] ) , composed of records of exchange data , social media activity , search trends , and user adoption of bitcoin .[ [ bitcoin - block - chain - and - software - client - user - base ] ] bitcoin block chain and software client : user base  the bitcoin block chain is a public ledger containing the full record of all public transactions in the history of the bitcoin currency .every node of the bitcoin network , running a bitcoin software client , keeps a copy of the block chain .our analysis of the block chain , as well as of the number of downloads of the software client , yields two approximations for the true number of new bitcoin users at any time . the number of new bitcoin users adopting the currency at time is represented by the variable in fig . [ fig : timeseries]b .  [ [ bitcoin - exchange - rates . ] ] bitcoin exchange rates .bitcoins ( btc ) are traded for other currencies at public internet exchanges . as of december 2013 ,the oldest public exchange , and the largest to trade btc for us dollars ( usd ) and euros ( eur ) , is mt .gox ; the largest exchange by btc volume is btc china , which trades btc for chinese renminbi ( cny ) . for our analysis, we use trading data from these two exchanges and a third exchange hosted in europe , btc - de , including exchange rates in three currencies usd , eur , and cny , using btc / usd as a historical reference .the trading price is represented by the variable in fig .[ fig : timeseries]b .[ [ information - search . ] ] information search .we measure the interest in obtaining information about bitcoin through the normalised search volume for the term `` bitcoin '' on the google search engine .search volume data have been shown useful to capture the information - gathering stage of the decision process of individuals , leading to insights about volume and volatility , as well as financial returns .alternatively , wikipedia usage can be used as an indicator of information gathering .both indicators have been shown to lead bitcoin prices , motivating the cross - validation of our results with the number of wikipedia views for the bitcoin page on the english wikipedia .the search volume is represented by the variable in fig .[ fig : timeseries]b .[ [ information - sharing . ] ] information sharing .  in our analysis ,information search is a private action that nee dnot be shared among individuals , while information sharing is strictly social .social interaction between individuals can be measured through their level of communication .previous works applied sentiment analysis on public messages on twitter ( tweets ) to predict stock price changes , or linguistic patterns in instant messages to predict stock volatility .in addition to sentiment , absolute on - line word - of - mouth levels , such as the total number of tweets or news articles , are useful in predicting price changes .we measure information sharing , or on - line word - of - mouth communication , through the daily number of bitcoin - related tweets per million messages in our twitter feed , calculated as . for additional validation , we compute an alternative by substituting the number of bitcoin - related tweets with the number of `` reshares '' of the messages posted on the oldest , regularly active public facebook page dedicated to bitcoin . on - line word of mouth is represented by the variable in fig .[ fig : timeseries]b ..sample sizes of the twitter , wikipedia , facebook , bitcoin network , and sourceforge data sets ( top ) , and btc exchange markets data sets ( bottom ) . [ cols="^,^,^,^",options="header " , ]   in addition to these two cycles , we find a negative relation from search to price ( ) .this is illustrated by a clear dyadic relation between the two variables extremes : 3 of the 4 largest daily price drops were preceded by the 1st , 4th , and 8th largest increases in google search volume the day before .table s2 of the esm presents the results of the same var with non - normalised time series ; in this way , relations between variables can effectively be quantified ( e.g. , an increase of 10000 client downloads leads to an increase of $ 3.80 in price )  these two feedback cycles and the negative role of search in price are consistent when fitting vector autoregressors using the number of users in the network instead of client downloads as user signal , wikipedia views instead of google search volume as search signal , and facebook bitcoin page reshares instead of bitcoin - related tweets as ( table s3 ) .all these results are also consistent when using other currency pairs ( btc / eur and btc / cny ) and data from other exchange platforms ( btc - china and btc - de ) for the analysis ( table s4 ) .  to further verify the connection between the variables in the social and user adoption cycles , we calculated the impulse response functions of the vector autoregression results shown in table [ tab : var ] .the impulse response function estimates the propagation of a shock of one standard deviation on a variable to the other variables . to control for our finite sample size, we performed 10000 boostrapped estimations , correcting for correlated noise with the standard hac method .the results of the estimation for the response of variables and their 95% confidence intervals are shown in fig .[ fig : irf ] .all the pairwise relations of the feedback cycles illustrated in fig .[ fig : timeseries ] are significant when analysed through impulse response functions .search levels experience significant increases two to four days after price increases , and both word of mouth and number of users increase one to two days after strong increases in search volume .price increases as a result of shocks on user adoption and word of mouth , and the negative influence of search in price is also evident .in contrast with findings in previous time periods , these response functions are similar for both google search trends and wikipedia page views . to further evaluate the significance of these responses , we measured the response estimate between each pair of signals for time increments of 1 and 2 days , testing for a significance of 97.5% , to achieve a 95% confidence level after bonferroni correction .all the relations in the feedback cycles have at least one significant change , including the negative effect of search trends to price ( more details in table s7 of the esm ) .  .]  [ [ reproducing - price - and - social - dynamics . ] ] reproducing price and social dynamics .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  the cycles presented above provide an explanation for the generation of bubbles in the bitcoin economy .recent findings indicate that the driving forces behind bitcoin prices changed since its invention , motivating our decomposition of the study period into characteristic time windows , each of which corresponds to a distinct bubble .we do this by estimating a lower bound for the fundamental value of bitcoin : we approximate the energy cost of producing one bitcoin , which is derived directly from the bitcoin difficulty ( see materials and methods ) . throughout our study period ,the price stayed almost always above the fundamental value ( fig .[ fig : fits]a : the trajectory of the weekly weighted mean price is almost exclusively on the left of the price / fundamental equality line ) .the trade of bitcoins at a much higher price indicates the possible presence of a bubble , and the events at which the market price starts diverging from the fundamental value mark the beginning of bubbles .we identify two such events : one around october 18th , 2011 , which marked the end of the 2011 bubble , and another one around november 28th , 2012 , which is the date at which mining rewards halved ( `` halving day '' ) , suddenly raising the fundamental value . during the study periodthe price never dropped significantly below the fundamental value , which validates our estimate for it : a drop of price below the fundamental value would introduce a paradox in which the maintenance of the public ledger by the miners is no more profitable for them .we thus define three characteristic periods , or bubbles ( fig .[ fig : timeseries]b ) : before the first event ( p1 ) , between the first and the second events ( p2 ) , and after the second event ( p3 ) .running independent var analyses on the three periods shows that the two feedback cycles exist in the system when considering the three periods together or p3 on its own , but not if we only take into account data until november 28th , 2012 ( table s5 of the esm ) . in the following , we focus on the last period to evaluate the quality of the var technique in reproducing changes in the four variables in the last bubble .the results shown in table [ tab : var ] , with a lag of one day , allow us to interpret our results in terms of feedback cycles .nevertheless , an extended var with a longer lag can have larger explanatory power , despite lacking a straightforward interpretation . by assessing the quality of the model through the schwarz criterion , we find the optimal explanatory power corresponds to a lag of four days ( fig .s4 of the esm ) . in this best fit ,the changes in each variable are calculated as linear combinations of the changes of all variables up to four days before .we use this optimised model to estimate daily changes in our four variables based on the empirical data . adding the daily changes yields a step - wise reconstruction of the time series of each variable .[ fig : fits]b , c show the overlaid times series of estimated and empirical price and word - of - mouth levels .while this method fits daily changes in the variables , we can assess the long - term quality of the var by correlating their cumulative time series with the reconstructed values .we find a positive significant pearson s correlation coefficient between the observed and estimated time series for all four variables , with the price comparison yielding , and the word - of - mouth comparison .additionally , reconstructed time series for the number of users ( ) and search volume ( ) are also accurate ( for all correlations . residuals from the fit of the changes are approximately normally distributed ( fig .s5 of the esm ) , which means that within period p3 there are no structural deficiencies in our model .finally , the var model correctly identifies the sign of all of the 10 largest daily price increases , and 9 of the 10 largest price drops during p3 ( table s6 of the esm ) .due do the decentralised character of the bitcoin currency , the dynamics of its economy largely depend on the behaviour of its users , who ( i ) mine new bitcoins and maintain the block chain , and ( ii ) influence the exchange rate by trading bitcoins to and from other currencies ; these interactions between users form the social backbone of bitcoin . in this paper, we used the digital traces of user activity in the bitcoin economy to disentangle the relationships between its different variables : user base , information search , information sharing , and price . while each of these four socio - economic signals can be studied independently , the novelty of our approach lies in the combination of all of them into a robust analytical approach .together , these signals provide a precise picture of bitcoin s growth over the three - year period we studied , from the first time bitcoins were publicly traded in the middle of 2010 through the successive price surges until the end of 2013 .these digital activity traces are a unique source of information to quantitatively describe the dynamics of large socio - economic systems ; our analysis combines for the first time social , economic , and technological factors into a unified perspective on one such system .this combined analysis reveals two positive feedback loops : a reinforcement cycle between search volume , word of mouth , and price ( social cycle ) , and a second cycle between search volume , number of new users , and price ( user adoption cycle ) .the social cycle provides evidence for interindividual influence in the decision to buy bitcoins .it translates as follows : bitcoin s growing popularity leads to higher search volumes , which in turn result in increased social media activity on the topic of bitcoin .more interest encourages the purchase of bitcoins by individual users , driving the prices up , which eventually feeds back on the search volumes .we hypothesise that the temporal correlation between price and search volume is mediated by the media reporting on price increases , thereby driving user curiosity and triggering their search activity .the second feedback loop we identify is the user adoption cycle , which complements the social cycle : new bitcoin users download the client and join the transaction network after acquiring information about the technology .this growth in the user base translates to a price increase , as the number of bitcoins available for trade does not depend on demand , but rather grows linearly with time .this is a direct consequence of the deflationary nature of bitcoin as a currency .another important result of the var is the negative weight of search on price .this marks sporadic connections between large price drops and the spikes in search volume that preceded them .in other words , user search activity responds faster to negative events , such as a security breach in a bitcoin exchange , than price . in thisregard , search spikes are early indicators of price drops .  we showed the robustness of our vector autoregression analysis , computing impulse response functions of the key couplings between variables in the feedback loops .furthermore , we validated that our findings are not a construct of the particularities of mt .gox as an on - line exchange , reproducing our results in three different currency markets ( usd , eur , and cny ) .our analysis of behavioural signals is also robust to the choice of data sources , as our results also appear when using alternative sources for search , word of mouth , and user adoption . in the case of user adoption, we found that the number of new users detected after processing the block chain yields information about price changes , which is not observed when using raw bitcoin addresses without pre - processing ( see table s3 of the esm ) .this indicates that identifying users from addresses is important to correctly characterise the bitcoin ecosystem , which has been overlooked in previous studies .we introduced a lower - bound estimate of bitcoin s fundamental value , based on the cost of the energy involved in mining .this allowed us to identify three characteristic time periods ; by studying the three periods independently , we found that the two feedback cycles are rooted in the last period of the data ( p3 , end of 2012 until end of 2013 ) , which may either denote that the appearance of these cycles is recent , or presumably that the larger quantity of data from the last period yields a higher signal - to - noise ratio , thus helping the characterisation of the cycles .this second hypothesis would also explain the more nuanced findings obtained by our analysis compared to earlier work on earlier data sets .we validated the results of the vector autoregression technique in this last period by comparing the output of the best var fit to the empirical time series of our four variables .it is important to note that the var provides a fit of the daily changes , but in this validation step we compare the cumulative levels .if the var errors were temporally correlated , this would create structural divergence in the var estimates of the levels . however , this is not what we find : the levels produced by the var are significantly correlated with the empirical values , and the residuals of the estimates are normally distributed ( fig .s5 of the esm ) .we also find that the dynamics of both qualitatively match ( fig .[ fig : fits ] ) , and that the var correctly classifies the sign of the largest price variations ( table s6 of the esm ) .the statistical technique we used in this paper thus proves to be a robust way of identifying the coupled dynamics of the socio - economic variables we study .it also produces accurate estimates of the future levels of any variable ( including price and word of mouth ) based on the past history of the system .the two positive feedback loops we identified imply a constant increase of the price , which should drive up the bitcoin exchange rate unsustainably .this provides an explanation for the successive periods of almost uninterrupted growth observed from mid-2010 to june 2011 , and early 2012 to april 2013 .it does not however explain the sudden drops observed at the end of these periods .these crashes can be attributed to external stimuli , such as the attack on the mt .gox platform of june 2011 .the relationship we found between price drops and preceding search spikes , while it does not explain the occurrence of the crashes , is consistent with past studies linking search volumes to price fluctuations in financial markets .while previous work indicated that news sources do not constitute good predictors of future price changes , our analysis suggests that the successive price surges in the bitcoin economy are largely due to its growing public attention .initially , bitcoins had a negligible exchange value and were only known by a small community of technically experienced users .our analysis suggests that the successive waves of growth of the bitcoin economy were driven by corresponding waves of new users from public circles gradually opening to the currency .the growing user base created more exchange opportunities for bitcoin which , at the time of writing , is accepted by a wide range of businesses , from hosting services to retail stores and illegal markets . as of december 2013 , the computational power of the bitcoin mining network sits at about floating - point operations per second roughly 300 times the combined power of the top 500 supercomputers .this striking figure illustrates the increasing importance of bitcoin in the technological , social , and economic landscapes , and motivates the design of policies to regulate bitcoin usage and exchange .for such regulations to be effective , policy makers require an empirical understanding of the dynamics of bitcoin adoption and trade .the digital traces left by the millions of users of the bitcoin network , exchange markets , on - line social networks , and search engines allowed us to systematically describe the dynamics of bitcoin adoption .our analysis of collective socio - economic signals can be applied beyond the study of cryptocurrencies , to understand other phenomena for which large amounts of data are available , such as adoption behaviour in on - line communities .dg s work was funded by the swiss national science foundation ( cr21i1_1464991 ) .we are grateful to frank schweitzer for supporting this project .hadas e. a prediction : bitcoin is doomed to fail ; 2013 .accessed : 2013 - 12 - 01 .the new york times .http://dealbook.nytimes.com/2013/11/27/a - prediction - bitcoin - is - doomed - t% o - fail[http://dealbook.nytimes.com/2013/11/27/a - prediction - bitcoin - is - doomed - t% o - fail ] .  .byron wien announces predictions for ten surprises for 2014 ; 2014 . accessed : 2013 - 12 - 07 .press release ./ news - views / press - releases / details / byron - wien-% announces - predictions - for - ten - surprises - for-2014[http://www.blackstone.com / news - views / press - releases / details / byron - wien-% announces - predictions - for - ten - surprises - for-2014 ] .                       ; 2013 . accessed : 2013 - 12 - 07 .http://www.eia.gov / electricity / monthly / epm_table_grapher.cfm?t = epmt_5_6% _ a[http://www.eia.gov / electricity / monthly / epm_table_grapher.cfm?t = epmt_5_6% _ a ] and http://epp.eurostat.ec.europa.eu / statistics_explained / index.php / energy_% price_statistics[http://epp.eurostat.ec.europa.eu / statistics_explained / index.php / energy_% price_statistics ] .        ; 2012 . accessed : 2013 - 12 - 08 .press release .http://www.prlog.org/12032578-bitcoin - community - celebrates - halving - day.% html[http://www.prlog.org/12032578-bitcoin - community - celebrates - halving - day.% html ] .garcia d , mavrodiev p , schweitzer f. social resilience in online communities : the autopsy of friendster . in : proceedings of the first acm conference on online social networks .cosn 13 ; 2013 .

Abstract: what is the role of social interactions in the creation of price bubbles ? answering this question requires obtaining collective behavioural traces generated by the activity of a large number of actors . digital currencies offer a unique possibility to measure socio - economic signals from such digital traces . here , we focus on bitcoin , the most popular cryptocurrency . bitcoin has experienced periods of rapid increase in exchange rates ( price ) followed by sharp decline ; we hypothesise that these fluctuations are largely driven by the interplay between different social phenomena . we thus quantify four socio - economic signals about bitcoin from large data sets : price on on - line exchanges , volume of word - of - mouth communication in on - line social media , volume of information search , and user base growth . by using vector autoregression , we identify two positive feedback loops that lead to price bubbles in the absence of exogenous stimuli : one driven by word of mouth , and the other by new bitcoin adopters . we also observe that spikes in information search , presumably linked to external events , precede drastic price declines . understanding the interplay between the socio - economic signals we measured can lead to applications beyond cryptocurrencies to other phenomena which leave digital footprints , such as on - line social network usage .  : social interactions , bubbles , socio - economic signals , bitcoin