fault - tolerant error correction promises to enable the reliable storage and manipulation of quantum information , but only if errors affecting qubits are not strongly correlated and occur with a probability below some threshold error rate . in practice , error correction should tolerate realistically high error rates and be compatible with experimentally feasible technology , without requiring excessive overhead in order to achieve a sufficiently reliable universal set of logical gates . in principle , topological error correction with the surface code appears to satisfy these criteria , enabling schemes that tolerate error rates of approximately one percent per gate for qubits constrained to a two - dimensional array with local interactions .however , the overhead of these schemes remains daunting , motivating the search for other approaches for example , see refs . . here , we are interested in error correction using topological color codes , which have many similarities with the surface code but in some cases also admit transversal implementation of logical gates that span the clifford group . this property may help to reduce the overhead required to achieve universality via state distillation , making these codes promising candidates for fault - tolerant error correction .one critical aspect of error correction is decoding , which is the task of inferring from the error syndrome a set of corrections that will return the system to the code space . in the first instance, a practical decoding algorithm should be computationally efficient .it should also tolerate uncertainty in the syndrome due to errors in other words , it should be fault tolerant .efficient fault - tolerant decoding of the surface code can be achieved by solving a particular graph matching problem , among other techniques .however , the equivalent approach for topological color codes introduces a hypergraph matching problem for which no efficient algorithm is known , although approximate solutions can still be used .recently , several efficient decoding algorithms for color codes have been proposed .these involve iteratively decoding using message passing or relating the error syndrome to syndromes across multiple copies of the surface code , which are then decoded using existing techniques .however , the performance of these algorithms remains unknown in the important , realistic case where the qubits used to measure the error syndrome are themselves affected by errors .  here, we investigate the efficient fault - tolerant decoding of color codes .we begin with the decoding algorithm due to delfosse , which relates the hypergraph matching problem associated with color codes to a set of three graph matching problems with efficient solutions .we modify this algorithm to account for syndrome errors , applying it to the family of codes defined by the 4.8.8 semiregular lattice embedded on a plane with a triangular boundary . for a three - dimensional bit - flip channel ,we report a threshold of 0.0208(1 ) , compared with 0.0305(4 ) previously reported for an integer - program - based decoding algorithm .this result indicates that decoding of topological color codes by graph matching is a feasible approach in the context of fault - tolerant error correction .when we account for correlated errors introduced by the syndrome measurement circuits , the threshold is reduced to 0.00143(1 ) per gate .this threshold is higher than others reported for the color code , but significantly lower than the equivalent threshold for the surface code under an identical noise model .a topological color code is defined by a three - colorable cubic ( trivalent ) graph embedded on a surface .the faces of a graph are three - colorable if every face can be assigned one of three colors such that no two faces that share an edge are the same color .data qubits reside at the nodes of the graph .each face is associated with two stabilizer generators , or check operators , given by where is a face in the embedding , is the set of qubits incident on , and and are the single - qubit pauli operators .the code space is the simultaneous + 1 eigenspace of the stabilizer generators .  .each node corresponds to a data qubit and each face corresponds to two stabilizer generators .errors ( black circles ) anticommute with stabilizer generators ( purple circles ) .hatched faces indicate where errors terminate on the boundaries , but the implied qubits and stabilizer generators are not actually present .logical pauli operators ( grey circles ) are associated with the boundaries . ]  here , we are particularly interested in the family of topological color codes defined by the 4.8.8 semiregular lattice embedded in a planar disc , which admit transversal implementation of the entire clifford group . if we restrict the number of logical qubits to equal one , then we are left with a family of codes defined by lattices with triangular boundaries , as illustrated in fig .the code parameters are $ ] for odd - integer code distance .logical pauli operators are connected chains of single - qubit pauli operators along any one of the three boundaries .additional logical qubits may be introduced by puncturing holes in the disc to modify its topology . in doing so ,the stabilizer generators remain local in two spatial dimensions , ensuring compatibility with most quantum computer technologies .in general , the error syndrome is obtained by measuring the eigenvalues of the complete set of stabilizer generators . however ,for now , we will focus on the -error syndrome , which is obtained by measuring the eigenvalues of only the -type stabilizer generators .the location of errors is implied by stabilizer generators whose eigenvalue is .for example , an isolated single - qubit error away from the boundaries will anticommute with the three adjacent stabilizer generators . however , more complicated chains of errors will only anticommute with the stabilizer generators at their terminals .errors may also terminate on the boundaries , and so , in keeping with the three - coloration of the graph , each boundary is assigned a color complementary to the colors of the adjacent faces .examples of errors and the corresponding syndromes are illustrated in fig .decoding is the procedure of inferring from the syndrome a set of corrections that will return the system to the code space .= .in contrast with fig .[ fig1 ] , each face corresponds to a data qubit and each node corresponds to a stabilizer generator .nodes connected by dashed edges correspond to the boundaries .the dual graph is partitioned into three colored subgraphs that connect nodes of the complementary set of colors .( b ) red subgraph for = where the syndrome is repeatedly measured in order from bottom to top .errors affecting data qubits and syndrome qubits correspond to spacelike and timelike edges respectively . ]  here , it is useful to introduce the dual graph , which can be partitioned into three colored subgraphs , as shown in fig .[ fig2](a ) .crucially , a single - qubit error will anticommute with only two adjacent stabilizer generators in each subgraph . in this way , when considered independently , each subgraph is equivalent to a surface code with its own syndrome .the minimum - weight set of errors consistent with the syndrome of a surface code can be found efficiently by solving a particular graph matching problem using edmonds perfect matching algorithm . with this in mind , assuming that the syndrome is perfectly reliable , decoding proceeds as follows :  1 .  for each subgraph ,find the minimum - weight set of errors consistent with the syndrome , corresponding to a set of edgescalculate the union of the edges identified in the three subgraphs , which partitions the data qubits into two sets .3 .  correct all qubits in the lowest - weight set .figures [ fig3 ] and [ fig4 ] illustrate this procedure , giving examples of success and failure respectively. it may be possible to exploit correlations between the three subgraphs to more accurately identify errors , but we have not investigated this possibility in detail .      in general , the error syndrome will be unreliable , as the circuits used to measure the eigenvalues of the stabilizer generators may introduce additional errors .this ambiguity is resolved by repeating the measurement circuits times .then , the location of errors is implied by stabilizer generators whose eigenvalue is found to change between successive measurements . in particular , an error affectinga data qubit will cause the eigenvalues of the adjacent stabilizer generators to change from their previous values , while an error affecting a syndrome qubit will cause the eigenvalue of a single stabilizer generator to change ( apparently ) from its previous value and then change back again .  again , our decoding strategy is to consider each subgraph as an independent surface code .however , to reflect the fact that the syndrome is repeatedly measured , each subgraph is extended along an axis which we identify with time .the minimum - weight set of errors consistent with the syndrome of each subgraph may include errors affecting data qubits and errors affecting syndrome qubits , which , respectively , correspond to spacelike and timelike edges in the subgraph , as illustrated in fig .[ fig2](b ) .therefore , we make the following modification to the second step of the decoding procedure :  * calculate the two - dimensional projection of the spacelike edges identified in the three subgraphs , which partitions the data qubits into two sets .edges cancel if they occur an even number of times in the projection .next , we specify circuits to measure the eigenvalues of the stabilizer generators . at the simplest level ,standard single - qubit operator measurements are sufficient for this purpose , as the structure of the code limits the propagation of errors . however , more intricate circuits may be useful to trade off various parameters such as threshold , circuit depth , and connectivity . here, circuits are chosen to ensure that high - weight correlated errors are relatively unlikely without introducing a stochastic verification procedure .measurements reveal any weight - two errors in the cat state caused by a single error occurring during its preparation . to execute these circuits , assuming that qubits are constrained to a two - dimensional array , each qubit must be allowed to interact with either three or five of its neighbors .]  weight - four stabilizer generators are measured with a two - qubit cat state , as shown in fig .[ fig5](a ) .weight - eight stabilizer generators are measured with a four - qubit cat state , which is prepared according to the left - hand side of fig .[ fig5](b ) . in preparing this state, a single error may propagate to several errors that may in turn propagate to several data qubits .however , the circuit is designed so that it is possible to determine the nature of such errors with a simple post - processing operation .referring to the right - hand side of fig .[ fig5](b ) , the measurements detect these errors while the measurement reveals the eigenvalue of the stabilizer generator , as required .when correlated errors are detected , appropriate corrections are applied to the affected data qubits .we simulate error correction with the color code using standard monte carlo methods . in each instancewe generate a set of errors based on some noise model , decode the corresponding syndrome , and apply corrections as required .we sample the logical error rate as a function of the physical error rate for odd code distances between 9 and 21 . following refs . , a universal scaling ansatz accounting for finite - size effects is fit to the data and the threshold is extracted ( where in all cases ) .where appropriate , we quote the lowest of the separate thresholds for and logical errors , which sets the overall threshold .   statistical error and curves are best fits accounting for finite - size effects .the value of the physical error rate at the intersection ( extracted from the fitting and indicated by the vertical dashed line ) is the threshold error rate .exact values are given in table [ tab1].,title="fig : " ]  statistical error and curves are best fits accounting for finite - size effects .the value of the physical error rate at the intersection ( extracted from the fitting and indicated by the vertical dashed line ) is the threshold error rate .exact values are given in table [ tab1].,title="fig : " ]  statistical error and curves are best fits accounting for finite - size effects .the value of the physical error rate at the intersection ( extracted from the fitting and indicated by the vertical dashed line ) is the threshold error rate .exact values are given in table [ tab1].,title="fig : " ]  llllc + lattice & surface & threshold & decoding algorithm & ref .+ & _ triangle _ & _ 0.0760(2 ) _ & _ graph matching _ & _ _ + 4.8.8 & triangle & 0.1056(1 ) & integer program & + 4.8.8 & torus & 0.087 & renormalization & + 4.8.8 & triangle & 0.0887 & hypergraph matching & + 4.8.8 & torus & 0.109(2 ) & optimal & + 4.8.8 & torus & 0.10925(5 ) & optimal & + 6.6.6 & torus & 0.078 & renormalization & + 6.6.6 & torus & 0.087 & graph matching & + 6.6.6 & torus & 0.109(2 ) & optimal & + 6.6.6 & torus & 0.1097(1 ) & optimal & +  llllc +  + lattice & surface & threshold & decoding algorithm & ref .+ & _ triangle _ & _ 0.0208(1 ) _ & _ graph matching _ & _ _ + 4.8.8 & triangle & 0.0305(4 ) & integer program & + 6.6.6 & torus & 0.045(2 ) & optimal & +  llllc +  + lattice & surface & threshold & decoding algorithm & ref .+ & _ triangle _ & _ 0.00143(1 ) _ & _ graph matching _ & _ _ + 4.8.8 & triangle & 0.00082(3 ) & integer program & + 4.8.8 & triangle & 0.001 & hypergraph matching & +  firstly , we consider _ code - capacity noise _ , where an error is applied to each data qubit with probability .this noise model assumes that the syndrome is perfectly reliable . in this case, we observe a threshold at .next , we consider _ phenomenological noise _ , where an error is applied to each data qubit and each syndrome bit value with probability .this noise model captures the behavior of errors arising in the syndrome measurement circuits without explicitly considering their correlations . in this case, we observe a threshold at .lastly , we consider _ circuit noise _ , where errors are applied to the individual gates in the syndrome measurement circuits .specifically , each gate fails independently with probability , where faulty preparation gives the incorrect eigenstate , a faulty gate introduces one- or two - qubit depolarizing noise , and faulty measurement reports the incorrect eigenstate . in this case, we observe a threshold at .the equivalent threshold for the surface code under this noise model is  .if we lower the probability of failure of identity gates to to reflect the fact that quantum memories may be significantly more reliable than other gates , then the threshold is increased to .numerical data for the three noise models are shown in fig .[ fig6 ] and our results are compared with previously reported thresholds for a range of topological color codes and decoding algorithms in table i.in summary , we have reported threshold error rates for the family of triangular 4.8.8 topological color codes , making use of an efficient fault - tolerant decoding algorithm based on graph matching . in the important case where one accounts for syndrome errors , our results accord well those in refs . , giving further evidence that thresholds for color codes are significantly lower than for the surface code .this would largely undermine the appealing features of color codes , namely their relatively high error - correcting rate and transversal implementation of the entire clifford group .on the other hand , thresholds for color codes are still significantly higher than the highest thresholds reported for concatenated codes assuming that qubits constrained to a one- or two - dimensional array with local interactions .  with respect to the decoding algorithm, there is still the prospect of improvement .for example , our simulations indicate that the full algebraic distance of the codes is not achieved .this is due to two reasons : firstly , the syndrome measurement circuits introduce correlated errors that are not accounted for in the decoding algorithm .secondly , the decoding algorithm fails for certain problematic error configurations , irrespective of these correlated errors .for example , fig .[ fig4 ] illustrates that fewer than errors may result in a logical error in some instances .addressing these failings will be necessary to determine the optimal performance of topological color codes for fault - tolerant error correction .nevertheless , our results indicate that decoding by graph matching is a feasible approach in this context .

Abstract: topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire clifford group , making them promising candidates for fault - tolerant quantum computation . recently , several efficient algorithms for decoding the syndrome of color codes were proposed . here , we modify one of these algorithms to account for errors affecting the syndrome , applying it to the family of triangular 4.8.8 color codes encoding one logical qubit . for a three - dimensional bit - flip channel , we report a threshold error rate of 0.0208(1 ) , compared with 0.0305(4 ) previously reported for an integer - program - based decoding algorithm . when we account for circuit details , this threshold is reduced to 0.00143(1 ) per gate , compared with 0.00672(1 ) per gate for the surface code under an identical noise model .


to make an application personal or more secure , we need to be able to distinguish a person from everyone else .biometric features are a popular way for authentication , which can not be imitated by any other than the desired person himself .many works revolve around identification and verification of biometric data including , but not limited to , faces , fingerprints , iris patterns , and palmprints - .iris recognition systems are widely used for security applications , since they contain a rich set of features and do not change significantly over time .there have been many algorithms for iris recognition in the past .one of the early algorithms was developed by john daugman that used 2d gabor wavelet transform . in a more recent work ,kumar proposed to use a combination of features based on haar wavelet , dct and fft to achieve high accuracy . in , farouk proposed a scheme which uses elastic graph matching and gabor wavelet .each iris is represented as a labeled graph and a similarity function is defined to compare the two graphs . in , belcher used region - based sift descriptor for iris recognition and achieved a relatively good performance .pillai proposed a unified framework based on random projections and sparse representations to achieve robust and accurate iris matching . in a more recent work ,minaee proposed an algorithm based on textural and scattering transform features , which achieved a high accuracy rate compared to the previous schemes .this approach also extracts features from a multi - layer representation , but with predefined filters .the reader is referred to for a comprehensive survey of iris recognition .  inmost of iris recognition works , first the iris region is segmented and then it is mapped to a rectangular region in polar coordinates .after that , the features are extracted from this region .most of these features are hand - crafted to work well for a given type of data or biometrics .the main issue with the traditional approaches is that they require a lot of pre - processing and parameter tuning to work reasonably well for a given dataset , and their efficiency on other biometrics or even a different dataset of the same biometric is not guaranteed . to overcome this issue ,there has been a lot of effort in the past few years to learn some general features which can be transferred across many tasks .along this direction , deep neural networks have achieved state - of - the - art results on various datasets , most notably alexnet , which is trained on imagenet competition ( which contains around 1.2 million hand - labeled images for 1000 categories ) . in the deep learning framework ,the images are fed as the input of the multi - layer neural network and the network discovers the best way to combine the pixels for maximizing the recognition accuracy . in , through an experimental study , it is shown that the features learned by training a deep network on image recognition , can be transferred to other tasks and datasets and achieve a remarkable performance . since then , features from different networks , such as alex - net , zf - net , vgg - net and res - net - have been used for various tasks , such as texture synthesis , object detection and , image segmentation .  in this work, we explored the application of deep features extracted from vgg - net for iris recognition task .we treat the trained model as a feature extraction engine and use it for feature extraction from iris images , without any fine - tuning to see if the general features are applicable for biometric recognition .then pca is applied to reduce feature dimensionality and then multi - class svm is used to perform recognition .we have provided extensive experimental study on two well - known iris datasets , casia 1000 iris dataset and iit iris dataset .four sample iris images from casia-1000 dataset are shown in figure 1 .it is worth mentioning that in our framework , we have skipped the segmentation step to see the robustness of these features to intra - class variation , and despite the existence of many variations in casia-1000 dataset , this algorithm has achieved a very high - accuracy rate .it is quite interesting that , although vgg - net is trained to classify objects from different categories , the cnn feature from this network works reasonably well for iris recognition which is to classify iris images of different subjects ( i.e. all images belong to the same object category , here iris ) .  the rest of the paper is organized as follows .section [ sectionii ] describes the deep features that are used in this work , and gives a quick overview of vgg - net architecture .section [ sectioniii ] provides a brief description of the dimensionality reduction and classification scheme used in this work .the experimental results and comparisons with other works are presented in section [ sectioniv ] , and the paper is concluded in section [ sectionv ] .extracting good features and image descriptors is a very important step in many computer vision and object recognition algorithms .many features have been designed during the past years , which provide good representation for many image categories , to name a few , scale - invariant feature transform ( sift ) , histogram of oriented gradient ( hog ) , and bag of words ( bow ) - .these features are mostly designed by computer vision experts , and are usually referred to as hand - crafted features .different applications , such as medical image analysis - , may use a very different set of hand - crafted features .recently , feature - learning algorithms and multi - layer representation have drawn a lot of attention , most notably convolutional neural networks , where the image is fed directly as the input to the deep neural network and the algorithm itself finds the best set of features from the image .deep architecture based on convolutional neural networks achieved state - of - the - art results in many computer vision benchmarks during past few years , including alexnet , zf - net , googlenet , vgg - net and resnet , which achieved promising results in imagenet competition in 2012 and 2015 .the features learned from these networks are shown to be well transferred to many other datasets and tasks , which means the trained network on imagenet images is used to extract features from other dataset and then a classifier is trained on top of these features to perform recognition in other datasets , such as caltech101 or caltech256 .they have also been explored for a wide range of other computer vision tasks , such as segmentation , object tracking , texture synthesis , colorization , and super - resolution -.  it would be interesting to analyze the application of deep features learned by training a convnet , for the iris recognition problem , which is different from object recognition in the sense that the goal is not to distinguish one type of object from another , but to distinguish images of the same kind belonging to different people . in this work ,the application of deep features , extracted from vgg - net , is explored for iris recognition , which to the best of our knowledge has not been studied yet . herewe give a quick overview of vgg - net .vgg - net was the runner - up in ilsvrc 2014 ( imagenet large scale visual recognition competition ) , proposed by karen simonyan and andrew zisserman .figure 2 shows the overall architecture of vgg - net .its showed that the depth of the network is a critical component for good performance .their final best network contains 16 conv / fc layers , and 5 pooling layers . in total, it has around 138 million parameters ( which most of it is in the first fc layer that contains 102 m weights ) .one of the nice thing about vgg - net is its extremely homogeneous architecture , which only performs 3x3 convolutions with stride 1 and pad 1 , and 2x2 pooling ( with no padding ) from the beginning to the end .we refer the reader to and for further details about the architecture and number of parameters of each layer of this network .we extract features from different layers of this network and evaluate their performance for iris recognition .we further perform pca on the deep features to reduce their dimensionality , and evaluate the performance of the proposed algorithm for different feature dimensions .principal component analysis ( pca ) , is a simple yet powerful algorithm for dimensionality reduction . given a set of correlated variables ,pca transforms them into another domain such the transformed variables are linearly uncorrelated .pca has many applications in computer vision , most notably in eigenface , where pca is used for face recognition .pca can be thought of as fitting a -dimensional ellipsoid to a set of data , where each axis of the ellipsoid represents a principal component .there are also others dimensionality reduction algorithms which are designed based on pca , such as kernel - pca and sparse - pca , .  without going into too much detail ,let us assume we have a dataset of iris images and denote their features .also let us assume that each feature has dimensionality of . to apply pca , all features need to be centered first by removing their mean : where .then the covariance matrix of the centered images is calculated as .+ next the eigenvalues and eigenvectors of the covariance matrix are computed .suppose s are ordered based on their values .we can reduce the dimensionality of the data by projecting them on the first principal vectors as : by keeping principal components , the percentage of retained variance can be found as : .we would like to mention that there are also other kinds of deep architectures , which do not learn the filters and use predefined weights .scattering convolutional network is one such network that decomposes the signal into multiple layers of different scales and orientations .then some statistical features are extracted from each transform image , and features from all transformed images are concatenated to form the overall feature representation .the scattering network has also been used for biometric recognition in the past - . to illustrate how this network works , the output of the second layer of scattering network for a sample iris imageis shown in figure 3 .as it can be seen , each image contains the edge information along some orientation in some scale .however , since the filters are not learned and are fixed across different tasks , this architecture may not be the optimum one for some of the visual applications .yet , they could be very useful for the cases the amount of training data is very limited and is not enough to train and convolutional neural net .after feature extraction , a classifier is needed to find the corresponding label for each test image .there are various classifiers which can be used for this task , including softmax regression ( or multinomial logistic regression ) , support vector machine and neural network . in this work ,multi - class svm has been used which is quite popular for image classification .a brief overview of svm for binary classification is presented here . for further details and extensions to multi - class settings and nonlinear version ,we refer the reader to and .let us assume we want to separate the set of training data , , ... , into two classes where is the feature vector and is the class label .assuming the two classes are linearly separable , we can separate them with a hyperplane . among all possible choices of hyperplanes svmfinds the one with the maximum margin .one can show that the maximum margin hyperplane can be found by the following optimization problem : the above optimization is convex , and therefore it can be solved with convex optimization techniques .one popular way is to solve the dual problem by introducing lagrange multipliers , which results in a classifier as , where s and are calculated by the svm learning algorithm .there is also a soft - margin version of svm which allows for mislabeled examples . to perform multi - class svm for a set of data with classes, we can train binary classifiers which can discriminate each class against all other classes , and to choose the class which classifies the test sample with the largest margin ( one - vs - all ) . in another approach, we can train a set of binary classifiers , each of which separates one class from another one and then choose the class that is selected by the majority of classifiers .there are also other approaches to perform svm for multi - class classification .in this section , a detailed description of our experimental results is provided . before presenting the results ,let us describe the parameter values of our algorithm . for each image , we extract features from different layers of vgg - net , from fc6 and some of the previous layers .the output of fc6 layer is a 4096 dimensional vector , but in the layers before that , we have 256/512 filter outputs of different sizes .we take the average of each filter output and form a feature vector from each layer to evaluate the performance of features extracted from different layers . for svm, we have used the libsvm library , and linear kernel is used with the penalty cost .we have tested our algorithm on two iris databases , casia - iris-1000 and iit delhi .casia - iris-1000 contains 20,000 iris images from 1,000 subjects , which were collected using an ikemb-100 camera .the main sources of intra - class variations in casia - iris-1000 are eyeglasses and specular reflections .the iit delhi database contains 2240 iris images captured from 224 different people .the resolution of these images is 320x240 pixels .we resize all images to 224x224 to be suitable for vgg - net input .  for each person , around half of the imagesare used for training and the rest for testing .we first evaluate the recognition accuracy of features from the fc6 layer .first pca is applied to all features and the recognition accuracy for different number of pca features evaluated .figures 4 and 5 show the recognition accuracy using different numbers of pca features for iit delhi and casia datasets respectively .interestingly , even by using few pca features , we are able to get a very high accuracy rate . as it can be seen , using 100 pca features results in an accuracy rate above 98% for iit database , which only increases around 1% by using more pca features .in another experiment , we evaluated the performance of deep features extracted from different layers of vgg - net . to have a fair comparison , we restrict the number of features from each layer to be 256 ( by taking first 256 pca , if that layer has more than 256 output filters ) . figure 6shows the recognition accuracy achieved by features from different layers of vgg - net on iit delhi database .  as it can be seen , by extracting features from any layer after the 7th , an accuracy rate of above 98% can be achieved .the recognition accuracy achieves its peak by using features from the 10th layer and then drops .one possible reason could be that by going into the higher layers of the deep network , they start to capture more abstract and high - level information which does not discriminate much between different iris patterns , whereas the mid - level features in the previous layers have more discriminating power for same - class recognition .we have also evaluated the robustness of the proposed scheme to the number of training samples used for training .we varied the number of training samples for each person from 1 to 5 ( out of 10 samples ) and found the recognition accuracy .figure 7 shows the accuracy of this algorithm for different training samples for iit database .as it can be seen , the recognition accuracy by using 3 samples out of 10 , has a large gain over using 1 or 2 samples , and it remains relatively constant by increasing the number of training samples .table 1 provides a comparison of the performance of the proposed scheme and those of other recent algorithms on iit database .the scattering transform scheme [ 10 ] also uses a multi - layer representation and achieves a very high accuracy rate . by using the deep features, we were able to achieve the highest accuracy rate on this dataset .this is mainly due to the richness of deep features which are able to capture many of the information lost in hand - crafted features , providing a very high discriminating power .one main advantage of this scheme is that , it does not require segmentation of iris from eye images although the segmentation could improve the results for some difficult cases .[ h ]   haar wavelet &  96.6% + log gabor filter by kumar & 97.2% + fusion &  97.4% + elastic graph matching & 98% + texture+scattering features & 99.2% + * proposed scheme * &  * 99.4% * +  [ tblcomp ]  the experiments are performed using matlab 2015 on a laptop with core i5 cpu running at 2.2ghz . for deep learning features ,the matconvnet package is used to extract the features from vgg - net .in this work , we evaluated the application of deep features , extracted from vgg - net , followed by a simple classification algorithm for the problem of iris recognition .deep features have been in the center of attention during these years and are being used for many different applications .although the original convolutional network used in this work is trained for a very different task ( object recognition ) , it is shown that the features can be well transferred to biometric recognition .this algorithm is tested on two well - known datasets and achieved promising results , which outperforms the previous best results on one of the databases .we want to note that most of the previous algorithms for iris recognition involve a lot of pre - processing and parameter tuning , but in our framework , no pre - processing and architecture optimization is performed .this result could be further improved by training a deep network specifically for iris recognition , which is left for future research .the authors would like to thank vadaldi s group at oxford university for providing the software implementation of convolutional neural network , and also csie group at ntu for providing the libsvm package .we would also like to thank casia center and iit delhi for providing the iris databases .w. zhao , r. chellappa , pj .phillips and a. rosenfeld , `` face recognition : a literature survey '' , acm computing surveys ( csur ) 35 , no . 4 : 399 - 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Abstract: iris is one of the popular biometrics that is widely used for identity authentication . different features have been used to perform iris recognition in the past . most of them are based on hand - crafted features designed by biometrics experts . due to tremendous success of deep learning in computer vision problems , there has been a lot of interest in applying features learned by convolutional neural networks on general image recognition to other tasks such as segmentation , face recognition , and object detection . in this paper , we have investigated the application of deep features extracted from vgg - net for iris recognition . the proposed scheme has been tested on two well - known iris databases , and has shown promising results with the best accuracy rate of 99.4% , which outperforms the previous best result .


multivariate time series modeling is an important component for the quantitative assessment of relationships between variables in many applied areas .this issue is essential in financial applications , for example , enabling optimal portfolio allocation , setting trading strategies over sectors of the market , or exchanging rates .in addition , the vector autoregressive model ( var ) is widely used in many fields such as economics , geophysics , bioinformatics and neuroscience .the main reasons for the attractiveness of the var model in applied areas are its simplicity and relation with the concept of granger causality .granger causality has become a prominent concept in connectivity networks modeling , because it provides inferences about the direction of information flow between different time series .several studies in biological systems emphasize the importance of identification and description of gene regulator networks , mainly in the study of tumors or structural diseases . introduced the utilization of var - based models to study these issues by applying these models to gene expression datasets . in neuroscience , the _ functional integration _theories highlight that brain functions heavily depend on neural connectivity networks .several neuroimaging studies suggested that var models and granger causality are suitable to identify the information flow between neural structures .nevertheless , it is well known that most biological measurements are subject to error , since the precision of acquisition equipments is never absolute . actually , this limitation is present in most studies involving experimental data , such as chemistry , physics , biometrics , etc .although technically incorrect , the most common procedure is simply to ignore the measurement errors , i.e. : assume that the variables of interest are the observed variables .it is important to highlight that this assumption has serious implications .the utilization of conventional var model in this case would not identify correctly the relationships between the variables of interest ( latent variables ) .it happens because the model white noise will not be independent which leads to misestimations of the model parameters .the usual assumption is acceptable when the errors are negligible .however , it is known that due to acquisition processes limitations , the measurement errors in biology ( e.g. : gene expressions or brain signals ) are not negligible . in these cases ,the utilization of conventional var models may result in biased parameter estimation and as a consequence , unreliable granger causality detection .  in the following ,we define the usual var model ( for a more detailed description , see for instance , * ? ? ?let denotes a vector of time series variables .the usual var(r ) model has the form where is the sample size , for are coefficient matrices and is a unobservable zero mean white noise vector process with covariance matrix .for convenience , we consider that for all .we are assuming throughout this paper that model ( [ var ] ) satisfies the stability condition defined in on page 12 .therefore , under stationarity conditions , the mean and the autocovariance function are given , respectively , by =\sum_{j=1}^r \bm{b}_j \bm{\gamma}(h - j ) , \quad \mbox{for } \ \  h = 1,2,3 , \ldots\ ] ] and where denotes the identity matrix and .model ( [ var ] ) can be written in short as where is a matrix and .  therefore , if the white noise has normal distribution , the conditional maximum likelihood ( ml ) estimators of , and are equal to the ordinary least squares estimators .they are given , respectively , by where , , , and .the consistence of those conditional ml estimators is assured under the stationary conditions ( see * ? ? ? * for further details ) .the consistence is shown using the fact that   where " denotes convergence in probability when the sample size increases , denotes the kronecker product , is a column vector of ones , and the covariance function of is given by \\ & = & \left [ \begin{array}{cccc } \bm{\gamma}(h )   & \bm{\gamma}(h+1 )  & \ldots & \bm{\gamma}(h+r-1)\\ \bm{\gamma}(h-1 )  & \bm{\gamma}(h )   & \ldots & \bm{\gamma}(h+r-2)\\\vdots       & \vdots       & \ddots & \vdots \\ \bm{\gamma}(h - r+1 ) & \bm{\gamma}(h - r+2 ) & \ldots & \bm{\gamma}(h ) \end{array}\right].\end{aligned}\ ] ]  as described previously , var modeling is commonly applied for detecting granger causality relationships .the basic idea of granger causality is the evaluation of temporal information founded on the assumption that the cause always precedes its effect .let and be two time series . from the statistical perspective , is said to granger - cause if the prediction error of , conditioning on the past values of both series , is less than considering solely the past values of .in other words , the past values of contains relevant information to improve the predictions of .note that granger causality concept is not equivalent to classical aristotelian causality , since the former is based solely on prediction errors .however , due to its simplicity , it may be applied to identify possible effective causalities .one possible approach of using var models for granger causality detection is by performing statistical tests on s coefficients .considering equation , if there is at least one coefficient multiplying the past values of which is not equal to zero , then is said to granger - cause .thus , this procedure involves the estimation of , their respective covariance matrices , and the application of hypothesis testing .  in general, many physical , biological and chemical variables have the measurement process subject to random effects and it is very common analyze them by using models assuming that these measurement errors are negligible. it may bring up undesirable features as biased estimates as well as their standard errors and , as a consequence , dangerously false confidence intervals and hypotheses testing will often be obtained using such approach .thus , it is necessary to consider the measurement error on the modeling of these type of time series .  in this paper, we study a var model with main concern on including measurement errors .let be the true variable that is not directly observed , instead a substitute variable is observed which has an additive structure given by    where is the observed vector and is the measurement error vector with mean zero and variance - covariance matrix . in most cases , if the usual conditional ml estimator is adopted for the observations subject to errors , i.e. , replacing with in the equation ( [ var ] ) , the estimator of will be biased ( as can be seen in ( [ olsp ] ) ) .therefore , in order to overcome this limitation the measurement errors should be included in the estimation procedure . nevertheless , the model ( [ var ] ) with the equation ( [ erro ] ) is not identifiable , since the covariance matrices of and are confounded when .it is easy to see that in the univariate ar(1 ) , note that when and we have : with , and for all .it is impossible to estimate and separately by observing only .this problem can be avoided by using previous knowledge about the variance of .this paper is organized as follows .section 2 proposes consistent estimators for the var model with measurement errors and also presents the asymptotic distribution of the estimator of the elements of . in section 3, simulation studies are undertaken to investigate some aspects of the proposed estimators ( rejection rates for a test of hypothesis , biases and mean square errors ) also it is verified the impact by erroneously considering the usual model .we applied the models in a functional magnetic resonance imaging dataset in section 4 and we finish the paper with conclusions and remarks in section 5 .in the presence of measurement errors , the conventional ml estimation of var models produces biased estimators and they can lead to wrong statistical inference ( see * ? ? ? * in which it is found a discussion over errors - in - variables in regression models ) .there are some studies about measurement errors in times series ( e.g. , * ? ? ?* ; * ? ? ?those studies use kalman filtering methodology and an expectation and maximization algorithm that requires intensive iterative procedures . provided a careful expose of the identifiability of some time series models with errors in variables . describes approaches based on state space modeling and kalman filtering and demonstrates the usefulness of these tools in dynamic models . show the efficiency gains adopting an errors - in - variables model , and the precision of kalman filter estimates in the face of autocorrelation .these measurement techniques have been applied to a variety of substantive problems , including dynamic representation , social problems ( such as racial inequality ) , monetary policy and public entrepreneurship ( * citar * ) .these state space models can be attractive alternatives to conventional var modeling .however , in practice , the implementation of the estimators are not described in analytical form , but by interactive algorithms or numerical optimization solutions .in addition , the derivation of estimators convergence , standard errors , consistence and asymptotic distribution may be complex in these cases . in ,the section on state space methods shows an alternative procedure for how to estimate , and under model ( [ var ] ) with the error equations ( [ erro ] ) , using the em algorithm . another iterative procedure to estimate these parameters .nevertheless , as the main goal of this paper is to test granger causality , these approaches can not be used , since the model becomes unidentifiable under the hypothesis .  in this study, we provide simple and closed forms for the estimators when is known , which allows the direct derivation of their respective asymptotic properties .since the main concern of several practical applications is granger causality testing , this information is essential to data analysis . in this section ,the main concern is the parameter estimation and its asymptotic properties .theorem [ esthomo ] states consistent estimators for the model parameters and theorem [ asympesthomo ] establishes the asymptotic distribution for the estimator of given in theorem [ esthomo ] , where is an operator that heaps the columns of the matrix .the methodology presented in this section is based on correcting the asymptotic bias of conventional ml estimator caused by the measurement error effect .the outcome is a consistent estimator with good asymptotic properties such as normality .the estimators and the asymptotic covariance matrix for the proposed estimator of are computed easily and no iterative procedure is required .we must remark that those estimators are not the conditional ml estimators nor the ml estimators taking into account the measurement errors which are very complicated to reach by maximizing the likelihood , even under normality of the errors .[ esthomo ] if with known .then , the parameters of the model ( [ var ] ) under measurement errors as in ( [ erro ] ) have consistent estimators given by  ^{\top}\ ] ]  and   where , , and .  the proof of theorem [ esthomo ] can be found in appendix [ a ] .notice that , if , that is , when there is no measurement error , then the estimators of theorem [ esthomo ] become the conditional ml estimators presented in ( [ vest ] ) .also , it can be seen that the conditional ml estimator of from the model ( [ var ] ) , without considering the errors ( [ erro ] ) , is given by ^{\top},\ ] ] which is not consistent , since  ^{-1}.\ ] ]  the main steps to demonstrate ( [ olsp ] ) is given in appendix [ a ] , in which is sufficient to compute the limit of and .the quantity has two sources of variations , one that refers to the unobservable variable and another one that refers to the measurement error . if the measurement error is huge and the sample size is not large enough , the quantity may not be positive definite and the estimator , presented in ( [ beta ] ) , will be inadmissible. if the quantity has at least one eigen value close to zero the estimator , presented in [ beta ] , will be unstable ( because the computation of a matrix inverse requires all eigen values to be different from zero ) .if the matrix is well specified , one way to avoid such inadmissibility and instability is increasing the sample size .  in many practical applications ,there is some interest on testing some elements of the matrix ( e.g. , the so called granger causality test ) .however , the exact distribution of is hard to compute .thus , one can use its asymptotic distribution to build confidence regions and hypothesis testing as an approximation when the sample size is finite .the theorem below gives us the asymptotic distribution of .[ asympesthomo ] if with known and for all , where is the element of .then , the asymptotic distribution of obtained in theorem [ esthomo ] is given by    where the matrix is given by    where   + \\ & &  - \sum_{h=1}^{r}\bigg\ { ( \bm{b}_{h } \bm{\sigma}_{e})\otimes\bm{\gamma}_r(h ) + ( \bm{\sigma}_{e}\bm{b}_{h}^{\top})\otimes\bm{\gamma}_r(-h)\bigg\ } + \\ & & +  \sum_{h=1-r}^{r-1 } [ \bm{b}(\bm{j}_{-h } \otimes \bm{\sigma}_e)\bm{b}^{\top}]\otimes \bm{\gamma}_r(h).\end{aligned}\ ] ]  and , where is a ( ) matrix of zeros with one s in the diagonal above ( below ) the main diagonal if ( ) and is a ( ) matrix of zeros .the proof of theorem [ asympesthomo ] can be seen in appendix [ b ] . for all and we have , as given in . the normal distribution assumption for the measurement erroris required to compute the expectation of polynomial functions ( until forth degrees ) of the elements of .notice that , if we have the model and the asymptotic covariance simplifies to   where  .\ ] ]  the element of , is asymptotically normally distributed with standard error given by the square root of diagonal element of .thus , we can obtain hypotheses tests on the individual coefficients , or more general form of contrasts   which involves coefficients across different equations of the var model .thus , granger causality testing can be carried out by adequately specifying this contrasts matrix .an illustrative example is the case of series and , in which we are interested in evaluating the granger causality from to in an -order var model .the matrix has rows , one for each coefficient related to the past values of in the equation .considering that each column of refers to each var coefficient , the contrast matrix is specified by simply setting 1 to the cell at the respective column and row for the coefficients in equation .this may be tested using the wald - type statistic conveniently expressed as  ^{-1}(\bm{c}\mbox{vec}(\widehat{\bm{b}}^{\top})-\bm{d})\ ] ]  under the null hypotheses , ( [ wald ] ) has a distribution in the limit , where gives the number of linear restrictions .the above study can also be developed to the intercept model estimator , it can be found by applying the delta method in the asymptotic distribution of ( , , ) , since .although , this asymptotic distribution is important to test hypotheses regarding the model intercept , it is outside the main scope of this article and does not have any impact on the granger causality , for this reason we skip it .in this section we conduct some simulation studies in order to evaluate the adequacy of the asymptotic distribution of for small and moderate samples sizes .computations were performed using the software r ( www.r-project.org ) . for each setup of parameters and sample sizes, we considered monte carlo samples generated from a var(1 ) model with measurement errors , given by  {z_{1,t-1 } \choose z_{2,t-1 } } + { q_{1t}\choose q_{2t}},\\ { z_{1,t } \choose z_{2,t } } & = & { z_{1,t } \choose z_{2,t } } + { e_{1t}\choose e_{2t}}.\end{aligned}\ ] ]  in all samples , we have considered the following setup of parameters : , ,  ,\ ] ] where the vector parameters values of ( ) were the values of the set , where , the variance of the measurement error was , and the size samples .the rejection rates of the hypothesis ( i.e. , does not help to explain and does not help to explain ) are shown in table [ tab : n ] , in which the test sizes are the rejection rates under the null hypothesis ( that appears in bold ). the wald - type statistics ( [ wald ] ) is used at 5% nominal level . from this tablewe have that , the test sizes from the proposed model are closer to the nominal level ( 5% ) as compared to the usual approach for all sample sizes .furthermore , when increases the test sizes for the usual model also increase and , consequently , they do not converge to the adopted nominal level .this is a somewhat expected behavior because the usual approach produces biased estimates and standard errors .table [ tab : n ] depicts the power of the test in each methodology , which shows a good performance of the proposed approach .nevertheless , it is not possible to compare the power between the two methods because they have different empirical test sizes .  [ [ table [ tab : n ] ] ]  we observe that ,the results shown in table [ tab : n ] are similar for other values of the parameters and , if we maintain the same proportionality of and as defined above .but , our simulations suggest that the larger the measurement error , the larger the sample size required to have a good asymptotic approximation for the wald - type statistics ( [ wald ] ) .  we also conduct simulation studies for testing the simple hypothesis at 5% nominal level . in this study , we keep fixed the value of .others simulations were built considering others values for , however , the results are close to each other and , for this reason , we omit them . as can be seen , tables [ tab : n ] and [ reject ] present similar behaviors , i.e. , the proposed model has always empirical size test closer to the nominal level than the usual model .[ [ table [ reject ] ] ]  in table [ tab : n ] and [ reject ] , the usual approach seems to be most powerful than the proposed approach when and .however , as aforementioned , they can not be compared directly , just because the real nominal level used to compute that powers are not the same .thus , we used a descriptive measure in order to analyze both methodologies around the null hypothesis .let be the probability of the error type i using the true distribution of ( [ wald ] ) when the sample size is and is the adopting nominal level based on its asymptotic distribution .for instance , in table [ reject ] we have estimated for the proposed approach and for the usual approach .an expected behavior for good statistics is which means that the quantiles of the true distribution of ( [ wald ] ) will be close to the quantiles of the asymptotic distribution , , when the sample size is sufficiently large .thus , the relation tell us how far is the -quantil of the asymptotic distribution from the true distribution of ( [ wald ] ) for each .therefore , we can define a sort of corrected power as   where is the power using the true probability of the error type i , namely .we are just penalizing the power by the distance between and .notice that , the power under the null hypothesis has to be the nominal level and for comparing powers from different statistics it must be done using the same nominal level .let and be the true probability of the error type i for two different statistics when the sample size is . then , under the null hypothesis , we have   and hence , the corrected powers and are comparable . moreover , under an alternative hypothesis and when increases , an expected behavior of is to converge to one .although , this corrected power is not a monotonic function of the sample size nor of the nominal level , we believe that it is a kind of descriptive measure to evidence how unsuitable is the usual model when compared with the proposed one outside the null hypothesis . furthermore , the proposed corrected power varies between and infinity .figure [ fig:1 ] shows the corrected power for both approaches , the null hypothesis was .the full line refers to the proposed approach and the dashed line refers to the usual one .the panels ( a.1 ) , ( b.1 ) , ( c.1 ) and ( d.1 ) refer to the corrected power when the alternative hypothesis are , , and , respectively at .the panels ( a.2 ) , ( b.2 ) , ( c.2 ) and ( d.2 ) refer to the corrected power when the alternative hypothesis are , , and , respectively at . the panels ( a.3 ) , ( b.3 ) , ( c.3 ) and ( d.3 ) refer to the corrected power when the alternative hypothesis are , , and , respectively at .we observe in all graphs that , the usual approach has the worst performance ( going to zero when the sample size increases ) while the proposed one have an expected behavior for a good statistic ( going to one when the sample size increases ) . in general, the corrected power under the usual methodology goes to zero because the distance between and increases much faster than the uncorrected power , , when increases .this behavior is still true for another setup of parameters .[ [ figure [ fig:1 ] ] ]  [ [ table [ vicioeqm ] ] ]  table [ vicioeqm ] shows that , for this set of parameters , the biases of the estimators of ( ) from the proposed model is smaller than the value supplied by the usual model .the larger the sample size , the smaller the bias and mse under the proposed model ( this does not happen for the usual approach ) .as previously described , the models with measurement errors have great relevance in applied sciences , since equipment imprecisions are inherent to data acquisition .actually , the usual models are commonly applied ignoring these errors .nowadays , the scientific community started to pay enough attention to the fact that these procedures may lead to spurious results . in this section, we illustrate the concepts introduced in the present study with an application embedded in neuroscience research , with the utilization of var modeling for the characterization of brain networks .the dataset explored in this application is proceeding from a functional magnetic resonance imaging ( fmri ) experiment .basically , fmri acquisition is based on monitoring the bold signal ( blood oxygenation level dependent ) at several brain regions through time .one of the main advantages of fmri over other imaging techniques is its non - invasive protocol and relative high spatial resolution .the bold signal is related to oxygen consumption and blood flow , being considered as an indirect measure of local neural activity ( ) . regarding this property ,this signal is used to quantify and locate the brain activity in humans .  in this study ,the bold signals at four brain regions from a subject in a resting state ( eyes closed ) condition were considered .the data was collected in a siemens 3tesla mr system ( tr=1800ms , ta=900ms , te=30ms ) .the selected brain regions were : left primary motor cortex ( left m1 ) , right primary motor cortex ( right m1 ) , supplementary motor area ( sma ) and right cerebellum .the anatomical location of this areas is shown in figure [ figareas ] .these areas are frequently involved in active and planned right hand fingertapping .we aim to evaluate the information flow between these areas in a resting state condition by using var models for granger causality identification .a well described limitation inherent to all fmri acquisition is the high level of scanner noise .thus , the signals observed mirror not only the physiological variations but also includes measurement errors . for this specific dataset, it was estimated that the error composed approximately of the observed time series standard deviation . for simplicity , each observed series were normalized to have mean zero and variance one .thus , the measurement error was considered to be serially uncorrelated , independent of the latent variables and with a standard deviation of .the model considered for the latent variable is given by where is the time series length , with the _ left m1 _ bold signal , the _ sma _ bold signal , the _right m1 _ bold signal and the _ right cerebellum _bold signal ; is the autoregressive coefficients matrix    and is an unobservable zero mean white noise vector .the observed variables are given by    where and is the measurement error vector .the time series plots corresponding to the respective observed bold signal at each brain region are represented in figure [ figsignal ] .since we are interested in identifying the links of connectivity networks using granger causality , the statistical inferences are related to the parameters  .if , then there is a information flow from brain area to area ( ) .the coefficient estimates , standard errors and p - values ( vs ) for both usual and proposed approaches are shown in tables [ app1 ] and [ app2 ] , respectively .[ [ figure [ figareas ] ] ]  [ [ figure [ figsignal ] ] ]  [ [ figure [ fignetwork ] ] ]  [ [ figure [ figqqapp ] ] ]  [ [ table [ app1 ] ] ]  [ [ table [ app2 ] ] ]  the results described in tables [ app1 ] and [ app2 ] suggest the existence of bidirectional information flow between left m1 and cerebellum .however , the application of usual approach indicates also that left m1 sends information to sma and right m1 , and that the latter sends to sma .for both usual and proposed approaches , the diagrams of the networks at the significance level of 5 are shown in figure [ fignetwork ] .as highlighted by the simulations results , the utilization of usual var estimation , ignoring the measurement errors , may result in wrong test nominal sizes . in this context , it is important to mention that the main differences between the usual and proposal results were on standard deviation estimates .further , the proposal estimates are almost twice the values resulting from usual approach .the theory and simulations suggest the existence of biases in the latter .consequently , the p - values from the usual method tend to be underestimated , resulting in high rejection rates .note that this connections may possibly exist , but since the nominal level of the test is `` incorrect '' , the type i error is not under control .in addition , note that some coefficients were considerably underestimated , for example , and . finally , the qq - plots represented in figure [ figqqapp ] suggest that the probability density of residuals are reasonably approximated by the normal distribution .some studies ( ) suggest the existence of functional networks between motor areas even in resting state condition .these studies are based on correlation analysis between the bold signal at different brain sites .first , it is important to note that granger causality is conceptually different from correlation , which is symmetric ( it does not provide the direction of information flow ) , evaluated in a pairwise fashion ( and not in the full multivariate sense ) and it does not take into account temporal information .in fact , correlation analysis is more closely related to instantaneous granger causality concept , which can be useful to quantify simultaneity between time series but it is unsuitable in the context of information flow detection .second , the usual correlation analysis does not consider the presence of measurement errors , which may also affect the statistical significance of results .the nature of functional networks in resting state is still unclear and is the subject of several studies ( ) .nevertheless , we have demonstrated in this study that the inclusion of measurement errors can considerably influence the final results .thus , the development of novel approaches dealing with this artifact is necessary .  in summary ,since the proposal and usual results differ , we conclude that the presence of measurement error can not be ignored .an adequate treatment for this artifact is essential for the adequate description and modeling of brain networks .it is surprising that this important limitation received proper attention only recently .we believe that a preliminary analysis of this problem points toward the demand for the development of new estimation procedures regarding scanner noise characterization , physiological noise and computational implementation .this paper has introduced a new approach to model multivariate times series when measurement errors are present .the simulation studies indicate that the proposed approach gives coherent results ( test size close to the nominal level even for small samples , power increasing with the sample size under alternative hypotheses , biases and mean square errors decreasing when the sample size increases ) under small and moderate measurement error .such features seem no to be shared by the conventional maximum likelihood estimators which presents a much poorer performance .furthermore , the proposal is easily attained and iterative procedures are not required .the theory , simulations and application showed that the presence of measurement error can not be neglected and a proper model has to be considered for the adequate description and modeling of brain networks .we expect to report generalizations of the proposed model ( for elliptical errors and heteroscedasticity situations ) , a residual study and more simulation studies for large measurement errors on incoming papers . we must study the limits of the quantities , , and when the sample size goes to infinity . note that , where , and under the stationary conditions of a model we have that   where , and means limited in probability even multiplying by ( it happens with the crossing product in the above expression ) .that is , . following the same scheme , we have that      the proof idea has three steps .the first step consists in show that can be written as linear combinations of a vectorial mean .the second one , we must demonstrate that this vectorial mean has an asymptotic normal distribution .the last step must conclude that also has an asymptotic normal distribution . in order to prove theorem [ asympesthomo ], we need some auxiliary results , which are exposed in two propositions below .* proof : * define as a vector ( ) of coefficients associated with the element of the vector , that is thus , we have that and the estimator of theorem [ esthomo ] for it can be written as , where and for . moreover , the model ( [ varcol ] ) may be rewritten in terms of the observed variables as           [ prop2 ] if with known and for all , where is the element of .the mean , , of proposition [ prop1 ] has an asymptotic distribution given by    + \\ & &  - \sum_{h=1}^{r}\bigg\ { ( \bm{b}_{h } \bm{\sigma}_{e})\otimes\bm{\gamma}_r(h ) + ( \bm{\sigma}_{e}\bm{b}_{h}^{\top})\otimes\bm{\gamma}_r(-h)\bigg\ } + \\ & & +  \sum_{h=1-r}^{r-1 } [ \bm{b}(\bm{j}_{-h } \otimes \bm{\sigma}_e)\bm{b}^{\top}]\otimes \bm{\gamma}_r(h).\end{aligned}\ ] ]   * proof : * notice that the expectation of is equal to zero for all . state a central limit theorem to a univariate m - dependent sequence of random variables with mean zero .we say that a time series is m - dependent if the set of values is independent of the set of values ( * ? ? ?66 ) . then , assuming that for all where is the element of and defining , where we have that , and   + e[\bm{f}_{ih } \otimes \bm{e^*}_{i-1}\bm{e}_{i - h-1}^{\bm{*}\top } ] + \\ & + & e[\bm{f}_{ih } \otimes \bm{e^*}_{i-1 } ( \bm{z^*}_{i - h-1}-\bm{\mu^*})^{\top } ] + e[\bm{f}_{ih } \otimes(\bm{z^*}_{i-1}-\bm{\mu^*})\bm{e}_{i - h-1}^{\bm{*}\top } ] - \\ & -&\bm{\psi}\bm{\psi}^{\top}\end{aligned}\ ] ]     \otimes \bm{\gamma}_r(h ) - ( \bm{b}_{h } \bm{\sigma}_{e})\otimes\bm{\gamma}_r(h ) \qquad \mbox{for } \quad h = 1,\ldots , r-1,\ ] ] \otimes \bm{\gamma}_r(h ) - ( \bm{\sigma}_{e}\bm{b}_{|h|}^{\top})\otimes\bm{\gamma}_r(h ) \qquad \mbox{for } \quad h = -1,\ldots , 1-r,\ ] ] \qquad \mbox{for } \quad h=0,\ ] ] where is a ( ) matrix of zeros with one s in the diagonal above ( below ) the main diagonal if ( ) and is a ( ) matrix of zeros .that is , is a strictly m - dependent sequence of random variables with mean zero ( where ) and , therefore , we can use the result stated in , which says that where    + \\ & &  - \sum_{h=1}^{r}\bigg\ { ( \bm{b}_{h } \bm{\sigma}_{e})\otimes\bm{\gamma}_r(h ) + ( \bm{\sigma}_{e}\bm{b}_{h}^{\top})\otimes\bm{\gamma}_r(-h)\bigg\ } + \\ & & +  \sum_{h=1-r}^{r-1 } [ \bm{b}(\bm{j}_{-h } \otimes \bm{\sigma}_e)\bm{b}^{\top}]\otimes \bm{\gamma}_r(h).\end{aligned}\ ] ]       abler , b. , roebroeck , a. , goebel , r. , hose , a. , schonfeldt - lecuona , c. , hole , g. , walter , h. 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Abstract: this paper develops a method for estimating parameters of a vector autoregression ( var ) observed in white noise . the estimation method assumes the noise variance matrix is known and does not require any iterative process . this study provides consistent estimators and shows the asymptotic distribution of the parameters required for conducting tests of granger causality . methods in the existing statistical literature can not be used for testing granger causality , since under the null hypothesis the model becomes unidentifiable . measurement error effects on parameter estimates were evaluated by using computational simulations . the results show that the proposed approach produces empirical false positive rates close to the adopted nominal level ( even for small samples ) and has a good performance around the null hypothesis . the applicability and usefulness of the proposed approach are illustrated using a functional magnetic resonance imaging dataset . + _ _ key words : _ asymptotic property , errors - in - variables model , granger causality , multivariate analysis . _  0.3 cm  * vector autoregressive models with measurement errors + for testing ganger causality *  0.3 cm  * alexandre g. patriota , joo r. sato and betsab g. blas achic *  0.3 cm   de estatstica , universidade de so paulo - sp - brasil  caixa postal 66281 - cep 05314 - 970 , so paulo - sp - brasil   institute of radiology - hospital das clnicas , so paulo - brasil  cep 05403 - 001 , so paulo - sp - brasil