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how many data points can be missed off ( set to zero ) from a 1-d discrete , regularly spaced time series record and still be recoverable ?
such a question is prompted by many situations , often associated to the automated collection of data .
we are motivated to address this question by our experimental field work to record the behaviour of the strength of optical turbulence parameter , @xmath0 , over time intervals of several weeks , under different climate conditions @xcite .
the data of interest to us are path integrated measures of @xmath0 measured across the visible to near infrared region .
the measurement path is a 600 m stretch of free space , approximately 1.5 m altitude above sea water in the caribbean .
the probe beam originated from an led centered on 0.9 @xmath1 m ; the instrumentation are two identical optical scientific inc .
loa-004 systems , which employ aperture averaging to estimate the value of @xmath0 .
the loa-004s are generally left unattended over the course of up to a few days during field operation , which mean that they are susceptible to spurious events which result in gaps in the data record .
although for some types of data analysis techniques , data gaps of a limited size can be tolerated , this is not universal .
moreover , there exists a new class of very powerful techniques based on empirical mode decomposition @xcite that are exceedingly sensitive to lossy datasets .
it is for this reason we are exploring different methodologies to synthetically fill the data holes ; we describe the results from our studies using principal component analysis in this paper .
the principal components of any ensemble can be used to identify the members of that ensemble .
this idea forms the foundation of face recognition and tracking through _ eigenfaces _ ( see , for example , turk and pentland 1991 @xcite ) . we may extend this method to reconstruct the missing data for any data record under given restrictions .
the key point is that the gappy data record must have the same , or similar , salient features as all the members of the ensemble .
the principal components are the eigenvectors of the covariance matrix of the data and represent the features of the dataset . provided that a reference library can be created
, each member of the library will contribute to each eigenvector , more or less .
as such , each member can be exactly represented by a linear combination of eigenvectors .
any similar data record external to the library will also be represented by a linear combination of eigenvectors , within a margin of error .
the first step for the filling procedure must therefore be to define the reference library .
we may do so by collecting a family of turbulence data series that share certain specific characteristics ; a difficult task since the definition of such is an open question .
moreover , since the mean value of the family of reference data plays a key part , all the members of the library would require normalisation . again how to do this is an open question .
alternatively we may use the neighbouring record around a data hole , sectioning this information to provide the ensemble members .
we opt for the latter technique since the record pre and post the data hole ( within a certain time interval ) ought to be similar in nature to the missing data .
the mean value of this type of library would probably not differ greatly from the mean of the missing data , so normalisation would not be so crucial .
how does one determine the principal components of a reference library ?
following sirovich and kirby @xcite , let the @xmath2 members ( each of length @xmath3 ) of the reference ensemble be @xmath4 .
thus , the average data record of this ensemble will be @xmath5 it is very reasonable to assume that departures from the mean record will provide an efficient procedure for extracting the primary features of the data .
therefore , we define @xmath6 now , if we consider the dyadic matrix @xmath7 where each term of the sum signifies a second rank tensor product , we can recognize this as the ensemble average of the two point correlation of the deviations from the mean . here , @xmath8 is the transpose of @xmath9 .
we require eigenvectors @xmath10 of the matrix @xmath11 . for ensembles
whose members have a large number of points @xmath12 , matrix @xmath11 issingular and its order can not exceed @xmath2 .
to find those eigenvectors of @xmath11 corresponding to nonzero eigen values , turk and pentland used a standard singular value decomposition technique , as described below .
@xmath13 where @xmath14 are the eigenvalues .
this deduction can be equated to @xmath15 where @xmath16 .
thus @xmath17 and @xmath18 have the same eigenvalues and their eigenvectors are related through @xmath16 , provided that @xmath19 .
the treatment described is recognizable as the karhunen 
love ( kl ) method @xcite .
the implication is that a dataset @xmath20 can be obtained from a limited summation @xmath21 where the coefficients @xmath22 are obtained through the inner product @xmath23 we emphasise that @xmath20 is not considered to be part of the @xmath24 , although it is similar in features .
to demonstrate the validity of the assertion of the previous section , we take a perfect data record of @xmath0 measurements over a 7 hour period starting from midnight , smoothed by a forward moving rolling average of 5 minutes interval ( 60 data points ) .
the data contain 2492 points in total .
the test data , split into 21 sections .
the data are padded before division with a set of points taken from the tail of the time series and mirrored outward . ]
we split up the test data into 21 sections , where all but 1 are members of the reference library , as shown in fig .
[ fig : cn2data ] .
the exclusive section is set to zero , and the algorithm described in sec . [
sect : pca ] is employed on the 20 library elements .
the reconstructions are created from the kl coeffiecients of the eigenvectors equivalent to the sections adjacent to the missing data .
hence we will talk of a prior and posterior reconstruction meaning e.g. for omitted section 5 , we use for the prior the kl coefficient equivalent to section 4 and for the posterior reconstruction we use the coefficient equivalent to section 6 of the test data set .
note that this does not reconstruct those sections , since the eigenvectors are generated from the entire reference library .
the reconstructions shown in the left hand set of fig .
[ fig : klresult ] represent the best level of error , while the right hand set shows the worst .
it is clear that the greater the difference between the ( masked off ) original data section and its neighbours , the poorer the reconstruction will be .
nevertheless , the reconstruction errors for the full set of test data are all 1 order of magnitude less than the mean value of the reconstruction and the original data segment .
evidently the end points have not been synthesised to be continuous with the adjacent segments of the reference library signal , as can be seen from the error .
a continuity condition in terms of the both the function and its derivative has to be imposed on both ends of the reconstructed segment in order to achieve smoothness .
the most effective way to do so is currently being investigated .
\(a )   best and worst reconstruction result using pca for ( a ) section 5 and ( b ) section 13 respectively .
the layout in each set is : ( top left ) segment prior to the selection .
( top middle ) the original selected data .
( top right ) segment after selection .
( centre left ) the difference between the reconstruction using the coefficient of eigenvector related to the prior segment and the original .
the value shown is the mean absolute difference per pixel .
( centre right ) the ( prior ) reconstruction .
( bottom left ) the difference between the reconstruction using the coefficient related to the posterior segment and the original .
( bottom middle ) the ( posterior ) reconstruction .
( bottom right ) the eigenvalue spectrum.,title="fig : " ] + ( b )   best and worst reconstruction result using pca for ( a ) section 5 and ( b ) section 13 respectively .
the layout in each set is : ( top left ) segment prior to the selection .
( top middle ) the original selected data .
( top right ) segment after selection .
( centre left ) the difference between the reconstruction using the coefficient of eigenvector related to the prior segment and the original .
the value shown is the mean absolute difference per pixel .
( centre right ) the ( prior ) reconstruction .
( bottom left ) the difference between the reconstruction using the coefficient related to the posterior segment and the original .
( bottom middle ) the ( posterior ) reconstruction .
( bottom right ) the eigenvalue spectrum.,title="fig : " ]    the eigenvalues determined through this method show a similar distribution in both cases , with minor variations only at the upper end of the spectrum , implying that the kl differences between best and worst section for reconstruction is not very large .      in the absence of a workable continuity condition
, we present here the effects of crudely patching the data hole with reconstructions determined from the prior and posterior terms with respect to the gap .
we apply the empirical mode decomposition ( emd ) algorithm @xcite to the reconstructions .
emd is a novel adaptive method for separating a nonlinear time series into components based on instantaneous frequency .
basically it acts as a dyadic filter bank @xcite ; the set for the best reconstruction case are illustrated in figs .
[ fig : emd ] .
we refer to these sets as @xmath25 and @xmath26 .
the original data s intrinsic mode functions ( imfs ) and residuals ( set @xmath27 ) are also shown and for comparison , we present the effect of a simple minded linear interpolation between the endpoints of the known data on the imfs .    upon visual inspection , we see that the original data generates 9 components : 8 intrinsic modes and 1 residual ( the stopping criterion for our emd implementation is the same as in huang _
et al _ @xcite ) .
on the other hand , the interpolated data have only 8 components . numbering the imfs from highest instantaneous frequency to lowest ,
starting from imf 1 , we can see by inspection that both @xmath25 and @xmath26 are strongly similar to @xmath27 in imfs 4,5,6 and 7 .
the differences appear in the higher frequency components , due to the discontinuity between the inserted segment and the unadulterated data .
the endpoint discontinuities evidently modify the variances of imfs 1,2 and 3 , although it seems that they are only affected in the area local to the discontinuity , per imf .    by way of comparison ,
a linear interpolant between the edges of the known data show that there is contamination all through the imfs .
it is so strong that imf 7 , which in the other sets clearly distinguishes the baseline rise and fall of the turbulence over the interval under study , is unable to pick out a clean pedestal .    [
cols="^,^ " , ]     we applied an arima(0,@xmath28,0 ) model to predict the behaviour of the final section of the split test data .
the length of each section is 119 points long ; by basing the data behaviour on the previous 2373 points and by setting the difference operator to be @xmath29 we generated the imf results shown in fig .
[ fig : emdarima ] .    as can be seen
, the effect of the arima(@xmath30 ) extrapolation is to fill in the higher order imfs in a smooth fashion ( apparently , by visual inspection , imfs 5 to 8 are smoothly filled at the tail ) .
it is only untrustworthy when considering the higher frequency components , because arima considers only the conditional first moment .
note that the number of imfs and residual are the same as the unadulterated signal @xmath27 .
we have discussed a data hole filling method for stochastic data , a necessary step to be able to use the new technique of empirical mode decomposition upon a time series record .
the karhunen 
love eigenvectors from an ensemble of neighbouring sections of complete data around a data hole can be used to reconstruct the missing segment to a reasonable degree of accuracy , at least for the purposes of applying emd .
we have shown that the edge continuity is important , although the effect of discontinuities is not universal through all the intrinsic modes of the data .
the quality of the reconstruction is much better using pca than a simplistic linear interpolant ( or merely ignoring the data gap ) .
we compared this to a simplified arima(0,@xmath28,0 ) model , which performed better than the linear interpolant but less effectively than the kl algorithm , disregarding edge effects .    in summary ,
when a data hole is present , merely ignoring the data hole or filling it with a linear interpolant is a poor technique from the point of view of emd .
the result of doing so leads to leakage of spurious effects both laterally in time ( per imf ) and longitudinally across all the imfs . by reconstructing the datagap with arima(@xmath31 )
one can limit the leakage laterally and longitudinally , although the lowest imfs remain strongly contaminated with artificial structure .
kl reconstruction limits the leakage best , and even without considering the continuity between the adjacent sections and the reconstructed datagap , it promises to provide the best reconstruction of the three methods described in this paper .
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measurements of optical turbulence time series data using unattended instruments over long time intervals inevitably lead to data drop - outs or degraded signals . 
 we present a comparison of methods using both principal component analysis , which is also known as the karhunen  
 love decomposition , and arima that seek to correct for these event - induced and mechanically - induced signal drop - outs and degradations . 
 we report on the quality of the correction by examining the intrinsic mode functions generated by empirical mode decomposition . 
 the data studied are optical turbulence parameter time series from a commercial long path length optical anemometer / scintillometer , measured over several hundred metres in outdoor environments .