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a significant number of studies were devoted to the estimation of power spectra and modelling of irregularly sampled time series ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) .    the descriptive statistics such as the mean , variance , higher central moments and robust equivalents : they can summarize essential features of signals in a few numbers employing straightforward computations , which makes them excellent precursors of more detailed analyses like modelling and classification . herein , a simple effects of irregular sampling on the characterization of a signal in terms of statistical parameters is presented . separate works describe corrections of biases induced by small sample sizes and gaussian uncertainties in the calculation of weighted moments and cumulants @xcite . time series sampled irregularly or with varying candences may lead to very different estimates of statistical parameters for the same signal . for example , if a sinusoid is sampled mostly at maximum or minimum , the mean is offset by about the amplitude of the signal , the variance might be much smaller than expected ( since most points sample the same region of the signal ) , and the distribution appears very skewed ( the few non - clumped measurements form an asymmetric tail in the distribution of measurements ) . if the same clump of data was in proximity of the average level , instead , the mean and skewness values could be close to the correct value ( by serendipity ) , but the variance would be much smaller than the true one . sampling - induced biases do not arise from sparsely sampled data only and they might manifest independently of the number of measurements . while time series with more measurements are generally associated with better coverage of signal features , the importance implicitly assigned to different parts of the signal by unweighted estimators is related to the relative frequency of measurements .    in principle , the most accurate and precise statistical parameters could be inferred from the model of a signal . alas , models are often complicated , in the attempt to describe the signal features under many circumstances , they might require lengthy processing and their accuracy can not be guaranteed in all cases . for example , fourier series are well fitted to model periodic signals , but the description of sharp features , like those present in ea - type eclipsing binaries , requires a high number of harmonics , which can overfit smoother parts of the signal and cause unrealistic excursions in large intervals with no data @xcite . linear interpolation is one of the simplest methods to approximate a model , assuming the signal features are sufficiently sampled . this work estimates statistical parameters by averaging the linear interpolation of functions of time series measurements ( depending on the specific estimator ) . if the signal can be recognized in time domain , interpolation can be performed in time without requiring further information . if a sparsely sampled signal is primarily mono - periodic , it can be interpolated in phase by folding the time series with the corresponding period . while the interpolated function might include profiles with spiky artefacts , its average is more robust and can be expressed as a weighted mean which assigns more relevance to scattered than clumped measurements . statistical parameters weighted by such a scheme were tested on data from the _ hipparcos _ periodic catalogue and led to a significant general improvement in the accuracy and precision of estimator values and automated classification results , with respect to those obtained with the unweighted or error - weighted counterparts . this paper is organized as follows . after the definition of the notation and terminology in sec . [ sec : notation ] , the description of weighted estimators as averages of linear interpolations is presented in sec . [ sec : method ] , with weights defined in time and phase , including adaptations for low signal - to - noise regimes and small sample sizes . the new weighting scheme is applied to time series from the _ hipparcos _ periodic catalogue in sec . [ sec : hip ] , which includes a comparison of the values of statistical parameters computed with different weighting schemes and their effect on automated classification . the conclusions are drawn in sec . [ sec : concl ] , followed by a series of appendices with more details on the interpolation of ( mono-)periodic signals ( appendix  [ app : periodicinterpolation ] ) , the illustration of modelled light curves to verify the accuracy of estimators ( appendix  [ app : data ] ) , the definitions of the statistical parameters employed ( appendix  [ app : def ] ) and additional scatter plots of estimators ( appendices  [ app : scatter ] and [ app : scatterunweighted ] ) . for a set of @xmath1 measurements @xmath2 , the following quantities are defined :    * the population central moments of order @xmath3 around the mean @xmath4 are denoted by @xmath5 and respective cumulants @xmath6 , @xmath7 and @xmath8 ( e.g. , * ? ? ? sample central moments @xmath9 , with respective cumulants @xmath10 , where @xmath11 and @xmath12 . * standardized skewness and kurtosis are @xmath13 and @xmath14 , with population values @xmath15 and @xmath16 , respectively . * noise - unbiased estimates ( sometimes called ` denoised ' for brevity ) of central moments and cumulants @xcite are denoted by an asterisk superscript . * no systematics or instrumental errors are considered herein and uncertainties are often referred to as errors . * the accuracy of an estimator is related to its distance from the true value and thus combines the concepts of bias and precision , while the classification accuracy rate is the ratio between the number of true positives and the total number of ( of a given type ) . * the precision of an estimator is quantified by its dispersion , while the classification precision rate is the ratio between the number of true positives of a given type and the total number of classified as such a type . the proposed weighting scheme is derived from linear interpolation and thus all of its pros and cons are inherited by definition . strictly speaking , statistical estimators like the mean and moments are not defined for deterministic signals . however , such estimators can still give acceptable approximations when the time scale of features ( or period ) is much smaller than the time series duration . in the case of mono - periodic signals , the proposed weighting scheme leads to estimator values of the signal as if the latter was measured on a complete cycle . ( population ) value of a statistical parameter @xmath17 of a continuous signal @xmath18 in time  @xmath19 is computed by integrating function of such a signal : @xmath20 in case of regular sampling in fine intervals , sum of discrete elements tends to the population result for infinitely small intervals . this limit is not necessarily satisfied when a continuous signal is sampled irregularly . the weighting scheme described in sec . [ sec : weights ] recovers the property that the sum of discrete ( weighted ) terms approaches the result of the continuous function , in the limit of very dense ( not necessarily uniform ) sampling . linear interpolation is a simple method to approximate a function with a broken line connecting the data points , provided the function is sufficiently sampled in time or phase . in this section , such a function is represented by the expression of the additive terms of a statistical estimator . for example , a central moment of order @xmath3 is defined by the average of terms of the form @xmath21 . results from linearly interpolating such a function are equivalent to those obtained from an effectively infinite regular sampling of the interpolated function . in sec . [ sec : weights ] , it is shown that the average of a linear interpolation can be expressed as a weighted mean . central moments of a sinusoidal signal are illustrated in fig . [ fig : illustration ] and they are related to the areas enclosed by @xmath22 , which are approximated by terms ( ) with weights ( ) adapting to the sampling density in phase .     is sampled unevenly by 10 measurements @xmath23 ( red circles ) and shown in the panel on the top - left hand side . the other panels illustrate different powers of deviations from the mean of the true signal ( blue curve ) and of the measurements ( red circles ) . in particular , @xmath24 identifies terms associated with the central moments of order @xmath3 : these estimator values are related to the areas enclosed by the blue curves and the zero level ( shaded in blue ) . the weights defined by eqs  ( [ eq : pwi])([eq : pwn ] ) define the variable widths of the shaded bars , which reduce the contribution of clustered measurements and increase the one of scattered data . in this case , although most measurements sample the first half of the signal , the increased weights assigned to the remaining data provides a better estimate of the area enclosed by the second half of the signal than using unweighted schemes . ]    since more importance is given to some measurements at the expense of other ones ( e.g. , a large sample with only a few important elements will be similar to one with only these few elements ) and exploiting correlations in the data with weights ( e.g. , assigning small weights to measurements separated by small intervals because their values are expected to be similar ) might introduce biases if applied to expressions assuming independent data . however , small biases could be justified by significant improvements in precision and a mixed weighting scheme which balances precision and accuracy depending on the signal - to - noise ( @xmath25 ) level and sample size is described in sec . [ sec : mixed ] .      if the targeted features of signals are sufficiently sampled in time , as for the data from the _ kepler _ @xcite and _ corot _ @xcite missions , linear interpolation is expected to provide reliable approximations of and weights might be expressed in time domain . if typical time intervals between measurements are larger than the resolution needed to sample the signal ( as it is often the case in the _ hipparcos _ and _ gaia _ surveys ) but the latter is mono - periodic or dominated by a single period , sampling can be improved by folding the light curve with the value of the fundamental period and computing interpolations ( or weights ) in phase .    the interpolation - based weighting scheme . if gaps in time or phase occasionally cover a relevant fraction of a signal , measurements at the gap boundaries might not provide sufficient information on the features of the signal and the importance of such data could be limited , for example , to some maximum weight value . this strategy can extend the applicability of the weighting scheme to under - sampled signals , periodic or non - periodic , to mitigate statistical biases due to data clumps on scales smaller than the signal features          if sampling in time is sparse but the data exhibit periodicity , the time series can be folded with the dominant period to increase the average sampling rate in phase ( by a factor the duration of the time series divided by the period ) .    since interpolation of phase - sorted data is carried over from the last to the first point in phase , weights in phase @xmath35 corresponding to eqs  ( [ eq : twi_delta])([eq : twn_delta ] ) become ( see appendix  [ app : periodicinterpolation ] ) : @xmath36 with @xmath37 .    [ [ sec : mixed ] ]    in order to avoid interpolating large noise fluctuations , weights could be set to reduce to inverse - squared uncertainties at low @xmath25 , which proved more precise in the simulations described by @xcite . the transition between high and low @xmath25 regimes could be pursued with function of the form @xmath38 could be used to reduce the relevance of interpolation - based weights for small sample sizes @xmath1 @xmath39/n , \end{aligned}\ ] ] where @xmath40 is defined by eqs  ( [ eq : twi])([eq : twn ] ) or ( [ eq : pwi])([eq : pwn ] ) . tuning parameters @xmath41 offer the possibility to reach a compromise solution between precision and accuracy at high and low values of @xmath1 and @xmath25 ratios , according to the specific set of estimators , signals , sampling , errors , sample sizes and their distributions in the data . the effect of interpolation - based weights on statistical parameters was explored for a realistic distribution of signals , represented by time series from the _ hipparcos _ catalogue of periodic variable stars . the _ hipparcos _ mission @xcite performed astrometric and photometric measurements of the brightest sources in the sky . the full catalogue @xcite contains 118204 sources with photometry . among the 11597 stars identified as variables , a reliable period could be computed ( or was consistent with the one from literature ) for 2712 objects , which were published in the periodic catalogue ( vol . * ; * ? ? ? this set of sources , which contained light variations dominated mostly by single periods , was chosen to illustrate the application of one of the weighting schemes described in sec . [ sec : method ] .      in order to assess the accuracy of estimators with respect to the ( unknown ) real signal , the latter was assumed to be represented by a model of the time series and were simulated employing uncertainties as gaussian random variables ( around the model ) . only good quality measurements were accepted , flagged by the field ht4 as zero or one ( see vol .  1 of * ? ? ? * ) , which reduced the number of sources considered to 2683 . only ) were : 1196 , 10027 , 17878 , 20570 , 24019 , 25673 , 39084 , 42715 , 42726 , 46502 , 52538 , 53937 , 58112 , 60904 , 61997 , 63125 , 69582 , 72583 , 88905 , 90026 , 93595 , 93724 , 96007 , 99675 , 102246 , 102409 , 112317 , 112470 and 118188 . ] time series were folded with the period provided by the _ hipparcos _ catalogue and modelled by a cubic smoothing spline ( the ` smooth.spline ` function from the ` stats ` package in r , * ? ? ? the condition of periodicity at the boundaries was approximated by replicating a whole cycle of folded data before and after the cycle considered as reference for the ` true ' statistical parameters and for the generation of simulated data . the smoothing parameter was estimated from the data with a generalized cross - validation method ( e.g. , * ? ? ? * ) and the degrees of freedom were adjusted in special cases , to mitigate over - fitting highly clumped data with large gaps as well as under - fitting the profile of eclipsing binaries . of the data , which identified light variations typical of eclipsing binaries , and the ratio @xmath42 between the median and the third largest gaps in phase , to better deal with clumped data with large gaps . denoting by @xmath1 the number of measurements in a time series , ` df ` equalled 15 , 24 and 36 for @xmath42 less than @xmath43 , @xmath44 and @xmath45 , respectively , provided @xmath46 . for greater values of @xmath47 , ` df ` was set to the smallest integer not less than @xmath48 . the quoted values of ` df ` take into account the replicated data at each of the extremes of the folded light curve ( to induce quasi - periodic boundary conditions on the model of the central cycle ) . ] the combination of large gaps and sharp features did not lead to accurate models and other methods might help avoid modelling artefacts in gapped data , although these are out of the scope of this application . also , cases in which sparse sampling missed important features could not be improved . nevertheless , the smoothed best - fitting curves seemed to capture the relevant shapes of true signals in most cases . the resulting models are presented together with the original and simulated data in appendix  [ app : data ] . the difficulty to achieve accurate models for all sources and the need of more complex modelling techniques were confirmed . while models were sometimes not ideal , most of them were of sufficient quality to supply a realistic distribution of the relevant features of the _ hipparcos _ periodic variable stars . differences from data of other surveys were expected to be greater than those due to modelling inaccuracies . less than one per cent of all sources ( 23 time series ) with significant modelling artefacts in gaps with no data were removed ( many of them were ea - type eclipsing binaries with large data gaps ) , thus 2660 objects were included in the assessment of statistical parameters . the number of measurements per time series ranged from 18 to 331 , with @xmath25 ratios from 0.2 to 116 , according to the definition in eq . ( [ eq : snr ] ) . the _ hipparcos _ light curves were folded with the catalogue period and interpolation - based weights were computed in phase . for brevity , such weights are referred to as ` phase weights ' , while inverse - squared uncertainties are called ` error weights ' . the estimators employed herein included weighted moments and cumulants corrected for biases from gaussian uncertainties ( called  noise - unbiased  or  denoised  estimators ; see * ? ? ? * ) and some robust weighted measures , as defined in appendix  [ app : def ] . simulations in @xcite suggested that noise - unbiased phase - weighted sample moments can be more accurate for @xmath49 and sample sizes @xmath50 with respect to other schemes , while error weighting appeared the most appropriate option for noisy signals . thus , the weighting scheme chosen for this application combined error and phase information as described by eqs  ( [ eq : mixed ] ) and ( [ eq : pwi])([eq : pwn ] ) . the balance between phase weights ( at high @xmath25 ) and error weights ( at low @xmath25 ) was controlled by the parameters @xmath51 and @xmath52 , which provided a satisfactory overall accuracy for the set of estimators and light - curve shapes of the _ hipparcos _ periodic variable stars . the signal - to - noise ratio @xmath25 was estimated from the true ( model ) signal variance @xmath53 and the average of squared measurement uncertainties @xmath54 as follows : @xmath55^{1/2}. \label{eq : snr}\ ] ]      the deviations of error - weighted sample statistical parameters from population values ( computed from a fine regular sampling of the models ) are compared to the noise - unbiased error - phase weighted counterparts as a function of @xmath25 ratio in figs  [ fig : noise1][fig : noise2 ] . error - phase weighted ( and noise - unbiased , when applicable ) estimators were generally more accurate than error - weighted estimators . since larger light variations were correlated with higher @xmath25 levels , the effect of the correlation between errors and magnitudes ( e.g. , see * ? ? ? * ) was visible when weighting by errors at high @xmath25 : the mean was biased towards brighter measurements , the variance tended to be smaller than the real value and the scatter of higher moments around the true values was greater . such effects were significantly alleviated by error - phase weights . also , the accuracy of the noise - unbiased variance @xmath56 and kurtosis moment @xmath57 were much improved at @xmath58 . in the particular case of the standardized kurtosis , the value of @xmath59 equalled the one of the cumulant @xmath60 by definition , for any weighting scheme ( unlike the noise - unbiased counterparts ) . involves a term in @xmath61 , which does not simplify after the normalization by @xmath62 . ] ( a ) +   +  ( c ) +   +  ( e ) +     ( b ) +   +  ( d ) +   +  ( f ) +     ( a ) +   +  ( c ) +     ( b ) +   +  ( d ) +     estimators which provide similar information ( such as the mean and median , variance and interquartile range , skewness and kurtosis standardized by the estimated or true variance ) are compared in scatter plots in appendix  [ app : scatter ] . although contributions from different @xmath25 ratios are not distinguished , error - phase weighted ( and noise - unbiased , when applicable ) estimators are always associated with more strongly peaked distributions around the true values than the error - weighted analogues .    in order to quantify the effect of error - phase weighting and additional denoising with respect to simple error weighting , the top panels in figs  [ fig : quantile1][fig : quantile3 ] indicate the fraction of sources associated with estimators improved by phase weights and noise correction . the accuracy of a generic estimator @xmath63 is assessed by its distance @xmath64 from the true value @xmath65 . the abscissas in figs  [ fig : quantile1][fig : quantile3 ] represent the difference between the distance of error - phase weighted ( and noise - unbiased , when applicable ) estimators and the one of error - weighted estimators from the correct values : negative differences indicate smaller distances to the true values and thus improved accuracy with respect to error - weighted estimators . the accuracy of error - phase weighted and noise - unbiased estimators improved in 70-to-90 per cent of the cases and deteriorations were typically smaller in frequency and magnitude than improvements . in the singular case of the standardized skewness @xmath66 , denoising worsened almost 8 per cent of the estimates with respect to error - phase weighting . as apparent in fig . [ fig : quantile2]a , the accuracy of the standardized noise - unbiased skewness worsened at low @xmath25 ratios . this was expected because the noise - unbiased variance might be easily underestimated when uncertainties are of the same order as the signal , and the normalization of the skewness by a much smaller number than the correct one could rapidly degrade the precision of the estimator . a similar ( less pronounced ) tendency at low @xmath25 ratios was also observed in the case of the standardized kurtosis . ( a ) +   +  ( c ) +     ( b ) +   +  ( d ) +     ( a ) +   +  ( c ) +     ( b ) +   +  ( d ) +     ( a ) +     ( b ) +     as noted in @xcite , the noise - unbiased variance can become too small at low @xmath25 levels , overestimating the standardized noise - unbiased skewness and kurtosis , or leading to an undetermined skewness value in case of non - positive variance . on the other hand , when the variance is not corrected by noise biases , it is generally overestimated , and if the true values of skewness or kurtosis are sufficiently close to zero , the standardized noise-_biased _ skewness and kurtosis become more accurate ( by serendipity ) . the effect of statistical parameters weighted by different schemes on automated classification was assessed by comparing the classification accuracy and precision as a function of variability type employing unweighted , error - weighted , phase - error weighted and noise - unbiased phase - error weighted estimators . automated classification of stellar variability types was pursued with a set of attributes which characterized features of different classes . some studies employed only information from light - curve modelling @xcite , with additional statistical parameters @xcite or colour information @xcite , while attributes from modelling , statistical and astrophysical quantities were used in @xcite . the list of attributes employed for classification herein was restricted to descriptive statistics ( mean , median , variance , interquartile range , normalized and non - normalized skewness and kurtosis ) , in order to test the effect of different weighting schemes on classification . only variability types which could be identified by the distribution of measurements were included in the training set , such as eclipsing binaries , rr lyrae , mira , @xmath67 cephei and alpha@xmath0 canum venaticorum stars , accounting for over 60  per cent of all sources in the _ hipparcos _ periodic catalogue .    in these classification experiments , statistical parameters did not depend on modelling and were computed on the original ( not simulated ) data , which thus included the sources previously excluded because of modelling issues . the same quality flags , periods and weighting schemes were applied as described in sec . [ sec : pars ] . since the true signal variance was unknown , the @xmath25 ratio was estimated by substituting the unknown @xmath53 in eq . ( [ eq : snr ] ) with the noise - unbiased phase - weighted sample variance @xmath68 and by weighting the average of squared errors in the denominator with the same weights for consistency : @xmath69^{1/2}.\ ] ] the @xmath25 ratios of the sources selected for classification ( as described in sec . [ sec : dataselection ] ) spanned a range from 0.4 to 140 . sources from the _ hipparcos _ periodic catalogue were cross - matched with classifications from literature as available in the variable star index @xcite http://www.aavso.org/vsx/help/variablestartypedesignationsinvsx.pdf[aavso.org/vsx/help/variablestartypedesignationsinvsx.pdf ] ] of the american association of variable star observers ( aavso ) with the nearest object within 1  arcsec . uncertain classifications ( with class labels followed by the mark ` :') were excluded , unless the uncertainty referred to properties of eclipsing binaries other than light - curve shapes , such as the physical characteristics , the luminosity class of the components , or the degree of filling of the inner roche lobes . objects associated with combinations of different classes ( with labels joined by the symbol ` + ' ) were not addressed herein . the subset of variability types chosen to test classification with statistical parameters are listed in table  [ tab : ts ] and include 1605 sources associated with class labels ea , eb , ew , acv , m , rrab , rrc , dcep and dceps , defined in table  [ tab : ts ] together with the corresponding sample sizes . sources which poorly represented their class were not removed from the training set in order to avoid the introduction of selection biases ( e.g. , by favouring objects with higher @xmath25 ratios ) and because the focus of this application was related to the _ relative _ classification accuracy employing attributes with different weighting schemes . llr variability type&label&number + eclipsing binary : algol type&ea&409 +  beta lyrae type&eb&208 +  w ursae majoris type&ew&170 + alpha@xmath0 canum venaticorum&acv&183 + mira ceti&m&218 + rr lyrae : asymmetric light curve&rrab&139 +  nearly symmetric light curve&rrc&28 + delta cephei & dcep&220 +  first overtone pulsators&dceps&30 +   + & total:&1605 +      the distributions of a selection of error - phase weighted ( and noise - unbiased , when applicable ) statistical parameters as a function of variability type are presented in fig . [ fig : scatter ] . the mean or median magnitudes of sources at broadly different distances from the observer do not provide information on the intrinsic source properties . however , the correlation between magnitude and noise , coupled with the interquartile range or variance ( as shown in fig . [ fig : scatter]a ) , is related to the @xmath25 level and thus to an observational selection . figure  [ fig : scatter]b illustrates clearly the difference between robust and non - robust estimators for the same quantity , such as the interquartile range versus the variance , which proves effective at separating eclipsing binaries ( mostly ea and some fraction of eb types ) . the interquartile range and standardized skewness are presented in fig . [ fig : scatter]c : the skewness separates the eclipsing binary subtypes from most other classes , which are better distinguished by the interquartile range . a similar scatter plot is shown in fig . [ fig : scatter]d in terms of normalized and non - normalized kurtosis moments , with the difference that some classes occupy different relative loci ( such as the ea types and the rrab with respect to dcep variables ) .    the estimators and variability types illustrated in fig . [ fig : scatter ] are also presented for the unweighted case in appendix  [ app : scatterunweighted ] . the distributions of most unweighted estimators in fig . [ fig : scatterunweighted ] tend to be more scattered than the ones shown in fig .  [ fig : scatter ] . in the case of acv stars , instead , the unweighted skewness and kurtosis are less scattered , because these sources have very low @xmath25 ratios ( typically @xmath70 ) and the correction of noise biases at such @xmath25 levels can decrease significantly the precision of higher moments @xcite . another noticeable difference between figs  [ fig : scatter ] and [ fig : scatterunweighted ] is related to the larger skewness @xmath66 of mira stars in the unweighted case , which can be understood by the systematic difference in the uncertainties associated with faint and bright measurements ( e.g. , * ? ? ? * ) . such a difference is enhanced by the large amplitudes of light variations of mira types , so that greater uncertainties on the faint side of the signal generate more faint than bright ` outliers ' , biasing the unweighted skewness towards greater values . ( a ) +   +  ( c ) +     ( b ) +   +  ( d ) +      automated classification tests were performed employing random forests @xcite . random forest is an accurate tree - based classification method ( e.g. , see * ? ? ? * ) , quite robust to outliers and strongly correlated attributes . the accuracy of the classifier was estimated from a subset of sources ( about one - third of all objects ) randomly omitted from the learning process and thus called ` out - of - bag ' sources . the importance of an attribute for classification was measured by the mean decrease in accuracy after permuting the values of that attribute in the out - of - bag sources ( see * ? ? ? * for more details ) . the random forest classifier was trained with estimators employing a single weighting scheme per run , i.e. , unweighted , error - weighted , error - phase weighted or noise - unbiased error - phase weighted . for each weighting scheme , random forest was executed 1000 times with 500 trees and the classification accuracy and precision rates were aggregated after each run in order to assess their mean and dispersion . the mean classification accuracy and precision rates are listed as a function of variability type and attribute weighting scheme in tables  [ tab : res1 ] and  [ tab : res2 ] , and illustrated in figs  [ fig : classresults]a and [ fig : classresults]b , respectively . training - set sources were reclassified by sole statistical parameters with an overall accuracy from 77.0 to 82.6 per cent on average , depending on the weighting scheme adopted . unweighted estimators led to the smallest accuracy level , closely followed by error - weighted estimators at 77.6 per cent . the introduction of phase weights improved the accuracy to 82.3 per cent and noise - unbiased error - phase weighted estimators achieved the best overall classification accuracy of 82.6 per cent ( with uncertainty at the level of 0.2 to 0.3 per cent ) . the overall trends were generally reflected by single variability types with only a few exceptions . in particular , error - weighted estimators of rrab types led to significantly worse classification accuracy than unweighted estimators . this was explained by the relatively large amplitude of rrab variables , which enhanced the systematic difference of uncertainties between bright and faint measurements in the light curve : weighting by errors decreased the importance of faint measurements , increasing the similarity ( and thus confusion ) with dcep and m - type light curves . in the case of dceps variables , denoising was marginally counterproductive for accuracy , although still significantly better with respect to the unweighted or error - weighted schemes . this was expected to be related to the increased confusion with the noise - unbiased estimators of ew types , which were more numerous and extended to lower @xmath25 ratios than the dceps stars : while the distributions of the standardized error - phase weighted kurtosis moments of dceps and ew types could just be separated , the additional denoising decreased the values of standardized kurtosis moments more for the ew than the dceps stars , leading to overlapping distributions dominated by the ew types .    @l@  c@  c@  c@  c@ var . & & error & error - phase & err .- ph . w. + type&unweighted&weighted&weighted & & denoised + ea & 87.7 @xmath71 0.3 & 87.7 @xmath71 0.4 & 88.9 @xmath71 0.4 & 89.0 @xmath71 0.3 + eb & 53.7 @xmath71 1.1 & 56.4 @xmath71 1.0 & 58.7 @xmath71 0.9 & 59.8 @xmath71 1.0 + ew & 59.2 @xmath71 1.4 & 60.9 @xmath71 1.2 & 68.5 @xmath71 1.0 & 70.2 @xmath71 1.0 + acv & 84.9 @xmath71 0.9 & 85.9 @xmath71 0.9 & 88.8 @xmath71 1.0 & 87.6 @xmath71 0.7 + m & 96.7 @xmath71 0.2 & 96.5 @xmath71 0.3 & 98.0 @xmath71 0.3 & 98.2 @xmath71 0.3 + rrab & 83.1 @xmath71 0.8 & 79.4 @xmath71 1.0 & 88.3 @xmath71 0.6 & 87.8 @xmath71 0.8 + rrc & 41.2 @xmath71 3.0 & 47.0 @xmath71 4.0 & 41.8 @xmath71 3.1 & 50.8 @xmath71 3.3 + dcep & 73.9 @xmath71 0.8 & 74.7 @xmath71 0.7 & 86.6 @xmath71 0.6 & 87.1 @xmath71 0.6 + dceps & 31.3 @xmath71 3.5 & 35.1 @xmath71 3.8 & 56.6 @xmath71 2.2 & 51.4 @xmath71 1.9 +   + all & 77.0 @xmath71 0.3 & 77.6 @xmath71 0.3 & 82.3 @xmath71 0.2 & 82.6 @xmath71 0.2 +    precision rates of classification per variability type showed successive improvements ( consistent within uncertainties ) from the unweighted to the error - weighted , error - phase weighted and noise - unbiased schemes . a single exception was related to mira stars : their classification precision with error - weighted estimators was about one per cent worse than with unweighted attributes , as a consequence of the increased contamination by rrab stars ( as explained above ) . @l@  c@  c@  c@  c@ var . & & error & error - phase & err .- ph . w. + type&unweighted&weighted&weighted & & denoised + ea & 88.5 @xmath71 0.4 & 89.4 @xmath71 0.4 & 91.2 @xmath71 0.4 & 91.0 @xmath71 0.3 + eb & 58.4 @xmath71 1.0 & 59.9 @xmath71 1.0 & 63.1 @xmath71 0.9 & 64.5 @xmath71 0.8 + ew & 58.5 @xmath71 1.1 & 59.4 @xmath71 1.0 & 65.5 @xmath71 0.9 & 65.2 @xmath71 0.9 + acv & 72.4 @xmath71 0.5 & 74.8 @xmath71 0.6 & 76.3 @xmath71 0.5 & 76.6 @xmath71 0.5 + m & 98.2 @xmath71 0.2 & 97.2 @xmath71 0.2 & 98.6 @xmath71 0.3 & 98.8 @xmath71 0.3 + rrab & 80.8 @xmath71 0.7 & 80.5 @xmath71 0.9 & 89.8 @xmath71 0.8 & 90.5 @xmath71 0.9 + rrc & 52.5 @xmath71 3.2 & 56.5 @xmath71 3.3 & 82.4 @xmath71 4.9 & 88.4 @xmath71 4.3 + dcep & 74.7 @xmath71 0.8 & 74.7 @xmath71 0.8 & 85.3 @xmath71 0.6 & 85.4 @xmath71 0.6 + dceps & 37.9 @xmath71 3.5 & 39.7 @xmath71 3.3 & 55.4 @xmath71 1.9 & 56.2 @xmath71 1.8 +   + all & 77.0 @xmath71 0.3 & 77.6 @xmath71 0.3 & 82.3 @xmath71 0.2 & 82.6 @xmath71 0.2 +     and [ tab : res2 ] . class labels are defined in table  [ tab : ts ] and colour coded as shown in the legend of each panel . , title="fig : " ] +   and [ tab : res2 ] . class labels are defined in table  [ tab : ts ] and colour coded as shown in the legend of each panel . , title="fig : " ]    the importance of phase - weighted ( and noise - unbiased , when applicable ) attributes as a function of variability type from a run of random forest is depicted in fig . [ fig : importance ] , with darker cells indicating more important attributes for a given class . for example , the skewness confirmed to be particularly useful to distinguish eclipsing binaries and the asymmetric light curves of rrab stars . also , large and small amplitude variables , such as acv and m types , could easily be separated by the variance .    . class labels are described in table  [ tab : ts ] . ]    the confusion matrix from a run of random forest , employing error - phase weighted ( and noise - unbiased , when applicable ) estimators , is shown in fig . [ fig : cm ] . the overall classification accuracy was 83 per cent and reached 92 per cent after aggregating the subtypes of eclipsing binary , rr lyrae and delta cephei into their superclasses . the improvement in classification accuracy , with respect to results employing unweighted estimators , was about 6 per cent in both cases of separated and aggregated variability subtypes . the level of confusion between eclipsing binaries of ea , eb and ew types was expected from the natural overlap in their definitions . the misclassification of many acv types as eclipsing binaries was attributed to the restricted set of attributes employed herein ( e.g. , the attribute  qso variations  in * ? ? ? * proved to be useful in this context ) . similarly , the confusion between rr lyrae and delta cephei variables would not have occurred if the period was included in the set of classification attributes . on the other hand , simple amplitude estimators such as variance and interquartile range were sufficient to separate mira stars from most other classes with high accuracy ( as apparent in fig . [ fig : scatter ] ) .    . a simple weighting scheme based on linear interpolation , applicable in phase or time , was proposed in order to improve the accuracy of descriptive statistics of unevenly sampled time series . weights represented by time intervals could be applied to well - sampled signals . for sparsely sampled but periodic time series ( dominated by a single period ) , light curves could be folded with the fundamental periods and weights expressed in terms of phase intervals . the _ hipparcos _ catalogue of periodic variables represented a suitable test bed of unevenly sampled light curves with realistic distributions of variability types to investigate the accuracy of estimators weighted by different schemes . noise - unbiased estimators weighted by phase intervals ( or inverse - squared uncertainties at low @xmath25 ratios ) improved the accuracy for 70-to-90 per cent of the sources with respect to the values of the error - weighted counterparts . deteriorations in accuracy were observed in 10-to-30 per cent of the cases , with magnitudes typically smaller than the ones corresponding to improvements . automated classification experiments confirmed that the best overall results were achieved employing the set of the most accurate attributes , i.e. , the noise - unbiased estimators weighted by phase intervals and inverse - squared uncertainties at high and low @xmath25 ratios , respectively . the overall improvement in classification accuracy with respect to the result employing unweighted estimators was about 6 per cent and most of it was related to the introduction of phase weights which adapted to the varying sampling density of the light curve . the author is grateful to m. sveges , p. dubath for useful comments on the original manuscript . 99 auvergne m. et al . , 2009 , a&a , 506 , 411 blomme j. et al . , 2010 , apj , 713 , l204 blomme j. et al . , 2011 , mnras , 418 , 96 borucki w.j . et al . , 2010 , science , 327 , 977 breiman l. , 2001 , machine learning , 45 , 5 carbonell m. , oliver r. , ballester j.l . , 1992 , a&a , 264 , 350 debosscher j. , sarro l.m . , aerts c. , cuypers j. , vandenbussche b. , garrido r. , solano e. , 2007 , a&a , 475 , 1159 debosscher j. et al . , 2009 , a&a , 506 , 519 dubath p. et al . , 2011 , mnras , 414 , 2602 european space agency , 1997 , the hipparcos and tycho catalogues , esa  sp-1200 eyer l. , 1998 , les toiles variables de la mission hipparcos , phd thesis n^o^  3002 , universit de genve eyer l. , mignard f. , 2005 , mnras , 361 , 1136 eyer l. , grenon m. , falin j .- l . , froeschle m. , mignard f. , 1994 , solar physics , 152 , 91 koen c. , 2005 , mnras , 361 , 887 liaw a. , wiener m. , 2002 , r news , 2 ( 3 ) , 18 perryman m.a.c . et al . , 1997 , a&a , 323 , l49 perryman m.a.c . et al . , 2001 , a&a , 369 , 339 r development core team , 2013 , r : a language and environment for statistical computing , r foundation for statistical computing , vienna , austria richards j.w . et al . , 2011 , apj , 733 , 10 rimoldini l. et al . , 2012 , mnras , 427 , 2917 rimoldini l. , 2013a , preprint ( http://xxx.lanl.gov/abs/1304.6564[arxiv:1304.6564 ] ) rimoldini l. , 2013b , preprint ( http://xxx.lanl.gov/abs/1304.6715[arxiv:1304.6715 ] ) ruppert d. , wand m.p . , carroll r.j . , 2003 , nonparametric regression , cambridge university press sarro l.m . , debosscher j. , aerts c. , lpez m. , 2009 , a&a , 506 , 535 stuart a. , ord j. , 1969 , kendall s advanced theory of statistics , charles griffin & co.  ltd , london van leeuwen f. , 1997 , sp . , 81 , 201 vio r. , strohmer t. , wamsteker w. , 2000 , pasp , 112 , 74 watson c. , henden a.a . , price a. , 2012 , vizier online data catalogue , aavso international variable star index , version 2012 - 09 - 23 light curves of the _ hipparcos _ periodic variables are illustrated in fig . [ fig : lc ] . panels on the left - hand side show the _ hipparcos _ data folded with the catalogue period , while the right - hand panels present data simulated around the model according to the measurement uncertainties ( assumed gaussian ) and the observed phases . simulated data were employed to assess the accuracy of statistical estimators ( with respect to the model ) , while classification was performed on the original data . figure  [ fig : lc ] provides sample light curves for 4 of the 2683 _ hipparcos _ sources available online ( see supporting information ) . only ) were not included in fig . [ fig : lc ] . ] the statistical estimators employed to test different weighting schemes are defined below . sample noise - unbiased weighted moments and cumulants are defined following @xcite . denoting the sample weighted central moments of order @xmath3 by @xmath79 and representing the gaussian uncertainties by @xmath54 , the noise - unbiased estimators ( starred ) are defined as follows : @xmath80   -\frac{3}{w^4}\left(\sum_{i=1}^{n } w_i^2\epsilon_i^2\right)^2\\ ( m_2 ^ 2)^ * = & \left(m^*_2\right)^2 -\frac{4}{w^2}\sum_{i=1}^{n } w_i^2\epsilon_i^2 \left [ \left(x_i-\bar{x}\right)^2 -\frac{\epsilon_i^2}{2}\left(1-\frac{2w_i}{w } \right)\right]+\nonumber\\ & + \frac{2}{w^4}\left(\sum_{i=1}^{n } w_i^2\epsilon_i^2\right)^2\\ k^*_4   = & \,m^*_4 - 3\,(m_2 ^ 2)^ * \\ g^*_1=&\,k^*_3/(k^*_2)^{3/2 } \\ g^*_2=&\,k^*_4/(k^*_2)^2.\end{aligned}\ ] ]      percentiles depend on the rank of sorted values , thus they are less sensitive to extreme values than moments and cumulants which involve powers of deviations from the mean and average over all elements . the @xmath81-th percentile @xmath82 is defined as the ( interpolated ) value such that @xmath81 per cent of the data are smaller than @xmath82 . two common percentiles are the median @xmath83 and interquartile range @xmath84 .    denoting the list of measurements @xmath23 sorted in increasing values by @xmath85 , associated with weights @xmath86 , respectively , the @xmath81-th weighted percentile @xmath82 is defined as follows : @xmath87 where @xmath88 figures  [ fig : scatter1][fig : scatter2 ] compare statistical parameters which provide similar information , such as robust versus non - robust or normalized versus non - normalized estimators , as a function of the weighting scheme . error - phase weighted ( and noise - unbiased , when applicable ) estimators correspond to the most strongly peaked distributions around the correct values . \(a ) +   + ( c ) +     \(b ) +   + ( d ) + figure  [ fig : scatterunweighted ] presents a selection of unweighted estimators for stars of different variability types . this figure is intended to be compared to fig . [ fig : scatter ] , which illustrates the same information for phase - error weighted and noise - unbiased estimators .
unevenly spaced time series are common in astronomy because of the day - night cycle , weather conditions , dependence on the source position in the sky , allocated telescope time , corrupt measurements , for example , or be inherent to the scanning law of satellites like _ hipparcos _ and the forthcoming _ gaia_. weighting scheme , adapting to the sampling density and noise level , time series from the _ hipparcos _ periodic catalogue led to significant improvements in the overall accuracy and precision of the estimators with respect to the unweighted counterparts and those weighted by inverse - squared uncertainties . 
 automated classification procedures employing statistical parameters weighted by the suggested scheme confirmed the benefits of the improved input attributes . 
 the classification of eclipsing binaries , mira , rr lyrae , delta cephei and alpha@xmath0 canum venaticorum stars employing exclusively weighted descriptive statistics achieved an overall accuracy of 92 per cent , about 6  per  cent higher than with unweighted estimators . 
 [ firstpage ]    methods : data analysis  methods : statistical  stars : variables : general .