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Paper text:
periodicity analysis of unevenly collected data is a relevant issue in several scientific fields .
classical spectral analysis methods are unsatisfactory to solve the problem . in this paper
we present a neural network based estimator system which performs well the frequency extraction from unevenly sampled signals .
it uses an unsupervised hebbian nonlinear neural algorithm to extract the principal components of the signal auto - correlation matrix , which , in turn , are used by the music frequency estimator algorithm to extract the frequencies @xcite , @xcite , @xcite .
we generalize this method to avoid an interpolation preprocessing step , which generally adds high noise to the signal , and improve the system performance by using a new stop criterion to avoid overfitting problems .
the experimental results are obtained comparing our methodology with the others known in literature ( see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) .
in what follows , we assume @xmath0 to be a physical variable measured at discrete times @xmath1 .
@xmath2 can be written as the sum of the signal @xmath3 and random errors @xmath4 : @xmath5 .
the problem we are dealing with is how to estimate fundamental frequencies which may be present in the signal @xmath6 @xcite , @xcite , @xcite .
if @xmath7 is measured at uniform time steps ( even sampling ) there are a lot of tools to effectively solve the problem which are based on fourier analysis @xcite , @xcite , @xcite .
these methods , however , are usually unreliable for unevenly sampled data @xcite . for instance , the typical approach of resampling the data into an evenly sampled sequence , through interpolation , introduces a strong amplification of the noise which affects the effectiveness of all fourier based techniques which are strongly dependent on the noise level . to solve the problem of unevenly sampled data , we consider two classes of spectral estimators :    spectral estimators based on fourier trasform ( least squares methods ) ;    spectral estimators based on the eingevalues and eingevectors of the covariance matrix ( maximum likelihood methods ) .
classic periodogram @xcite , @xcite , @xcite , lomb s periodogram @xcite , scargle s periodogram @xcite , dcdft @xcite are the methods of the first class that we use , while music @xcite , @xcite , and esprit @xcite belong to second class .
the methods based on the covariance matrix are more recent and have great potentiality .
starting by this consideration , we develop a method based on the music estimator .
it is compared with classic methods to highlight the results .
in the last years several papers dealed with learning in pca neural nets @xcite , @xcite , @xcite , @xcite , @xcite finding advantages , problems and difficulties of such neural networks . in what follows we shall use a robust hierarchical learning algorithm @xmath8    where @xmath9 is the weight vector of the @xmath10 output neuron at step @xmath11 , @xmath12 is the corresponding output , @xmath13 is the learning rate and @xmath14 is learning function because it has been experimentally shown that it is the best performing one in our problem @xcite , @xcite .
our neural estimator ( ne ) can be summarized as follows :    * preprocessing : calculate and subtract the average pattern to obtain zero mean process with unity variance .
interpolate input data if it is the case .
* initialize the weight matrix and the other neural network parameters ; * input the @xmath15 pattern @xmath16 $ ] where @xmath17 is the number of input components . *
calculate the output for each neuron @xmath18 . *
modify the weights @xmath19 . *
if convergence test is true then goto * step 8*. * @xmath20 .
goto * step 3*. * end .
* frequency estimator : we use the frequency estimator music .
it takes as input the weight matrix columns after the learning .
the estimated signal frequencies are obtained as the peak locations of the function of following equation @xcite , @xcite , @xcite : @xmath21    where @xmath22 is the @xmath23th neural network weight vector after learning , and @xmath24 is the pure sinusoidal vector . in the case of an interpolation preprocessing @xmath25^{h}$ ] . in the generalization to
non interpolated input data , @xmath26^{h}$ ] where @xmath27 are the first @xmath28 components of the temporal coordinates of the uneven signal .
when @xmath29 is the frequency of the @xmath23th sinusoidal component , @xmath30 , we have @xmath31 and @xmath32 . in practice
we have a peak near and in corrispondence of the component frequency .
estimates are related to the highest peaks    furthermore , to optimize the performance of the pca neural networks , we stop the learning process when @xmath33 @xmath34 , so avoiding overfitting problems .
in fact leaving the stop condition used in the ne causes to the ne to find periodicities not present in the signal , while the new condition preserves it from this problem .
a simple example is illustrated in figure 2 where we can see how the frequency identification varies depending on the stop condition ( see next section for signal information ) .
in fact , without the early stopping ( see figure 2.b ) , @xmath35 after @xmath36 epochs remains at a value between @xmath37 and @xmath38 and we can not know when the system reaches the best performance .
the new stopping criterion , instead , permits to @xmath35 to have a final value about @xmath39 just after @xmath40 epochs ( see figure 2.a ) .
many experiments on synthetic and real signals were made , and in this paper we present the results obtained with one specific real signal , which highlights the main features of our problem .
the real signal is related to the cepheid su cygni @xcite .
the sequence was obtained with the photometric technique ubvri and the sampling made from june to december 1977 .
the light curve is composed by 21 samples , and has a period of @xmath41 , as shown in figure 1 .
the first experiment is concerning the interpolation . in this case
we apply three different methods by using the signal processing mathlab @xmath42 toolbox : linear , cubic and spline , because they are quite simple and the most used ones . in figure 3
there is a plotting of the interpolating functions and the frequency estimates obtained by the ne with the spline interpolated signal as input .    in this case , the parameters of the ne are : @xmath43 , @xmath44 , @xmath45 , @xmath46 .
the estimate frequency interval is @xmath47 $ ] .
the estimated frequency without interpolation is @xmath48 ( 1/jd ) .
a comparison is made with the other methods cited in a previous section and the experimental results are shown in figure 4 and in table 1 .
only the lomb s periodogram is in agreement with the right periodicity , but showing some spurious peaks .
furthermore , if we enlarge the frequency window for the two best performing methods , while the ne continues to work well , the lomb s periodogram does not work at all as illustrated in figure 5 .
in this paper we have illustrated an improved technique based on pca neural networks and music to estimate the frequency of unevenly sampled data .
it has been shown that it obtains good results on real data ( here we used the su cygni light curve ) compared with other well - known methods .
in fact , it obtains a good estimate of the signal frequency also with few unevenly sampled inputs , it reduces the noise problems related to input data interpolation , it optimizes the convergence by introducing an early stopping criterion , and , finally , it is more resistant to the dimension of the frequency windows .
future research lines regard the introduction of genetic algorithms to optimize the weight initialization of the pca neural networks and to use filters to extract and identify one frequency at each time when dealing with multi - frequency signals .
m. rasile , l. milano , r. tagliaferri , g. longo , periodicity analysis of unvenly spaced data by means of neural networks , in neural nets wirn vietri 97 , m. marinaro and r. tagliaferri editors , springer verlag , pag.201 - 212 , 1997 .
r. roy , t. kailath , esprit- estimation of signal parameters via rotational invariance tecniques , in f.a .
grnbaum , j. w. helton and p. khargonear editors , signal processing part ii : control theory and applications , springer verlag , new york , pag.369 - 411 , 1990 .
r. tagliaferri , a. ciaramella , l. milano , f. barone , g. longo , spectral analysis of stellar light curves by means of neural networks , accepted for publication , astronomy & astrophysics supplement series , vol.137 , june , 1999 .
in this paper we present a neural network based estimator system which performs well the frequency extraction from unevenly sampled signals . 
 it uses an unsupervised hebbian nonlinear neural algorithm to extract the principal components which , in turn , are used by the music frequency estimator algorithm to extract the frequencies .    we generalize this method to avoid an interpolation preprocessing step and to improve the performance by using a new stop criterion to avoid overfitting .    
 the experimental results are obtained comparing our methodology with the others known in literature .