# A Hybrid Technique for the Periodicity Characterization of Genomic Sequence Data

- Julien Epps
^{1, 2}Email author

**2009**:924601

https://doi.org/10.1155/2009/924601

© Julien Epps. 2009

**Received: **29 May 2008

**Accepted: **21 January 2009

**Published: **2 March 2009

## Abstract

Many studies of biological sequence data have examined sequence structure in terms of periodicity, and various methods for measuring periodicity have been suggested for this purpose. This paper compares two such methods, autocorrelation and the Fourier transform, using synthetic periodic sequences, and explains the differences in periodicity estimates produced by each. A hybrid autocorrelationâ€”integer period discrete Fourier transform is proposed that combines the advantages of both techniques. Collectively, this representation and a recently proposed variant on the discrete Fourier transform offer alternatives to the widely used autocorrelation for the periodicity characterization of sequence data. Finally, these methods are compared for various tetramers of interest in *C. elegans* chromosome I.

## 1. Introduction

The detection of structure within the DNA sequence has long captivated the interest of the research community. Among the various statistical characterizations of sequence data, one measure of structure within sequences is the degree of correlation or periodicity at various displacements along the sequence. Periodicity characterization of sequence data provides a compact and informative representation that has been used in many studies of structure within genomic sequences, including DNA sequence analysis [1], gene and exon detection [2], tandem repeat detection [3], and DNA sequence search and retrieval [4].

To measure such periodicity, autocorrelation has been widely employed [1, 5–11]. Similarly, Fourier analysis and its variants have been used for periodicity characterization of sequences [4, 9, 12–24]. In some cases [25, 26], the Fourier transform of the autocorrelation sequence has also been computed, however using existing symbolic-numeric mappings such as binary indicator sequences [27], this transform can also be calculated without first determining the autocorrelation. Other recent promising approaches to periodicity characterization for biological sequences include the periodicity transform [28], the exactly periodic subspace decomposition [3], and maximum-likelihood statistical periodicity [29], however these techniques have yet to be adopted by biologists for the purposes of sequence structure characterization.

Studies of structure within sequences, such as those referenced above, have tended to use either the autocorrelation or the Fourier transform, and to the author's knowledge, the limitations of each have not been compared in this context. In this paper, the limitations of both approaches are investigated using synthetic symbolic sequences, and caveats to their characterization of sequence data are discussed. A hybrid approach to periodicity characterization of symbolic sequence data is introduced, and its use is illustrated in a comparative manner on a study of tetramers in *C. elegans*.

## 2. Periodicity Measures for Symbolic Sequence Characterization

### 2.1. Definition of Periodicity

While this expression of in terms of a binary impulse train is perhaps not so common in signal processing of numerical sequences, the reverse is true for DNA sequences, which have been represented numerically using binary indicator sequences [27] in many studies (e.g., [13, 19, 23, 24, 30]).

### 2.2. Autocorrelation

*n*is the sequence index,

*ρ*is the lag, and

*N*is the length of the sequence. The application of the autocorrelation as defined in (5) to a symbolic sequence requires a numerical representation . The binary indicator sequences [27], which are sufficiently general as to form the basis for many different representations of DNA sequences, are employed in this analysis to represent in terms of

*M*binary signals:

*M*is the number of symbols (or patterns of symbols, such as a polynucleotide) , to which the numerical values are assigned, respectively, resulting in

*M*components . Assuming , the numerical representation can thus be unambiguously expressed as

Note that applying the decomposition in (2) to an exactly periodic sequence results in comprising a sequence of the numerical values that correspond to the repeated pattern of symbols.

so that the autocorrelation at a lag, or period,
for a symbol (or pattern of symbols) is simply the count of the number of instances of that symbol at a spacing of *ρ*.

*p*, so that the numerical representation of the sequence has a component . The autocorrelation of this component , for a segment of finite length

*N*, has the following expression:

*N*. Thus a shortcoming of the autocorrelation for sequence characterization is that an exactly

*p*-periodic sequence will show not only a peak at , but also peaks at values of that are integer multiples of

*p*(an example is given in Figure 1(a)). Note that similar artifacts can be found in other periodicity detection methods (e.g., [29]).

### 2.3. Fourier Interpretation of Periodicity

where *k* is the discrete frequency index. Since the DFT has sinusoidal basis functions, the notion of periodicity in the Fourier sense is described in terms of the frequencies of those basis functions onto which the projections of
are the largest in magnitude. That is, the magnitude of the DFT at a frequency *k*,
, is often taken as an estimate of the relative amount of that frequency component occurring in
[13, 19, 23, 24], from which the relative contribution of a particular period
can be estimated.

where the are determined according to (10).

*N*and

*k*need to be carefully chosen to allow fast Fourier transform-based calculation of for periods , where

*P*is the longest period to be estimated. Secondly, calculating the DFT at other frequencies is unnecessary. For these reasons, the integer period DFT (IPDFT) was proposed as an alternative to the DFT [19]:

Using a similar process to that described above in (10) and (11), the numerical representation of a symbolic sequence
can also be transformed using the IPDFT to produce a spectrum
that is linear in period (*ρ*) rather than in frequency (*k*). For the periodicity characterization of sequences, usually the magnitude
is of greatest interest. Some care is needed in the interpretation of the IPDFT, since for a binary periodic sequence such as
of fixed length *N*,
will decrease for longer periods due to the fact that the energy of
is
.

where
. That is,
is relatively large for
, and relatively small for
. From this, we see that a shortcoming of Fourier transform approaches such as the IPDFT for sequence characterization by periodicity is that they produce not only a peak at
, but also peaks at values of
that are integer divisors of the period *p* (see example in Figure 1(b)). For the DFT, this effect is also seen, but instead for indices whose value is
(i.e., harmonics of the frequency
with integer frequency indices).

### 2.4. Periodicity of a Synthetic Sequence Using Autocorrelation and DFT

To illustrate the shortcomings of the autocorrelation and DFT discussed in Sections 2.2 and 2.3, consider the periodicity characterization of an example signal (i.e., exact monomer periodicity ), where and . The autocorrelation and IPDFT are shown in Figures 1(a) and 1(b), respectively, from which the ambiguities in period estimate discussed in Sections 2.2 and 2.3 can be clearly seen.

## 3. Hybrid Autocorrelation-IPDFT Periodicity Estimation

### 3.1. Hybrid Autocorrelation-IPDFT

For the simple example signal from Section 2.4, the calculation of results in a single, unambiguous periodicity estimate, as seen in Figure 1(c).

where , which may be helpful for biologists who have conventionally used either the autocorrelation ( ) or the Fourier transform ( ). For the purpose of sequence periodicity visualization, for example, could be represented as a parameter available for real-time control, so that a biologist viewing a periodicity characterization of a sequence might subjectively assign a relative weight to each of the autocorrelation and Fourier transform components. Care is needed, however, with the application of (15), since is only well defined for for all . Note that this is satisfied by the autocorrelation defined in (8), in addition to a number of DNA numerical representations (several example representations are discussed in [30]). It is further noted that (14) and (15) do not have a straightforward physical interpretation, in contrast to and .

### 3.2. Evaluation of Periodicity Estimation in Noise

where is calculated using (14) throughout both this section and Section 4.

*p*in Figure 3 (

*p*small,

*p*larger and prime, and

*p*larger and highly composite), and as a function of the period in Figure 4. These results confirm earlier observations that the IPDFT provides more robust period estimates for prime periods than the autocorrelation, while the reverse is true for highly composite periods. The results also show that the hybrid technique is often able to provide a lower period error rate than either the autocorrelation or the IPDFT. Exceptions to this occur for some prime periods (see Figure 4), where the poorer performance of the autocorrelation seems to slightly adversely affect the hybrid estimate relative to the IPDFT-only estimate .

### 3.3. Evaluation of Multiple Periodicity Estimation

It is noted that the signal processing literature includes examples of methods for detecting multiple periodic signal components, such as the MUSIC algorithm [31]. For comparative purposes, the above experiment was repeated employing MUSIC to estimate the strengths of the periodic components. Results indicated that MUSIC was unable to consistently estimate either the periods or the relative strengths of the three components, returning no instances of all three periods correct and in the correct order. The dominant period estimate often contained the common factors of two or more of the true periodic components, an artifact attributable to the superposition of harmonic spectra reinforcing multiples of the individual component fundamentals that coincide in frequency. Two assumptions of MUSIC are not valid for this application: (i) the periodic components are not sinusoidal (although they can be represented as a harmonic series of sinusoids), (ii) the periodic components and noise may not be uncorrelated.

## 4. Application to DNA Sequence Data

Having discussed the differences between the autocorrelation and DFT for synthetic sequences, we now investigate the effect of using the IPDFT and hybrid autocorrelation-IPDFT in place of the autocorrelation on real sequence data. Numerous researchers have used autocorrelation [1, 5–10, 32]; here we compare with examples from the study of tetramer periodicity in the *C. elegans* genome using autocorrelation by Kumar et al. [1].

*C. elegans*tetramers analyzed in [1].

Note also that the IPDFT reveals a strong period-25 component, not at all evident in the autocorrelation. This surprising result was verified by constructing a synthetic sequence with perfect periodic components at
and
, and examining its autocorrelation and IPDFT. The autocorrelation of the sequence did not display visually any significant peak at
until the period-2 component had been eroded by at least 80%. In contrast, the IPDFT showed a clear peak at
with no period-2 erosion at all. The period-25 component has rarely been noted in previous literature, however in [11], a filtered distribution of distances between TA dinucleotides shows a strong peak at
, which Salih et al. attribute to a 5-base periodicity associated with the period-10 consensus sequence structure for *C. elegans*.

## 5. Conclusion

This paper has made two contributions to the periodicity characterization of sequence data. Firstly, the origins of ambiguities in period estimates for symbolic sequences due to multiples or sub multiples of the true period in the autocorrelation and Fourier transform methods, respectively, were explained. This is significant because these two methods account for perhaps the majority of the periodicity analysis seen in biology literature, and yet, to the author's knowledge, their limitations have not been discussed in this context. Secondly, a hybrid autocorrelation-IPDFT technique for periodicity characterization of sequences has been proposed. This technique has been shown to provide improved accuracy relative to the autocorrelation and IPDFT for period estimation in noise and multiple periodicity estimation, for synthetic sequence data. Comparative results from a preliminary investigation of tetramers in *C. elegans* chromosome I suggest that the proposed approach yields estimates that are consistently less prone to attribute significance to integer multiples or divisors of the true period(s). Thus, the hybrid autocorrelation-IPDFT is putatively advanced as a useful tool for biologists in their quest to reveal and explain structure within biological sequences. Future work will include studies of different types of periodicity in sequence data from other organisms, using IPDFT-based and hybrid techniques.

## Declarations

### Acknowledgments

The author would like to thank two anonymous reviewers for a number of helpful suggestions, which have certainly improved the quality of this paper. Thanks are also due to Professor Eliathamby Ambikairajah for helpful discussions. This research was supported by a University of New South Wales Faculty of Engineering Early Career Research Grant for genomic signal processing, 2009.

## Authors’ Affiliations

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