Both the exploratory and confirmatory approaches.Methods for detecting periodic fragments
Both the exploratory and confirmatory approaches.Methods for detecting periodic fragments Numerical Representation and PeriodicityIn this investigation, we represent DNA sequence fragments s[n], where n ? is a position index, using the binary indicator sequence numerical representation [36]. That is,x[n] = 1 (s[n], s[n + 1]) AA, TT, TA 0 otherwise(1)This representation is commonly employed and can be simply generalised to produce several other existing numerical representations. For the purposes of this paper we define perfect p-periodicity in a fragment of length N asN/pp [n] =k=[n – kp + n0 ],(2)where PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/27107493 n0 is the position of the first dinucleotide(s) of interest and [n] is the Kronecker delta. In practise perfect periodicity is very rare, and hence we also define (i) imperfect or eroded periodicity [30], which contains substitutions of the dinucleotide(s) of interest, and (ii) approximate periodicity, in which the period changes randomly from one cycle to the next but has expected value p, representing insertions or deletions of the dinucleotide(s) of interest.Significance Measures for Confirmatory Period EstimationResults and Discussion Figure 1 presents an overview of the process by which estimates of the periodicity of symbolic data were obtained in this study. Irrespective of whether the starting material was biological or synthetic, all subsequentIn this study, we employ four period UNC0642 chemical information estimation techniques, derived from the two most commonly employed groups of techniques in the literature: autocorrelation and Fourier-based methods. The integer period discrete Fourier transform (IPDFT) is a variant of the wellknown discrete Fourier transform (DFT) that is tailored for the integer-period problem of exploratory period estimation, however the two are virtually equivalent for the purposes of confirmatory period estimation. The Hybrid autocorrelation-IPDFT combines the strengths of the autocorrelation and DFT methods for exploratory period estimation. Four measures of periodicity significance are considered: Two are based on the g-statistic and resampling (blockwise bootstrap (BWB)) approaches discussed in the background, while the Chi-squared measure is new in the current context. The Cram -Rao bound (CRB) developed herein is applicable to Fourier-based period estimation, and although it is not strictly a test, it gives a measure of the strength of periodicity. To distinguishEpps et al. Biology Direct 2011, 6:21 http://www.biology-direct.com/content/6/1/Page 5 ofbetween exploratory period estimation and BWB based on a particular period estimation technique, we will refer to the use of period estimation methods within BWB as embedded. For instance, a BWB using the IPDFT to determine the peak values |S[p]| of resampled sequence fragments will be referred to as employing an embedded IPDFT. Formal definitions and full details of all measures are given in the Methods section.Statistical properties of period estimation (from simulated data)We evaluated the statistical properties of the different techniques by first applying them to simulated data. The striking dinucleotide periodicities reported for nucleosome associated DNA are typically presented as means of many sequences. Individual sequences are unlikely to perfectly match this signature due to the effect of mutation. We considered two different definitions of imperfectly periodic sequences: (i) imperfectly periodic due to erosion of a perfectly period-10 signal; (ii) imperfectly period.
