Idered.Pearson’s Chi-Squared TestP(g > gobs ) =m=(-1)m-(pmax
Idered.Pearson’s Chi-Squared TestP(g > gobs ) =m=(-1)m-(pmax – 1)! m!(pmax – 1 – m)! ?1 – mgobs )pmax -2 ,M= 1 gobs(9)where gobs is the observed value of g and, it ispossible to test whether a sequence is purely random or whether it has periodic behaviour. The g-statistic is applicable to numerical signals.Blockwise BootstrapIn sequence analysis, both of the permutation approaches discussed in the background section have the limitation that they disrupt the non-random background distribution of polynucleotides. In essence, neighbouring nucleotides cannot be considered to be independently distributed [34,35]. PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/27107493 Hence, we adopt a method we refer to as the blockwise bootstrap (BWB). As in Ying et al. [33], here we resample sequences of interest by building a resampled sequence sp[n] from p-length fragments of s[n] selected at random, i.e. (sp[lp], sp[lp + 1],…, sp[(l + 1)p – 1]) = (s[n], s[n + 1],…, s[n + p – 1]), l = 0, 1,…, N/p, where n ?0, 1,…, N p is selected at random for each l. An appealing feature of this approach is that it preserves the base rate of buy OPC-8212 occurrence of nucleotides (and polynucleotides up to length p) during the test. R synthetic sequences are produced by randomly resampling, with replacement, from the original sequence fragment, and the number of ^ times NG a peak greater than or equal to |S[p]| is mea^ sured at period p is recorded. Finally the p-value PBWB of the test sequence is determined as NG /R. A low pvalue, for example less than 0.01, corresponds to fewer than 1 in 100 resampled sequences exhibiting a peak ^ ^ greater than or equal to |S[p]| at period p. Of course this ^ test can be applied to other periods than p, to ascertain the significance of a secondary peak, for example. This type of test has been applied in a very wide range of applications, e.g. [42].Treating perfect periodicity as a model, whose fit to (or deviation from) the sequence data of interest is to be measured, a chi-squared test can be developed. In this case the deviation corresponds to a period estimation significance measure, while the test itself is a threshold for the measure. To calculate the deviation, it is necessary to first define the `model’. For a sequence fragment that is perfectly p-periodic with respect to symbol sm, m ?1, 2,…, M , the probability mass function (pmf) is 1 if k = m . Note that Psm (s[n + p] = sk | s[n] = sm ) = 0 otherwise this pmf says nothing about other symbols; they could be randomly occurring or also perfectly periodic, but most often we are interested in the periodicity of a particular symbol or symbols. Having determined the periodicity `model’, a count of the observed instances of periodicity is required. For each position in a sequence fragment s[n], note the value sm of the current symbol, look ahead by p, record the presence or absence of each symbol of interest, then aggregate these across all instances of sm and divide by the total number of occurrences of sm, to produce an empirical p-spaced pmf for each symbol. That is, for a sequence fragment s comprised of symbols s1, s2,…, sM, form the set Cm = s[n] = sm, n = 0, 1,…, N – 1 and the M sets Cm = s[n] , one per symbol s m . The empirical pmf for each symbol, E Psm (s[n + p] = sk | s[n] = sm ), is then |Cmsk | if k = m |C E Psm = |C m |- |C | , msk m| otherwise |Cm |skwhere |C| denotes the number of elements in C. The deviation measure can thus be constructe.
