The note derives an expression for the number of interchanges made by selection sort when the sorting elements are iid variates from geometric distribution. Empirical results reveal we can work with a simpler model compared to what is suggestive in t
heory. The morale is that statistical analysis of an algorithms complexity has something to offer in its own right and should be therefore ventured not with a predetermined mindset to verify what we already know in theory. Herein also lies the concept of an empirical O, a novel although subjective bound estimate over a finite input range obtained by running computer experiments. For an arbitrary algorithm, where theoretical results could be tedious, this could be of greater use.
The notes which play the most important and second most important roles in expressing a raga are called Vadi and Samvadi swars respectively in (North) Indian Classical music. Like Bageshree, Bhairavi, Shankara, Hamir and Kalingra, Rageshree is anothe
r controversial raga so far as the choice of Vadi-Samvadi selection is concerned where there are two different opinions. In the present work, a two minute vocal recording of raga Rageshree is subjected to a careful statistical analysis. Our analysis is broken into three phases: first half, middle half and last half. Under a multinomial model set up holding appreciably in the first two phases, only one opinion is found acceptable. In the last phase the distribution seems to be quasi multinomial, characterized by an unstable nature of relative occurrence of pitch of all the notes and although the note whose relative occurrence of pitch suddenly shoots is the Vadi swar selected from our analysis of the first two phases, we take it as an outlier demanding a separate treatment like any other in statistics. Selection of Vadi-Samvadi notes in a quasi-multinomial set up is still an open research problem. An interesting musical cocktail is proposed, however, embedding several ideas like melodic property of notes, note combinations and pitch movements between notes, using some weighted combination of psychological and statistical stability of notes along with watching carefully the sudden shoot of one or more notes whenever there is enough evidence that multinomial model has broken down.
A raga is a melodic structure with fixed notes and a set of rules characterizing a certain mood endorsed through performance. By a vadi swar is meant that note which plays the most significant role in expressing the raga. A samvadi swar similarly is
the second most significant note. However, the determination of their significance has an element of subjectivity and hence we are motivated to find some truths through an objective analysis. The paper proposes a probabilistic method of note detection and demonstrates how the relative frequency (relative number of occurrences of the pitch) of the more important notes stabilize far more quickly than that of others. In addition, a count for distinct transitory and similar looking non-transitory (fundamental) frequency movements (but possibly embedding distinct emotions!) between the notes is also taken depicting the varnalankars or musical ornaments decorating the notes and note sequences as rendered by the artist. They reflect certain structural properties of the ragas. Several case studies are presented.
