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ESTIMATION OF THE FORMULA FOR THE RANKINGS IN VOCALOID COLLECTION

SHANNON

ABSTRACT. We estimate the formula for the rankings in VOCALOID Collection by expanding it into a Laurent series and searching its optimal family of coefficients.

1. ASSUMPTIONS

We assume that the rankings in the VOCALOID Collection are determined solely by a value that depends on three explicit variables, namely view counts, the number of comments, and the number of ”My Lists”. We may assume that this function is defined everywhere except at the origin since it is supposed that all VOCALOID producers will never be abandoned, so we can omit (0, 0, 0) from the domain of f. Moreover, we also assume that the formula for this value is a real analytic function with three variables. Thus, we consider the formula f as expressed by a Laurent series:

Where the variables v, c, and m are respectively view counts, the number of comments, and the number of “My Lists” and each a is a real number.

2. METHODS

Based on the above assumptions, we search for the family of coefficients of f that yields the closest result to the actual ranking by comparing the hypothetical rankings obtained from substituting real data into f. Practically, we approximate f by a Laurent polynomial. Hereafter let f be a Laurent polynomial of degree d.

2. 1 Method of evaluation. To compare hypothetical and actual rankings, we employ the concept of inversion numbers. The inversion number of a permutation represents the count of index pairs that the permutation inverts. We view a hypothetical ranking as a permutation of the actual ranking, enabling us to calculate its inversion number.

2.2 Method of Searching for Coefficients. Let N be the number of terms appearing in the Laurent polynomial f. We regard a family of coefficients as an element of an N-dimensional R-vector space, V. To obtain rankings, it’s necessary only to consider the ratios of families of coefficients. We consider spheres in V of varying radii and select points from these spheres. By considering larger spheres, we derive a more diverse set of ratios. In order to avoid local optima, significant effort is required.

3. RESULTS