Evolving the best known approximation to the Q function

Abstract

The Gaussian Q-function is the integral of the tail of the Gaussian distribution; as such, it is important across a vast range of fields requiring stochastic analysis. No elementary closed form is possible, so a number of approximations have been proposed. We use a Genetic Programming (GP) system, Tree Adjoining Grammar Guided GP (TAG3P) with local search operators to evolve approximations of the Qfunction in the form given by Benitez [1]. We found more accurate approximations than any previously published. This confirms the practical importance of local search in TAG3P.

Better GP benchmarks: community survey results and proposals

“…For problems to include, several respondents suggested new variants of symbolic regression, including dynamic symbolic regression, the Q-function used by Phong et al [39], the chaotic flow function described by Sprott [44], symbolic regression for protein folding predictions, the Dow Chemical dataset used in previous symbolic regression EvoCompetitions (see Sect. 6.3), and the spline regression problems defined by Friedman [11].…”

New Tight Bounds for the Gaussian Q-Function and Applications

“…Next, we follow the same approach as in [5] to propose two novel tighter bounds (i.e., fourth and fifth) with almost the same complexity by multiplying U (1) (x) and U (3)…”

“…The six bounds presented in this paper ordered, except the lower bound, by tightness in increasing order are summarized in Table II. Although U (4) is the tightest upper bound among the proposed ones, its flexibility in numerous integrals still difficult compared to U (3) where only one exponential function is involved.…”

“…Scanning the open literature, in [1], the GQF has been approximated by the ratio of an exponential function to a polynomial, while in [2] it was approximated by the sum of either three or four exponential functions via the Trapezoidal Rule. To this end, an approximate expression for GQF using an exponential function of degree-six polynomial was proposed in [3], based on tree-adjoining grammar guided genetic programming. In a similar way, an approximate expression, in the form of an exponential of degree-2 polynomial with adaptive parameters, was investigated in [4].…”

“…• A refinement of U (1) (x) and U (3) (x) leads to two other tight bounds, U (4) (x) and U (5) (x). • The accuracy versus simplicity of the aforementioned bounds are discussed.…”