Research on the properties of separable binary functions

Purpose According to the fact that the single function transformation which can both reduce the class ratio dispersion and keep the relative error no enlargement after the inverse transformation does not exist, this paper provides the separable binary function transformation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo></m:mrow><m:mi>x</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>k</m:mi><m:mrow><m:mo>)</m:mo></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mrow><m:mo>(</m:mo><m:mi>k</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>⋅</m:mo><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>k</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> . The authors select the appropriate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo></m:mrow><m:mi>x</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>k</m:mi><m:mo>)</m:mo></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>k</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> to get <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo></m:mrow><m:mi>x</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>k</m:mi><m:mrow><m:mo>)</m:mo></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mrow><m:mo>(</m:mo><m:mi>k</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>⋅</m:mo><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo><m:mi>k</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> . The sequence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msubsup><m:mrow><m:mo>{</m:mo><m:mrow><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo></m:mrow><m:mi>x</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>k</m:mi><m:mrow><m:mo>)</m:mo></m:mrow><m:mo>,</m:mo><m:mi>k</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>}</m:mo></m:mrow><m:mrow><m:mi>k</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:mi>n</m:mi></m:msubsup></m:mrow></m:math> can not only improve the modeling accuracy but also ensure that the inverse transformation relative error has no enlargement. Design/methodology/approach First of all, to meet that the sequence reduces the class ratio dispersion after binary function transformation, the sufficient and necessary condition of binary function transformation with reduced class ratio dispersion is obtained. Secondly, to meet the condition that the inverse transformation relative error is not enlarged, the necessary condition of separable binary function transformation is obtained respectively for monotonically increasing and monotonically decreasing function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> . Finally, the feasibility and correctness of this method are illustrated by example analysis and application. Findings The sufficient and necessary condition of binary function transformation with reduced class ratio dispersion and the necessary condition of separable binary function transformation with the inverse transformation relative error no enlargement. Practical implications According to the properties of separable binary function transformation provided in this paper, the grey prediction function model is established, which can improve the modeling accuracy. Originality/value This paper provides a binary function transformation, and researches the sufficient and necessary condition of binary function transformation with reduced class ratio dispersion and the necessary condition of separable binary function transformation with the inverse transformation relative error no enlargement. It is easy for scholars to carry out the pretest before selecting the separable binary function transformation. The binary function transformation is the further extension of single function transformation, which broadens and enriches the choice of function transformation.