Resumen de Semi-parametric density estimation with TrioScale

Paul H. C. Eilers, Mark de Rooij

• The TrioScale display transforms three-dimensional compositions to unbounded twodimensional coordinates. Then it is easy to estimate a smooth density for better visualization of scatterplots. We use penalized least squares smoothing of a detailed twodimensional histogram and back-transformation for display of a density in a ternary diagram. Originally we derived TrioScale by representing logarithms of proportions as negative squared distances from anchor points. An equivalent but simpler interpretation uses cyclic log-ratios, resulting in an unbounded triangular diagram with linear log-ratio axes. We also establish an equivalence to the isometric log-ratio transform. We apply our smoothing procedure to real-world data.