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Appendix A: Kernel density and LOWESS

The kernel density is a non-parametric estimate of the probability density function from a discrete set of data. It can be viewed as a generalization of the histogram, with better theoretical properties (Härdle & Simar 2012). For a set of n observations x₁, x₂, ..., x_n, a kernel density with bandwidth h has the form: (A.1)where the kernel function K is chosen to be a probability density function. Several choices of kernel are available. In this work. we make use of a Gaussian kernel: (A.2)The kernel selection usually has a minor influence on the kernel estimate with respect to the bandwidth h. This parameter is selected balancing two effects since an increment of h increases the bias of , while it reduces its variance. An often adopted choice for a Gaussian kernel is given by the rule of thumb (Silverman 1986): (A.3)where is the sample standard deviation and R the sample interquartile range.

Other choices for the bandwidth, based on the asymptotic expansion of the mean integrated squared error, are reported in the literature. The different choices have an impact on kernel estimator for multi-modal distributions, which is not the case of the present work. We refer interested readers to Feigelson & Babu (2012), Härdle & Simar (2012), Venables & Ripley (2002), Sheather & Jones (1991).

A frequently used bivariate smoother is the local regression technique LOWESS, which combines a linear least squares regression with a robust nonlinear regression. It provides a generally smooth curve, whose value at a particular location along the x axis is only determined by the points in its neighbourhood. The first step is to fit a polynomial regression in a neighbourhood of x. A fraction f of the n sample points near x is selected. We define m =  ⌈ fn ⌉ the number of points used in the fit. Then the technique obtains the estimates by minimizing (A.4)where W_i(x) are the weights, usually obtained by the tricubic function: (A.5)where d is the maximum distance between x and the other predictor values x_i in the span. For LOWESS estimates the value p = 1 is usually adopted, implying a local linear regression. The model residuals and the scale parameter are computed. The median absolute deviation of the residuals is evaluated: . Then the algorithm computes the robustness weights , with R(u) = 15/16(1 − u)² for | u |  ≤ 1 and R(u) = 0 otherwise. The local regression of Eq. (A.4) is computed again, but with weights given by δ_iK_i(x). This procedure is iterated a variable number of times between one and five; this makes the local estimate robust even in presence of outliers. Further details on the technique and on the numerical methods used to speed up the computations are available in Cleveland (1981).

The computations outlined in this section were performed using the functions density and lowess, available in the R 2.15.2 (R Development Core Team 2012).

	Fig. 6 As in Fig. 2, but for data sampled from a grid with ΔY/ΔZ = 1, and recovered with the standard grid adopting ΔY/ΔZ = 2.
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	Fig. 8 As in Fig. 2, but for synthetic data sampled from a grid with αml = 1.50, and estimated with the standard grid (i.e. αml = 1.74).
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	Fig. 9 As in Fig. 2, but for synthetic data sampled from a grid with αml = 1.98 and estimated with the standard grid (i.e. αml = 1.74).
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	Fig. 10 As in Fig. 5, but for synthetic data sampled from grids with different values of radiative opacity and reconstructed with the standard grid.
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	Fig. 11 As in Fig. 2, but for synthetic data sampled from a grid with kr at its low value, estimated on the standard grid.
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	Fig. 13 As in Fig. 2, but for synthetic data sampled from the standard grid and recovered on the grid, which does not include diffusion in the computations.
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	Fig. 14 As in Fig. 2, but for synthetic data sampled from the standard grid and recovered on the same grid but without taking the surface [Fe/H] evolution into account.
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© ESO, 2014