Two methods for the light curves extrema determination

arXiv:1111.1142v1 [astro-ph.IM] 4 Nov 2011 TWO METHODS FOR THE LIGHT CURVES EXTREMA DETERMINATION Z. Mikul´aˇseka, b and T. Gr´afa aObservatory and Planetarium of J. Palisa, VˇSB–Technical University of Ostrava, Czech Republic tomas.graf@vsb.cz bDepartment of Theoretical Physics and Astrophysics, Masaryk University, Brno, Czech Republic mikulas@physics.muni.cz Abstract: Two methods for the determination of extrema timings and their uncertainties appropriate for the analysis of time series of variable stars using matrix calculus are presented. The method I is suitable for determination of times of extrema of non-periodical variables or objects, whose light curves vary. The method II is apt for O-C analyses of objects whose light curves are more or less repeating. Keywords: methods – variable stars 1. Linear regression using matrix calculus The determination of times of extrema and its uncertainty of light vari- ations belongs to the most common problems in the astrophysics of variable stars. The following instructive text develops the basic ideas outlined in (1) Mikul´aˇsek et al. (2006). 1.1. The data Let we have n observations {yi} creating the vector Y, Y = {yi} done in moments {ti} creating the vector t, t = {ti}, both of the orders of n × 1. It is demanding to adapt the time scale so that t ⇒t −t. The weight wi expresses the quality of an i-th measurement; w = {wi}. If we know the inner uncertainty (error) of the i-th observation δyi, we can determine the weight of it as follows: wi ∼(δyi)−2. If we assume that the quality of all measurements are more or less equal or if we know about the quality of measurements little or nothing, it would be honest to put wi ≡1, w = E(n, 1). The most of relations look simpler if we normalise weights so it is valid w = 1, then Pn i=1 wi = n. Frequently, we use instead of the w square matrix (n × n) W, W = diag(w). 1.2. Linear regression The observed relationship between the dependent variable (inaccurately me- asured quantity - mostly magnitude, radial velocity, temperature) y and the independent variable (precisely measured quantity – typically time) t can be fit by an appropriate model function F(t). The model function is determi- ned by g free parameters βj that create a column vector ⃗β = [β1, β2, ...βg]T. The upper index T denotes transposing the matrix. If the model function F(t) can be expressed as a linear combination of g different functions of time fk(t) (we speak here about the linear model function). Then f(t) = [f1(t), f2(t), ..., fg(t)], F(t, ⃗β) = g X k=1 βk fk(t) = f(t) ⃗β. (1) Let introduce the matrix X of rank n×g and the column vector Yp, (n×1): X =        f(t1) f(t2) ... f(tn)        ; Yp =        F(t1) F(t2) ... F(tn)        = X ⃗β. (2) As the objective measure of the success rate of the fit for an a ⃗β is used usually the sum of weighted squares of deflection of observed values and predicted ones S(⃗β): S(⃗β) = (Y −Yp)T W (Y −Yp) = YT W Y −2 βT U + βT V β, (3) where U = XT W Y; V = XT W X; H = V−1. (4) The least square method (LSM) considers the fit by the model function F(t, ⃗β) as the best one if the sum R = S(⃗β = b) is minimal. In the case of the linear model function F(t, ⃗β) we obtain for b and R: ∂S ∂⃗β ⃗β=b = 0 = −2 U + 2 V b, ⇒ b = H U; R = YTW Y −bTU. (5) 1.3. The example Standardly used linear regression models are polynomials or trigonometric functions of arbitrary orders. As an example we select the parabolic model - the simplest model we can use for fit of the real function in the form: F(t) = β1 t2 + β2 t + β3, f(t) = [t2, t, 1], X = [{t2 i } {ti} {1}]. 1.4. Standard deviation. Uncertainties The standard deviation σ can be estimated using relation σ = s R n −g . (6) The components of the column vector δb used to be considered as a rigou- rous estimate of the uncertainty of the particular parameters. Unfortuna- tely, they have this meaning only exceptionally, nevertheless it is sometimes required by referees. Contrary, very valuable is the following estimate of the uncertainty of the model predictions δF(t) δb = σ q diag(H); δF(t) = σ q f(t) H f(t)T. (7) 2 Method I Method I of the determination of the extrema times consists of several steps. • Plot the time series and select the appropriate linear model function which can fit the observed light curve sufficiently well with the mini- mum of free parameters. • Fit the observed light curve by the model function and check whether the dependence of deflections of observed and predicted light curves show only random scatter or some trends. In the latter case improve your model function • Find the extreme time on the fitted light curve and calculate its un- certainty. 2.1. When the minimum/maximum occurs? The moments of the fitted model function extrema occurs if it is fulfilled the condition that time derivative in that te equals to zero, especially 0 = ˙yp(te) = ˙F(te) = d (f(te) b) dt = ˙f(te) b; ˙f(t) = [ ˙f1(t), ˙f2(t), ... ˙fg(t)]. (8) There are many techniques how

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