Barely Significant

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When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$tau$$\end{document} tau >0, it indicates that the sample has an increasing trend; when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$tau$$\end{document} tau is negative, it indicates a decreasing trend.
Since the annual rate F' displays an increasing trend (Fig. 4 a,b) we consider the rate change F"(t) instead, and by ∀k in N we define the Fourier coefficients: Φ k , F ′ = [ φ 1 k 2 + φ 2 k 2 ] 1 / 2 / k , φ 1 k = sin k · F ′′ , φ 2 k = cos k F ′′ , where · is the integral product over the interval [t 0 , t 0 + Δt], and ∀t in the interval we have: sin k t = sin 2 π kt / Δ t , cos k t = cos 2 π kt / Δ t .