Table 1.

Description of equations used in estimating the peak timing.

Description	Equation
Phase shift, Ψ as a distance of peak from beginning of time series in radians	$-\arctan \left(\frac{{\beta }_{sin}}{{\beta }_{cos}}\right)$ (5)
Variance of phase shift, Var(Ψ)	$\frac{\text{{}{\left({\text{β}}_{\cos }{\text{*σ}}_{{\text{β}}_{\sin }}\right)}^{\text{2}}\text{+}{\left({\text{β}}_{\text{sin}}{\text{*σ}}_{{\text{β}}_{\cos }}\right)}^{\text{2}}-{\left({\text{2*β}}_{\text{sin}}{\text{*β}}_{\cos }\text{*}{\sigma }_{{\beta }_{\sin }}{}_{{\beta }_{\cos }}\right)}^{\text{2}}\text{}}}{{\left({\text{β}}_{\text{sin}}{}^2{\text{+β}}_{\text{cos}}{}^2\right)}^{\text{2}}}$ (6)
Confidence interval of phase shift in radians	$\Psi \pm (1.96*\sqrt{Var(\Psi )}$ (7)
Peak timing in months, where M is the length of one cycle	$\begin{array}{l}M*\left(-\frac{\Psi }{2\pi }\right)\text{, }{\beta }_{\cos }>0\text{ \& }{\beta }_{\sin }>0;\\ M*\left(1-\frac{\Psi }{2\pi }\right)\text{, }{\beta }_{\cos }>0\text{ \& }{\beta }_{\sin }<0;\\ M*\left(\frac{1}{2}-\frac{\Psi }{2\pi }\right)\text{, }{\beta }_{\cos }<0\end{array}$ (8)
Confidence interval for peak timing in months	Substitute equation 7 for Ψ in equation 8