5 Spurious periodicities

The period with the smallest $\chi^2$ is not always "real'', because "spurious'' periods are introduced by the interplay between $\bar{t}$ distribution and real periodicity. The phases of $\bar{t}$ folded with an arbitrary period are circular data. This $\bar{t}$ distribution can be studied with numerous methods for analysing circular data (e.g. Batshelet 1981; Jetsu & Pelt 1996). We apply the Deeming (1975) window function

$${\gamma}_n(f) = n^{-1} \sum_{k=1}^n {\mathrm{e}}^{~2{\pi}i ~ft_k},$$ (15)

where the n complex vectors $[\cos{2 \pi f t_k}, i \sin{2 \pi f t_k}]$ are more parallel for a significant window period $P_0\!=\!f^{-1}$. If $\bar{y}$ varies with P₁, then P₀ induces an important set of spurious periods

$$P'\!=\!P'(P_0\!:\!k_1\!:\!k_2)\!=\!\left[P_1^{-1}+k_1(k_2P_0)^{-1}\right]^{-1},$$ (16)

where $k_1\!=\!\pm 1$, $\pm 2,...$and $k_2\!=\!1,2,..$. This relation (Tanner 1948) states that t_i separated by k₂P₀ have an integer phase difference k₁ with the periods P₁ and P'. These $P'(P_0\!:\!k_1\!:\!k_2)$ do not significantly alter the $\bar{y}$ sequence in $\bar{\phi}$. However, $P'\!<\!0$ produce a "mirror image'' by reversing the $\bar{\phi}$ sequence.

But Eq. (16) cannot separate P₁ from $P'(P_0\!:\!k_1\!:\!k_2)$. The following modification of another method by Tanner (1948) may reveal, if a window P₀ induces a spurious period P₁: (1) Derive $\delta \phi_i'\!=\! {\mathrm{~FRAC}} [(t_i\!-\!t_1)P_0^{-1}]$. Transform $\delta \phi_i'\!>\!0.5$ to $\delta \phi_i'\!=\!\delta \phi_i'\!-\!1$. (2) Model the $\bar{y}$ with P₁, and measure the "phase residuals'' ( $\delta \phi _i$), i.e. the minimum $\phi$ distance for each y_i from the model $g(\phi)$. These $\delta \phi _i$ are unique, when the distance between data $(\phi _i, y_i)$ and model $(\phi ,g(\phi ))$ is defined as $d^2\!=\!d_1^2\!+\!d_2^2$, where $d_1\!=(\phi-\phi_i)$and $d_2\!= \mid y_i\! - \! g(\phi) \mid \{{\mathrm{max}}[g(\phi)]\! - \! {\mathrm{min}}[g(\phi)]\}^{-1}$. Transform $d_1 \! > \!0.5$ to $d_1\!=\!d_1\!-\!1$. The d₁ and d₂ are comparable, because d₂ is independent of the y_i units. The d minimum determines the point $(\phi ,g(\phi ))$ closest to $(\phi _i, y_i)$ uniquely, which yields the phase residuals $\delta\phi_i\!=\!\phi\!-\!\phi_i$. Transform $\delta \phi_i\!>\!0.5$to $\delta \phi_i\!=\!\delta \phi_i-1$. (3) The linear correlation coefficient ( $\mid \!\! r_0 \!\! \mid$) between $\delta \bar{\phi}'$ and $\delta \bar{\phi}$ is higher for "spurious'' than "real'' periods. Note that we subtract the means from $\delta \bar{\phi}'$ and $\delta \bar{\phi}$(Figs. 1q-u, 3q-u, 5q-u and 7q-u), since this does not influence $\mid \! r_0 \! \mid$. Our "correlation hypothesis'' is

H_R: The $\delta \bar{\phi}'$ and $\delta \bar{\phi}$are uncorrelated random samples.

Under H_R, the value of $P(r_0)\!=\!P(\mid\!r\!\mid \geq \mid \!r_0\!\mid)$is the probability (i.e. the critical level) that $\mid \!r\! \mid$ reaches $\mid \! r_0 \! \mid$, or an even larger, value.