6 Application examples

In this section we present detailed TSPA analysis examples for real data. All time units are days. Sections 6.1 and 6.2 describe the second order TSPA ($K\!=\!2$) analysis for the normalized magnitudes of V 1794 Cyg during subsets SET=42 and 114 from Jetsu et al. (1999b, Paper II). The next Sect. 6.3 clarifies our methodology in Paper I, where the second order TSPA analysis is performed for all subsets of normalized magnitudes of V 1794 Cyg. In Sect. 6.4 we apply the fifth order TSPA ($K\!=\!5$) to the V light curve of the cepheid variable BL Her (Moffett & Barnes 1984). One sample of the V magnitudes of the nonvariable primary comparison star SAO50205 from Paper II are analysed with the first order TSPA ($K\!=\!1$) in Sect. 6.5.

   
6.1 Normalized UBVR magnitudes of SET=42

The magnitudes of V 1794 Cyg within each passband ($\bar{y}$) have been normalized with

$$y_{\mathrm{norm}}(t_i) = \left (y(t_i)-m_y \right) \left( H s_y \right)^{-1},$$ (17)

where m_y and s_y are the mean and standard deviation of $\bar{y}$, and H is an arbitrary scaling factor. The $\bar{y}_{\mathrm{norm}}$ standard deviation $s_{y,{\rm norm}}\!=\!H^{-1}$ and $\bar{\sigma} s^{-1}_y \!=\! \bar{\sigma}_{\mathrm{norm}} s_{\mathrm{y,norm}}^{-1}$ give $\bar{\sigma}_{\mathrm{norm}} \!= \![H s_y]^{-1} \bar{\sigma}$. The accuracy of each individual photometric observation is usually unknown, like in Paper II that contains published and new photometry from several observatories. The $\bar{y}_{\mathrm{norm}}$ and $\bar{\sigma}_{\mathrm{norm}}$ were determined with $\sigma\!=\!0\hbox{$.\!\!^{\rm m}$ }015$ in BVR, $\sigma\!=\!0\hbox{$.\!\!^{\rm m}$ }030$ in U, and $H\!=\!4$ in Eq. (17). These normalized magnitudes are hereafter denoted by $\bar{y}$ and $\bar{\sigma}$, i.e. the "norm'' subscripts are omitted here, as well as in Sects. 6.2 and 6.3.

In SET=42, the deepest $\Theta'_{\mathrm{pilot}}(f)$ (Eq. 6) minimum between $P_{\mathrm{min}}\!=\!0.5$ and $P_{\mathrm{max}}\!=\!10$ is at $P_1\!=\!3.33$ (Fig. 1a: $n\!=\!149$). The best window period P₀=0.9997 detected with $\gamma_{\mathrm{n}}$ (Eq. 15) connects P₁ to $P_2\!=\!0.77 \!\approx \!P'(P_0\!:\!1\!:\!1)\!=\!0.77$ and $P_3\!=\!1.42\!\approx \mid\!P'(P_0\!:\!-1\!:\!1)\!\mid = \mid\!-1.43\!\mid$, while $P_4\!\approx\!2P_2$ and $P_5\!\approx\!2P_1$. The y_i curves resemble sinusoids with P₁, P₂ and P₃( $k_1\!=\!-1$: "mirror image''), and those with P₄ and P₅ double sinusoids (Figs. 1b-f). The $\Theta_{\mathrm{grid}}(f)$ minima provide more accurate P₁, ..., P₅ (Figs. 1g-k), and the RSch trial values $\bar{\beta}_0$. The RSch bootstrap with $S\!=\!200$ gives the final P₁, ..., P₅. Of these, $P_1\!=\!3.361\pm0.004$ has the smallest $\chi^2_0\!=165.9$, that for $P_5\!\approx\!2P_1$ being comparable (Figs. 1l-p). The critical levels for rejecting H₀ with m=55 are not significant, not even for P₁ (Fig. 1l: $P(\chi^2_0)\!=\!1.00$). Three alternatives could explain this: (1) The errors $\bar{\sigma}$ are not correct, and $\chi^2_0$ is unreliable. (2) The errors are correct, but the model is not, e.g. the order $K\!=\!2$ may be too low, or perhaps the light curve evolves during $\Delta T$. (3) A few erroneous observations can contaminate the $\chi^2_0$determined by four normalized magnitudes at each t_i (i.e. one in each UBVR passband). Hence we can not use $P(\chi_0^2)\!<\!0.01$ to reject H₀ for these data. Paper I (A_V changes in Sect. 3.2.) gives a counterexample, where TSPA modelling can apply this criterion. The lines connecting each $(\phi _i, y_i)$ to the closest point of the model $(\phi ,g(\phi ))$ determine the phase residuals $\delta \phi _i$ (Figs. 1l-p). The linear correlation coefficient $\mid \! r_0 \! \mid$between $\delta \bar{\phi}'$ and $\delta \bar{\phi}$ is smallest for P₁, but of the same order as that for P₅ (Figs. 1q and u). The highly improbable $\mid \! r_0 \! \mid$ for P₂, P₃ and P₄ reveal that these periods are indeed spurious. In conclusion, the best period is P₁ with the smallest $\chi^2_0$ and lowest $\mid \! r_0 \! \mid$. The double sinusoid with P₅ imitates the solution with P₁.

The $S\!=\!200$ RSch bootstrap not only provides the P error estimate, but also those of M (mean), A (total amplitude), t_min,1 and t_min,2 (epochs of primary and secondary minima). Figure 2 summarizes the earlier P₁ bootstrap of Fig. 1l. The F_S(u) (Eq. 12) for the $S\!=\!149(\!=\!n)$ residuals $\bar{\epsilon}$ does not cause H_G rejection with $\alpha \!=\!0.01$ (Fig. 2b, Eq. 14), i.e. this "empirical distribution'' of $\bar{\epsilon}$ (Eq. 11) is Gaussian. The M and P distributions are also Gaussian (Figs. 2c-f). The model gives these theoretical M and P estimates directly, but those of A and t_min,1 are measured from the model for each $\bar{y}^*$ to avoid the complicated theoretical solutions with $K\!\geq\!2$. That these A and t_min,1 distributions are also Gaussian, confirms the validity of our measurement approach (Figs. 2g-j). Note that SET=42 has no secondary minimum t_min,2with $P_1\!=\!3.361$, and that the primary minimum phase is $\phi_{\mathrm{min,1}}\!=\!{\mathrm{FRAC}} [t_{\mathrm{min,1}}\!-\!t_1]$ (Fig. 2i). In conclusion, the $\bar{\epsilon}$, $\bar{M}$, $\bar{P}$, $\bar{A}$ and $\bar{t}_{\mathrm{min,1}}$ bootstrap statistics for SET=42 are reliable.

The model parameter error estimates in this paper utilize $S\!=\!200$ bootstrap samples during the RSch. This ensures reliable statistics, because these estimates usually stabilize already at . For example, increasing the number of samples from $S\!=\!25$ to 200 in the RSch of Figs. 2c and e gave the following error estimates for the mean ( $\sigma_{\mathrm{M}}$) and the period ( $\sigma_{\mathrm{P}}$)

S	25	50	75	100	150	200
$\sigma_{\mathrm{M}}$	0.0058	0.0054	0.0050	0.0052	0.0051	0.0052
$\sigma_{\mathrm{P}}$	0.0042	0.0041	0.0040	0.0040	0.0042	0.0042

   

Table 1: TSPA results between $P_{\mathrm{min}}\!=\!0.5$ and $P_{\mathrm{max}}\!=\!10$ for 93 subsets of normalized photometry of V 1794 Cyg

	$P \! \sim \!3.33$	$P'(P_0\!:\! 1\!:\!1) \! \sim \!0.77$	$-P'(P_0\!:\!-2\!:\!1) \! \sim \!0.58$	$-P'(P_0\!:\!-1\!:\!1) \! \sim \!1.42$	$2P \! \sim \!6.66$
Among detected P1, ..., P5	${84 \over 93}$(90%)	${83 \over 93}$(89%)	${61 \over 93}$(66%)	${77 \over 93}$(83%)	${25 \over 93}$(27%)
Smallest $\chi^2_0$ of P1, ..., P5	${31 \over 84}$(37%)	${22 \over 83}$(27%)	${10 \over 61}$(16%)	${ 8 \over 77}$(10%)	${ 4 \over 25}$(16%)
Smallest $\mid \! r_0 \! \mid$ of P1, ..., P5	${32 \over 84}$(38%)	${12 \over 83}$(14%)	${ 7 \over 61}$(11%)	${13 \over 77}$(17%)	${ 7 \over 25}$(28%)
$P(\mid \!r_0\! \mid) \leq \alpha=0.01$	${ 4 \over 84}$(5%)	${11 \over 83}$(13%)	${15 \over 61}$(25%)	${16 \over 77}$(21%)	${ 2 \over 25}$(8%)

   
6.2 Normalized UBV magnitudes of SET=114

The TSPA for SET=42 posed no serious problems (Sect. 6.1), but "pathological'' cases are not rare for real data, and analysing the normalized UBV magnitudes of SET=114 serves as such a "good'' counterexample. The "real'' $P_2\!=\!3.33$ periodicity is not the best one detected by PSch (Fig. 3a), and TSPA reveals that $P_1\!\approx\!0.77$ has the smallest $\chi^2_0$(Figs. 3l-p). The P₁, ..., P₅ and $P_0\!=\!0.9981$ connections are obvious (Eq. 16). All five periods reach extreme critical levels ( ), although the residuals appear extraordinary large. These $P(\chi_0^2)$ are extreme, because the data accuracy is equal or below the UBVlight curve amplitudes $A_U\!=\!0\hbox{$.\!\!^{\rm m}$ }020$, $A_B\!=\!0\hbox{$.\!\!^{\rm m}$ }014$ and $A_V\!=\!0\hbox{$.\!\!^{\rm m}$ }025$ during SET=114 (Paper I, Table 1). But lowering the modelling order, e.g. $K\!=\!0\!\equiv\!\bar{g}$=constant, would confuse the time series analysis, it being crucial that the same model is used for all subsets to obtain M, A, P, t_min,1 and t_min,2. As expected, the $P_2\!=\!3.34~\pm ~0.05$ periodicity during SET=114 (Fig. 3m) is less accurate than that of $P_1\!=\!3.361~\pm ~0.004$ during SET=42 with high $A_V\!=\!0\hbox{$.\!\!^{\rm m}$ }192$ (Paper I, Table 1). Nevertheless, the 3.34 signal detection in SET=114 confirms that we are not modelling "noise''. The low UBV amplitudes in SET=114 also explain the absence of correlations in Figs. 3q-u. The $S\!=\!200$ RSch bootstrap reveals other "unpleasant'' results for SET=114 (Fig. 4): The P distribution is not Gaussian (Figs. 4e and f), and a secondary minimum ( t_min,2) is present, but only in $S\!=\!163$ samples (Figs. 4k and l). The next Sect. 6.3 clarifies how this type of apparently confusing results were interpreted in Paper I.

   
6.3 All normalized subsets

For $K\!=\!2$ and $S\!=\!200$ in Paper I, three rules were found sufficient for the TSPA modelling of any particular SET of normalized magnitudes of V 1794 Cyg:

• R_I: If H_G is rejected with $\alpha \!=\!0.01$(Eq. 14) for any of the $\bar{\epsilon}$, $\bar{M}$, $\bar{P}$, $\bar{A}$, $\bar{t}_{\mathrm{min,1}}$or $\bar{t}_{\mathrm{min,2}}$ distributions, then the P, t_min,1 and t_min,2 estimates are excluded from further analysis.

• R_II: The P, t_min,1 and t_min,2 of subsets with less than 10 observing nights (nts) are excluded from further analysis.

• R_III: A secondary minimum t_min,2not present in $S\!\geq\!190$ ( $\!\equiv\!$ 95%) bootstrap samples is excluded from further analysis. Note that P or t_min,1are not excluded.

For example, H_G is rejected for P in SET=114 (Fig. 4f), and R_I excludes the P, t_min,1 and t_min,2 estimates. R_III alone would exclude t_min,2 with $S\!=\!163$ (Figs. 4k and l). The mean (m_y) and the standard deviation (s_y) of the data in subsets with ${\mathrm{nts}}\!\geq\!7$ (i.e. all normalized subsets) are connected to the mean (M) and the total amplitude (A) of the TSPA model. The linear approximations $M \!\approx\!m_y$ and $A /s_y \!\approx\! 2.7$ are very accurate. Note that the criteria R_I, R_II and R_III are therefore not used to reject the M or A estimates. Furthermore, m_y and s_y are independent of P, but t_min,1 and t_min,2 are not. Thus the TSPA (or any other) M and A modelling could be substituted with the above m_y and s_y relations in subsets with an adequate phase coverage. These m_y and s_y can be used for the same purposes (e.g. Donahue et al. 1997) that we use M and A in Paper I. Our R_II states the experimental result that the R_I rejections occur mostly with ${\mathrm{nts}} \! < \! 10$. This is understandable for $2K\!+\!2\!=6$ free model parameters when one night covers only about P_phot/10 for V 1794 Cyg. Table 3 in Paper I summarizes the TSPA of all normalized subsets of V 1794 Cyg, and the R_I, R_II or R_III rejections.

Our Table 1 confirms that the $\sim\!3.33$ periodicity is real, which would appear far from trivial, were there several subsets similar to SET=114. We summarize the TSPA combined with the best window period P₀determined separately for each SET of Paper I. The first line shows that $\sim\!3.33$ is among the five best periods in 84/93 cases for TSPA between $P_{\mathrm{min}}\!=\!0.5$ and $P_{\mathrm{max}}\!=\!10$(nine $\sim\!3.33$ values in Table 3 of Paper I were detected with $P_{\mathrm{min}}\!=\!2$ and $P_{\mathrm{max}}\!=\!6$). The spurious $\sim\!0.77$ period is detected nearly as often (83/93), but the detection rates for the other spurious periods are lower, especially for $\sim\! 6.66$. The next lines in Table 1 rate each period, if detected. The 2nd and 3rd lines confirm that $\sim\!3.33$ is undoubtedly the best period with the $\chi_0^2$ (best model) and $\mid \! r_0 \! \mid$ (not spurious) criteria. The 4th line lists H_R rejections with $P(\mid r_0 \mid)\!\leq\!\alpha \!=\!0.01$, i.e. when a high $\mid \! r_0 \! \mid$ between $\delta \bar{\phi}'$and $\delta \bar{\phi}$ implies spurious periodicity. This occurs more frequently for periods other than $\sim\!3.33$. Only $\sim\!0.77$, $\sim \!0.58$, $\sim \!1.42$ or their multiples reach extreme critical levels (e.g. Figs. 1r-t: ). In conclusion, our Table 1 illustrates the probability for obtaining a spurious (i.e. false) period for some arbitrary object.

   
6.4 V light curve of BL Her

The above V 1794 Cyg light curve analysis was performed with a second order TSPA. Here we study the V light curve of the cepheid variable BL Her from Moffett & Barnes (1984). The complicated shape of this light curve required a higher order model. Several details of this shape are undoubtedly real, because the total amplitude of the brightness variations ( $0\hbox{$.\!\!^{\rm m}$ }84$) is nearly 100 times larger than the internal accuracy of these data. Our fifth order TSPA ($K\!=\!5$) between $P_{\mathrm{min}}\!=\!0.5$ and $P_{\mathrm{max}}\!=\!10$ gave the best period of $P_1\!=\!1.30747~\pm ~0.00003$ (Fig. 5), and confirmed the 1.307443 estimate in Moffett & Barnes (1984). This periodicity certainly reached the best $\chi^2_0$among all period candidates (Figs. 5l-p), but the $0\hbox{$.\!\!^{\rm m}$ }01$ internal error in Vyields $\chi_0^2\!=\!188.5 \!>\!n\!=\!69$. Considering that the external error of the transformation into the standard Johnson system was "nearly as good'' as the internal error (Moffett & Barnes 1984), the above $\chi^2_0$ should be divided by an unknown factor larger than two. Three of the other period candidates are nearly multiples of P₁, i.e. $P_2\!=\!2.00P_1$, $P_3\!=\!3.07P_1$ and $P_4\!=\!4.12P_1$. The P₅ periodicity is also not induced by the $P_0\!=\!0.9996$window period, but it may be connected to the 277^d gap within the data and/or the 446^d time span of the whole data. Since these spurious periodicities are not connected to P₀, the phase residual correlations are not significant (Figs. 5q-u). The RSch bootstrap with P₁ confirmed that the modelling statistics are reliable, because the $\bar{\epsilon}$, $\bar{M}$, $\bar{P}$, $\bar{A}$, $\bar{t}_{\mathrm{min,1}}$ and $\bar{t}_{\mathrm{min,2}}$ distributions are all Gaussian (Fig. 6). Note also that the secondary minimum t_min,2 is present in $S\!=\!190$ bootstrap samples (Figs. 6kl).

   
6.5 V light curve of SAO 50205

The nonvariable star SAO50205 (B8II) has been used as the primary comparison star in the differential photometry of V 1794 Cyg. The long-term constant brightness of this object was verified in Paper II, where it was also noted that no periodicity was detected in the short-term brightness. The V magnitudes of SAO50205 measured with the Automatic Photoelectric Telecope (APT) during four subsets SET=111-114 are studied here. If the selected model $\bar{g}$ for these $n\!=\!59$ values were their mean of 7.350, the $0\hbox{$.\!\!^{\rm m}$ }012$ external error of these APT data would give $\chi^2_0\!=\!34.5$ having $P(\chi^2_0)\!=\!6~10^{-3}$ (Eq. (10): $\nu\!=\!n-1$ and $m\!=\!1$). In other words, a constant brightness model would suffice. But to illustrate a case of "pure noise'', the analysis of these data with the first order TSPA ($K\!=\!1$) between $P_{\mathrm{min}}\!=\!0.4$ and $P_{\mathrm{max}}\!=\!50$ is presented in Fig. 7. The PSch periodogram $\Theta'_{\mathrm{pilot}}(f)$ is nearly featureless, except for the weak minima close to the frequencies of 1 and 2 (Fig. 7a). Such integer values usually indicate spurious periodicity. The five best period candidates were divided into two groups when inferring their origin.

The periods of P₁ and P₄ belong to the first group. They are window periods inducing long gaps in the phase distribution of the data (Figs. 7b and e). Both periodicities are connected to the best window period $P_0\!=\!0.9986$ and the whole $\Delta T\!=\!176.8$ time span of the data. The combinations are $P'(\Delta T\!\!:\!\!-1\!\!:\!\!1)\! = \!1.004 \! \approx \!P_4\!=\!1.003$and $P_4/2\!=\!0.5015 \!\approx\!P_1\!=\!0.5004$. The measurements follow a weak slope spanning about 0.5 in phase. The gaps in the phases of the data allow a reasonable fit of the $K\!=\!1$ model to these slopes.

The second group containing P₂, P₃ and P₅ is a more complicated case. These three periods are connected to $P''\!=\!16.8~\pm ~0.4$, which would have been the best period, if the TSPA had been performed from $P_{\mathrm{min}}\!=\!2$ to $P_{\mathrm{max}}\!=\!50$. The $\Theta_{\mathrm{pilot}}'(f)$ periodogram has one weaker minimum at $f''\!=\!1/P''\!\approx\!0.06$ (Fig. 7a). The model with P'' would reach $\chi^2_0\!=\!23.6$, which is comparable to $22.1\!\leq\!\chi^2_0\!\leq\!22.7$ with P₂, P₃ and P₅. Combining P'' to the window period $P_0\!=\!0.9986$ yields $P'(P_0\!\!:\!\!-2\!\!:\!\!1)\!=\!-0.5146\!\approx\!-P_2\!=\!0.5145$, $P'(P_0\!\!:\!\! 2\!\!:\!\!1)\!=\! 0.4849\!\approx\! P_3\!=\!0.4832$and $P'(P_0\!\!:\!\!-1\!\!:\!\!1)\!=\!-1.0617\!\approx\!-P_5\!=\!1.0616$. The low total amplitude of the corresponding models prevented us from utilizing the phase correlations in indentifying spurious periodicities (Figs. 7r, s and u). This problem was solved by performing the same TSPA to the 19 other available subsets of V photometry of SAO50205. These additional data revealed no signatures of $P''\!=\!16.8$. Hence P'' represents an artifact only present in SET=111-114, where it induces the "detection'' of the P₂, P₃ and P₅ periodicities.

Since this TSPA tested $m\!=\!438$ independent frequencies, the "best'' period of $P_2\!=\!0.5145 \pm 0.0002$ with $\chi^2_0\!=\!22.1$does not reach the significance level of the constant brightness model with $\chi_0^2\!=\!34.5$, which was discussed before performing the TSPA analysis. Although the accuracy of the data ( $0\hbox{$.\!\!^{\rm m}$ }012$) was comparable to the total amplitude of the model ( $0\hbox{$.\!\!^{\rm m}$ }015$), there was no need to reject the Gaussian hypothesis ( H_G) for any of the model parameter estimates in the RSch bootstrap with P₂ (Fig. 8). The above analysis of the V magnitudes of SAO50205 illustrates the difficulties encountered in confirming the case of "pure noise'' for real data with the $\chi^2_0$-criterion (Eq. 10). In conclusion, one must know the accuracy of the data and infer the contribution of the window period(s).