# 统计学 Statistics 1Z STATS1003_1

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An appropriate model for a series with a single cycle is then
$$y_{t}=A \cos (2 \pi f t \quad P) \quad u_{t}$$
where $A$ is the amplitude and $P$ the phase of the cycle, and period $1 / f$, namely the number of time units from peak to peak. To allow for several $(r)$ frequencies operating simultaneously in the same data, the preceding may be generalised to
$$y_{t}=\sum_{j=1}^{r} A_{j} \cos \left(2 \pi f_{j} t \quad P_{j}\right) \quad u_{t}$$

For stationarity to apply, the $A_{j}$ may be taken as uncorrelated with mean 0 and the $P_{j}$ as uniform on $(0,2 \pi)$. Because
$$\cos \left(2 \pi f_{j} t \quad P_{j}\right)=\cos \left(2 \pi f_{j} t\right) \cos \left(P_{j}\right)-\sin \left(2 \pi f_{j} t\right) \sin \left(P_{j}\right)$$

## STATS1003_1 COURSE NOTES ：

For discrete outcomes, dependence on past observations and predictors may be handled by adapting metric variable methods within the appropriate regression link. Thus for Poisson outcomes
$$y_{t} \sim \operatorname{Poi}\left(\mu_{t}\right)$$
an $\operatorname{AR}(1)$ dependence on previous values in the series could be specified
$$\log \left(\mu_{t}\right)=\rho y_{t-1} \quad \beta x_{t}$$
Here, non-stationarity or ‘explosive’ behaviour would be implied by $\rho>0$ (Fahrmeir and Tutz, 2001, p. 244), and in an MCMC framework stationarity would be assessed by the proportion of iterations for which $\rho$ was positive. Autoregressive errors lead to specification such as
$$\log \left(\mu_{t}\right)=\beta x_{t} \quad \varepsilon_{t}$$
with
$$\varepsilon_{l}=\gamma \varepsilon_{t-1} \quad u_{l}$$
for $t>1$, and $u_{t}$ being white noise.