## STAT510 Time-Series Analysis课程简介

Time series data are intriguing yet complicated information to work with. While this
Credits 3 course will provide students with a basic understanding of the nature and basic processes used to analyze such data, you will quickly realize that this is a small first step in being able to confidently understand what trends might exist within a set of data and the complexities of being able to use this information to make predictions or forecasts. Yet, whether it is financial, medical or weather related, this type of data is quite frequently found in much of our daily lives.

## PREREQUISITES

Topics typically covered in this graduate level course include:

• Understanding the characteristics of time series data
• Understanding moving average models and partial autocorrelation as foundations for analysis of time series data
• Exploratory Data Analysis – Trends in time series data
• Using smoothing and removing trends when working with time series data
• Understanding how periodograms are used with time series data
• Implementing ARMA and ARIMA time series models
• Identifying and interpreting various patterns for intervention effects
• Examining the analysis of repeated measures design
• Using ARCH and AR models in multivariate time series contexts
• Using spectral density estimation and spectral analysis
• Using fractional differencing and threshold models with time series data

## STAT510 Time-Series Analysis HELP（EXAM HELP， ONLINE TUTOR）

Problem-1: Consider MA $(q): y_t=\sum_{j=0}^q \beta_j \epsilon_{t-j}$, where $\beta_0=1$ and $\epsilon_t$ is a white noise with variance $\sigma^2$.
(a) Compute $\mu \equiv E\left[y_t\right]$.
(b) Compute $\gamma_0 \equiv E\left[\left(y_t-\mu\right)^2\right]$.
(c) Compute $\gamma_j \equiv E\left[\left(y_t-\mu\right)\left(y_{t-j}-\mu\right)\right]$ for $j \geq 1$. (Remark: It is important to observe that $\gamma_j=0$ for $j \geq q+1$.)
(d) Conclude that $\left{y_t\right}$ is covariance stationary whatever $\beta_1, \ldots, \beta_q$ are.

(a) The mean of the MA(q) process is $\mu = E[y_t] = E\left[\sum_{j=0}^q \beta_j \epsilon_{t-j}\right]$. Since $\epsilon_t$ is a white noise process, $E[\epsilon_t] = 0$ for all $t$, so we have:

$$\mu = E\left[\sum_{j=0}^q \beta_j \epsilon_{t-j}\right] = \sum_{j=0}^q \beta_j E[\epsilon_{t-j}] = 0$$

Therefore, the mean of the MA(q) process is always zero.

(b) To compute $\gamma_0$, we start by expanding the square in the definition of $\gamma_0$:

$$\gamma_0 = E[(y_t-\mu)^2] = E\left[\left(\sum_{j=0}^q \beta_j \epsilon_{t-j}\right)^2\right] = E\left[\sum_{j=0}^q \beta_j^2 \epsilon_{t-j}^2 + 2\sum_{j=0}^{q-1} \sum_{k=j+1}^q \beta_j \beta_k \epsilon_{t-j} \epsilon_{t-k}\right]$$

Since $\epsilon_t$ is a white noise process with variance $\sigma^2$, we have $E[\epsilon_t^2] = \sigma^2$ for all $t$, and $E[\epsilon_t \epsilon_s] = 0$ for $t \neq s$. Therefore:

$$\gamma_0 = \sum_{j=0}^q \beta_j^2 \sigma^2$$

(c) For $j \geq 1$, we have:

\begin{align*} \gamma_j &= E[(y_t-\mu)(y_{t-j}-\mu)] \ &= E\left[\left(\sum_{i=0}^q \beta_i \epsilon_{t-i}\right)\left(\sum_{k=0}^q \beta_k \epsilon_{t-j-k}\right)\right] \ &= E\left[\sum_{i=0}^q \sum_{k=0}^q \beta_i \beta_k \epsilon_{t-i} \epsilon_{t-j-k}\right] \ &= \sum_{i=0}^q \sum_{k=0}^q \beta_i \beta_k E[\epsilon_{t-i} \epsilon_{t-j-k}] \end{align*}

If $j > q$, then $t-j-k < t-q \leq t$, and $\epsilon_t$ is independent of $\epsilon_{t-j-k}$, so $E[\epsilon_{t-i} \epsilon_{t-j-k}] = 0$ for all $i$ and $k$, and we have $\gamma_j = 0$. If $j \leq q$, then we have:

$$\gamma_j = \sum_{i=0}^{q-j} \beta_i \beta_{i+j} \sigma^2$$

(d) To show that ${y_t}$ is covariance stationary, we need to show that $\mu$ and $\gamma_j$ are finite and do not depend on $t$. We have already shown that $\mu = 0$ and that $\gamma_j$ is finite and depends only on $\beta_0, \ldots, \beta_q$ for $j \leq q$. To see that $\gamma_j$ does not depend on $t$, note

Problem-2: In class we covered $\mathrm{AR}(1)$ and $\mathrm{MA}(1)$ processes. Here we consider a more general process called ARMA $(1,1)$ :
$$y_t=\phi y_{t-1}+\epsilon_t+\beta \epsilon_{t-1}$$
where $\epsilon_t$ is a white noise with variance $\sigma^2$. Assume $|\phi|<1$ so that $\left{y_t\right}$ is covariance stationary. (Remark: $\beta$ does not play any role for covariance stationarity.) Let us compute autocorrelation functions $\rho_j$ using the Yule-Walker equations.
(a) Show that $\gamma_0=\phi \gamma_1+\sigma^2+\beta(\phi+\beta) \sigma^2$.
(b) Show that $\gamma_1=\phi \gamma_0+\beta \sigma^2$.
(c) Combining (a) and (b), solve for $\gamma_0$ and $\gamma_1$.
(d) Show that
$$\rho_1=\frac{(\phi+\beta)(1+\phi \beta)}{1+\phi \beta+\beta(\phi+\beta)}$$
(e) Show that $\gamma_j=\phi \gamma_{j-1}$ for $j \geq 2$.

(a) We have \begin{align*} \gamma_0 &= \operatorname{Var}(y_t) \ &= \operatorname{Var}(\phi y_{t-1} + \epsilon_t + \beta \epsilon_{t-1}) \ &= \phi^2 \operatorname{Var}(y_{t-1}) + \operatorname{Var}(\epsilon_t) + \beta^2 \operatorname{Var}(\epsilon_{t-1}) + 2\phi \operatorname{Cov}(y_{t-1}, \epsilon_t) + 2\beta \operatorname{Cov}(\epsilon_{t-1}, \epsilon_t) \ &= \phi^2 \gamma_0 + \sigma^2 + \beta^2 \sigma^2 + 2\phi \gamma_1 \end{align*} using the fact that $\operatorname{Var}(\epsilon_t) = \sigma^2$ and $\operatorname{Var}(\epsilon_{t-1}) = \sigma^2$, and that $\operatorname{Cov}(y_{t-1}, \epsilon_t) = 0$ and $\operatorname{Cov}(\epsilon_{t-1}, \epsilon_t) = 0$ since $\epsilon_t$ is a white noise process.

(b) We have \begin{align*} \gamma_1 &= \operatorname{Cov}(y_t, y_{t-1}) \ &= \operatorname{Cov}(\phi y_{t-1} + \epsilon_t + \beta \epsilon_{t-1}, y_{t-1}) \ &= \phi \operatorname{Cov}(y_{t-1}, y_{t-1}) + \beta \operatorname{Cov}(\epsilon_{t-1}, y_{t-1}) \ &= \phi \gamma_0 + \beta \sigma^2 \end{align*}

(c) Rearranging the two equations above, we have

\gamma_0 = \frac{\sigma^2 + \beta^2 \sigma^2}{1 – \phi^2} – \frac{2\beta \phi \sigma^2}{1 – \phi^2} \gamma_1γ0​=1−ϕ2σ2+β2σ2​−1−ϕ22βϕσ2​γ1​

and

\gamma_1 = \frac{\phi \gamma_0 + \beta \sigma^2}{1 – \phi^2}γ1​=1−ϕ2ϕγ0​+βσ2​

Multiplying the first equation by $\phi$ and subtracting the second equation, we get

\gamma_0 = \frac{\sigma^2 + \beta^2 \sigma^2 – 2\beta \phi \sigma^2 \left(\phi + \frac{\beta}{1 – \phi^2}\right)}{1 – \phi^2 + 2\phi^2 \left(\phi + \frac{\beta}{1 – \phi^2}\right)}γ0​=1−ϕ2+2ϕ2(ϕ+1−ϕ2β​)σ2+β2σ2−2βϕσ2(ϕ+1−ϕ2β​)​

and

\gamma_1 = \frac{\phi \beta \sigma^2 + \phi \beta \phi \gamma_0}{1 – \phi^2 + 2\phi^2 \left(\phi + \frac{\beta}{1 – \phi^2}\right)}γ1​=1−ϕ2+2ϕ2(ϕ+1−ϕ2β​)ϕβσ2+ϕβϕγ0​​

which simplifies to

\gamma_0 = \frac{\sigma^2 + \beta^2 \sigma^2 + 2\beta\phi\sigma^2}{1 – \phi^2 – 2\beta\phi^2}γ0​=1−ϕ2−2βϕ2σ2+β2σ2+2βϕσ2​

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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## STAT510 Time-Series Analysis课程简介

Time series data are intriguing yet complicated information to work with. While this
Credits 3 course will provide students with a basic understanding of the nature and basic processes used to analyze such data, you will quickly realize that this is a small first step in being able to confidently understand what trends might exist within a set of data and the complexities of being able to use this information to make predictions or forecasts. Yet, whether it is financial, medical or weather related, this type of data is quite frequently found in much of our daily lives.

## PREREQUISITES

Topics typically covered in this graduate level course include:

• Understanding the characteristics of time series data
• Understanding moving average models and partial autocorrelation as foundations for analysis of time series data
• Exploratory Data Analysis – Trends in time series data
• Using smoothing and removing trends when working with time series data
• Understanding how periodograms are used with time series data
• Implementing ARMA and ARIMA time series models
• Identifying and interpreting various patterns for intervention effects
• Examining the analysis of repeated measures design
• Using ARCH and AR models in multivariate time series contexts
• Using spectral density estimation and spectral analysis
• Using fractional differencing and threshold models with time series data

## STAT510 Time-Series Analysis HELP（EXAM HELP， ONLINE TUTOR）

1. a. (3 pts)
Below are the plots of the lowess estimates of the first 400 observations of the EEG data using $\mathrm{f}=0.10,0.25,0.50,0.80$, and 1.00 , respectively, to smooth. The smoothing parameter does not seem to have any impact on the trend, as there is not one.b. ( 3 pts)
The regression line is
$$\mathrm{EEG}=-0.2968+0.0015 * \text { Time }$$
The slope is negligible and therefore, the regression line adequately represents the trend (as previously stated, there does not appear to be a trend).

a. The statement that “the smoothing parameter does not seem to have any impact on the trend” is incorrect. The plots show that the choice of smoothing parameter does have an impact on the trend. A smaller smoothing parameter (such as 0.10) leads to a more wiggly, jagged line that closely follows the individual data points, while a larger smoothing parameter (such as 1.00) leads to a smoother line that is less affected by individual data points and captures more of the overall trend.

b. While it is true that the slope of the regression line is small (0.0015), it does not necessarily follow that the regression line adequately represents the trend. The regression line only captures a linear relationship between time and EEG, and there may be more complex relationships that are not captured by this simple model. Additionally, the lowess plots suggest that there may be some non-linear patterns in the data that are not captured by the linear regression model. Therefore, it is important to consider both the lowess plots and the regression model when interpreting the data.

a. (2 pts)
For seasonality, the smoothing parameters chosen were $q=6,12,40$. The smoothing parameter of 12 is the best as the season is best represented by a yearly cycle. When the smoothing parameter is 6 , the average is rather sporadic. When the smoothing parameter is 12 , the cycle looks stable. When the smoothing parameter is 40 , the data is smoothed too much.

b. (2 pts) The statement is not entirely accurate. While the smoothing parameter of 12 may be the best for capturing a yearly cycle, it is not necessarily true that a yearly cycle is present in the data. It is important to also consider other factors, such as the nature of the data and any external factors that may be influencing it, before concluding that there is seasonality present. Additionally, while a smoothing parameter of 6 may lead to more sporadic results, it may still be worth considering if there are shorter-term patterns in the data that need to be captured. Finally, while a smoothing parameter of 40 may be too high and lead to oversmoothing, this also depends on the nature of the data and the level of detail that needs to be captured. Overall, the choice of smoothing parameter should be guided by a combination of statistical methods and subject matter expertise.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.