Lecture 7 Solutions

Author

Julia Schedler

Activity 1 Solutions: Fill out this table

Phrase	Symbols	Words	Bonus content
$x_t$ are white noise	$\E(x_t) = 0$\[\gamma(s,t) = \begin{cases} \sigma^2_w& \text{ if } s = t\\ 0& \text{ if } s \ne t \end{cases}\] $x_i$ are i.i.d.	“Marginally (for a given $t$), $x_t$ (”$x$ of $t$” or “$x$ sub $t$”) follows some distribution follows some distribution that is zero on average and are uncorrelated sigma w (sigma sub w) (except for when $s = t$, then the and $x_s$ is independent of $x_t$ for all $s, t4	“When we talk about white noise, an assumption is independence which necessarily implies we have multiple values of $t$, and for each value of $t$ we have a random vector $x_t$ which satisfies the distributional assumptions. Also \[\gamma(t,t) = \sigma^2_w = \text{marginal variance of }x_t \]”Gamma of t comma t is sigma squared w and is called the marginal variance of $x_t$ , where we consider $x_t$ to be a univariate random variable (think about how the word distribution was used in intro stats, that’s the setting)
$x_t$ is stationary in the mean	$\E(x_t)$ is not a function of $t$	The expected value of $x_t$ does not depend on time (have a $t$ in the formula)	This basically equates to $x_t$ having a horizontal trend, where the y-intercept can be any value. For white noise, for example, it’s 0.There is no temporal structure in the average behavior of $x$.
$x_t$ is stationary in the auto-covariance	$\gamma_x(s,t) = \gamma_x(h)$ for all $s,t$	The autocovariance function depends only on the distance between time points, not the specific values of the time points.	White noise is stationary, but if a time series $x_t$ is stationary it is not necessarily white noise. For example, if $v_t$ is a p-point moving average of white noise, the covariance function of $v_t$ for the case when p = 3 is:\[ \gamma_v(s,t) = \begin{cases} 3/9 \sigma^2_w & s-t = 0\\ 2/9 \sigma^2_w & \vert s-t \vert = 1\\ 1/9 \sigma^2_w & \vert s-t \vert =2 \\ 0 & \vert s-t \vert > 2 \end{cases} \]We can rewrite as a function of $h = s-t$, meaning $v_t$ is stationary in the autocovariance (or sometimes people just say covariance) This is not the same function as $\gamma_x(s,t)$ for white noise .Specifically, notice that the right hand side contains the “covariance structure”, the possible magnitudes of dependence (left of comma) for given combinations of two time points $s$ and $t$. This is the meaningful difference between the two functions. The left hand sides, while different because of the subscript, are just symbols for “covariance function of ” where is the letter part of the time series in question, i.e. $\gamma_v(s,t)$ is the autocovariance function of $v_t$.
$x_t$ is stationary	Both previous	Both previous	“In general, a stationary time series will have no predictable patterns in the long-term.”- Forecasting Principles and Practice“A stationary time series is one whose statistical properties do not depend on the time at which the series is observed”- Forecasting Principles and Practice
$x_t$ has temporal structure	$\E(x_t)$ is a function of $t$ $\gamma_x(s,t)$ is nonzero for some $s \ne t$	On average, the value of the time series $x_t$ varies with respect to $t$. $x_t$ exhibits temporal autocorrelation, or, knowing the value of $x_t$ tells us something about the likely value of $x_s$ when $\gamma(s,t) > 0$	Notice that when the time series is stationary in the mean, we say there is no temporal structure in the mean, but the autocovariance can still have temporal structure and be stationary. (If it didn’t it wouldn’t be a very interesting time series!)

Activity 2 Solutions

Code solutions

Download the data time_series.csv from Canvas and create a sub-folder in your Lecture7 folder called Data. Read in the data. Extract y1, the first time series, and name it x_t. Save it in a data frame called all_ts.

library(astsa)
library(ggplot2)
time_series <- read.csv("Data/time_series.csv")
x_t <- time_series$y1
all_ts <- data.frame(x_t, Time = time(x_t))

Plot the time series data set using both points and lines.

tsplot(all_ts$x_t, type = "b", ylab = "TS 1", main = "Observed Time Series", pch = 16)