Characteristics of time series (ts)

• What is a ts?

• Classifying ts

• Trends

• Seasonality (periodicity)

Classical decomposition

A set of observations taken sequentially in time

A ts can be represented as a set

\[ \{ x_1,x_2,x_3,\dots,x_n \} \]

For example,

\[ \{ 10,31,27,42,53,15 \} \]

Interval across real time; \(x(t)\)

• begin/end: \(t \in [1.1,2.5]\)

Discrete time; \(x_t\)

• Equally spaced: \(t = \{1,2,3,4,5\}\)
  
• Equally spaced w/ missing value: \(t = \{1,2,4,5,6\}\)
  
• Unequally spaced: \(t = \{2,3,4,6,9\}\)

Discrete (eg, total # of fish caught per trawl)

Continuous (eg, salinity, temperature)

Univariate/scalar (eg, total # of fish caught)

Multivariate/vector (eg, # of each spp of fish caught)

Integer (eg, # of fish in 5 min trawl = 2413)

Rational (eg, fraction of unclipped fish = 47/951)

Real (eg, fish mass = 10.2 g)

Complex (eg, cos(2π2.43) + i sin(2π2.43))

Univariate \((x_t)\)

Multivariate \(\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}_t\)

Time series objects have a special designation in R: ts

ts(data,
   start, end,
   frequency
   )

Time series objects have a special designation in R: ts

ts(data,
   start, end,
   frequency
   )

data should be a vector (univariate)

or a data frame or matrix (multivariate)

Time series objects have a special designation in R: ts

ts(data,
   start, end,
   frequency
   )

start and end give the first and last time indices

For monthly series, specify them as c(year, month)

Time series objects have a special designation in R: ts

ts(data,
   start, end,
   frequency
   )

frequency is the number of observations per unit time

For annual series, frequency = 1

For monthly series, frequency = 12

Time series objects have a special designation in R: ts

ts(data,
   start, end,
   deltat
   )

deltat is the fraction of the sampling period

For annual series, deltat = 1

For monthly series, deltat = 1/12

set.seed(507)

## annual data
dat_1 <- rnorm(30)
dat_yr <- ts(dat_1,
             start = 1991, end = 2020,
             frequency = 1)

## monthly data
dat_2 <- rnorm(30*12)
dat_mo <- ts(dat_2,
             start = c(1991, 1), end = c(2020, 12),
             frequency = 12)

There is a designated function for plotting ts objects: plot.ts()

plot.ts(ts_object)

We can specify some additional arguments to plot.ts

plot.ts(dat_yr,
        ylab = expression(italic(x[t])),
        las = 1, col = "blue", lwd = 2)

Most statistical analyses are concerned with estimating properties of a population from a sample

For example, we use fish caught in a seine to infer the mean size of fish in a lake

Time series analysis, however, presents a different situation:

• Although we could vary the length of an observed time series, it is often impossible to make multiple observations at a given point in time

Time series analysis, however, presents a different situation:

• Although we could vary the length of an observed time series, it is often impossible to make multiple observations at a given point in time

For example, one can’t observe today’s closing price of Microsoft stock more than once

Thus, conventional statistical procedures, based on large sample estimates, are inappropriate

A time series model for \(\{x_t\}\) is a specification of the joint distributions of a sequence of random variables \(\{X_t\}\), of which \(\{x_t\}\) is thought to be a realization

White noise: \(x_t \sim N(0,1)\)

Random walk: \(x_t = x_{t-1} + w_t,~\text{with}~w_t \sim N(0,1)\)

Model time series \(\{x_t\}\) as a combination of

1. trend (\(m_t\))
   
2. seasonal component (\(s_t\))
   
3. remainder (\(e_t\))

\(x_t = m_t + s_t + e_t\)

We need a way to extract the so-called signal from the noise

One common method is via “linear filters”

Linear filters can be thought of as “smoothing” the data

Linear filters typically take the form

\[ \hat{m}_t = \sum_{i=-\infty}^{\infty} \lambda_i x_{t+1} \]

For example, a moving average

\[ \hat{m}_t = \sum_{i=-a}^{a} \frac{1}{2a + 1} x_{t+i} \]

For example, a moving average

\[ \hat{m}_t = \sum_{i=-a}^{a} \frac{1}{2a + 1} x_{t+i} \]

If \(a = 1\), then

\[ \hat{m}_t = \frac{1}{3}(x_{t-1} + x_t + x_{t+1}) \]

For example, a moving average

\[ \hat{m}_t = \sum_{i=-a}^{a} \frac{1}{2a + 1} x_{t+i} \]

As \(a\) increases, the estimated trend becomes more smooth

Once we have an estimate of the trend \(\hat{m}_t\), we can estimate \(\hat{s}_t\) simply by subtraction:

\[ \hat{s}_t = x_t - \hat{m}_t \]

Seasonal effect (\(\hat{s}_t\)), assuming \(\lambda = 1/9\)

But, \(\hat{s}_t\) really includes the remainder \(e_t\) as well

\[ \begin{align} \hat{s}_t &= x_t - \hat{m}_t \\ (s_t + e_t) &= x_t - m_t \end{align} \]

So we need to estimate the mean seasonal effect as

\[ \hat{s}_{Jan} = \sum \frac{1}{(N/12)} \{s_1, s_{13}, s_{25}, \dots \} \\ \hat{s}_{Feb} = \sum \frac{1}{(N/12)} \{s_2, s_{14}, s_{26}, \dots \} \\ \vdots \\ \hat{s}_{Dec} = \sum \frac{1}{(N/12)} \{s_{12}, s_{24}, s_{36}, \dots \} \\ \]

Now we can estimate \(e_t\) via subtraction:

\[ \hat{e}_t = x_t - \hat{m}_t - \hat{s}_t \]

1. Log-transform data

2. Linear trend

Characteristics of time series (ts)

• What is a ts?
• Classifying ts
• Trends
• Seasonality (periodicity)

Classical decomposition