5 Jan 2021

Topics for today

Characteristics of time series (ts)

  • What is a ts?

  • Classifying ts

  • Trends

  • Seasonality (periodicity)

Classical decomposition

What is a time series?

A set of observations taken sequentially in time

What is a time series?

A ts can be represented as a set

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

For example,

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

Classification of time series

By some index set

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

  • begin/end: \(t \in [1.1,2.5]\)

Classification of time series

By some index set

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: \(t = \{2,3,4,6,9\}\)

Classification of time series

By the underlying process

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

Continuous (eg, salinity, temperature)

Classification of time series

By the number of values recorded

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

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

Classification of time series

By the type of values recorded

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))

Classification of time series

We will focus on integers & real-values in discrete time

Univariate \((x_t)\)


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

Time series objects in R

Time series objects have a special designation in R: ts

ts(data,
   start, end,
   frequency
   )

Time series objects in R

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 in R

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 in R

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 in R

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

Time series objects in R

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)

Plotting time series objects in R

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

plot.ts(ts_object)

Plotting time series objects in R

Plotting time series objects in R

We can specify some additional arguments to plot.ts

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

Plotting time series objects in R

Analysis of time series

Statistical analyses of time series

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

Statistical analyses of time series

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

Statistical analyses of time series

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

Descriptions of time series

Number of users connected to the internet

Number of users connected to the internet

Descriptions of time series

Number of lynx trapped in Canada from 1821-1934

Number of lynx trapped in Canada from 1821-1934

What is a time series model?

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

Joint distributions of random variables

We have one realization

Some simple time series models

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

Some simple time series models

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

Classical decomposition

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

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

\(x_t = m_t + s_t + e_t\)

Classical decomposition

1. The trend (\(m_t\))

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

Classical decomposition

1. The trend (\(m_t\))

Linear filters typically take the form

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

Classical decomposition

1. The trend (\(m_t\))

For example, a moving average

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

Classical decomposition

1. The trend (\(m_t\))

For example, a moving average

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

If \(a = 1\), then

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

Classical decomposition

1. The trend (\(m_t\))

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

Example of linear filtering

Monthly airline passengers from 1949-1960

Monthly airline passengers from 1949-1960

Example of linear filtering

Monthly airline passengers from 1949-1960

Monthly airline passengers from 1949-1960

Example of linear filtering

Monthly airline passengers from 1949-1960

Monthly airline passengers from 1949-1960

Example of linear filtering

Monthly airline passengers from 1949-1960

Monthly airline passengers from 1949-1960

Classical decomposition

2. Seasonal effect (\(s_t\))

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

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

Classical decomposition

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

Classical decomposition

2. Seasonal effect (\(s_t\))

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

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

Classical decomposition

2. Seasonal effect (\(s_t\))

So we need to estimate the mean seasonal effect as

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

Mean seasonal effect (\(s_t\))

Classical decomposition

3. Remainder (\(e_t\))

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

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

Remainder (\(e_t\))

Let’s try a different model

With some other assumptions

  1. Log-transform data

  2. Linear trend

Log-transformed data

Monthly airline passengers from 1949-1960

Monthly airline passengers from 1949-1960

The trend (\(m_t\))

Seasonal effect (\(s_t\)) with error (\(e_t\))

Mean seasonal effect (\(s_t\))

Remainder (\(e_t\))

Summary

Today’s topics

Characteristics of time series (ts)

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

Classical decomposition