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Intro to time series analysis

FISH 507 – Applied Time Series Analysis

Mark Scheuerell

8 Jan 2019

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))

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

Descriptions of time series

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

trend ( $m_t$ )
seasonal component ( $s_t$ )
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

One common method is via "linear filters"

$m_t = \sum_{i=-\infty}^{\infty} \lambda_i x_{t+1}$

Classical decomposition

1. The trend ( $m_t$ )

For example, a moving average

$m_t = \sum_{i=-a}^{a} \frac{1}{2a + 1} x_{t+i}$

If $a = 1$ , then

$m_t = \frac{1}{3}(x_{t-1} + x_t + x_{t+1})$

Example of linear filtering

Monthly airline passengers from 1949-1960

Example of linear filtering

Monthly airline passengers from 1949-1960

Example of linear filtering

Monthly airline passengers from 1949-1960

Example of linear filtering

Monthly airline passengers from 1949-1960

Classical decomposition

2. Seasonal effect ( $s_t$ )

Once we have an estimate of the trend $m_t$ , we can estimate $s_t$ simply by subtraction:

$s_t = x_t - m_t$

Classical decomposition

Seasonal effect ( $s_t$ ), assuming $\lambda = 1/9$

Classical decomposition

But, $s_t$ includes the remainder $e_t$ as well

Mean seasonal effect ( $s_t$ )

Classical decomposition

3. Remainder ( $e_t$ )

Now we can estimate $e_t$ via subtraction:

$e_t = x_t - m_t - s_t$

Remainder ( $e_t$ )

Let's try a different model

With some other assumptions

Log-transform data
Linear trend

Log-transformed data

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)

Intro to time series analysis

FISH 507 – Applied Time Series Analysis

Topics for today

Characteristics of time series (ts)

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

For example,

Classification of time series

By some index set

Classification of time series

By some index set

Classification of time series

By the underlying process

Classification of time series

By the number of values recorded

Classification of time series

By the type of values recorded

Statistical analyses of time series

Statistical analyses of time series

Statistical analyses of time series

Descriptions of time series

Descriptions of time series

What is a time series model?

Joint distributions of random variables

We have one realization

Some simple time series models

Some simple time series models

Classical decomposition

Model time series {xt}\{x_t\} as a combination of

Classical decomposition

1. The trend (mtm_t)

Classical decomposition

1. The trend (mtm_t)

Example of linear filtering

Example of linear filtering

Example of linear filtering

Example of linear filtering

Classical decomposition

2. Seasonal effect (sts_t)

Classical decomposition

Classical decomposition

Mean seasonal effect (sts_t)

Classical decomposition

3. Remainder (ete_t)

Remainder (ete_t)

Let's try a different model

With some other assumptions

Log-transformed data

The trend (mtm_t)

Seasonal effect (sts_t) with error (ete_t)

Mean seasonal effect (sts_t)

Remainder (ete_t)

Summary

Today's topics

Characteristics of time series (ts)

Classical decomposition

Model time series $\{x_t\}$ as a combination of

1. The trend ( $m_t$ )

1. The trend ( $m_t$ )

2. Seasonal effect ( $s_t$ )

Mean seasonal effect ( $s_t$ )

3. Remainder ( $e_t$ )

Remainder ( $e_t$ )

The trend ( $m_t$ )

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

Mean seasonal effect ( $s_t$ )

Remainder ( $e_t$ )