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))

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

One common method is via "linear filters"

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

For example, a moving average

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

If \(a = 1\), then

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

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

\[ s_t = x_t - m_t \]

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

But, \(s_t\) includes the remainder \(e_t\) as well

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

\[ e_t = x_t - m_t - s_t \]

1. Log-transform data

2. Linear trend

Characteristics of time series (ts)

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

Classical decomposition