Annotated Equations

A quick reference for the models introduced in the R Tutorial.

White noise model

The simplest possible time-series model: every observation is a random number drawn from a normal distribution (aka White Noise)

\[ y_t = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{mean}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

\[ \epsilon_t \sim \underbrace{\color{#6b46c1}{N}}_{\textstyle\color{#6b46c1}{\text{normal distribution}}}\!\left( \underbrace{\color{#2c7a7b}{0}}_{\textstyle\color{#2c7a7b}{\text{mean = 0}}},\; \quad \underbrace{\color{#b7791f}{\sigma^2}}_{\textstyle\color{#b7791f}{\text{variance}}} \right) \]

• \(y_t\) — the observed value at time \(t\) (e.g., NDVI in a given month)
• \(c\) — a constant: the mean of the observed (\(y\)) values
• \(\epsilon_t\) — the error at time \(t\)
• \(\sigma^2\) — the variance of the error distribution, i.e., how wide is the Normal distribution we are drawing random numbers from

Key assumption: each \(\epsilon_t\) is drawn independently — knowing \(\epsilon_{t-1}\) doesn’t inform \(\epsilon_t\). The model is just “mean plus noise”

AR(1) model

The last observed value (\(y_{t-1}\)) influences the current value (\(y_t\))

\[ y_t = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{constant}}} \;+\; \underbrace{\color{#c05621}{b_1 y_{t-1}}}_{\textstyle\color{#c05621}{\text{lag-1 effect}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

\[ \epsilon_t \sim \underbrace{\color{#6b46c1}{N}}_{\textstyle\color{#6b46c1}{\text{normal distribution}}}\!\left( \underbrace{\color{#2c7a7b}{0}}_{\textstyle\color{#2c7a7b}{\text{mean = 0}}},\; \quad \underbrace{\color{#b7791f}{\sigma^2}}_{\textstyle\color{#b7791f}{\text{variance}}} \right) \]

• \(y_t\) — the observed value at time \(t\)
• \(c\) — a constant, analogous to the intercept in a regression
• \(b_1 y_{t-1}\) — the lag-1 effect: \(b_1\) is the AR1 coefficient, determining how strongly (and in which direction) the previous value, \(y_{t-1}\), influences \(y_t\)
• \(\epsilon_t\) — random error at time \(t\), independent of past errors

Reading the coefficient: if \(b_1\) is large and positive, a high value last time step predicts a high value this time step (values persist). If \(b_1\) were negative, high values would tend to be followed by low values.

Ecological connection: if \(y\) is \(\log(N)\), this is essentially a Gompertz population model — the current abundance depends on the previous abundance.

AR(2) model

Like the AR(1) model but with 2 lags.

\[ y_t = \underbrace{\color{#2b6cb0}{c}}_{\textstyle\color{#2b6cb0}{\text{constant}}} \;+\; \underbrace{\color{#c05621}{b_1 y_{t-1}}}_{\textstyle\color{#c05621}{\text{lag-1 effect}}} \;+\; \underbrace{\color{#2f855a}{b_2 y_{t-2}}}_{\textstyle\color{#2f855a}{\text{lag-2 effect}}} \;+\; \underbrace{\color{#718096}{\epsilon_t}}_{\textstyle\color{#718096}{\text{random error}}} \]

\[ \epsilon_t \sim \underbrace{\color{#6b46c1}{N}}_{\textstyle\color{#6b46c1}{\text{normal distribution}}}\!\left( \underbrace{\color{#2c7a7b}{0}}_{\textstyle\color{#2c7a7b}{\text{mean = 0}}},\; \quad \underbrace{\color{#b7791f}{\sigma^2}}_{\textstyle\color{#b7791f}{\text{variance}}} \right) \]

• \(y_t\) — the observed value at time \(t\)
• \(c\) — a constant, analogous to the intercept in a regression
• \(b_1\) — the AR1 coefficient: the effect of the value one time step back (\(y_{t-1}\))
• \(b_2\) — the AR2 coefficient: the effect of the value two time steps back (\(y_{t-2}\))
• \(\epsilon_t\) — random error at time \(t\), independent of past errors

Reading the coefficients: \(b_1\) and \(b_2\) don’t have to point the same direction. For example, a large positive \(b_1\) combined with a smaller negative \(b_2\) means a high value last time step predicts a high value now, while a high value two time steps back predicts a slightly lower value now — the two lags can partially offset each other.