Activity3

Autocorrelation in Time Series

Autocorrelation, also known as serial correlation, measures the correlation of a time series with its own past and future values. Mathematically, the autocorrelation function (ACF) at lag \(k\) for a time series \(\{X_t\}\) is defined as:

\[ \rho_k = \frac{\text{Cov}(X_t, X_{t-k})}{\sqrt{\text{Var}(X_t) \text{Var}(X_{t-k})}} \]

where \(\rho_k\) measures the linear relationship between values \(k\) time periods apart.

  • Positive autocorrelation at lag \(k\): high values tend to follow high values and low values tend to follow low values after \(k\) periods.
  • Negative autocorrelation at lag \(k\): high values tend to follow low values and vice versa.
  • No autocorrelation: the series values \(k\) periods apart are uncorrelated, suggesting randomness at that lag.

Partial Autocorrelation in Time Series

Partial autocorrelation measures the correlation between a time series and its lagged values, after removing the influence of intermediate lags. In other words, it quantifies the direct relationship between \(X_t\) and \(X_{t-k}\), controlling for the effects of \(X_{t-1}, X_{t-2}, \dots, X_{t-k+1}\). Mathematically, the partial autocorrelation function (PACF) at lag \(k\) for a time series \(\{X_t\}\) is defined as:

\[ \phi_{kk} = \text{Corr}(X_t, X_{t-k} \mid X_{t-1}, X_{t-2}, \dots, X_{t-k+1}) \]

where \(\phi_{kk}\) represents the partial autocorrelation at lag \(k\).

  • Interpretation of PACF:

    • A significant partial autocorrelation at lag \(k\) suggests a direct relationship between \(X_t\) and \(X_{t-k}\), independent of the intermediate lags.
    • A non-significant partial autocorrelation indicates that the relationship between \(X_t\) and \(X_{t-k}\) is fully explained by the intermediate lags.

  • Comparison with ACF:

    • While the ACF measures the total correlation between \(X_t\) and \(X_{t-k}\), the PACF isolates the direct correlation, making it a more precise tool for model identification.

Practical Illustration with Real Data

We will use real datasets from the fpp3 package to compute and visualize autocorrelation. The following code snippets demonstrate how to plot the ACF and PACF for a time series, providing insights into its internal structure.

library(fpp3)

aus_airpassengers %>%
  ACF(Passengers) %>%             # Calculate autocorrelations for the Passengers series
  autoplot() +                    # Plot the ACF
  labs(title = "ACF of Australian Air Passengers", y = "ACF", x = "Lag")

# Example 2: Partial Autocorrelation of Australian Air Passengers
aus_airpassengers %>%
  PACF(Passengers) %>%            # Calculate partial autocorrelations for the Passengers series
  autoplot() +                    # Plot the PACF
  labs(title = "PACF of Australian Air Passengers", y = "PACF", x = "Lag")

White Noise

  1. White Noise

    • No autocorrelation: \(\mathbb{E}[X_t X_{t-k}] = 0, \quad \forall k \neq 0\)
    • Constant variance: \(\operatorname{Var}(X_t) = \sigma^2\)

    The white noise process is defined as:
    \[ X_t = \varepsilon_t, \quad \varepsilon_t \stackrel{iid}{\sim} \mathcal{N}(0, \sigma^2) \]

# White Noise Simulation
wn <- tsibble(time = 1:500, y = rnorm(500), index = time)

# Visualize and test for autocorrelation
wn %>% autoplot(y) + ggtitle("White Noise Process")

wn %>% ACF(y) %>% autoplot()

wn %>% features(y, ljung_box, lag = 10)  # Should retain H₀ (no autocorrelation)
# A tibble: 1 × 2
  lb_stat lb_pvalue
    <dbl>     <dbl>
1    7.71     0.658

  1. Portmanteau (Ljung-Box) Test

    • Null hypothesis (\(H_0\)): No autocorrelation up to lag \(h\)
    • Test statistic:
      \[ Q = n(n+2) \sum_{k=1}^{h} \frac{\hat{\rho}_k^2}{n-k} \sim \chi^2_h \]
wn %>% features(y, ljung_box, lag = 10)  # p > 0.05 ⇒ retain H₀ (white noise)
# A tibble: 1 × 2
  lb_stat lb_pvalue
    <dbl>     <dbl>
1    7.71     0.658
  1. Practical Verification
    • Simulate contaminated white noise to test robustness:
contaminated_wn <- wn %>% 
 mutate(y = y + 0.3*lag(y, 5) %>% replace_na(0))
  • Visualize and test the contaminated series:
contaminated_wn %>% 
 ACF(y) %>% 
 autoplot() + 
 ggtitle("ACF of Contaminated Series")

contaminated_wn %>% 
 features(y, ljung_box, lag = 10)  # p < 0.05 ⇒ reject H₀
# A tibble: 1 × 2
  lb_stat  lb_pvalue
    <dbl>      <dbl>
1    44.7 0.00000251

Lab Activity: White Noise Series, ACF, and Portmanteau Test

  1. Simulate a White Noise Series

    Generate a white noise series of length \(n = 500\) using the equation: \[ X_t = \varepsilon_t, \quad \varepsilon_t \stackrel{iid}{\sim} \mathcal{N}(0, 1) \] Plot the series.

  2. Plot the ACF

    Compute and plot the autocorrelation function (ACF) of the white noise series. Interpret the results.

  3. Contaminate the Series

    Introduce contamination into the white noise series by adding a lagged component: \[ Y_t = X_t + 0.4 \cdot X_{t-3} \] Replace missing values with 0. Plot the contaminated series.

  4. Plot the ACF of the Contaminated Series

    Compute and plot the ACF of the contaminated series. Compare it with the ACF of the original white noise series.

  5. Perform the Portmanteau (Ljung-Box) Test

    • Apply the Ljung-Box test to the original white noise series. Interpret the results.
    • Apply the Ljung-Box test to the contaminated series. Interpret the results.