1. Classical Protocol Game
Before modeling, we must visually inspect the series and check for stationarity. Drag the correct labels from the pool to the corresponding time series plots.
Plot A
Plot B
Plot C
2. Process Simulator
Generate various stochastic processes and observe their time series, scatter, and lag-1 behavior.
Quick Quiz
Change the process to "Random Walk". What happens to the Lag-1 plot?
3. Stationarity Explorer
Analyze these three theoretical models. eps_t denotes IID Gaussian White Noise.
| Process Formula | Constant Mean? | Constant Variance? | Cov depends only on h? | Weakly Stationary? |
|---|---|---|---|---|
| Model A: X_t = eps_t | ||||
| Model B: Y_t = 0.5 * t + eps_t | ||||
| Model C: Z_t = Z_{t-1} + eps_t |
4. Autocorrelation Function (ACF)
The ACF, denoted rho(h), normalizes autocovariance to stay between -1 and 1. It measures the linear correlation between a series and a lagged version of itself.
Prediction Time
Generate a Gaussian White Noise series. What do you expect the ACF plot to look like for lags h > 0?
5. Lag Plot Investigation
A lag plot scatters X(t) against X(t-h). It's a great tool to detect non-random structure (like seasonality or autoregression).
Based on the plots above, which lag shows the strongest positive linear dependence?
6. Detrending via Regression
Often, a series is non-stationary because of a deterministic trend. We can estimate this trend using Ordinary Least Squares (OLS) and subtract it (residuals = original - trend).
Which detrending model produced residuals that look most like stationary white noise?
7. Detrending via Differencing
Instead of regression, we can apply the differencing operator: ∇X_t = X_t - X_{t-1}. This is highly effective at removing trends and random walks.
To completely flatten a quadratic trend (making the mean constant), how many times do you need to difference the series?
8. Random Walk Casino (Gambler's Ruin)
A classic application of random walks. You start with $M. At each step, you win $1 (prob p) or lose $1 (prob 1-p). The game stops if you hit $0 (ruin) or $Target (success).
9. Final Challenge
I have generated a mystery process. Look at its Time Series, ACF, and Lag-1 plot. Identify what kind of process it is.