Questions & explanations
1. Compare kriging variance with the variance of the sample data. How are they different?
Kriging variance is a measure of prediction uncertainty at a specific location, while sample variance describes the spread of observed data values. Sample variance is calculated from the data values themselves, ignoring spatial locations. Kriging variance depends on the spatial configuration of sample points and the variogram, but not on the data values. For example, if you have clustered samples, kriging variance may be low near clusters but high far away, even if sample variance is constant. Sample variance is a single number for the whole dataset, whereas kriging variance varies across space. They serve different purposes: sample variance summarizes data variability, kriging variance quantifies prediction confidence.
2. How do you choose between a first-order and second-order polynomial trend surface?
A first-order polynomial (plane) is suitable when the data show a simple linear gradient, like elevation increasing from east to west. A second-order polynomial (quadratic) can model a single peak or trough, such as a hill or basin. To choose, compare the goodness-of-fit (e.g., R-squared) of both models. If the second-order significantly improves the fit, it may be better. However, higher-order polynomials can overfit and create unrealistic oscillations. Cross-validation can help: compare prediction errors for both orders. Also, consider the physical process: if you expect a simple slope, use first-order; if a dome shape, use second-order. Avoid orders higher than third unless you have a strong reason.
3. How does bootstrap resampling work in spatial interpolation?
Bootstrap resampling in spatial interpolation involves repeatedly drawing random samples (with replacement) from the original data to create many new datasets. For each bootstrap sample, you fit the interpolation model and make predictions. Then you compute the variability of predictions across bootstrap samples to estimate uncertainty. For example, you can get a 95% confidence interval for the predicted value at a location. However, standard bootstrap ignores spatial correlation, so it may underestimate uncertainty. Spatial bootstrap methods, like block bootstrap, resample entire blocks to preserve spatial structure. Bootstrap is useful for assessing model stability and prediction intervals.
4. What is narrative analysis and Labov's model of narrative structure?
Narrative analysis studies how people tell stories to make sense of their experiences. Labov's model, developed by linguist William Labov, breaks a story into six parts: abstract (summary), orientation (who, when, where), complicating action (what happened), evaluation (why it matters), resolution (how it ended), and coda (return to present). For example, a person telling about a flood might start with 'You won't believe what happened' (abstract), then describe the setting (orientation), the water rising (complicating action), their fear (evaluation), rescue (resolution), and 'now we have a new house' (coda). This structure helps researchers analyze how storytellers create meaning.
5. How does the variogram model affect the kriging variance and prediction interval width?
The variogram model describes how spatial correlation decays with distance. A model with a large nugget (nonspatial variance) increases kriging variance everywhere because there is more random noise. A model with a long range (correlation persists over large distances) reduces kriging variance near sample points but may increase it far away if the sill is high. The shape of the variogram (e.g., spherical vs exponential) also influences how quickly variance changes with distance. Choosing a wrong variogram model can lead to incorrect kriging variances and thus misleading prediction intervals. Cross-validation helps assess whether the variogram and resulting variances are realistic.
6. Compare trend surface analysis with kriging. What are the main differences?
Trend surface analysis models only the deterministic trend (large-scale variation) and treats residuals as independent noise. Kriging models both the trend and the spatial correlation of residuals (stochastic component). Trend surface is a global, non-spatial regression, while kriging is a local, spatial interpolation method. Trend surface does not use the spatial structure of residuals, so it may give poor predictions if residuals are spatially correlated. Kriging accounts for that correlation, producing more accurate predictions and uncertainty measures. Trend surface is simpler and faster, but kriging is more flexible and generally more accurate for spatially dependent data.
7. How does regularized spline differ from thin-plate spline?
Regularized spline adds a smoothing parameter that allows the surface to not pass exactly through the sample points. This reduces overfitting and makes the surface smoother. While thin-plate spline exactly interpolates the data, regularized spline trades off between fitting the data and minimizing curvature. The smoothing parameter controls this trade-off: a small value gives a surface close to the data, a large value produces a very smooth surface. Regularized spline is more robust to noisy data. In contrast, thin-plate spline can create unrealistic wiggles if data have errors. Both are based on radial basis functions, but regularized spline includes a regularization term.
8. Compare leave-one-out cross-validation with k-fold cross-validation for spatial data.
Leave-one-out cross-validation (LOOCV) uses all but one point for training and predicts the left-out point, repeating for each point. It is computationally expensive for large datasets but gives a nearly unbiased estimate of prediction error. K-fold cross-validation splits data into k groups, trains on k-1 groups, and tests on the remaining group. It is faster but can have higher variance. For spatial data, both methods need to account for spatial dependence. LOOCV may still be optimistic if left-out points are near training points. Spatial k-fold cross-validation with contiguous blocks reduces this bias. In practice, spatial k-fold is often preferred for large datasets.
9. Compare IDW with power=1 and power=4. How do the resulting surfaces differ?
With power=1, IDW gives a relatively smooth surface because distant points still have considerable influence. The interpolated values are more averaged across the area, and local variations are subdued. With power=4, the surface becomes very local: only the nearest points matter, and the surface can have sharp peaks or pits near sample points, resembling a nearest-neighbor effect. High power can lead to a 'bullseye' pattern around sample points. Low power produces a more global, smoother surface. The choice depends on whether you believe the underlying process is smooth or has sharp local changes. Cross-validation can help decide which is more appropriate for your data.
10. How would you choose the best power parameter for IDW using cross-validation?
To choose the best power parameter, perform cross-validation: for each candidate power value (e.g., 0.5, 1, 2, 3, 4), repeatedly leave out one sample point, predict its value using IDW with that power, and compute the prediction error. Then calculate the overall mean squared error (MSE) or root mean squared error (RMSE) for each power. The power that gives the smallest RMSE is considered optimal. You can also use other criteria like mean absolute error. This process helps select a power that minimizes prediction errors for your specific dataset. It is important to test a range of powers, not just integers. The optimal power may vary for different variables or regions.
11. What does kriging variance tell us about a prediction?
Kriging variance measures the uncertainty of a kriging prediction at a location. It is not the actual prediction error but the expected squared deviation of the predicted value from the true value. A higher kriging variance means we are less certain about the prediction. This variance depends on the spatial arrangement of sample points and the variogram model. It does not depend on the observed data values themselves, only on their locations. Kriging variance is used to create prediction intervals, which give a range of likely values. For example, a 95% prediction interval is roughly the predicted value plus or minus two times the square root of the kriging variance.
12. What is spatial cross-validation and why is it used?
Spatial cross-validation is a method to evaluate how well a spatial interpolation model predicts at unsampled locations. Unlike regular cross-validation, it accounts for spatial autocorrelation by not randomly splitting data but using spatial blocks or leaving out entire clusters. This avoids overly optimistic error estimates because nearby points are similar. For example, k-fold cross-validation with random splits may give low error because training and test points are close. Spatial cross-validation uses techniques like leave-one-out or spatial blocking to get a more realistic assessment. It helps compare different interpolation methods and select model parameters.