PS2 (Due Nov 19)
Module 2 Questions
This problem set is designed to test your conceptual understanding of the modeling procedures and assumptions we’ve covered in Module 2. It is entirely “pen and paper.” You will not write or run any code, although you may use code or a calculator to help with arithmetic.
Your goal is to articulate why we choose certain methods, how those methods work procedurally, and what their critical limitations are.
Question 1
We’ve focused on QQ-mapping to correct GCM bias. A simpler, older method is Linear Scaling (or “delta-change” for variance). This method works as follows:
- You calculate the historical
GCM_Mean,GCM_Std,Obs_Mean, andObs_Std. - To “correct” a future GCM value (
GCM_Future_Val), you “shift” and “scale” it:Corrected_Val = Obs_Mean + (GCM_Future_Val - GCM_Mean) * (Obs_Std / GCM_Std)
(a): You have the following statistics:
- GCM (Hist) Mean = 20^\circC, GCM (Hist) Std = 3^\circC
- Obs (Hist) Mean = 15^\circC, Obs (Hist) Std = 2^\circC
A future GCM projects a day with a temperature of 26^\circC. What is the corrected temperature using Linear Scaling? Show your calculation.
(b) In 1-2 sentences, describe the one bias this method corrects and the one bias it scales.
(c): Imagine a GCM is “too Gaussian”—it has a perfect bell-curve shape, but observations are heavily skewed (e.g., lots of low values, long tail of high values). Why would the Linear Scaling method fail to create a realistic distribution of corrected values? In your answer, consider what aspects of the distribution’s shape (beyond mean and standard deviation) are important. Include a sketch showing the GCM distribution, the observed distribution, and the corrected distribution after Linear Scaling to illustrate your explanation.
(d): In 2-3 sentences, explain why QQ-mapping would successfully fix this “skew” problem when Linear Scaling fails. In your answer, explicitly discuss the role of the inverse CDF (or quantile function) in the QQ-mapping procedure.
Question 2
You used HMMs to model weather, which have “memory” (the next state depends on the current state). A simpler alternative is to treat each day as independent.
You use a simple K-Means Clustering model on large-scale atmospheric data. You find two clusters, “Cluster 0” (Dry) and “Cluster 1” (Wet). You find that “Cluster 0” occurs 70% of the time and “Cluster 1” occurs 30% of the time.
(a) Using this model, what is the probability of getting a 3-day wet spell (i.e., Day 1 is “Wet”, Day 2 is “Wet”, AND Day 3 is “Wet”)? Show your calculation.
(b) Now, you use a 2-state Hidden Markov Model (HMM). You get this transition matrix:
To: Dry To: Wet
From: Dry [ 0.95, 0.05 ]
From: Wet [ 0.10, 0.90 ]Assuming the “stationary” probability of being in the “Wet” state is 30% (just like in 2a), what is the probability of a 3-day wet spell using the HMM? Show your calculation.
(c) Compare your answers from (a) and (b). In 2-3 sentences, explain why the HMM is critically important for modeling persistence (like droughts or wet spells) in a way the K-Means model is not.
Question 3
You conduct the following weather typing analysis:
- You have n = 1000 days of sea-level pressure data at p = 500 grid cells, stored in matrix \mathbf{X} (dimensions: n \times p)
- You perform PCA and retain k = 4 principal components
- You transform your data to PC space: \mathbf{Z} = \mathbf{X} \mathbf{V}_k, where \mathbf{V}_k contains the first k eigenvectors (PC loadings)
- You run K-means clustering with m = 3 clusters on the PC scores \mathbf{Z}
- K-means returns cluster centroids \mathbf{C} in PC space (dimensions: m \times k)
You want to visualize what Cluster 1 “looks like” by creating a composite pressure pattern. You have two options:
- Option A: Transform the centroid from PC space back to the original 500-dimensional space
- Option B: Calculate the mean of all days assigned to Cluster 1 in the original space
(a) For Option A, write the mathematical formula to transform centroid \mathbf{c}_1 (the first row of \mathbf{C}, dimensions: 1 \times k) back to the original pressure space. Your answer should use the notation \mathbf{c}_1, \mathbf{V}_k, and the mean vector \boldsymbol{\mu} (dimensions: 1 \times p) that was subtracted before PCA. What are the dimensions of the resulting reconstructed pattern?
(b) In 2-3 sentences, explain why Option A (reconstructing from the centroid) and Option B (averaging original days) will give different results. What information is lost when you use only k = 4 PCs instead of all p = 500 dimensions?
(c) In 2-3 sentences, explain when you would prefer Option A versus Option B for interpreting your weather types. Consider what each option reveals about the cluster structure.
Question 4
You fit two models to precipitation data and calculate the following information criteria:
- Simple Model (2 parameters): AIC = 450, BIC = 458
- Complex Model (6 parameters): AIC = 442, BIC = 466
(a) Which model would you select based on AIC? Which would you select based on BIC?
(b) In 1-2 sentences, explain why AIC and BIC disagree in this case. What does each criterion prioritize?
(c) You plan to use the selected model for climate change impact assessment 50 years into the future. In 2-3 sentences, explain why the “M-open” perspective suggests that neither criterion may identify the “true” model, and how this affects your interpretation of the model selection results.
Question 5
You are bias-correcting precipitation using two methods: the Delta Method and Quantile Delta Mapping (QDM).
Historical data:
- GCM 95th percentile: 40 mm/day
- GCM 50th percentile: 10 mm/day
- Observed 95th percentile: 50 mm/day
- Observed 50th percentile: 12 mm/day
Future GCM projects:
- 95th percentile increases to 48 mm/day (20% increase from 40)
- 50th percentile increases to 11 mm/day (10% increase from 10)
(a) Using the multiplicative Delta Method, what are the corrected future values for both the 50th and 95th percentiles? Show your calculations.
(b) Using QDM (which preserves the relative change at each quantile), what are the corrected future values? Show your calculations.
(c) In 2-3 sentences, explain what assumption QDM makes that the Delta Method doesn’t. Why might this be important for extreme precipitation (high percentiles) versus typical precipitation (median)?
Question 6
Consider the comparison between the deep learning approach in Vandal et al. (2017) and the process-informed stochastic weather generator in Steinschneider et al. (2019).
(a) In 2-3 sentences, explain what Vandal et al. (2017) assumes to be stationary (unchanged between historical and future climate) and what it allows to be non-stationary (able to change). Consider both spatial patterns and statistical relationships.
(b) Steinschneider et al. (2019) explicitly separates changes in weather regime frequencies (dynamical changes) from changes in within-regime precipitation intensities (thermodynamic changes). In 2-3 sentences, explain why this separation is useful for climate change applications and what physical understanding motivates this approach.
(c) In 2-3 sentences, describe one advantage and one disadvantage of the deep learning approach compared to the process-informed approach. Consider factors such as: amount of training data needed, interpretability, physical constraints, and ability to extrapolate beyond training conditions.