Tutorial: Introduction to Unsupervised Learning with a Focus on PCA

Tutorial: Introduction to Unsupervised Learning with a Focus on PCA#

Tutorial to the class Introduction to Unsupervised Learning with a Focus on PCA based on the same case study as in Tutorial: Regularization, Model Selection and Evaluation.

Tutorial Objectives

Apply Principal Component Analysis (PCA) to climate data to analyze patterns of variability
(Combine PCA reduction/\(k\)-means clustering to Ordinary Least Squares (OLS) to predict climate variables)
(Use cross-validation to regularize the OLS with the number of retained Empirical Orthogonal Functions (EOFs) or clusters).

Dataset presentation#

Input:
- Geopotential height at 500hPa
  - Domain: North Atlantic
  - Spatial resolution: \(0.5° \times 0.625°\)
  - Time resolution: monthly
  - Period: 1980-2021
  - Units: m
  - Source: MERRA-2 reanalysis
Target:
- Onshore wind capacity factors
  - Domain: Metropolitan France
  - Spatial resolution: regional mean
  - Time resolution: daily
  - Period: 2014-2021
  - Units:
  - Source: RTE

Getting ready#

Reading the wind capacity factor and geopotential height data#

We follow the same procedure as in # Tutorial: Regularization, Model Selection and Evaluation.

# Import modules
from pathlib import Path
import numpy as np
import matplotlib.pyplot as plt
plt.rc('font', size=14)

# Data directory
DATA_DIR = Path('data')

# Filename to geopotential height at 500hPa from MERRA-2 reanalysis
START_DATE = '19800101'
END_DATE = '20220101'
filename = 'merra2_analyze_height_500_month_{}-{}.nc'.format(START_DATE, END_DATE)
z500_label = 'Geopotential height (m)'

Question

Read the geopotential height data using xarray.load_dataset and print it.

# answer cell

Question (optional)

Coarsen the grid resolution of the geopotential height field to reduce the number of variables.

# answer cell

Representing the first moments of the geopotential height field#

Question

Compute the mean and the variance of the geopotential height with the mean and var methods.

Plot the mean with the plot method.

Do a filled-contour plot of the variance with the plot.contourf method.

# answer cell

Question

Plot the variance of the geopotential height.

# answer cell

Question

Scale the geopotential-height deviations to account for variations in the area represented by each grid point.

Plot the variance of the scaled geopotential height.

Qualitatively describe the mean and variance of the geopotential height.

# answer cell

Answer:

PCA of the geopotential height field#

Question

Estimate the covariance matrix of the scaled geopotential height using the stack method of data arrays.

# answer cell

Question

Compute EOFs and corresponding variances using np.linalg.eigh.

# answer cell

Question

Sort the EOFs and corresponding variances by decreasing variances.

Plot the fraction of variance “explained” by the leading 20 EOFs.

Interpret your results.

# answer cell

Answer:

Question

Plot the leading EOF on a map.

To what physical phenomenon could this pattern be associated to?

# answer cell

Answer:

Question

Compute the principal component associated with the leading EOF.

Compare its variance to the corresponding eigenvalue and explain your result.

Plot this principal component.

Confirm or reconsider you previous answer on the physical phenomenon that could be associated to the leading EOF.

# answer cell

Answer:

Dealing with the seasonal cycle#

Question (optional)

Use the scipy.signal.welch to estimate the Power Spectral Density (PSD) of the leading principal component and plot it.

Confirm or reconsider you previous answer on the physical phenomenon that could be associated to the leading EOF.

# answer cell

Question

Compute the seasonal cycle of the geopotential height (averages over all years of the same month of the year for each month) with the groupby of data arrays.

Plot all 12 months. You can use the col option of the plot method of data arrays.

Also plot the variance of the seasonal cycle on a map.

Confirm or reconsider you previous answer on the physical phenomenon that could be associated to the leading EOF.

# answer cell

Answer:

Question

Compute seasonal anomalies (deviations from the seasonal cycle) of the geopotential height with groupby.

Plot the variance of the seasonal anomalies on a map.

How does it compare to the variance of the data with the seasonal cycle?

# answer cell

Answer:

Representing and interpreting the EOFs#

Question

Estimate the covariance matrix of the anomalies (with the seasonal cycle subtracted).

Compute the EOFs and corresponding variances.

Plot the explained variances associated with the EOFs together with the cumulative sum of the explained variances.

What is the minimum number of EOFs that one needs to keep to explain at least 90% of the variance.

# answer cell

Answer:

Question

Plot the leading 4 EOFs and principal components.

Can you associate these patterns to known climate phenomena?

# answer cell

Reconstructing the geopotential height field from the EOFs and PCs#

Answer:

Question

Reconstruct the inputs from the leading 4 EOFs only.

Compare the original time series at a few arbitrary locations to the corresponding reconstructed time series.

Plot the variance of the reconstruction on a map.

Same question but keeping more EOFs

Interpret your results in terms of filtering.

# answer cell

Answer:

Using PCA to extract features for prediction#

Question (optional)

Design a linear model that best predicts present (not future) wind capacity factors in data/reseaux_energies_capacityfactor_wind-onshore.csv using geopotential-height principal components as inputs. To do, use cross-validation to regularize based on the number of leading principal components retained.

# answer cell

Question (optional)

Use \(k\)-means clustering with sklearn.cluster.KMeans to detect “atmospheric regimes” from the geopotential-height data and compare the result to the EOFs obtained above.

Design a linear model as above but based on clusters rather than EOFs.