Tutorial: Supervised Learning Problem and Least Squares

Tutorial: Supervised Learning Problem and Least Squares#

Tutorial to the classes Supervised Learning Problem and Least Squares and Ordinary Least Squares.

Tutorial Objectives

Read, plot and analyze train data
Use supervised learning to predict the regional electricity consumption of France in response electric heating based on temperature data
Test the linear least squares (OLS) model
Evaluate their performance by estimating their Expected Prediction Errors (EPE) using test data

Dataset presentation#

Input:
- 2m-temperature
  - Domain: Metropolitan France
  - Spatial resolution: regional average
  - Time resolution: hourly
  - Period: 2014-2021
  - Units: °C
  - Source: MERRA-2 reanalysis
Target:
- Electricity demand
  - Domain: Metropolitan France
  - Spatial resolution: regional sum
  - Time resolution: hourly
  - Period: 2014-2021
  - Units: MWh
  - Source: RTE

Reading and pre-analysis of the input and output data#

Import data-analysis and plot modules and define paths#

# Path manipulation module
from pathlib import Path
# Numerical analysis module
import numpy as np
# Formatted numerical analysis module
import pandas as pd
# Plot module
import matplotlib.pyplot as plt
# Default colors
RC_COLORS = plt.rcParams['axes.prop_cycle'].by_key()['color']
# Matplotlib configuration
plt.rc('font', size=14)

# Set data directory
data_dir = Path('data')

# Set keyword arguments for pd.read_csv
kwargs_read_csv = dict()

# Set first and last years
FIRST_YEAR = 2014
LAST_YEAR = 2021

# Define temperature filepath
temp_filename = 'surface_temperature_merra2_{}-{}.csv'.format(
    FIRST_YEAR, LAST_YEAR)
temp_filepath = Path(data_dir, temp_filename)
temp_label = 'Temperature (°C)'

# Define electricity demand filepath
dem_filename = 'reseaux_energies_demand_demand.csv'
dem_filepath = Path(data_dir, dem_filename)
dem_label = 'Electricity consumption (MWh)'

Reading and plotting the raw temperature data#

Question (code cells below)

Use pd.read_csv with the filepath and appropriate options to make sure to get the column names and the index as dates (DatetimeIndex).

Use the resample method from the data frame to compute daily means.

Plot the 'Île-de-France' daily-mean temperature time series for (a) the whole period, (b) one year, © one month in winter and (d) one month in summer on 4 different figures (use plt.figure) using plt.plot or the plot method from data frames (preferably).

Use the mean and var methods to get mean and variance of the daily-mean temperature.

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Reading and plotting the demand data#

Question

Same question for the demand but with daily sums instead of daily means

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Analyzing the input and target data and their relationships#

Question (write your answer in text box below)

Describe the seasonality of the temperature in Île-de-France.

Are all years the same?

Describe the seasonal and weakly demand patterns.

Answer:

Question

Select the temperature and demand data for their largest common period using the intersection method of the index attribute of the data frames.

Represent a scatter plot of the daily demand versus the daily temperature using plt.scatter.

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Question

Compute the correlation between the daily temperature and the daily demand in Île-de-France using np.corrcoef.

Compute the correlation between the monthly temperature and the monthly demand using the resample method.

What do you think explains the difference between the daily and the monthly correlation?

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Answer:

Ordinary Least Squares#

Question

Perform an OLS with intercept using the entire dataset from the temperature using the formula for the optimal coefficients derived in Supervised Learning Problem and Least Squares (without Scikit-Learn). To do so:

Prepare the input matrix and output vector with the np.concatenate function (for instance);

Use the matrix-multiplication operator seen in Introduction and the np.linalg.inv function to compute the optimal coefficients and print them.

Use the estimated coefficents to predict the target from the input train data.

Overlay your prediction to the scatter plot of the train data.

Compute the train Mean Squared Error (MSE) and the train coefficient of determination (\(R^2\)) and print them.

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Question

Compute the optimal coefficients using centered input temperatures.

Compute the optimal intercept alone using a single-column input matrix.

Compare the resulting two estimations of the intercept with the sample mean of the target train data.

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Question

Perform an OLS fit with intercept using the entire dataset to predict the demand from the temperature using Scikit-learn. To do so:

Import the linear_model module from sklearn (Scikit-Learn)

Define a regressor using linear_model.LinearRegression (by default, the regressor is configured to fit an intercept in addition to the features, see fit_intercept option)

Prepare the input matrix and output vector for the fit method of the regressor

Apply the fit method to the input and output

Print the fitted coefficients using the coef_ attribute of the regressor.

Compute the train \(R^2\) coefficient using the score method of the regressor.

Compare the resulting coefficients and score to those obtained above by applying the formulas yourself.

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Answer:

Question

Define and array of 100 temperatures ranging from -5 to 35°C with np.linspace.

Make a prediction of the demand for these temperatures using the trained OLS model with the predict method of the regressor.

Plot this prediction over the scatter plot of the train data.

Does the demand prediction seem satisfactory over the whole range of temperatures?