Artificial Intelligence and the Environment

Contents

• Introduction and overview
• 0. Review: energy, uncertainty and time-series basics
• 1. Environmental impacts of AI and life cycle assessment
• 2. Measuring the footprint of an AI system
• 3. Climate and environmental data
• 4. Machine learning on infrasound data in polar regions
• 5. Sensor and IoT time series: preparation, segmentation, anomalies, prediction
• 6. AI for a smaller footprint: overview and the water cycle
• 7. Complements: sensitivity, EOFs, exact segmentation, quantiles and decisions
• 8. Research directions and open problems
• Exercises
• Projects and assessment
• References

Introduction and overview

Audience: a student with a good M1 in mathematics, computer science or physics who can train a model in Python and wants to put numbers, with their uncertainty, on the environmental side of machine learning.

Empirical first: the final training of the 176-billion-parameter language model BLOOM used about 433 MWh of electricity. On the low-carbon French grid this is about 25 tonnes of CO₂-equivalent, but the estimate doubles to about 50 tonnes once the idle power of the cluster and the manufacturing of the hardware are counted (Luccioni, Viguier and Ligozat, 2023; ch. 2). Where the factor of two comes from, and what the same run would emit elsewhere, is the first question of this course.

The course links machine learning to climate change and to the impact of human activity on the environment, from three angles. First, the footprint of AI itself: life cycle assessment, energy measurement, the carbon footprint of a digital service (ch. 1–2). Second, AI to observe the Earth: climate records, reanalyses, simulations and infrasound arrays (ch. 3–4). Third, AI to use resources better, through sensor time series and the water cycle (ch. 5–6).

As listed on master-mva.com, lecture 1 is given by Anne-Laure Ligozat (ENSIIE), lecture 2 by Sentia Oger and lecture 3 by Gilles Faÿ (CentraleSupélec); the number of places is limited to 35. Related MVA pages: Machine Learning for Time Series, Statistical Learning for Scientific Imaging and Physical Measurements: Geosciences, Training and Deploying Large-Scale Models and Remote Sensing Data.

What you will be able to do

• Distinguish the direct, indirect and systemic environmental effects of an AI system, and name the impact categories beyond carbon: water, abiotic resources, energy.
• Set up a life cycle assessment with a functional unit and system boundaries, and compute its inventory by the matrix method.
• Explain when an attributional or a consequential assessment answers the question asked, and compute average and marginal footprints.
• Measure or model the electricity of a training run or an inference service, and turn it into a carbon footprint with PUE, carbon intensity and embodied emissions.
• Propagate the uncertainty of a footprint estimate and rank its sources.
• Compute area-weighted climate averages, anomalies and trends with autocorrelated noise.
• Estimate the direction and speed of an infrasound wave from an array, and classify detections without leakage.
• Prepare raw sensor series, segment them exactly, and detect anomalies with robust statistics and CUSUM.
• Build and evaluate forecasts against seasonal baselines, from rating curves and rainfall–runoff models to demand.
• Weigh the savings of an AI application against its own footprint, category by category.

Official objectives and syllabus (2026–2027)

The course description on master-mva.com, repeated in the Université Paris-Saclay course detail, sets the objective of linking the technology that is transforming every science and every sector, artificial intelligence, to two major stakes for humanity: climate change and the impact of human activity on the environment.

It announces three complementary subjects: the impact of AI itself on the environment, the use of AI to measure these impacts, and its use to actively reduce the effects of humans on the planet. The course is organised as three lectures of 3 h and three labs of 3 h, each lecture followed by its lab.

Every official topic is taught on this page. Its extensions beyond the syllabus are the review chapter 0, the matrix form of life cycle assessment and the rebound model, array processing for infrasound, exact change-point algorithms, rating curves and rainfall–runoff models, the complements of ch. 7 (sensitivity indices, EOFs, PELT, CUSUM optimality, conformal quantiles, cost–loss decisions) and the research chapter 8.

Where this course fits in the programme

Official prerequisites: notions of modelling and of machine learning and/or deep learning, and Python programming with numpy, pandas, scikit-learn and pytorch or tensorflow. It is an Opening course of semester 2, period 2, with 2.5 ECTS and 18 h.

• Preparation — the refreshers Probability and Statistics (variance, estimation), Linear Algebra, Analysis and Optimization (least squares, the SVD) and Scientific Python, NumPy and PyTorch (every code block).
• Semester 1 courses that prepare it — Machine Learning for Time Series (ch. 5), Statistical Learning for Scientific Imaging and Physical Measurements: Geosciences (ch. 3–4), Introduction to Statistical Learning, Statistical Learning with Extreme Values (floods and heat waves), Fundamentals of Reproducible Research and Free Software and the Turing Seminar (societal costs and benefits).
• Related courses of semester 2 — Training and Deploying Large-Scale Models (the compute behind ch. 2), Remote Sensing Data (satellites, ch. 3), Learning Physical Phenomena: Metamodelling and Uncertainty Quantification (emulators, sensitivity indices), Audio Signal Processing: Time–Frequency Analysis (spectra, ch. 4), Interactions: Complexity, Stochasticity, Extrema and Rare Events, Regulating AI and Responsible Machine Learning.
• Afterwards — an environmental topic, in a laboratory or a utility, is a natural Research Internship.

Course outline

• Review: energy, uncertainty and time-series basics — units of power and energy, products of uncertain factors, least squares, autocorrelation, honest validation (self-study).
• Environmental impacts of AI and life cycle assessment — orders of effects, impact categories, the ISO life cycle method in matrix form, attributional against consequential, rebound.
• Measuring the footprint of an AI system — power models and meters, PUE and carbon intensity, embodied emissions, training against inference, digital services, uncertainty and trade-offs.
• Climate and environmental data — observations, satellites, simulations and reanalyses; averages on the sphere, anomalies, trends; the uses of machine learning.
• Machine learning on infrasound data in polar regions — spectra, array processing for direction and speed, classification of detections without leakage.
• Sensor and IoT time series — preparing meter data, exact segmentation, robust and sequential anomaly detection, forecasting and its evaluation.
• AI for a smaller footprint: overview and the water cycle — what machine learning does for energy, transport, water and waste; rating curves, rainfall–runoff forecasts, sensor validation, net balance.
• Complements — sampling error of power traces, Sobol indices, EOFs, PELT, the optimality of CUSUM, quantile forecasts with conformal calibration, cost–loss decisions (self-study).
• Research directions and open problems — measurement standards, indirect effects, machine-learning weather and climate models, ungauged basins, label-scarce sensing (self-study).

Session plan

18 h in class = 6 sessions of 3 h. The volume is the Université Paris-Saclay programme table (9 h of lectures, 9 h of labs); master-mva.com announces the same organisation, three lectures of 3 h and three labs of 3 h. The plan keeps the official order (lecture 1, lab 1, lecture 2, lab 2, lecture 3, lab 3); the split of the chapters between sessions is a proposal.

Level 1 = foundations (definitions, orders of magnitude, first computations), 2 = core methods, 3 = case study close to research; the level never decreases.

Self-study, outside the 18 h (about 8 h in total): ch. 0 Review: energy, uncertainty and time-series basics, before session 1 (≈ 2 h) · ch. 7 Complements, section by section after the session of the chapter it completes (≈ 4 h, with Ex. 16) · ch. 8 Research directions and open problems, after session 6 (≈ 2 h, with Ex. 18). The mini-project or critical reading comes on top (see Projects and assessment).

How to study this module

• Read ch. 0 before session 1: every later chapter multiplies uncertain factors, fits least squares or works with autocorrelated series.
• Read each chapter before its session. The labs (sessions 2, 4 and 6) assume the lecture before them: ch. 2 uses ch. 1, ch. 4 uses ch. 3, and ch. 6 uses ch. 5.
• Read each section of ch. 7 after the chapter it completes: 7.1–7.2 after ch. 2, 7.3 after ch. 3, 7.4–7.6 after ch. 5, 7.7 after ch. 6.
• Keep a notebook of orders of magnitude (watts of a GPU, PUE of a data centre, grams of CO₂e per kWh) with their source and uncertainty. A footprint without its assumptions is not a result.
• After each session, do the exercises of its row: ★ checks definitions, ★★ asks for a short proof or a computation.
• Run every Python block: NumPy, SciPy, pandas and scikit-learn on a CPU, fixed seeds, synthetic data, a few seconds each.
• Write each lab as a short report, as for the continuous assessment: question, method, numbers with uncertainty, one limitation.

0. Review: energy, uncertainty and time-series basics

Four M1 tools return in every chapter: physical units of power and energy (ch. 1–2, 6), products of uncertain factors (ch. 1–2), least squares with its standard errors (ch. 3–6), and autocorrelated series with honest validation (ch. 3–5). This chapter states them in the form used later. The refreshers Probability and Statistics and Linear Algebra, Analysis and Optimization give more detail.

0.1 Power, energy and carbon units

Power is a rate and energy a quantity; most errors in footprint reports come from mixing them. A GPU that draws 300 W for 24 h uses

0.3\ \text{kW} \times 24\ \text{h} = 7.2\ \text{kWh}.

At an illustrative 50 g CO₂e/kWh this is 0.36 kg CO₂e; at 400 g CO₂e/kWh it is 2.9 kg. The factor of eight between the two grids is larger than most hardware or software optimisations.

0.2 Products of uncertain factors

A footprint is almost always a product: energy × overhead × carbon intensity, or hours × power × lifetime share. Each factor is uncertain, often by tens of percent. Products are best handled on the logarithmic scale.

The share of factor i in the variance of \log C is \sigma_i^2 / \sum_j \sigma_j^2. This ranks the assumptions worth checking first; ch. 2.5 uses it on a training run.

import numpy as np

rng = np.random.default_rng(0)
n = 200_000
# energy (kWh), PUE, carbon intensity (kg/kWh): medians and log-sd
med = np.array([500.0, 1.3, 0.06])
sig = np.array([0.10, 0.08, 0.30])
logs = np.log(med) + sig * rng.standard_normal((n, 3))
C = np.exp(logs.sum(axis=1))
print(f"median {np.median(C):.1f} kg CO2e, 90% interval "
      f"[{np.quantile(C, .05):.1f}, {np.quantile(C, .95):.1f}]")
print("variance shares of log C:", np.round(sig**2 / (sig**2).sum(), 3))

The median is 500 \times 1.3 \times 0.06 = 39 kg CO₂e. The carbon intensity carries about 85 % of the variance of the logarithm, so it deserves the first check.

0.3 Least squares and standard errors

The first formula follows from setting the gradient of \|y - X\beta\|^2 to zero. The covariance follows from \hat\beta - \beta = (X^\top X)^{-1} X^\top \varepsilon. The last equality uses \sum_{t=1}^n (t - \bar t)^2 = n(n^2 - 1)/12.

0.4 Stationarity and autocorrelation

With \phi = 0.6, a hundred daily values are worth only 25 independent ones. Chapter 3 applies the same correction to climate trends.

0.5 Honest validation

The rule is simple: the test set must look like the situation in which the model will be used. A flood forecaster will be used on future years, so it is tested on later years. A leak detector will be installed on new districts, so it is tested on districts it has not seen.

1. Environmental impacts of AI and life cycle assessment

Before measuring anything, one must say what is measured: which effects, over which part of the life of the equipment, for which question. This chapter gives the vocabulary of impacts and the standard accounting method, life cycle assessment, in matrix form.

1.1 Direct and indirect effects, impact categories

Kaack and coauthors (2022) use a parallel split for machine learning: computing-related, immediate application and system-level impacts. A model that speeds up oil exploration has a small direct footprint but a large indirect one, of the opposite sign to a model that helps integrate wind power.

For climate change the factors are global warming potentials over 100 years: 1 for CO₂, roughly 30 for methane and roughly 270 for nitrous oxide in the latest IPCC assessment. The score is then in kg CO₂-equivalent (CO₂e). Other categories used for digital equipment are water use, abiotic resource depletion (metals and minerals, in kg of antimony-equivalent), and primary energy.

For resource depletion, the CML method (Guinée, 2002) characterises an element i by its yearly extraction \mathrm{DR}_i and its ultimate reserve R_i, relative to antimony: \mathrm{ADP}_i = \frac{\mathrm{DR}_i / R_i^2}{\mathrm{DR}_{\mathrm{Sb}} / R_{\mathrm{Sb}}^2}\ \text{kg Sb-eq per kg}. Scarce metals, such as the gold of circuit boards, therefore weigh orders of magnitude more per kilogram than iron.

Reporting a single category hides burden shifting: a liquid-cooled data centre may use less electricity but more water. A complete assessment is multi-criteria.

1.2 Life cycle assessment in matrix form

Processes are linked by their products: a process uses electricity, itself made by a process, and so on up the supply chain. Linear algebra solves this loop exactly.

The series has a direct reading: f is the demand, Mf the inputs needed to make it, M^2 f the inputs of those inputs, and so on up the supply chain. The condition \rho(M) < 1 says that the economy is productive: making a product does not consume more of it than it delivers.

import numpy as np

# products / processes: electricity [kWh], GPU-hour [h], training run [run]
A = np.array([[1.0, -0.48, -100.0],
              [0.0,  1.0, -1000.0],
              [0.0,  0.0,     1.0]])
f = np.array([0.0, 0.0, 1.0])
s = np.linalg.solve(A, f)
for ci in (0.05, 0.40):
    B = np.array([[ci,  0.01, 0.0],     # kg CO2e per unit of each process
                  [1.0, 0.20, 0.0]])    # litres of water
    g = B @ s
    emb = 0.01 * s[1] / g[0]
    print(f"CI={ci}: activity {s}, CO2e {g[0]:.0f} kg, water {g[1]:.0f} L, "
          f"embodied share {emb:.0%}")

On a decarbonised grid, the manufacturing of the hardware becomes the main lever: longer lifetimes and higher utilisation, rather than lower power alone (Gupta and coauthors, 2021).

1.3 Attributional and consequential assessment

The two answer different questions. Consider a grid of average intensity 60 g CO₂e/kWh whose evening demand is met at the margin by gas plants at about 450 g CO₂e/kWh (illustrative values).

• The attributional footprint of the 580 kWh of Example 1.5 is about 35 kg CO₂e.
• If the run is added in the evening, the consequential footprint is about 261 kg CO₂e.
• If the decision is long-term (building a data centre that triggers new wind farms), the marginal source may be the new capacity, and the consequential footprint can be lower than the average one.

The time horizon of the decision decides which marginal source is relevant (Ekvall and Weidema, 2004).

1.4 Rebound, trade-offs and indirect effects

Efficiency gains are the main promise of AI for the environment: less energy per trip, per tonne, per query. The rebound effect says that the saving can be partly or wholly eaten by extra demand.

With r = 0.3 and \varepsilon = 0.5, energy falls by only 16 % instead of 30 %, a rebound of about 46 %. Jevons (1865) observed that more efficient steam engines increased coal consumption.

Weighing environmental and societal costs against benefits has no single formula, but three principles help.

• Compare with the situation without the AI system, under the same functional unit, rather than reporting the footprint alone (Ligozat and coauthors, 2022).
• Count indirect effects on both sides: avoided emissions, and induced consumption.
• Keep categories separate when they cannot be traded: a model that saves CO₂ but uses scarce water in a dry region has not simply “paid back”.

2. Measuring the footprint of an AI system

Chapter 1 said what to count; this chapter says how to measure it for a training run, an inference service and a digital service, and how sure one can be of the result. It ends with the lab sheet of the first case study.

2.1 Electricity consumption of digital equipment

Fan, Weber and Barroso (2007) found this model accurate for whole servers. Idle power is large: a server idling at half its peak power wastes energy whenever it waits for data.

Power is measured at three levels, from coarse to fine.

• A wattmeter at the wall plug sees a whole node, including fans and power supply losses.
• On-chip counters report component energy: RAPL for Intel processors and memory, NVML for NVIDIA GPUs. They miss the rest of the node.
• Software estimators, such as CodeCarbon, Carbontracker (Anthony, Kanding and Selvan, 2020), experiment-impact-tracker (Henderson and coauthors, 2020) and the Green Algorithms calculator (Lannelongue, Grealey and Inouye, 2021), read these counters or use catalogue power values, then apply PUE and carbon intensity.

Sampled power misses short spikes between two samples. A cumulative energy counter, as RAPL provides, integrates in hardware and avoids the problem; ch. 7.1 bounds the error of sampling.

2.2 Data-centre overhead, carbon intensity and water

Water is used twice: on site for evaporative cooling, and off site to cool the power plants that make the electricity (Li and coauthors, 2023). Like carbon intensity, both depend on place and season.

The operational and embodied parts combine into the footprint of a computation that runs devices for T hours at IT energy E_{\mathrm{IT}}, with e_{\mathrm{emb}} the embodied rate of Proposition 1.6:

2.3 Training against inference

A model is trained once and queried many times. Luccioni, Jernite and Strubell (2024) measured inference across tasks and models. Generative tasks cost far more energy per query than classification, and image generation is the most expensive; general-purpose models cost more than task-specific ones for the same task.

With 433 MWh of training and an illustrative 3 Wh per query, N^\star \approx 1.4 \times 10^8 queries. A service answering 10^7 queries a day reaches it in two weeks, so for deployed models the footprint of inference, and of the terminals that call it, cannot be ignored.

2.4 The carbon footprint of a digital service

Allocation is a modelling choice and must be stated. For networks, it hides a real difficulty.

Over the long term the picture changes: traffic growth triggers new capacity, whose embodied and fixed power become marginal. This is Definition 1.7 again: the question (short-term reduction or long-term trend) fixes the right factor.

2.5 Uncertainty and trade-offs

Every factor of C is uncertain: power is measured on some components only, the PUE is a yearly average, carbon intensity varies by hour and country, and manufacturing data are scarce. A Monte Carlo simulation turns these ranges into an interval and a ranking (Proposition 0.2).

import numpy as np

rng = np.random.default_rng(1)
n = 100_000
gpu_hours = 1000.0
names = ["GPU power", "PUE", "carbon int.", "embodied"]
med = np.array([0.40, 1.2, 0.06, 0.010])   # kW, -, kg/kWh, kg/GPU-h
sig = np.array([0.05, 0.10, 0.40, 0.50])   # log-sd of each factor
X = np.exp(np.log(med) + sig * rng.standard_normal((n, 4)))
oper = gpu_hours * X[:, 0] * X[:, 1] * X[:, 2]
emb = gpu_hours * X[:, 3]
C = oper + emb
print(f"median {np.median(C):.1f} kg, 90% interval "
      f"[{np.quantile(C, .05):.1f}, {np.quantile(C, .95):.1f}] kg")
# first-order sensitivity: share of Var(C) explained by each factor
for j, name in enumerate(names):
    bins = np.quantile(X[:, j], np.linspace(0, 1, 21))
    idx = np.clip(np.digitize(X[:, j], bins) - 1, 0, 19)
    cond_means = np.array([C[idx == b].mean() for b in range(20)])
    print(f"{name:12s} {cond_means.var() / C.var():.2f}")

The final loop estimates the share of the variance of C explained by each factor, the sensitivity indices of ch. 7.2. Carbon intensity and embodied emissions dominate; measuring GPU power more precisely would hardly change the interval.

The benefit side matters as much. Schwartz and coauthors (2020) asked that papers report the computational cost of their results next to the accuracy, so that efficiency becomes a criterion.

Only models on the front are reasonable choices; choosing among them is a value judgement that the numbers make explicit.

2.6 Lab sheet: environmental assessment of an AI system

The first case study follows the four phases of Definition 1.3.

1. Goal and scope: one training run plus one year of inference; functional unit “one thousand answered queries”; boundaries: GPUs, servers, storage, network, data-centre overhead.
2. Inventory: power, hours, PUE, location and hour of the carbon intensity, embodied factors, each with its source.
3. Impacts: carbon split into dynamic, idle and embodied parts (Example 2.3), and water through the WUE.
4. Interpretation: Monte Carlo interval, sensitivity ranking, break-even point (Proposition 2.4), Pareto front of the candidate models.

import time
import numpy as np
from sklearn.datasets import load_digits
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.ensemble import RandomForestClassifier

X, y = load_digits(return_X_y=True)
Xtr, Xte, ytr, yte = train_test_split(X, y, test_size=0.3, random_state=0)
P_cpu = 15.0  # assumed power of one CPU core under load, in W (illustrative)
models = {"logreg": LogisticRegression(max_iter=2000),
          "rf-10": RandomForestClassifier(10, random_state=0, n_jobs=1),
          "rf-100": RandomForestClassifier(100, random_state=0, n_jobs=1),
          "rf-400": RandomForestClassifier(400, random_state=0, n_jobs=1)}
rows = []
for name, m in models.items():
    t0 = time.process_time()
    m.fit(Xtr, ytr)
    acc = m.score(Xte, yte)
    energy_J = P_cpu * (time.process_time() - t0)
    rows.append((name, acc, energy_J))
for name, acc, e in rows:
    dominated = any(a2 >= acc and e2 <= e and (a2 > acc or e2 < e)
                    for n2, a2, e2 in rows if n2 != name)
    print(f"{name:7s} acc={acc:.3f} energy={e:6.2f} J  "
          f"{'dominated' if dominated else 'Pareto'}")

The timings vary between machines, which is itself a lesson: an energy estimate from time and an assumed power is a model, and must say so.

3. Climate and environmental data

Climate science has more data than most fields, but the data are uneven in space, in time and in kind. This chapter describes the sources, then the operations that every analysis of climate data starts with: averages on the sphere, anomalies, and trends with honest error bars. Empirical orthogonal functions, the climate name of principal component analysis, are in the self-study ch. 7.

3.1 Observations, satellites, simulations and reanalyses

Better projections and predictions are what society needs to plan its adaptation. Four kinds of data feed them.

• In situ observations: weather stations, ships and buoys. Some records go back to the end of the 19th century, but coverage is dense in Europe and North America and sparse over the oceans, deserts and poles.
• Satellite data, since the 1970s: radiances turned into temperature, sea ice, sea level or vegetation. Coverage is global, but successive instruments must be intercalibrated.
• Climate simulations: numerical models of atmosphere, ocean, ice and land, run under common protocols (the CMIP intercomparison projects). An ensemble of runs samples the uncertainty.
• Reanalyses, such as ERA5 (Hersbach and coauthors, 2020): a weather model assimilates past observations into a complete, gridded, hourly history of the atmosphere over several decades.

ERA5 is distributed by the Copernicus Climate Data Store and CMIP outputs by the Earth System Grid Federation. With free machine-learning libraries, anyone can investigate natural processes from these data.

3.2 Averages on the sphere

Gridded data come on a regular latitude–longitude grid. Cells near the poles are much smaller than cells near the equator, so the plain mean of all cells gives the poles too much weight.

import numpy as np

lat = np.arange(-89.5, 90, 1.0)                 # 1-degree cell centres
lon = np.arange(0.5, 360, 1.0)
warming = np.where(lat[:, None] > 60, 3.0, 1.0) * np.ones((lat.size, lon.size))
w = np.cos(np.deg2rad(lat))[:, None] * np.ones_like(warming)
print(f"plain mean {warming.mean():.2f} C, area-weighted {np.average(warming, weights=w):.2f} C")

3.3 Anomalies and trends

Anomalies remove the seasonal cycle and the differences of altitude between stations. This is why temperature records are averaged as anomalies, not as absolute values.

A trend is then fitted by least squares (Theorem 0.3). The residuals of climate series are autocorrelated, so the white-noise standard error is too small.

In practice \phi is estimated from the residuals, and the effective sample size n(1 - \phi)/(1 + \phi) replaces n in the test (Santer and coauthors, 2000).

import numpy as np

rng = np.random.default_rng(3)
n, phi, b = 45 * 12, 0.6, 0.02 / 12             # 45 years of months, 0.2 C per decade
u = np.zeros(n)
for t in range(1, n):
    u[t] = phi * u[t - 1] + 0.12 * rng.standard_normal()
t = np.arange(n)
x = b * t + u
X = np.column_stack([np.ones(n), t])
coef, res, *_ = np.linalg.lstsq(X, x, rcond=None)
r = x - X @ coef
s2 = r @ r / (n - 2)
se_white = np.sqrt(s2 / ((t - t.mean()) ** 2).sum())
phi_hat = np.corrcoef(r[:-1], r[1:])[0, 1]
se_ar1 = se_white * np.sqrt((1 + phi_hat) / (1 - phi_hat))
print(f"trend {coef[1] * 120:.3f} C/decade, se white {se_white * 120:.3f}, "
      f"se AR(1) {se_ar1 * 120:.3f}, phi_hat {phi_hat:.2f}")

The corrected standard error is twice the naive one here, so every white-noise t-statistic is twice too large.

3.4 Machine learning on climate data

Machine learning enters climate science at four places, each with its own pitfall.

• Downscaling and bias correction learn a map from coarse model output to local observations. The relation learnt on the past must still hold in a warmer climate.
• Emulators are cheap surrogates of climate models, trained on simulations; ClimateBench (Watson-Parris and coauthors, 2022) emulates the response of temperature and precipitation to emission scenarios. They must not be trusted outside the scenarios they were trained on.
• Data-driven weather forecasting: trained on ERA5, GraphCast (Lam and coauthors, 2023) matched or exceeded the operational medium-range forecast of the European Centre on most of the variables and lead times evaluated.
• Detection and classification of events in observations: cyclones, heat waves, and the infrasound signals of ch. 4.

Validation follows Definition 0.6: samples close in space and time are correlated, so test sets are whole years and whole regions. Under climate change the test period is warmer than the training period, which is precisely the situation of use.

4. Machine learning on infrasound data in polar regions

Infrasound is sound below about 20 Hz, inaudible to humans, which can travel thousands of kilometres in the atmosphere. Polar regions are rich in infrasound sources and poor in other observations. This chapter turns raw pressure records into features, then classifies detections: the programme of the second lab.

4.1 Sources and stations

• The International Monitoring System of the Comprehensive Nuclear-Test-Ban Treaty runs a global network of infrasound stations, several at high latitudes, Antarctica included.
• Each station is an array of a few microbarometers spread over at most a few kilometres.
• Sources at high latitudes include microbaroms from ocean swell (about 0.1 to 0.5 Hz, peaking near 0.2 Hz), ice calving, aurorae, volcanoes and meteors.
• The main nuisance is wind noise, strong under katabatic winds and incoherent between sensors.

Beyond monitoring, these records inform on winds and temperatures of the middle atmosphere (Le Pichon, Blanc and Hauchecorne, 2010).

4.2 Spectra

The price is resolution: a segment of length L resolves frequencies about 1/L apart.

import numpy as np
from scipy.signal import welch, lfilter

rng = np.random.default_rng(5)
fs, T = 20.0, 1800.0                          # 20 Hz, 30 minutes
t = np.arange(0, T, 1 / fs)
amp = 1 + 0.3 * np.sin(2 * np.pi * t / 300)   # slowly modulated microbarom
micro = 0.5 * amp * np.sin(2 * np.pi * 0.21 * t)
wind = lfilter([1.0], [1.0, -0.98], rng.standard_normal(t.size)) * 0.3
p = micro + wind
f, S = welch(p, fs=fs, nperseg=4096)
band = (f > 0.05) & (f < 2)
print(f"spectral peak at {f[band][np.argmax(S[band])]:.3f} Hz "
      f"(resolution {fs / 4096:.4f} Hz, {2 * t.size // 4096 - 1} segments)")

4.3 Array processing: direction and speed

A plane wave from back-azimuth \theta (direction of the source, clockwise from north) crosses the array at apparent speed c. Its slowness vector is u = -\frac{1}{c}\,(\sin\theta, \cos\theta). The wave reaches the sensor at position r_i (east, north) at time u \cdot r_i, so the delay of sensor j after sensor i is \tau_{ij} = u \cdot (r_j - r_i).

Each delay is the lag that maximises the cross-correlation of two sensors. For a true plane wave the delays around a triangle cancel, \tau_{ij} + \tau_{jk} + \tau_{ki} = 0: this closure separates coherent signals from wind noise. Detectors such as PMCC (Cansi, 1995) slide windows in time and frequency bands and keep those with small closure and high correlation. An apparent speed near 0.34 km/s is acoustic; seismic arrivals are much faster.

import numpy as np
from scipy.signal import butter, filtfilt

rng = np.random.default_rng(6)
fs, c, theta = 100.0, 340.0, np.deg2rad(135.0)       # true speed and back-azimuth
r = np.array([[0, 0], [800, 100], [200, 900], [-600, 400]], float)
u = -np.array([np.sin(theta), np.cos(theta)]) / c
t = np.arange(0, 60, 1 / fs)
b, a = butter(4, [0.5, 3.0], btype="band", fs=fs)
src = filtfilt(b, a, rng.standard_normal(t.size + 2000))
P = np.array([np.interp(t + 10 - ri @ u, np.arange(src.size) / fs, src) for ri in r])
P += 0.3 * src.std() * rng.standard_normal(P.shape)
pairs = [(i, j) for i in range(4) for j in range(i + 1, 4)]
taus, D = [], []
for i, j in pairs:
    xc = np.correlate(P[j], P[i], "full")
    taus.append((np.argmax(xc) - (t.size - 1)) / fs)
    D.append(r[j] - r[i])
u_hat = np.linalg.lstsq(np.array(D), np.array(taus), rcond=None)[0]
baz = np.degrees(np.arctan2(-u_hat[0], -u_hat[1])) % 360
print(f"back-azimuth {baz:.1f} deg, apparent speed {1 / np.linalg.norm(u_hat):.0f} m/s")
print("closure 0-1-2:", round(taus[0] + taus[3] - taus[1], 3), "s")

4.4 Classifying detections without leakage

Each detection becomes a feature vector: back-azimuth, apparent speed, dominant frequency, duration, amplitude, closure residual, time of day and season. Labelling detections as microbarom, wind or other signals is then supervised learning, with three pitfalls.

• One event yields many near-identical detections. A random split puts copies of the same event in training and test: this is leakage (Definition 0.6). Split by event or by day.
• Classes are imbalanced, so report balanced accuracy or per-class recall.
• Back-azimuth is an angle: encode it as (\sin\theta, \cos\theta).

import numpy as np
from sklearn.ensemble import RandomForestClassifier
from sklearn.model_selection import KFold, GroupKFold, cross_val_score

rng = np.random.default_rng(7)
X, y, g = [], [], []
for e in range(150):                                  # 150 events, 15 detections each
    k = rng.choice(3, p=[0.6, 0.25, 0.15])            # microbarom, wind, other
    centre = np.array([[0.2, 340, 200], [0.6, 300, 120], [1.5, 360, 60]])[k]
    centre = centre + rng.normal(0, [0.45, 40, 100])   # classes overlap at event level
    for _ in range(15):
        f0, cel, baz = centre + rng.normal(0, [0.02, 3, 4])
        X.append([f0, cel, np.sin(np.deg2rad(baz)), np.cos(np.deg2rad(baz))])
        y.append(k); g.append(e)
X, y, g = np.array(X), np.array(y), np.array(g)
clf = RandomForestClassifier(200, random_state=0, n_jobs=1)
rand = cross_val_score(clf, X, y, cv=KFold(5, shuffle=True, random_state=0),
                       scoring="balanced_accuracy")
grp = cross_val_score(clf, X, y, cv=GroupKFold(5), groups=g, scoring="balanced_accuracy")
print(f"balanced accuracy: random split {rand.mean():.3f}, grouped by event {grp.mean():.3f}")

The random split reports a balanced accuracy of about 0.99, the split by event about 0.73. Only the second number predicts the accuracy on next winter’s events.

5. Sensor and IoT time series: preparation, segmentation, anomalies, prediction

Communicating sensors measure drinking water, energy, traffic and waste continuously. Their raw output is rarely ready for learning: timestamps are irregular, values are cumulative, some are missing or wrong. This chapter follows the sequence of the official themes: preparation, segmentation, anomaly detection, prediction.

5.1 Preparing sensor data

Typical defects are irregular or missing transmissions, clock drift, counter resets, quantisation (one pulse per litre), stuck values, outliers and duplicated messages. Preparation sorts and deduplicates, corrects resets, resamples to a regular grid, and flags gaps rather than filling them silently. Unstructured data (images of meters, operator notes) exist too, but this chapter, like the official course, focuses on time series.

Averaging a flow over each grid interval has no such guarantee when transmissions are missing, which is why counters are preferred for water and energy billing.

import numpy as np
import pandas as pd

rng = np.random.default_rng(8)
fine = np.arange(0, 72, 0.01)                              # hours, fine time grid
rate = 20 + 25 * np.exp(-((fine % 24 - 8) / 2) ** 2) + 30 * np.exp(-((fine % 24 - 20) / 2) ** 2)
V = np.cumsum(rate) * 0.01                                 # true counter, litres
hrs = np.sort(rng.uniform(0, 72, 60))                      # irregular reading times
hrs = hrs[(hrs < 30) | (hrs > 42)]                         # 12 h without transmission
idx = np.interp(hrs, fine, V)                              # transmitted counter values
grid = np.arange(np.ceil(hrs[0]), np.floor(hrs[-1]) + 1)   # whole hours inside the record
v = np.interp(grid, hrs, idx)
hourly = pd.Series(np.diff(v), index=pd.Timestamp("2027-01-04") + pd.to_timedelta(grid[1:], unit="h"))
print(f"{hourly.size} hours, min {hourly.min():.1f} L, total {hourly.sum():.0f} L, "
      f"counter change {v[-1] - v[0]:.0f} L")
gap = hourly["2027-01-05 08:00":"2027-01-05 17:00"]
true_gap = np.diff(np.interp(np.arange(31, 42), fine, V))
print(f"in the gap: estimated {gap.min():.1f} to {gap.max():.1f} L/h, "
      f"true {true_gap.min():.1f} to {true_gap.max():.1f} L/h")

The total is exact, but inside the gap the estimate is a flat 27 L/h while the true use swung between 20 and 43 L/h. The volume is right; its timing is invented, so those hours must be flagged.

5.2 Segmentation

A consumption series is a succession of regimes: occupied or empty building, normal network or network with a leak. Segmentation finds the times of change.

With cumulative sums of y and y^2, each quadratic cost takes O(1). For Gaussian noise of known variance \sigma^2, a common penalty is \beta = 2\sigma^2 \log n. The pruning that makes the algorithm linear in practice (PELT) is in ch. 7.

import numpy as np

def segment(y, beta):
    n = y.size
    c1 = np.concatenate([[0], np.cumsum(y)]); c2 = np.concatenate([[0], np.cumsum(y**2)])
    cost = lambda a, b: c2[b] - c2[a] - (c1[b] - c1[a]) ** 2 / (b - a)
    F = np.full(n + 1, np.inf); F[0] = -beta; last = np.zeros(n + 1, int)
    for t in range(1, n + 1):
        vals = [F[s] + cost(s, t) + beta for s in range(t)]
        last[t] = int(np.argmin(vals)); F[t] = vals[last[t]]
    cps, t = [], n
    while t > 0:
        t = last[t]; cps.append(t)
    return sorted(cps)[1:]

rng = np.random.default_rng(9)
mean = np.r_[np.full(90, 12.0), np.full(60, 7.0), np.full(100, 12.5), np.full(50, 15.5)]
y = mean + rng.normal(0, 1.2, mean.size)        # daily consumption, m3
print("change points:", [int(c) for c in segment(y, 2 * 1.2**2 * np.log(y.size))],
      "(true: 90, 150, 250)")

The third change is placed at 256, not 250: the first six days of the new regime happen to be low, and a change point is located only as precisely as the noise allows.

5.3 Anomaly detection

Two kinds of anomaly matter. Point anomalies are isolated wrong or unusual values; collective anomalies are lasting shifts, such as a new leak.

A leak adds a roughly constant flow. It is best seen in the minimum night flow of a district, around 3 a.m., when legitimate use is lowest (Puust and coauthors, 2010). Point tests miss a small shift; accumulating evidence finds it.

After a shift of \delta > k, the increments of S_t have mean \delta - k > 0, so S_t climbs like a random walk with drift. The expected delay is roughly h / (\delta - k), and Wald’s identity makes this precise up to the overshoot. Before the shift the drift is -k < 0, so false alarms are rare when h is a few standard deviations.

import numpy as np

rng = np.random.default_rng(11)
mnf = 8.0 + 0.8 * rng.standard_normal(120)       # minimum night flow, m3/h, 120 nights
mnf[80:] += 1.0                                  # a leak of 1 m3/h appears on night 80
mu0, sd = np.median(mnf[:60]), 1.4826 * np.median(np.abs(mnf[:60] - np.median(mnf[:60])))
k, h = 0.5, 5 * sd                               # k = delta / 2
S, alarm = 0.0, None
for t, x in enumerate(mnf):
    S = max(0.0, S + x - mu0 - k)
    if S > h and alarm is None:
        alarm = t
r = (mnf - mu0) / sd
print(f"CUSUM alarm on night {alarm}; nights with |robust z| > 3.5: {np.flatnonzero(np.abs(r) > 3.5)}")

The robust z-score flags a single night, 37 nights after the leak started. The CUSUM alarms on night 89, close to the predicted delay h/(\delta - k) \approx 8 nights.

5.4 Forecasting and its evaluation

Forecasts of demand let a utility pump water when electricity is cheap and low-carbon, and size reserves. A forecast is useful only if it beats a simple baseline.

Evaluation uses a rolling origin: train on the past up to a date, forecast the next day, move the date forward, and average the errors. A random split of hours would leak the future into the past.

import numpy as np
from sklearn.linear_model import Ridge

rng = np.random.default_rng(11)
H = 24 * 7 * 20                                  # 20 weeks of hourly demand
t = np.arange(H); hour, dow = t % 24, (t // 24) % 7
temp = 15 + 8 * np.sin(2 * np.pi * t / (24 * 365)) + 4 * np.sin(2 * np.pi * (hour - 9) / 24) \
       + rng.normal(0, 1.5, H)
y = 100 + 30 * np.exp(-((hour - 8) / 2.0) ** 2) + 25 * np.exp(-((hour - 20) / 2.5) ** 2) \
    - 15 * (dow >= 5) + 1.5 * np.maximum(temp - 18, 0) + rng.normal(0, 5, H)

def features(i):
    return np.r_[np.eye(24)[hour[i]], np.eye(7)[dow[i]], max(temp[i] - 18, 0), y[i - 168]]

mae_sn, mae_ml, scale = [], [], []
for origin in range(H - 28 * 24, H, 24):         # 28 rolling origins, one day ahead
    tr = np.arange(168, origin); te = np.arange(origin, origin + 24)
    model = Ridge(alpha=1.0).fit(np.array([features(i) for i in tr]), y[tr])
    pred = model.predict(np.array([features(i) for i in te]))
    mae_ml.append(np.abs(pred - y[te]).mean())
    mae_sn.append(np.abs(y[te - 168] - y[te]).mean())
    scale.append(np.abs(y[tr] - y[tr - 168]).mean())
print(f"MAE seasonal naive {np.mean(mae_sn):.2f}, ridge {np.mean(mae_ml):.2f}, "
      f"MASE ridge {np.mean(np.array(mae_ml) / np.array(scale)):.2f}")

Interval forecasts, which a utility needs to size its reserves, come from the quantile loss of ch. 7.

6. AI for a smaller footprint: overview and the water cycle

Chapters 1–2 counted the cost of AI; this chapter turns to its benefits. It surveys where machine learning saves resources, then follows the third case study, sensor data of the large water cycle, from the sensor to a forecast and a net balance; the decision that a forecast must change is in ch. 7.

6.1 Overview: machine learning for a smaller footprint

Rolnick and coauthors (2022) survey these applications. All of them are second-order effects in the sense of Definition 1.1: they are real only if they change what happens, and they can be eaten by rebound (Proposition 1.8). Section 6.4 turns this into a balance.

6.2 The large water cycle and its sensors

The large cycle is observed by rain gauges, weather radar, river stage (water level) gauges and groundwater piezometers. Discharge, the volume per second that matters for floods and droughts, is rarely measured continuously. It is derived from the stage through a rating curve fitted on occasional manual gaugings, Q = a\, (h - h_0)^b, with h_0 the stage of zero flow.

import numpy as np

rng = np.random.default_rng(13)
h = np.sort(rng.uniform(0.4, 2.5, 25))                  # stages of 25 gaugings, m
Q = 12 * (h - 0.2) ** 1.6 * np.exp(0.05 * rng.standard_normal(h.size))   # m3/s

def fit(h0):
    X = np.column_stack([np.ones(h.size), np.log(h - h0)])
    beta, rss, *_ = np.linalg.lstsq(X, np.log(Q), rcond=None)
    return float(rss[0]), np.exp(beta[0]), beta[1]

grid = np.linspace(-0.3, 0.38, 69)
h0 = grid[np.argmin([fit(g)[0] for g in grid])]
_, a, b = fit(h0)
print(f"h0 = {h0:.2f} m, a = {a:.1f}, b = {b:.2f}")
for g in (h0 - 0.15, h0, h0 + 0.1):
    _, a_g, b_g = fit(g)
    print(f"h0 = {g:.2f}: Q(4.5 m) = {a_g * (4.5 - g) ** b_g:.0f} m3/s")

The three curves fit the gaugings with a scatter of 4 to 8 %, yet at a flood stage of 4.5 m, far above the highest gauging, they give 104 to 150 m³/s. Extrapolation is the main uncertainty of flood discharges.

6.3 Rainfall–runoff forecasting

The simplest model of a catchment is a linear reservoir: storage S fills with rain P and empties at the rate Q = S/k, so that dS/dt = P - S/k.

Recurrent networks trained on many catchments, such as the LSTM of Kratzert and coauthors (2018), rival calibrated conceptual models; the linear model below is the baseline they must beat.

import numpy as np

rng = np.random.default_rng(12)
n = 6 * 365                                            # six years of daily data
P = np.where(rng.random(n) < 0.35, rng.gamma(0.8, 8.0, n), 0.0)   # rain, mm/day
phi = np.exp(-1 / np.array([2.0, 30.0]))               # fast and slow reservoirs (k = 2 and 30 days)
share = np.array([0.15, 0.25])                         # runoff coefficients of the two paths
Qr = np.zeros((n, 2))
for t in range(1, n):
    Qr[t] = phi * Qr[t - 1] + (1 - phi) * share * P[t]
Q = Qr.sum(1) * np.exp(0.1 * rng.standard_normal(n))   # observed discharge, mm/day

def nse(obs, sim):
    return 1 - ((obs - sim) ** 2).sum() / ((obs - obs.mean()) ** 2).sum()

X = np.column_stack([Q[1:-1], Q[:-2], P[2:], P[1:-1]])  # Q_{t-1}, Q_{t-2}, P_t, P_{t-1}
yv, cut = Q[2:], 4 * 365                                # train on 4 years, test on 2
coef = np.linalg.lstsq(X[:cut], yv[:cut], rcond=None)[0]
print("AR coefficients:", np.round(coef[:2], 3), " noise-free theory:",
      np.round([phi.sum(), -phi.prod()], 3))
print(f"test NSE: ARX {nse(yv[cut:], X[cut:] @ coef):.3f}, "
      f"persistence {nse(yv[cut:], Q[1:-1][cut:]):.3f}")

Driven by the observed rain of the day, the one-day-ahead ARX forecast reaches an NSE of about 0.92 on the two test years, against 0.34 for persistence. Its coefficients are far from the theoretical ones: noisy lagged discharges bias least squares (errors in variables). A good forecaster need not identify the physics.

The same residuals validate sensors: a gauge that drifts away from the forecast of its neighbours, with a large robust z-score or a CUSUM alarm (ch. 5.3), is more likely faulty than the river.

6.4 The net balance

Illustrative numbers for leak detection in a district (ch. 5.3). Twenty leaks a year, each of 2 m³/h, found ten days earlier, save 9600 m³ of water. At 0.5 kWh/m³ for production and distribution, this avoids 4800 kWh, that is 288 kg CO₂e at 60 g CO₂e/kWh.

On the other side, fifty loggers of 10 kg CO₂e each over ten years, plus 100 kWh a year of transmission and computation, cost about 56 kg CO₂e a year. The carbon balance is positive, about 232 kg CO₂e a year. The main benefit, 9600 m³ of water, lies in another category and is reported separately.

7. Complements: sensitivity, EOFs, exact segmentation, quantiles and decisions

This self-study chapter gives the proofs and tools that the sessions only cite. Each section depends on one earlier chapter, named in its first line, and can be read on its own.

7.1 Sampling a power trace

This section completes ch. 2.1: how wrong is an energy computed from sampled power?

The bound fails for power traces with sharp steps, where P'' is unbounded: short spikes between two samples are simply missed. This is why ch. 2.1 prefers cumulative energy counters, which integrate in hardware.

7.2 Variance-based sensitivity indices

Chapter 2.5 ranked the factors of a footprint by the share of variance they explain. This section defines that share and gives a simple estimator.

For a non-additive model the first-order indices sum to less than 1; the remainder measures interactions (Sobol’, 1993).

The pick-freeze estimator uses two independent samples A and B of the inputs, and the matrix A_B^{(j)} equal to A except for column j, taken from B.

Since f(B) and f(A_B^{(j)}) share only X_j, their covariance is the numerator of S_j (Saltelli and coauthors, 2010).

import numpy as np

rng = np.random.default_rng(1)
N, gpu_hours = 100_000, 1000.0
med = np.array([0.40, 1.2, 0.06, 0.010])          # the model of ch. 2.5
sig = np.array([0.05, 0.10, 0.40, 0.50])
draw = lambda: np.exp(np.log(med) + sig * rng.standard_normal((N, 4)))
f = lambda X: gpu_hours * (X[:, 0] * X[:, 1] * X[:, 2] + X[:, 3])
A, B = draw(), draw()
fA, fB = f(A), f(B)
V = np.var(np.r_[fA, fB])
for j, name in enumerate(["GPU power", "PUE", "carbon int.", "embodied"]):
    ABj = A.copy(); ABj[:, j] = B[:, j]           # A with column j taken from B
    Sj = np.mean(fB * (f(ABj) - fA)) / V          # Saltelli (2010) estimator
    print(f"{name:12s} S = {Sj:.2f}")
logs = sig ** 2 / (sig ** 2).sum()
print("log-scale shares of the operational product (exact):", np.round(logs[:3] / logs[:3].sum(), 2))

The indices sum to about 1, so interactions are weak in this model. They are slightly higher than the binned estimates of ch. 2.5, which lose the variation inside each bin.

7.3 Empirical orthogonal functions

This section completes ch. 3. Empirical orthogonal functions (EOFs), the climate name of principal component analysis, summarise a field of anomalies (Definition 3.4) by a few spatial patterns and their time series.

So the first k EOFs are the best k-dimensional summary of the field. The leading EOF of tropical Pacific sea-surface temperature anomalies, for example, is the El Niño pattern. EOFs are statistical modes, not physical ones: orthogonality can mix two physical patterns into one EOF (Hannachi, Jolliffe and Stephenson, 2007).

import numpy as np

rng = np.random.default_rng(4)
lat = np.linspace(-60, 60, 25); lon = np.linspace(0, 350, 36)
LA, LO = np.meshgrid(lat, lon, indexing="ij")
w = np.sqrt(np.cos(np.deg2rad(LA))).ravel()
p1 = np.exp(-((LA / 20) ** 2 + ((LO - 200) / 50) ** 2)).ravel()   # tropical "El Nino-like" blob
p2 = np.sin(np.deg2rad(LA) * 3).ravel()                           # meridional dipole
n = 480                                                           # 40 years of months
pc1 = np.convolve(rng.standard_normal(n + 11), np.ones(12) / 12, "valid") * 4   # slow index
pc2 = 0.5 * rng.standard_normal(n)
F = np.outer(pc1, p1) + np.outer(pc2, p2) + 0.3 * rng.standard_normal((n, p1.size))
X = (F - F.mean(0)) * w
U, s, Vt = np.linalg.svd(X, full_matrices=False)
print("explained variance of EOFs 1-4:", np.round(s[:4] ** 2 / (s ** 2).sum(), 3))
for k in range(2):
    c = [abs(np.corrcoef(Vt[k], w * p)[0, 1]) for p in (p1, p2)]
    print(f"EOF {k + 1}: |corr| with blob {c[0]:.2f}, with dipole {c[1]:.2f}")

The dipole, spread over the whole grid, comes first although its time series has the smaller amplitude: EOFs rank patterns by explained variance, not by physical importance.

7.4 Exact segmentation in linear time: PELT

Optimal partitioning (Theorem 5.4) costs O(n^2). Killick, Fearnhead and Eckley (2012) showed that most candidate change points can be discarded for good without losing optimality.

The quadratic cost satisfies the assumption with K = 0, because splitting a segment never increases the residual sum of squares. When the number of changes grows linearly with n, the expected cost of PELT is O(n) under conditions given by the authors.

import time
import numpy as np

def make_cost(y):
    c1 = np.r_[0, np.cumsum(y)]; c2 = np.r_[0, np.cumsum(y ** 2)]
    return lambda s, t: c2[t] - c2[s] - (c1[t] - c1[s]) ** 2 / (t - s)   # s: array of starts

def op(y, beta, prune=False):
    n, cost = y.size, make_cost(y)
    F = np.empty(n + 1); F[0] = -beta; last = np.zeros(n + 1, int)
    cand, evals = np.array([0]), 0
    for t in range(1, n + 1):
        s = cand if prune else np.arange(t)
        v = F[s] + cost(s, t) + beta; evals += s.size
        j = np.argmin(v); F[t], last[t] = v[j], s[j]
        if prune:                                   # PELT rule with K = 0
            cand = np.r_[s[F[s] + cost(s, t) <= F[t]], t]
    cps, t = [], n
    while t > 0:
        t = last[t]; cps.append(int(t))
    return sorted(cps)[1:], evals

rng = np.random.default_rng(14)
n = 4000
mean = np.repeat(rng.normal(10, 3, 40), n // 40)    # 39 true changes, every 100 points
y = mean + rng.normal(0, 1.0, n)
beta = 2 * np.log(n)
for prune in (False, True):
    t0 = time.perf_counter(); cps, ev = op(y, beta, prune)
    print(f"{'PELT' if prune else 'OP  '}: {len(cps)} changes, {ev:,} cost evaluations, "
          f"{time.perf_counter() - t0:.2f} s")
print("identical segmentations:", op(y, beta)[0] == op(y, beta, True)[0])

Both versions return the same 37 change points, but pruning cuts the number of cost evaluations by a factor of about 30. Truong, Oudre and Vayatis (2020) review this and other segmentation methods.

7.5 How good is CUSUM?

Chapter 5.3 chose k = \delta/2 by intuition. Likelihood ratios explain the choice and its optimality.

Moustakides (1986) proved that the likelihood-ratio CUSUM minimises Lorden’s worst-case delay among all stopping rules with the same \mathrm{ARL}_0. The threshold h is set from the false-alarm rate one can afford, then the delay follows: about h/(\delta - k) steps, as in ch. 5.3.

7.6 Quantile forecasts and conformal intervals

This section completes the forecasts of ch. 5.4. A reserve is sized on a high quantile of demand, not on its mean, and quantiles are learnt with a loss of their own.

This is quantile regression (Koenker and Bassett, 1978): any model trained with \rho_\tau estimates a conditional quantile. A 90 % interval uses \tau = 0.05 and \tau = 0.95. Its coverage must be checked on test data, and it is often too low. Split conformal prediction corrects it.

Time series are not exchangeable, so the guarantee is only approximate here; calibrating on the most recent weeks keeps it close.

import numpy as np
from sklearn.ensemble import HistGradientBoostingRegressor

rng = np.random.default_rng(15)
H = 24 * 7 * 20
t = np.arange(H); hour, dow = t % 24, (t // 24) % 7
base = 100 + 30 * np.exp(-((hour - 8) / 2.0) ** 2) + 25 * np.exp(-((hour - 20) / 2.5) ** 2) - 15 * (dow >= 5)
y = base + (2 + 0.08 * (base - 85)) * rng.standard_normal(H)      # noise grows with demand
X = np.column_stack([hour, dow])
tr, ca, te = np.arange(0, 12 * 168), np.arange(12 * 168, 16 * 168), np.arange(16 * 168, H)
lo, hi = [HistGradientBoostingRegressor(loss="quantile", quantile=q, max_iter=200,
                                        random_state=0).fit(X[tr], y[tr]) for q in (0.05, 0.95)]
def cover(Q):
    l, h = lo.predict(X[te]) - Q, hi.predict(X[te]) + Q
    return np.mean((y[te] >= l) & (y[te] <= h)), np.mean(h - l)
E = np.maximum(lo.predict(X[ca]) - y[ca], y[ca] - hi.predict(X[ca]))   # conformity scores
k = int(np.ceil((ca.size + 1) * 0.9))
Q = np.sort(E)[k - 1]
for name, q in [("quantile model", 0.0), ("conformalised", Q)]:
    c, w = cover(q)
    print(f"{name:15s}: coverage {c:.3f}, mean width {w:.1f}")

The raw quantile model covers only about 81 % of the test hours; after conformal calibration the coverage is about 90 %, at the price of wider intervals.

7.7 From forecast to decision

A forecast (ch. 5.4, ch. 6.3) saves resources only if it changes a decision: close a flood barrier, release water from a dam, restrict abstraction.

8. Research directions and open problems

The three parts of the course are active research areas. This self-study chapter points to the questions that remain open, with work one can start from.

8.1 Measuring and reporting the footprint of AI

Strubell, Ganesh and McCallum (2019) drew wide attention to the energy of training language models. Patterson and coauthors (2021) then showed how strongly such estimates depend on the hardware, the data centre and the grid, and argued that some earlier figures were much too high. The lesson of ch. 2 follows: a footprint without its assumptions cannot be compared.

• Luccioni and Hernandez-Garcia (2023) surveyed the reported emissions of 95 models and found the carbon intensity of the energy to be the main source of variation.
• Dodge and coauthors (2022) measured the carbon intensity of cloud instances and showed that the choice of region and of the time of day can cut emissions substantially.
• Inference at scale (Luccioni, Jernite and Strubell, 2024), embodied emissions of accelerators (Gupta and coauthors, 2021; Wu and coauthors, 2022) and water (Li and coauthors, 2023) remain poorly documented, mostly for lack of manufacturer and operator data.

Open problems. A common, auditable method for the footprint of a model across its whole life; per-query inference measurements for deployed services; manufacturing inventories of accelerators.

8.2 Indirect and systemic effects

Kaack and coauthors (2022) argue that the largest effects of machine learning on climate are not its own emissions but those of its applications and of the system-level changes it causes. Quantitative evidence for these effects is scarce. Ligozat and coauthors (2022) apply life cycle assessment to AI solutions designed for the environment, and show that their own footprint and their indirect effects are rarely counted.

Open problems. A consequential assessment (Definition 1.7) of the adoption of an AI technology in a whole sector; measuring rebound (Proposition 1.8) for digital services, whose demand elasticities are unknown.

8.3 Machine learning for weather and climate

Data-driven medium-range weather models trained on reanalyses now compete with physics-based forecasts: GraphCast (Lam and coauthors, 2023) and Pangu-Weather (Bi and coauthors, 2023). Hybrid models that keep a differentiable dynamical core and learn the rest, such as NeuralGCM (Kochkov and coauthors, 2024), aim at climate time scales. Emulators of climate models are benchmarked by ClimateBench (Watson-Parris and coauthors, 2022).

Open problems. Stability and physical consistency over decades; skill on rare extremes, which are by definition under-represented in training data; extrapolation to a climate warmer than any training year (ch. 3.4).

8.4 Hydrology and ungauged basins

Kratzert and coauthors (2019) trained an LSTM on 531 US catchments and showed that, in catchments it has never seen, it predicts discharge better than a conceptual model (SAC-SMA) calibrated on each catchment and than the US National Water Model. Nearing and coauthors (2024) extended this to global forecasts of extreme floods, reporting reliability up to five days ahead in ungauged watersheds comparable to that of a current global system at zero lead time.

Open problems. Uncertainty that remains calibrated in regions with few gauges; non-stationarity from land-use and climate change; rating-curve errors (ch. 6.2) in the training targets themselves.

8.5 Environmental signals with few labels

Infrasound arrays, hydrophones and seismometers in polar regions record far more than experts can label, and stations differ in geometry and noise (ch. 4). Learning representations that transfer from one station to another, with few labels and physically meaningful features such as back-azimuth and apparent speed, is an open problem shared with many environmental sensors.

Open problems. Transfer between stations and seasons; evaluation by event rather than by detection; detection of rare sources against a changing noise background.

Exercises

Levels: ★ apply the course, ★★ combine or prove a short result, ★★★ open-ended. The session plan says which exercises follow which session.

1. ★ From GPU-days to carbon. A model is trained on 8 GPUs drawing 300 W on average for 5 days, in a data centre of PUE 1.4. Compute the IT energy, the facility energy, and the emissions on grids at 50 and 450 g CO₂e/kWh (ch. 0.1, ch. 2.2).

   Solution: the IT energy is 8 \times 0.3 \times 120 = 288 kWh and the facility energy 288 \times 1.4 = 403.2 kWh. The emissions are 403.2 \times 0.05 \approx 20.2\ \text{kg}, \qquad 403.2 \times 0.45 \approx 181.4\ \text{kg CO}_2\text{e}. The grid changes the result by a factor of 9; the embodied part is not included.

2. ★ A two-process inventory. Electricity is made with 0.05 kWh of own use per kWh and emits 0.08 kg CO₂e per kWh. One server-hour uses 0.5 kWh and carries 0.02 kg CO₂e of embodied emissions. Compute the activity levels and the emissions for 100 server-hours with Theorem 1.4, then check the result with the series \sum_k M^k.

   Solution: the technology matrix has columns (0.95, 0) for electricity and (-0.5, 1) for the server-hour. Solving As = (0, 100) gives s_2 = 100 and 0.95\, s_1 = 50, so s_1 \approx 52.6\ \text{kWh}, \qquad g = 0.08 \times 52.6 + 0.02 \times 100 \approx 6.21\ \text{kg CO}_2\text{e}. With M having entries 0.05 and 0.5 in its first row, the series gives first-row entries 1/0.95 \approx 1.053 and 0.5/0.95 \approx 0.526, the first row of A^{-1}.

3. ★ Embodied share of a GPU server. A server has a manufacturing footprint of 2500 kg CO₂e, a life of 5 years, a utilisation of 60 %, and draws 3 kW in a facility of PUE 1.2. Compute its embodied rate and its embodied share at 50 and at 500 g CO₂e/kWh (Proposition 1.6).

   Solution: the server is used 0.6 \times 43\,800 = 26\,280 hours, so the embodied rate is 0.095 kg/h. The operational rate is 3 \times 1.2 \times \mathrm{CI}, that is 0.18 kg/h or 1.8 kg/h. The embodied shares are \eta \approx 0.35 \quad \text{and} \quad \eta \approx 0.05.

4. ★ When does inference win? Training a model used 50 MWh; one query costs 0.5 Wh; the service answers two million queries a day. After how many queries, and how many days, does inference dominate (Proposition 2.4)?

   Solution: N^\star = 5 \times 10^7 / 0.5 = 10^8 queries, reached after 50 days. Over a year of service, inference uses about 7 times the training energy.

5. ★ An area-weighted mean. Three latitude bands of equal width are centred at 0°, 45° and 75° and have temperature anomalies 0.8, 1.2 and 2.5 °C. Compare the plain mean with the area-weighted mean (Proposition 3.2).

   Solution: the plain mean is 1.5 °C. With weights \cos 0^\circ = 1, \cos 45^\circ \approx 0.707 and \cos 75^\circ \approx 0.259, \bar x = \frac{0.8 + 0.849 + 0.647}{1.966} \approx 1.17\,^\circ\text{C}. The plain mean overweights the fast-warming high latitudes.

6. ★ Robust z-scores by hand. Nine night flows (m³/h) are 7.9, 8.1, 8.0, 7.8, 8.3, 8.0, 12.5, 8.2, 7.9. Compute the median, the MAD and the robust z-score of 12.5; compare with the usual z-score (Definition 5.5).

   Solution: the median is 8.0 and the MAD 0.1, so the robust score of 12.5 is 4.5 / 0.148 \approx 30. The mean is 8.52 and the standard deviation 1.50, inflated by the outlier itself, so the usual z-score is only 2.65 and would miss it at a threshold of 3.

7. ★ CUSUM by hand. With \mu_0 = 8, k = 0.5 and h = 2, run the CUSUM of Definition 5.7 on 8.2, 7.6, 8.9, 9.1, 8.7, 9.4. When is the alarm raised?

   Solution: the increments x_t - 8.5 are -0.3, -0.9, 0.4, 0.6, 0.2, 0.9. The statistic is S = 0,\ 0,\ 0.4,\ 1.0,\ 1.2,\ 2.1, so the alarm is raised at the sixth observation. No single value is far from 8, but the sum of small excesses is.

8. ★ Persistence on a flashy river. Observed daily discharges are 2, 4, 9, 5, 3 and the previous day was 2. Compute the NSE of the persistence forecast (Definition 6.4).

   Solution: the forecasts are 2, 2, 4, 9, 5, the errors 0, 2, 5, −4, −2, with squares summing to 49. The observed mean is 4.6 and the sum of squared deviations 29.2. Hence \mathrm{NSE} = 1 - 49/29.2 \approx -0.68, worse than predicting the mean: persistence lags every peak.

9. ★★ Rebound for small gains. Using Proposition 1.8, show that the rebound tends to the elasticity \varepsilon when the efficiency gain r tends to 0. Compute R for r = 0.2 with \varepsilon = 0.3 and with \varepsilon = 1.2.

   Solution: to first order, (1 - r)^{1 - \varepsilon} \approx 1 - (1 - \varepsilon) r, so the saving is (1 - \varepsilon) r and R \to \varepsilon.

   Numerically, 0.8^{0.7} \approx 0.855 gives R \approx 0.28. For \varepsilon = 1.2, 0.8^{-0.2} \approx 1.046, so energy rises by 4.6 % and R \approx 1.23 > 1 \quad \text{(backfire)}.

10. ★★ Attributional or consequential? A data centre adds a constant 10 MW load to a grid whose average intensity is 60 g CO₂e/kWh. In the short term the extra demand is met by gas plants at 450 g CO₂e/kWh. Compute the yearly emissions in both views. Which number belongs in the company’s annual report, and which in the decision to build the data centre (Definition 1.7)?

    Solution: the load uses 10 \times 8760 = 87\,600 MWh a year. The attributional footprint is about 5 260 t CO₂e and the short-term consequential one about 39 400 t. The report allocates existing burdens, so it uses the attributional figure. The siting decision changes the system, so it calls for the consequential figure. In the long term, the margin may be new wind or solar capacity, so the time horizon must be stated.

11. ★★ A trend that is not significant. Annual anomalies over 30 years have AR(1) residuals with standard deviation 0.15 °C and \phi = 0.5. The fitted slope is 0.010 °C per year. Test it at the 5 % level with the white-noise and with the AR(1) standard errors (Proposition 3.5). What is the effective sample size?

    Solution: the white-noise standard error is \sqrt{12 \times 0.0225 / (30 \times 899)} \approx 0.0032, so t \approx 3.2, significant. The AR(1) factor (1 + \phi)/(1 - \phi) = 3 multiplies the variance, giving a standard error of 0.0055 and t \approx 1.8 < 1.96, not significant. The effective sample size is 30 \times 0.5/1.5 = 10 years.

12. ★★ Slowness from three sensors. Sensors sit at (0, 0), (1000, 0) and (0, 1000) m (east, north). The delays of sensors 2 and 3 after sensor 1 are -1.47 s and -2.55 s. Find the slowness vector, the apparent speed and the back-azimuth (Proposition 4.3). Then show that three sensors on a line cannot give the back-azimuth.

    Solution: here D is 1000 times the identity, so u = (-1.47, -2.55) \times 10^{-3} s/m. Then c = 1/\|u\| \approx 340\ \text{m/s}, \qquad \theta = \operatorname{atan2}(1.47, 2.55) \approx 30^\circ. For collinear sensors every row of D is a multiple of the same vector e, so D has rank 1 and the component of u along e^\perp is not identifiable. The array cannot tell a wave from one side of the line from its mirror image.

13. ★★ Measuring leakage (Python). Modify the code of ch. 4.4: vary the event-level spread of the features (multiply it by 0.5, 1, 2) and the number of detections per event (5, 15, 40). Plot the gap between the random-split and the grouped balanced accuracy.

    Hints and expected result: the grouped score falls as the event-level spread grows, because the classes overlap more, while the random-split score stays near 0.99, because every test detection has near-copies in training. The gap grows with both the spread and the number of detections per event. Only the grouped score is a fair estimate.

14. ★★ Optimal partitioning by hand. Apply the recursion of Theorem 5.4 to y = (1, 1, 5, 5, 5) with the quadratic cost and \beta = 1. Give F(1), \dots, F(5) and the optimal segmentation.

    Solution: F(0) = -1, then F(1) = 0 and F(2) = 0 (one segment).

    For t = 3 the best last change is s = 2, so F(3) = 0 + 0 + 1 = 1; the alternatives cost 9 and 10.7.

    Likewise F(4) = 1 and F(5) = 1, each with s = 2. The optimum is one change after the second point: two segments of zero cost plus one penalty.

15. ★★ Carbon-aware scheduling (Python). A flexible 6-hour job uses 120 kWh. The grid intensity follows a daily profile with a solar dip at midday and an evening peak. Find the start time that minimises the emissions, and compare with a start at midnight.

    Solution:

    import numpy as np
    
    h = np.arange(48)                                     # two days, hourly
    ci = 300 - 150 * np.clip(np.sin(np.pi * ((h % 24) - 7) / 12), 0, None) \
         + 60 * np.exp(-(((h % 24) - 19) / 2.0) ** 2)    # g CO2e/kWh: solar dip, evening peak
    E_job, L = 120.0, 6                                   # kWh, spread evenly over 6 hours
    cost = np.array([E_job / L * ci[s:s + L].sum() / 1000 for s in range(24)])   # start in day 1
    print(f"start at 00:00 -> {cost[0]:.1f} kg; best start {np.argmin(cost):02d}:00 -> {cost.min():.1f} kg; "
          f"worst start {np.argmax(cost):02d}:00 -> {cost.max():.1f} kg")

    The best start is 10:00 with about 19.9 kg CO₂e, against 36.0 kg at midnight and 38.9 kg at 18:00. Shifting in time halves the emissions here, without any change to the job: this is the lever measured by Dodge and coauthors (2022).

16. ★★ Who should use a yes/no flood warning? A warning has hit rate 0.7 and false-alarm rate 0.05 for an event of frequency 0.1. Compute its relative value (Example 7.13) for a user with C/L = 0.2 and for one with C/L = 0.02.

    Solution: per unit of L, for C/L = 0.2 the expenses are 0.1 (climatology: never protect), 0.02 (perfect) and 0.053 (warning), so the value is about 0.59. For C/L = 0.02, climatology already says “always protect” (0.02 per day). The warning costs 0.0323, so its value is (0.02 - 0.0323)/(0.02 - 0.002) \approx -0.68 < 0. A user with very cheap protection should ignore the yes/no warning; a probabilistic forecast with threshold C/L serves every user.

17. ★★★ The net balance under uncertainty (Python). Redo the leak-detection balance of ch. 6.4 with log-normal uncertainty on every factor. Estimate the probability that the net carbon effect is negative, and rank the factors with the indices of ch. 7.2.

    Solution (one possible set of assumptions):

    import numpy as np
    
    rng = np.random.default_rng(17)
    N = 200_000
    ln = lambda med, s: np.exp(np.log(med) + s * rng.standard_normal(N))
    leaks, days, flow = ln(20, 0.3), ln(10, 0.4), ln(2.0, 0.3)       # per year, days earlier, m3/h
    kwh_m3, ci = ln(0.5, 0.3), ln(0.06, 0.4)                          # kWh per m3, kg CO2e per kWh
    emb, ops = ln(10, 0.5), ln(100, 0.5)                              # kg per logger, kWh per year
    water = leaks * days * 24 * flow
    net = water * kwh_m3 * ci - (50 * emb / 10 + ops * ci)
    print(f"water saved: median {np.median(water):.0f} m3; net carbon: median {np.median(net):.0f} kg/yr, "
          f"90% interval [{np.quantile(net, .05):.0f}, {np.quantile(net, .95):.0f}], "
          f"P(net < 0) = {np.mean(net < 0):.3f}")
    factors = {"leaks": leaks, "days": days, "flow": flow, "kWh/m3": kwh_m3,
               "carbon int.": ci, "embodied": emb, "operation": ops}
    for name, x in factors.items():                                   # binned estimator of ch. 2.5
        idx = np.clip(np.digitize(x, np.quantile(x, np.linspace(0, 1, 21))) - 1, 0, 19)
        S = np.array([net[idx == b].mean() for b in range(20)]).var() / net.var()
        print(f"{name:11s} S = {S:.2f}")

    With these assumptions the median net effect is about 227 kg CO₂e a year, the 90 % interval is roughly 15 to 950 kg, and the probability of a negative balance is about 3 %.

    The first-order indices rank the days gained and the carbon intensity first (about 0.2 each), then the number of leaks, their flow and the energy per m³ (about 0.1 each). The footprint of the loggers and of their operation hardly matters (0.01 or less). The indices sum to about 0.75: the avoided emissions are a product, so interactions carry the rest.

    The answer is open: the ranking depends on the ranges chosen, which the report must justify. The water saved, about 9 600 m³, is reported separately.

18. ★★★ A critical reading. Write a two-page critical reading of Luccioni, Viguier and Ligozat (2023), the format of the written-report option of the official assessment. State the functional unit and the boundaries, classify each assumption as measured, modelled or taken from the literature, and say whether the assessment is attributional or consequential. Then say how the conclusion would change on a grid at 400 g CO₂e/kWh.

    Hints and expected result: the functional unit is the final training run; the boundaries include dynamic and idle power and the embodied emissions of the hardware, but neither the experiments before the final run nor deployment, which the paper discusses separately. The assessment is attributional (average carbon intensity of the French grid). On a 400 g grid, dynamic emissions grow about sevenfold and the embodied share falls (Proposition 1.6). A good reading ends with the one measurement that would most reduce the uncertainty.

Projects and assessment

The official assessment (master-mva.com) is 50 % continuous assessment on the labs and 50 % for a mini-project or a critical reading of a paper, graded on a written report; the Université Paris-Saclay course detail calls it non-integral continuous assessment. No written exam is announced. The lab sheets of ch. 2.6, 4 and 6 prepare the lab part; the subjects below are sized as the mini-project, and solo project 2, like Ex. 18, follows the critical-reading option. The written exam is only a self-assessment proposal.

Solo projects

1. [Solo · ≈ 20 h] Paper study: the carbon footprint of BLOOM (Luccioni, Viguier and Ligozat, 2023). Question: how robust is the conclusion that idle power and manufacturing double the footprint of training?
   • Rebuild the dynamic, idle and embodied terms from the inputs stated in the paper (Example 2.3).
   • Put a log-normal range on each factor, compute a 90 % interval and the Sobol indices (ch. 7.2), and recompute the total for three grids and for a marginal intensity.
   • Deliverables: code, a 4–6 page report, a 10-minute talk. Chapters 1, 2 and 7.
   • Success: the central total within 5 % of the published 50.5 t, an interval, a ranking of the factors, and a figure of the footprint against carbon intensity.
2. [Solo · ≈ 20 h] Critical reading: Strubell, Ganesh and McCallum (2019) against Patterson and coauthors (2021). Question: why do published estimates of the footprint of the same kind of training differ so much?
   • List every assumption of both papers (hardware, hours, PUE, carbon intensity, what is counted) and classify it as measured, modelled or borrowed.
   • Recompute one estimate of each paper from its stated assumptions, then swap the assumptions that differ, one at a time.
   • Deliverables: the written report of the official option (4–6 pages) and a short notebook. Chapters 1, 2 and 8.
   • Success: each recomputed figure within 10 % of the published one, and a table attributing the gap between the papers to named assumptions.
3. [Solo · ≈ 20 h] Paper study: LSTM rainfall–runoff models (Kratzert and coauthors, 2018). Question: does an LSTM beat the linear reservoir of Proposition 6.3 out of sample?
   • Use five catchments of the public CAMELS-GB data set (Coxon and coauthors, 2020), or the synthetic catchment of ch. 6.3 if no download is possible.
   • Fit the ARX baseline and a one-layer LSTM in simulation mode, driven by rain and temperature only; split by years (Definition 0.6).
   • Deliverables: code, a 4–6 page report, a 10-minute talk. Chapters 5 and 6.
   • Success: test NSE per catchment for both models, with the median difference and a discussion of the peaks.

Group projects

1. [Group of 2–3 · ≈ 30 h each] Environmental assessment of a machine-learning pipeline. Question: what does one thousand predictions of a small model cost, and which choices matter?
   • Train three to five models on a public data set (scikit-learn digits, or CIFAR-10 on a free GPU tier) and serve one of them for a simulated day of queries.
   • Measure energy with a software tracker and with a hardware counter or wall-plug meter where available; add PUE, carbon intensity and embodied emissions (ch. 2).
   • Split: one member measures, one builds the life cycle model and its uncertainty (ch. 1, 7.2), one benchmarks accuracy and draws the Pareto front (Definition 2.8).
   • Deliverables: code, a 6–8 page report, a 15-minute talk. Success: footprint per thousand queries with a 90 % interval, the gap between two measurement methods explained, and the break-even point of Proposition 2.4.
2. [Group of 2–3 · ≈ 30 h each] An infrasound detection and classification pipeline. Question: how well can sources be separated with array features when validation is honest?
   • Extend the simulator of ch. 4 to several sources (microbaroms, transient events, wind noise) with varying signal-to-noise ratio, back-azimuth and season.
   • Build a detector on correlation and closure, extract features, and classify detections with grouped validation.
   • Split: one member writes the simulator, one the detector and slowness estimation (Proposition 4.3), one the classifier and the evaluation.
   • Deliverables: code, a 6–8 page report, a 15-minute talk. Success: back-azimuth error below 5° at signal-to-noise ratio 1, and balanced accuracy by event with a confidence interval over folds.
3. [Group of 2–3 · ≈ 30 h each] Leaks and demand in a simulated water district. Question: which detector finds leaks earliest at a fixed false-alarm rate, and is the system worth its footprint?
   • Simulate a year of hourly district inflow with daily and weekly cycles, temperature effects, sensor gaps and leaks of random size.
   • Compare the robust z-score, CUSUM (ch. 5.3, ch. 7.5) and penalised segmentation (ch. 5.2), and forecast demand with a rolling origin (ch. 5.4).
   • Split: one member writes the simulator and the data preparation, one the detectors, one the forecasts and the net balance (ch. 6.4).
   • Deliverables: code, a 6–8 page report, a 15-minute talk. Success: detection delay against false alarms per year for each detector, a MASE below 1, and a net balance with an interval.

Written exam (proposal)

Two hours, one handwritten A4 sheet allowed, a calculator; four questions.

1. Question (5 points). Define the three orders of environmental effects of a digital technology and give one example of each for a model that forecasts electricity demand. Explain the difference between an attributional and a consequential life cycle assessment and say which one answers “should we deploy this model?”.

   Expected: Definition 1.1 with fitting examples (servers; better scheduling of power plants; rebound or changed consumption); Definition 1.7; consequential, because the question is a decision, with the time horizon stated.

2. Question (6 points). A model is trained for 2000 GPU-hours at 400 W in a data centre of PUE 1.2 on a grid at 100 g CO₂e/kWh; embodied emissions are 0.01 kg CO₂e per GPU-hour. Each query then costs 0.2 Wh at the same PUE and grid. Compute the training footprint, its embodied share, and the number of queries after which inference dominates.

   Expected: 960 kWh, hence 96 kg operational and 20 kg embodied, 116 kg in total with an embodied share of 17 %; one query costs 0.024 g, so inference dominates after about 4.8 million queries (Proposition 2.4 with the embodied part included).

3. Question (5 points). Prove that a linear reservoir dS/dt = P - S/k, with Q = S/k and rain constant over each step \Delta, gives Q_t = \phi Q_{t-1} + (1 - \phi) P_t with \phi = e^{-\Delta/k}. Then show that forecasting the observed mean gives an NSE of 0.

   Expected: solve the linear equation on one step (Proposition 6.3); substitute \hat Q_t = \bar Q in Definition 6.4.

4. Question (4 points). A district’s minimum night flow has in-control mean 6 m³/h and standard deviation 0.5 m³/h. Choose the CUSUM parameters to detect a leak of 1 m³/h and estimate the detection delay with h = 2.5. Explain why a random split would overestimate the accuracy of a leak classifier trained on hourly data.

   Expected: k = 0.5 (Definition 5.7), delay about h/(\delta - k) = 5 nights; neighbouring hours of the same leak are near-duplicates, so the split must be by leak or by period (Definition 0.6).

Grading rubric (proposal)

References

Official references. The course pages on master-mva.com and the Université Paris-Saclay course detail list no bibliography for 2026–2027. The works below are standard references for each chapter; the two papers coauthored by an instructor (A.-L. Ligozat) come first.

Environmental impacts of AI and life cycle assessment (ch. 1–2, 8)

• Luccioni, A. S., Viguier, S. and Ligozat, A.-L. (2023). Estimating the carbon footprint of BLOOM, a 176B parameter language model. Journal of Machine Learning Research, 24(253), 1–15.
• Ligozat, A.-L., Lefèvre, J., Bugeau, A. and Combaz, J. (2022). Unraveling the hidden environmental impacts of AI solutions for environment life cycle assessment of AI solutions. Sustainability, 14(9), 5172.
• Kaack, L. H., Donti, P. L., Strubell, E., Kamiya, G., Creutzig, F. and Rolnick, D. (2022). Aligning artificial intelligence with climate change mitigation. Nature Climate Change, 12, 518–527.
• ISO 14040:2006, Environmental management — Life cycle assessment — Principles and framework, and ISO 14044:2006, Requirements and guidelines. International Organization for Standardization.
• Heijungs, R. and Suh, S. (2002). The Computational Structure of Life Cycle Assessment. Kluwer Academic Publishers.
• Guinée, J. B. (ed.) (2002). Handbook on Life Cycle Assessment: Operational Guide to the ISO Standards. Kluwer Academic Publishers.
• Ekvall, T. and Weidema, B. P. (2004). System boundaries and input data in consequential life cycle inventory analysis. The International Journal of Life Cycle Assessment, 9(3), 161–171.
• Hilty, L. M. and Aebischer, B. (2015). ICT for sustainability: an emerging research field. In ICT Innovations for Sustainability, Advances in Intelligent Systems and Computing 310, Springer.
• Jevons, W. S. (1865). The Coal Question. Macmillan, London.
• Strubell, E., Ganesh, A. and McCallum, A. (2019). Energy and policy considerations for deep learning in NLP. Proceedings of ACL 2019, 3645–3650.
• Patterson, D., Gonzalez, J., Le, Q., Liang, C., Munguia, L.-M., Rothchild, D., So, D., Texier, M. and Dean, J. (2021). Carbon emissions and large neural network training. arXiv:2104.10350.
• Schwartz, R., Dodge, J., Smith, N. A. and Etzioni, O. (2020). Green AI. Communications of the ACM, 63(12), 54–63.
• Henderson, P., Hu, J., Romoff, J., Brunskill, E., Jurafsky, D. and Pineau, J. (2020). Towards the systematic reporting of the energy and carbon footprints of machine learning. Journal of Machine Learning Research, 21(248), 1–43.
• Anthony, L. F. W., Kanding, B. and Selvan, R. (2020). Carbontracker: tracking and predicting the carbon footprint of training deep learning models. ICML Workshop on Challenges in Deploying and Monitoring Machine Learning Systems; arXiv:2007.03051.
• Lannelongue, L., Grealey, J. and Inouye, M. (2021). Green Algorithms: quantifying the carbon footprint of computation. Advanced Science, 8(12), 2100707.
• Fan, X., Weber, W.-D. and Barroso, L. A. (2007). Power provisioning for a warehouse-sized computer. Proceedings of the 34th International Symposium on Computer Architecture (ISCA 2007).
• Gupta, U., Kim, Y. G., Lee, S., Tse, J., Lee, H.-H. S., Wei, G.-Y., Brooks, D. and Wu, C.-J. (2021). Chasing carbon: the elusive environmental footprint of computing. IEEE International Symposium on High-Performance Computer Architecture (HPCA 2021).
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Climate and environmental data (ch. 3, 7.3, 8)

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Infrasound and spectral analysis (ch. 4)

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Sensor time series, segmentation, anomalies and forecasting (ch. 5, 7)

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Water cycle and decisions (ch. 6, 7.7, 8)

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