Data-driven Lake Water Quality Forecasting for Time Series with Missing Data using Machine Learning

AI summary

Overview

• Research area: Machine learning for environmental time-series forecasting, specifically forecasting lake water clarity (Secchi Disk Depth, SDD) from irregular, gap-ridden citizen-science monitoring records.
• Technical level: Intermediate. The paper relies on well-established tools (MICE imputation, ridge regression, Mean Decrease in Impurity feature ranking, normalized MAE), so no novel deep-learning architecture is required to follow it, but some familiarity with imputation and forecast evaluation metrics helps.
• Scope (one sentence): Using three decades of Maine lake monitoring records — narrowed to the 30 lakes with the fewest gaps out of 793 — the authors ask how much recent training data and how many measured predictors are minimally needed to forecast water clarity within 5% of a full-data, full-feature model.

What This Paper Is About

Volunteer-collected lake records are sparse and irregular: ice cover, bad weather, access limits, and human error leave large gaps, and sampling is often skipped entirely. That makes it hard to forecast water clarity and to give early warning of harmful algal blooms. This paper builds a simple forecasting pipeline for Secchi Disk Depth, fills the gaps statistically, and then works out a practical rule for how many recent samples and which measurements a monitoring program actually needs.

Key Contributions

1. Minimal sample size. Using a backward (recent-history-first) training protocol, the authors quantify the smallest training set that still lands within 5% of a complete-record, full-feature reference model: on average about 176 training samples per lake.
2. Minimal feature set. They rank predictors by Mean Decrease in Impurity and run greedy forward selection with a ridge forecaster, finding that a compact four-feature subset — TPEC, OXIC, SCHMIDT, and CONDUCT — matches the thirteen-feature baseline within the same 5% tolerance.
3. A unified selection rule. They introduce a joint feasibility function that simultaneously picks the shortest recent history and the smallest predictor set meeting a fixed 5% accuracy target, rather than varying history length or feature set in isolation as prior work does.
4. A cross-lake comparable metric. They replace scale-dependent MAE/RMSE with a normalized MAE (average absolute error divided by the test block's mean SDD) so lakes with very different baseline clarity can be compared fairly.

Main Findings

• Ridge regression wins among six candidates. Compared across the 30 lakes on MICE-imputed covariates, ridge achieved the best mean test performance: MAE 0.621, nMAE 0.165, R² −0.214. The next best was random forest (MAE 0.647, nMAE 0.173, R² −0.238), followed by TSMixer (0.748, 0.204, −0.578), Transformer (0.829, 0.223, −0.699), ARIMA (0.883, 0.169, −0.401), and SARIMA (0.921, 0.180, −0.577). Ridge also outperformed the ARIMA/SARIMA statistical baselines and the two deep-learning models.
• Imputation helps every model. Moving from complete-case (unimputed) data to MICE-completed covariates improved MAE, nMAE, and R² for all six candidates — for example ridge went from MAE 0.695 / nMAE 0.190 / R² −0.280 to 0.621 / 0.165 / −0.214, and ARIMA from 0.950 / 0.185 / −0.450 to 0.883 / 0.169 / −0.401.
• Diminishing returns on training history. Test nMAE falls rapidly over roughly the first 50–100 samples per lake and then flattens; beyond about 150–200 samples, added history yields limited improvement for most lakes. The mean minimal sample count across lakes is approximately 176.
• Four features are enough. The subset {TPEC, OXIC, SCHMIDT, CONDUCT} reached nMAE within 5% of full-feature performance; adding further variables gave no gain beyond the sacrificed 5%.
• The joint rule yields very small configurations. The minimally sufficient configuration had a median of n̂ = 64 recent observations (IQR 129) and a median of k̂ = 1 predictor (IQR 1). Ten of the 30 lakes had no feasible pair and reverted to the full-data/full-feature baseline.
• OXIC dominates as a lone predictor. When the joint rule returned a single-predictor model (20 lakes), OXIC was selected in 90% of those lakes and TPEC in 10% (Figure 3).
• Normalization changes the picture across lakes. The paper reports that Sabattus Pond and Upper Narrows Pond had similar test MAE (0.80 m vs. 0.74 m), but their test-set mean SDDs differed widely (2.14 m vs. 6.43 m). Normalizing gives nMAE of 0.37 for Sabattus (average absolute error is 37% of typical clarity) versus 0.12 for Upper Narrows (12%).
• Train–test gaps are mostly small. For most lakes, train MAE and nMAE were less than or equal to test values. Four lakes showed test ≤ train — Cochnewagon Pond, Woodbury Pond, Indian Pond, and Long Pond — which the authors attribute to higher variance within the training block.

Methodology in Plain English

The authors merged monitoring archives from the Maine Department of Environmental Protection and Lake Stewards of Maine using MIDAS lake IDs, producing records for 793 lakes. Chlorophyll was dropped to avoid target leakage (it closely governs SDD and acts as a proxy for it), and "SDD-to-bottom" observations were dropped because a Secchi depth cannot be determined when the disk is still visible at the bottom. They ranked lakes by how complete their data was and kept the 30 with the least missingness as a feasibility-oriented, information-rich subset.

Missing values in the predictor columns were filled with Multiple Imputation by Chained Equations (MICE), which models each gapped variable conditional on the others and iterates to produce a completed matrix. The target SDD was never imputed. The authors chose MICE over MissForest because it handles both continuous and categorical values, and they note that deep generative imputation methods brought no significant performance gain for their added computational cost.

For forecasting, they held out the most recent five years of each lake as a test block and used everything before it as the training pool — chosen because five years spans multiple seasonal cycles and matches common five-year water-quality review cycles. Rather than training on the earliest data, they train on the most recent n pre-test samples (a "backward-expanding" protocol), which keeps recent climate and weather patterns in view. They then find the smallest n whose nMAE is within 5% of the full-history nMAE.

For features, they train a reference ridge model on all thirteen features, rank features by Mean Decrease in Impurity, and greedily add them one at a time in rank order, again finding the smallest k that stays within 5% of the full-feature nMAE. Finally, they combine both ideas into a feasibility function f(n, k) that is 1 when a given (sample count, feature count) pair hits the 5% tolerance. Pairs where n < k + 1 are excluded as ill-posed. The chosen configuration is the lexicographically smallest feasible pair — minimize n first, then k — and if nothing is feasible the lake defaults to the full-data, full-feature configuration. ARIMA and SARIMA were fit with statsmodels, the deep models (TSMixerModel, TransformerModel) with Darts, and Ridge and RandomForestRegressor with scikit-learn.

Why This Matters

Impact on research. The paper reframes a common but often vague question — "how much data do we need?" — as a joint, parameter-free decision rule tied to an explicit accuracy target. It also shows that a scale-normalized metric like nMAE exposes cross-lake differences that raw MAE hides, which matters when lakes differ greatly in baseline clarity and depth. Because most prior SDD machine-learning studies were evaluated on only a few lakes, the 30-lake scope is a step toward generalizable, cross-lake conclusions.

Real-world applications:

• Citizen-science monitoring programs can use the target of roughly 64 recent samples and one predictor as a concrete sampling-effort goal instead of guessing.
• State and environmental agencies designing monitoring networks can prioritize OXIC and TPEC measurements, cutting laboratory and field costs while retaining near-full accuracy.
• Early warning for harmful algal blooms, since SDD transparency is linked to bloom risk and the tolerance-based rule gives a defensible accuracy floor for forecasts.
• Resource-limited or data-poor regions, where the four-feature model (TPEC, OXIC, SCHMIDT, CONDUCT) offers a sampling-efficient path to deploying forecasting tools.

Industry relevance. The rule translates directly into operational targets: maintain roughly n̂ most-recent observations, and prioritize only k̂ measurements per visit. If the selected predictors are sensor-friendly (temperature, conductivity), effort shifts toward instrument deployment and maintenance with dense temporal coverage; if they are laboratory analytes, fieldwork becomes targeted seasonal campaigns, supplemented by frequent low-cost clarity checks from trained volunteers. The authors note the ridge model also beat the deep learning alternatives, which favors cheap, interpretable models for this kind of deployment.

Future Directions

1. Physics-informed ML that encodes lake morphological information such as depth into the model.
2. Imputation quality testing by injecting artificial missingness to see how well MICE-recovered values hold up.
3. Causal analysis of relationships among features and water quality (SDD), moving beyond correlational feature ranking.
4. Sensitivity analysis of the 5% accuracy tolerance and the lexicographic tie-breaking rule, to quantify how stable the selected (n̂, k̂) configurations really are.
5. A leakage-safe nested train–validation–test design in which imputation is fit only on the training fold and applied to validation and test, with threshold choices made on validation and final metrics reported on the held-out test set.

The authors also flag limitations in the current study: the nMAE normalization by test-set mean SDD, MICE's Missing-At-Random assumption, MDI's tendency to bias rankings toward high-variance or high-cardinality features, and the fact that the 30-lake subset may skew toward lakes with longer records. They state they are actively expanding to additional lakes from the full database.

Target Audience

This paper is most useful to limnologists and lake researchers setting sampling protocols; environmental agencies and community-science coordinators allocating limited monitoring budgets; and machine-learning practitioners working on environmental time series with structured missingness. It is also a good fit for readers interested in applied ML where simple, interpretable models (ridge regression) outperform heavier deep-learning alternatives on small, gappy datasets.

Authors’ abstract

Volunteer-led lake monitoring yields irregular, seasonal time series with many gaps arising from ice cover, weather-related access constraints, and occasional human errors, complicating forecasting and early warning of harmful algal blooms. We study Secchi Disk Depth (SDD) forecasting on a 30-lake, data-rich subset drawn from three decades of in-situ records collected across Maine lakes. Missingness is handled via Multiple Imputation by Chained Equations (MICE), and we evaluate performance with a normalized Mean Absolute Error (nMAE) metric for cross-lake comparability. Among six candidates, ridge regression provides the best mean test performance. Using ridge regression, we then quantify the minimal sample size, showing that under a backward, recent-history protocol, the model reaches within 5% of full-history accuracy with approximately 176 training samples per lake on average. We also identify a minimal feature set, where a compact four-feature subset matches the thirteen-feature baseline within the same 5% tolerance. Bringing these results together, we introduce a joint feasibility function that identifies the minimal training history and fewest predictors sufficient to achieve the target of staying within 5% of the complete-history, full-feature baseline. In our study, meeting the 5% accuracy target required about 64 recent samples and just one predictor per lake, highlighting the practicality of targeted monitoring. Hence, our joint feasibility strategy unifies recent-history length and feature choice under a fixed accuracy target, yielding a simple, efficient rule for setting sampling effort and measurement priorities for lake researchers.