Multiple Linear Regression Outperforms Machine Learning for Monthly Temperature Forecasting across the Peruvian Altiplano

INTRODUCTION

Monthly temperature forecasts carry measurable economic weight in high-altitude agricultural regions. In the Peruvian Altiplano, where Puno lies above 3,800 m a.s.l., planting calendars, irrigation scheduling and frost protection all hinge on anticipating thermal conditions weeks ahead, and a frost outside the expected window can eliminate a season of quinoa or potato production for smallholder farmers with no insurance. The operational requirement follows directly: a forecast earns its place only if it beats what a farmer already infers from decades of seasonal experience.

The regional climate makes that requirement harder to satisfy than it first appears. Temperature in the southern Peruvian Andes couples a pronounced annual cycle with modest interannual variability, and station-based analyses report warming trends whose magnitude shifts with elevation and season (Imfeld et al., 2021; Anishchenko et al., 2026). Bolivian records across the wider Altiplano show comparable behaviour with strong spatial heterogeneity (Seiler et al., 2013), and Andean temperature departed from the global mean in its response to large-scale forcing during the early 2000s (Vuille et al., 2015; Wong et al., 2025). A model built for one Andean station therefore confronts a signal whose seasonal component is large and easy to reproduce, and whose interannual component is small and genuinely difficult to predict.

Machine learning has been applied widely to this problem, and studies in Latin America report encouraging accuracy. Feed-forward neural networks are the reference non-linear approach in forecasting (Zhang et al., 1998), although comparative work on air temperature prediction finds their advantage over simpler regression modest (Chevalier et al., 2011; Morales et al., 2025; Novak & Dvorak, 2025). Within the region, neural networks have modelled climatic factors in Nicaragua (Tirado Picado, 2024) and long short-term memory networks have supported reservoir early-warning systems in Peru (Marín Vilca & Pineda Torres, 2019). Closest to the study site, an agroclimatic prediction system coupling machine learning with Internet of Things sensors was developed for the agricultural sector of Puno (Yana Puma & Quispe Merma, 2020), and a comparison of twelve models for frost prediction across thirteen Altiplano stations found an ensemble to outperform the best statistical method by 35% (Almeida & Moreira, 2025; Torres-Cruz et al., 2025). Together these studies establish that machine learning can represent climatic relationships in the region, and they consistently report low error metrics.

What they do not establish is whether those low error metrics represent predictive information at all. Monthly mean temperature in a strongly seasonal regime is dominated by the annual cycle, so a model predicting the long-term average of the target calendar month already reproduces most of the variance; a coefficient of determination near 0.90 is thus attainable by a model that has learned nothing beyond seasonality. Reported without a climatological comparison, such a value cannot be interpreted, because the reader has no way to separate genuine skill from the trivial reproduction of the seasonal cycle.

A second practice compounds the first. Splitting a temporal series at random, a convention inherited from cross-sectional machine learning, lets information from future months shape the fitted model and inflates apparent accuracy (Bergmeir & Benítez, 2012). An absent baseline and a shuffled split reinforce each other: together they make an uninformative model appear skilful and leave the deficiency invisible. Establishing whether machine learning adds value here requires controlling both at once.

This study therefore evaluates one-month-ahead forecasts of monthly mean temperature at the Puno Principal Climatological Station under a verification protocol built to close both gaps. Multiple linear regression, random forest, and a multilayer perceptron are compared against monthly climatology and persistence on a chronologically held-out test period, with hyperparameters tuned by blocked time-series cross-validation confined to the training years. Differences in accuracy are tested with the Diebold-Mariano statistic (Diebold & Mariano, 1995) rather than inferred from point estimates, and skill is expressed as the mean square error skill score relative to climatology (Murphy, 1988; Abdullah et al., 2025; Salem et al., 2025).

Five contributions follow: the improvement over climatology is quantified as a skill score rather than an unreferenced coefficient of determination; the apparent advantage of the neural model is shown to be an artefact of a single random initialisation; the protocol is replicated across eleven independent Altiplano series; learning curves are extended to 2,475 training months, testing the sample-size explanation by measurement rather than extrapolation; and a reproducible quality-control and gap-handling procedure for an incomplete Andean record is documented.

MATERIALS AND METHODS

Study area, data source and replication dataset

The Puno Principal Climatological Station records daily observations on the western shore of Lake Titicaca at approximately 3,820 m a.s.l. The dataset comprises 7,279 daily records spanning 1 January 2003 to 31 December 2024, with mean relative humidity (%), total precipitation (mm), and mean, maximum, and minimum temperature (°C). The workbook also contained a sheet labelled as the Los Uros Ordinary Climatological Station, whose 65,538 cells proved identical to those of the Puno sheet apart from the station label, so it was excluded and every analysis rests on the Puno record alone.

Because conclusions drawn from one station cannot be separated from its peculiarities, the protocol was replicated on an independent source. Daily records for thirteen Altiplano locations, the same used in a previous regional frost-prediction study (Torres-Cruz et al., 2025), were retrieved from NASA POWER, which distributes MERRA-2 reanalysis output as open data without registration (NASA POWER, 2025). The retrieval covered 1 January 2000 to 31 December 2024 and returned 9,132 daily records per location with no missing values.

Two checks preceded any modelling. First, pairwise comparison of the thirteen series revealed two pairs, Ayaviri with Cojata and Juli with Yunguyo, whose values were numerically identical because each pair falls inside the same MERRA-2 grid cell of roughly 0.5° × 0.625°. The redundant members were removed, leaving eleven independent series with pairwise daily correlations between 0.690 and 0.963. Second, the reanalysis was compared against the station record at Puno over 6,499 matched days. Reanalysis mean temperature is systematically colder by 2.577 °C (paired t = −178.3, p < 0.001), a bias consistent with a grid cell whose mean terrain elevation differs from that of the station site, yet it tracks variability closely. The reanalysis is therefore unsuitable as ground truth but well suited to replication, because each series is modelled and evaluated against itself and a constant offset cancels within that comparison.

Quality control and monthly aggregation

Daily records were screened before aggregation. Sentinel values below −99 were converted to missing, relative humidity outside 0–100% was rejected, and the ordering constraint Tmin ≤ Tmean ≤ Tmax was verified; no record violated it. Calendar completeness exposed 757 absent dates concentrated in six blocks rather than scattered at random (Table 1). The distinction matters: dispersed losses are absorbed by averaging, whereas block losses remove entire months and sever the lag structure any forecasting model depends on.

Monthly values were computed as the mean of the available daily observations for temperature and humidity, and as the sum for precipitation. A month was admitted only when it satisfied the completeness rule of the World Meteorological Organization (WMO, 2017), which rejects a month with more than 10 missing daily values or 5 or more consecutive ones. Of 264 candidate months, 236 (89.4%) met the rule, and 26 of the 28 rejections corresponded to entire absent blocks (Bello & Musa, 2026).

Reconstruction of missing daily mean temperature

Daily mean temperature was missing on 10.72% of records while maximum and minimum temperature were missing on only 1.85% and 1.26%, an asymmetry that made reconstruction from the daily extremes worth attempting. Evaluated against the 6,499 days on which the observed mean and both extremes were available, the midpoint estimator (Tmax + Tmin)/2 underestimated the observed mean by 0.687 °C (paired t = −65.21, p < 0.001). The bias is systematic rather than random, because the recorded mean derives from synoptic-hour observations; substituting the raw midpoint would have injected an offset equal to 42% of the standard deviation of the monthly series.

The midpoint was therefore calibrated by ordinary least squares,

with coefficients estimated exclusively on training-period days (2003–2016, n = 4,541), giving Tmean = 1.5230 + 0.9241 · (Tmax + Tmin)/2. Ten-fold cross-validation on the fitting sample returned a mean absolute error of 0.691 °C, an R² of 0.801, and a residual bias of −2 × 10⁻⁵ °C; applied to the held-out years it retained a mean absolute error of 0.609 °C and a residual bias of 0.284 °C, confirming that the correction transfers across time without reintroducing the original offset. The procedure recovered 606 daily values across the record, of which 604 fall inside the retained months and represent 8.4% of their days.

Forecasting task and predictors

The task is a one-month-ahead point forecast: given information observable up to and including month t, predict the mean temperature of month t + 1. Twelve predictors evaluable at time t were constructed. Three lagged temperature terms (t, t−1, t−2) and a three-month rolling mean carry thermal memory; the diurnal range, relative humidity and precipitation of month t carry concurrent atmospheric state. The mean temperature of the target calendar month, computed exclusively from training-period observations, supplies the climatological expectation, and the departure of month t from its own climatological mean supplies the current anomaly. Sine and cosine transforms of the target month index encode the annual cycle without a December-to-January discontinuity, and a linear time index represents the secular trend.

Longer lags were deliberately excluded. A twelve-month lag would have been the most autocorrelated predictor, but requiring twelve consecutive valid months cut the usable sample from 186 to 145 observations, because each absent block spans several months; the climatological predictor conveys the same seasonal information at no such cost. After removing rows with any missing predictor, 186 supervised observations remained, covering April 2003 to November 2024.

Models, reference forecasts and evaluation

Three regression models were compared. Multiple linear regression supplies the parametric benchmark and has no free hyperparameters; random forest aggregates decorrelated regression trees and captures non-additive structure (Breiman, 2001); and the multilayer perceptron is a feed-forward neural network with a single hidden layer trained by backpropagation (Rumelhart et al., 1986), the standard non-linear forecasting model for three decades (Zhang et al., 1998). Predictors were standardised inside the modelling pipeline for the linear and neural models, so that scaling statistics were estimated on training folds alone; all models were implemented in scikit-learn (Pedregosa et al., 2011; Sa-Couto et al., 2025). Two reference forecasts define the standard the models must clear: climatology predicts the mean of the target calendar month computed from training-period data only, and persistence predicts the value observed in month t. Without these references a reader cannot judge whether any stated accuracy exceeds what the seasonal cycle already delivers (Jolliffe & Stephenson, 2011).

The 186 observations were divided chronologically: target months up to December 2016 formed the training set (n = 137) and those from January 2017 onward the test set (n = 49). Hyperparameters were selected by grid search under blocked time-series cross-validation with five folds, which trains on past folds and validates on the immediately following fold, preserving temporal order (Bergmeir & Benítez, 2012). The search evaluated 54 configurations for random forest and 32 for the perceptron, scored by cross-validated mean absolute error. The test set was consulted once, after hyperparameter selection had closed.

Four accuracy measures were computed on the test set: mean absolute error and root mean square error quantify typical and outlier-sensitive error magnitude, the coefficient of determination quantifies explained variance, and mean bias error exposes systematic over- or under-forecasting:

Skill was expressed as the mean square error skill score (Murphy, 1988),

which equals zero for a forecast equivalent to climatology, approaches one for a perfect forecast, and turns negative for one worse than climatology. Because point estimates may reflect sampling variation, pairwise differences were tested with the Diebold-Mariano statistic on squared-error loss differentials (Diebold & Mariano, 1995), and 95% confidence intervals came from 2,000 bootstrap resamples of the test set (Mulyani et al., 2026).

Complementary analyses

Five further analyses test whether the headline comparison holds under scrutiny. An ablation refitted every model on five nested predictor sets, isolating the contribution of seasonal information, thermal memory, and atmospheric covariates. A seasonal breakdown partitioned the test set into wet (December–March), dry (May–August), and transition (April, September–November) months. Residual diagnostics assessed normality with the Shapiro-Wilk test and serial independence with the Ljung-Box statistic at five lags, since residuals that retain structure indicate predictable signal left unused. Learning curves refitted each model on training subsets of 30 to 137 months, using up to six contiguous windows per size and five initialisations per window.

The replication applied the same pipeline to each of the eleven independent series, with training months up to December 2018 and test months from January 2019 onward, yielding 2,475 pooled training and 792 pooled test observations. Pooled learning curves were measured at sizes from 100 to 2,475 months, extending the range examined at Puno eighteenfold so that the sample-size explanation could be tested against data instead of extrapolated.

The fifth analysis addresses a reporting practice that small samples make hazardous. Random forest and the perceptron are stochastic, whereas ordinary least squares has a closed-form solution, so reporting a single seed conflates model performance with initialisation luck. Both stochastic models were refitted under 50 seeds, and the resulting distributions of test error are reported alongside the single-seed values.

RESULTS AND DISCUSSION

Characteristics of the temperature record

The retained monthly series has a mean of 10.72 °C and a standard deviation of 1.62 °C, ranging from 6.59 °C to 14.21 °C (Table 1). The record warms significantly at +0.54 ± 0.17 °C decade⁻¹ (p = 0.0017), which exceeds the long-term rates reported for the tropical Andes across the twentieth century (Vuille et al., 2003) and agrees in sign with station-based trends for the southern Peruvian Andes (Imfeld et al., 2021). Figures 1a, 1b present the series alongside the monthly count of missing days, and show the losses clustering after 2020 rather than spreading evenly across the record.

The seasonal cycle dominates the variance, and this single fact governs the interpretation of every result that follows. Monthly climatological means range from 8.17 °C in July to 12.85 °C in November, an amplitude of 4.68 °C, whereas the within-month standard deviation never exceeds 0.94 °C and averages 0.71 °C (Figure 1c). The autocorrelation function confirms the structure, with coefficients of 0.763 at lag 1, 0.878 at lag 12, and 0.847 at lag 24, and a pronounced −0.521 at lag 6 (Figure 1d). The consequence is unavoidable: a forecast reproducing only the seasonal cycle will already look accurate, which is why the climatological reference is indispensable rather than optional.

Table 1. Characteristics of the daily record, the absent periods, and the retained monthly series at the Puno Principal Climatological Station, 2003–2024.

Note. The World Meteorological Organization completeness rule rejects a month with more than 10 missing daily values or 5 or more consecutive ones. The climatological statistics describe the 236 retained months.

Forecast accuracy against the references

All three machine learning models outperformed climatology, though by a moderate rather than a transformative margin (Table 2, Figure 2a). Against the 0.597 °C mean absolute error of climatology, multiple linear regression reached 0.421 °C, the multilayer perceptron 0.415 °C, and random forest 0.476 °C, giving mean square error skill scores of 0.505, 0.489, and 0.310. The best models therefore remove approximately half of the mean square error climatology leaves behind: a real improvement with a clear ceiling.

Persistence performed markedly worse than climatology, at 0.866 °C and a skill score of −0.958, a degradation the Diebold-Mariano test confirms as significant (t = 3.057, p = 0.004). The result is physically coherent: where the seasonal cycle spans 4.68 °C and month-to-month change routinely exceeds the interannual variability of any individual month, carrying the previous month forward is worse than ignoring it. Persistence alone therefore cannot serve as the reference in seasonal climates, and climatology must anchor the comparison.

Statistical testing tempers the ranking the point estimates suggest. Against climatology, linear regression (t = −2.255, p = 0.029) and random forest (t = −2.222, p = 0.031) improved significantly at the 5% level, whereas the perceptron fell marginally short (t = −1.998, p = 0.051), and the bootstrap intervals overlap across the three models (Table 2). With 49 test months the evidence does not support fine distinctions, and reporting them as a strict ordering would overstate what the data can bear.

Table 2. Forecast accuracy, skill and generalisation on the test period (January 2017 – November 2024, n = 49 months).

Note. MLR = multiple linear regression; RF = random forest; MLP = multilayer perceptron. MAE = mean absolute error; RMSE = root mean square error; MBE = mean bias error; MSSS = mean square error skill score relative to climatology; DM = Diebold-Mariano test against climatology. Confidence intervals use 2,000 bootstrap resamples; the test/train ratio is test MAE divided by training MAE; cross-validated MAE was obtained under blocked time-series cross-validation with five folds on the training period.

Model complexity brings no measurable benefit

The central result is that the simplest model performs at least as well as the most complex one. Under the single fit reported in Table 2, the perceptron achieved the lowest mean absolute error, 0.415 °C against 0.421 °C for linear regression, yet the Diebold-Mariano test detects no significant difference (t = 0.104, p = 0.918), and linear regression returns the lower root mean square error (0.532 °C against 0.541 °C), the higher coefficient of determination, and the higher skill score. The 0.006 °C margin lies far below the resolution that matters operationally, and the two scatters are visually indistinguishable (Figures 2c, 2e).

Generalisation behaviour explains why complexity earns nothing here. Random forest reached a training error of 0.226 °C against a test error of 0.476 °C, a ratio of 2.11 indicating substantial overfitting even after tuning had restricted tree depth to six (Table 2), while the perceptron shows a milder ratio of 1.37 and linear regression essentially none, at 1.01. With 137 training observations and 12 predictors, flexible learners have ample capacity to fit sampling noise, and the strong regularisation selected by cross-validation (α = 1.0) reflects that pressure directly.

The neural advantage does not survive re-seeding

Refitting the stochastic models under 50 random initialisations dissolves the perceptron's apparent lead (Table 3). Across seeds it averages 0.495 °C with a standard deviation of 0.048 °C and a range from 0.407 °C to 0.647 °C, so the 0.415 °C reported for a single fit sits at the 6th percentile of its own sampling distribution. Only 4 of 50 initialisations beat the deterministic 0.421 °C of linear regression, so 92% of the time an analyst repeating this experiment with a different seed would conclude that linear regression wins. Random forest behaves differently, with a standard deviation of 0.005 °C and no initialisation reaching the linear model, confirming that seed sensitivity is specific to the neural architecture rather than general to stochastic learners.

Averaging the 50 perceptrons into an ensemble does not recover the advantage either: it attains 0.4209 °C against 0.4208 °C for linear regression, with a higher root mean square error (0.564 °C against 0.532 °C). Fifty neural networks combined therefore reproduce, at fifty times the cost and with no interpretable coefficients, what a twelve-parameter closed-form regression delivers in a single fit.

Table 3. Residual diagnostics, sensitivity to random initialisation, predictor ablation and seasonal breakdown on the test period.

Note. Model abbreviations as in Table 2. Shapiro-Wilk tests normality and Ljung-Box Q, computed at five lags, tests serial independence; p-values above 0.05 indicate residuals consistent with normality and white noise respectively. Seed statistics summarise 50 random initialisations; ordinary least squares is deterministic. In the ablation, k = number of predictors, seasonal = target-month climatology and the two cyclical encodings, memory = lagged temperature terms and the three-month rolling mean, and covariates = relative humidity, precipitation and diurnal temperature range. Wet = December–March, dry = May–August, transition = April and September–November. Climatology attains 0.597 °C overall.

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