A Methodology Based on Random Forest to Estimate Precipitation Return Periods: A Comparative Analysis with Probability Density Functions in Arequipa, Peru

Abstract

Precipitation within specific return periods plays a crucial role in the design of hydraulic infrastructure for water management. Traditional analytical approaches involve collecting annual maximum precipitation data from a station followed by the application of statistical probability distributions and the selection of the best-fit distribution based on goodness-of-fit tests (e.g., Kolmogorov-Smirnov). However, this methodology relies on current data, raising concerns about its suitability for outdated data. This study aims to compare Probability Density Functions (PDFs) with the Random Forest (RF) machine learning algorithm for estimating precipitation at different return periods. Using data from twenty-six stations located in various parts of the Arequipa department in Peru, the performance of both methods was evaluated using MSE, RMSE, R² and MAE. The results show that RF outperforms PDFs in most cases, having more precision using the metrics mentioned for precipitation estimates at return periods of 2, 5, 10, 20, 50, and 100 years for the studied stations.

1. Introduction

Floods are one of the most frequent and destructive types of disasters, impacting communities worldwide. They are responsible for up to 40% of global natural disasters, causing almost half of all natural hazard-related fatalities, and are currently exhibiting a worrying increase [1]. As a result, the necessity for sustainable urban planning that integrates demand-driven hydraulic infrastructure becomes paramount.

According to Bertilsson, et al. [2], climate change and rapid urbanization present significant obstacles to sustainable urban planning, while the proliferation of megacities exacerbates flood risk by increasing impervious surfaces and altering hydrological pathways, with increased urbanization and population concentration increasing exposure to natural hazards [3]. It is important to consider that effective flood risk management necessitates a well-informed and accurate assessment of flood hazards, acknowledging the likelihood of increased extreme flood events due to global warming [1]. Making accurate estimations of intense precipitation is key to improving early warnings and protecting the population; therefore, the analysis of data from climate stations is of great relevance, as noted by Padji, et al. [4].

Furthermore, this analysis is essential in areas such as in planning infrastructure resilient to extreme conditions and efficient water resources management. In this regard, an effective approach to improve flood planning relies on the ability to anticipate extreme values of rainfall [5].

The theory of statistical extremes shows that the frequency of extreme events is more closely linked to changes in climate variability than to fluctuations in the mean climate state [6]. Furthermore, the distribution fitting process involves matching a statistical probability distribution (PDF) to a dataset derived from random processes and is a critical step in capturing underlying patterns. Therefore, since probability distributions are essential for quantifying uncertainty, selecting the wrong distribution can lead to erroneous conclusions [7].

RF is presented as a powerful alternative to predict missing data within a data series or dataset, providing greater accuracy in the estimates. According to Breiman [8] and Lin, et al. [9], Random Forest is a machine learning model that employs an ensemble of decision trees to make predictions. Each tree in the forest is built using a random subset of data and features. Predictions are made by averaging the results of all the trees in the ensemble, which improves the overall accuracy of the model and reduces the risk of overfitting. This approach allows for more robust results, since the combination of several decision trees corrects individual errors and increases resistance to noise in the data. In this context, this article proposes the application of Random Forest (RF) in the city of Arequipa, a region that faces a notable shortage of updated meteorological data. This limitation represents a significant challenge for the accurate estimation of extreme rainfall, essential for the planning and management of resilient infrastructures that can mitigate the risks associated with floods and other extreme weather phenomena.

Despite the widespread use of machine learning (ML) techniques in hydrological studies worldwide, a significant research gap exists in the application of ML methods such as Random Forest in regions like Arequipa, Peru. This gap is characterized by the limited availability and quality of hydrological data, the lack of adaptation of ML models to local climatic conditions, insufficient computational infrastructure, and minimal practical implementation in water management projects.

In this context, this study aims to address a significant gap in the application of machine learning methods, particularly Random Forest (RF), to estimate precipitation return periods in regions with limited hydrological data, such as Arequipa, Peru. Specifically, the objectives of this research are to evaluate the performance of RF as a predictive tool for precipitation return periods in data-scarce environments, to compare RF with traditional probability density functions (PDFs) in terms of predictive accuracy and reliability, to establish a robust methodology that addresses local challenges such as data variability and computational constraints, and to provide actionable insights for public policy, infrastructure planning, and disaster risk management. By achieving these goals, this study aims to contribute both to hydrology through innovative methodological approaches and to machine learning by exploring its applications in unique climatic and logistical contexts.

This study is structured as follows: The Background section provides similar cases to the application of predictive tools around the world and the use of RF in different research fields in Peru, while the Data and Study Area section provides an overview of the region and the data used. The Methodology section describes the PDFs used and the RF algorithm, including the comparison metrics. The Results section presents precipitation estimates for various return periods using both methods, along with a comparative analysis. The Discussion section delves into the differences between the two methods, explaining their performances. Finally, the conclusion section summarizes the key findings and suggests avenues for future research.

2. Background

There are numerous studies that highlight the importance of identifying the best-fitting distribution for data from a weather station, as demonstrated by Mandal and Choudhury Mandal and Choudhury [10], who investigated the annual, seasonal, and monthly daily maximum rainfall patterns at Sagar Island, situated on the continental shelf of the Bay of Bengal. The study revealed that the normal distribution provided the best fit for the annual, post-monsoon, and summer seasons. Furthermore, in Asia, Zhai, et al. [11] applied methodologies to analyze rainfall predictions, focusing on probability distribution models such as Pearson Type III, Pareto–Burr–Feller, generalized extreme value (GEV), and Weibull. These models were calibrated with historical data to predict extreme rainfall. In addition, they implemented the Mann–Kendall test to detect long-term trends in rainfall. These techniques helped to improve the prediction of extreme event recurrence and assess long-term trends in rainfall frequency and magnitude, providing a solid basis for designing drainage infrastructure and flood management systems. In addition, Si, et al. [12] addressed the issue by constructing regional weather stations, allowing for a more in-depth study of extreme rainfall, even with short datasets. They employed peak-over-threshold sampling methods and the generalized Pareto distribution to optimize spatial interpolation parameters, improving the accuracy of daily extreme rainfall forecasts at regional stations. For example, in the United Arab Emirates, Branch, et al. [13] highlighted the need for improved extreme event forecasts, particularly in arid regions. They employed high-resolution solutions validated with weather station data, despite the limitations posed by the scarce information provided.

As an alternative to the previously described mathematical methods based on probability density functions (PDFs), the integration of Big Data and Artificial Intelligence has allowed Machine Learning (ML) to emerge as a promising tool in weather prediction, leveraging its strengths in addressing non-linear complexities and uncovering previously unknown relationships within the Earth’s climate system [14]. In this sense, this method could be useful in determining the return period, which is an essential metric that quantifies the probability of extreme events such as floods and droughts that can inflict significant damage on society and the environment. This probabilistic concept is widely used in hydrological studies and has gained increased attention due to the need to effectively manage complex processes in an evolving environmental landscape [15]. In the domain of water resources design and management, return period analysis is a fundamental tool for risk assessment and communication [5]. However, access to up-to-date data is not always possible due to various obstacles, including a shortage of weather stations in the region, the inadequate maintenance of measuring equipment, and restrictions on access to updated data due to privacy policy restrictions.

In Peru, these traditional methods are still fundamental in the calculation of the return period; that is to say that the applicability of ML-based methods still has no place at the implementation level within the field of Peruvian hydrology. However, some research carried out within the framework of this study can be found, such as with Aragón, et al. [16], who analyzed the capacity of the IMERG satellite product of the Global Precipitation Measurement (GPM) and the observed data from meteorological stations through a mixed approach to estimate the distribution of extreme rainfall in the Cusco region. The present study highlights its unique and innovative approach that differentiates it from the proposed cases since Arequipa faces data availability problems, which is a variable that is essential for the application of the RF algorithm to estimate precipitation for a specific return period instead of relying on PDFs. This aspect was highlighted by Sun, et al. [17], since RF is effective in capturing non-linear relationships between precipitation and predictive variables, such as the normalized difference vegetation index, land surface temperature, and topographic features, amongst others. Table 1 reinforces this idea, analyzing the application of the RF algorithm in different study areas in Peru.

The RF algorithm has proven to be a versatile tool in various disciplines in Peru, such as landslide prediction in high Andean grasslands, modeling contaminants in cocoa soils, analyzing the relationship between dengue and climate variables, and investigating deforestation patterns in the tropical Andes. These applications highlight how RF can effectively address environmental, agricultural, climate, and health issues.

However, the Arequipa case study is particularly unique, as it proposes the use of RF as an innovative alternative to traditional PDF-based methods for estimating extreme rainfall and return periods. This approach is especially relevant in the context of scarcity of updated meteorological data, a common challenge in vulnerable regions such as Arequipa. Unlike previous work, such as that of Mandal and Choudhury Mandal and Choudhury [10], who identified specific distributions in an island environment, or that of Si, Shao, Zhao, Wei, Hou and Huang [12], who optimized spatial interpolation parameters in regional studies, this research directly addresses the limitations associated with the lack of reliable meteorological stations and data in a mountainous and arid region. Furthermore, compared to high-resolution solutions employed in the United Arab Emirates, this study demonstrates that the use of RF can be a robust and replicable tool in similar local contexts, providing a novel methodological solution for regions with high exposure to flood and landslide risks. In this way, it contributes significantly to the development of resilient infrastructure, sustainable planning, and adaptation to climate change in Peru.

3. Data and Study Area

The case study presented in this article focuses on the department of Arequipa. Located in southern Peru, Arequipa is a diverse region nestled in the Andean highlands. It is bordered by the Pacific Ocean to the west and the Andes Mountain range to the east. The climate is characterized by mild temperatures and low humidity throughout the year. It is important to recognize that Arequipa is a prosperous economic center, driven by mining, agriculture, tourism, and manufacturing. The region is rich in mineral resources, including copper, zinc, and gold. Agriculture is another important activity, with crops of potatoes, corn, and wheat. Tourism attracts visitors from around the world with its stunning natural landscapes, including the Colca Canyon, Misti Volcano, and the Andagua Valley. Arequipa is a region where 70% of the population is exposed to flood and landslide risks according to ANA reports. These events can cause significant economic losses, such as the more than PEN 40 million in damage recorded during extreme rainfall in 2019. With accurate estimates, resilient infrastructures can be designed, sustainable agricultural practices implemented, and key landscapes protected, thus ensuring sustainable development and economic security in the region against the impacts of climate change.

The preceding description provides an overview of some characteristics of the Arequipa department. However, to perform a comparative analysis of methods, it is necessary to have access to the corresponding data. To this end, the data were obtained from the Peruvian National Meteorology and Hydrology Service (SENAMHI) for 26 stations distributed across the Arequipa department. However, these datasets vary in length for each station and are not updated to the last year, with most stations having data only up to 2014. The analysis was conducted considering the following years for each station: 1965 to 2013; the maximum annual precipitation was determined for each year to initiate the subsequent comparison of the methods. The time series of the stations can be found in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5.

4. Methodology

When selecting the most appropriate distribution for a specific location, it is crucial to consider the variety of distribution models available. This section outlines the distribution models, the RF algorithm, the goodness-of-fit test, and the metrics used in the study, MSE, RMSE, R² and MAE.

4.1. Commonly Used Probability Distributions

Statistical distributions are essential tools for modeling and analyzing complex phenomena and enable researchers to describe and predict extreme events. In the next lines, the mathematical expressions are found with some applications of these distributions.

4.1.1. Normal

The distribution is a symmetric probability model characterized by its mean and standard deviation parameters. The probability density function (PDF) is as follows:

(1)$f\left(x\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{1}{2{\sigma }^2}}{\left(x-\mu \right)}^2\right]$

for the range of $-\infty <x<\infty$. Annual precipitation and runoff analyses often rely on the normal distribution [28].

4.1.2. Log-Normal

Log-Normal is a continuous probability model suitable for skewed data. Its logarithmic transformation ensures positive values, making it ideal for modeling variables with lower bounds.

(2)$f\left(x\right)={\displaystyle \frac{1}{x{\sigma }_Y\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{1}{2{\sigma }_Y^2}}{\left(\ln \left(x\right)-{\mu }_Y\right)}^2\right]$

where the range of random variable is $x>0$. The two parameters are expressed as follows:

(2a)${\sigma }_Y={\left[\ln \left(1+{\displaystyle \frac{{\sigma }_X^2}{{\mu }_X^2}}\right)\right]}^{{\displaystyle \frac{1}{2}}}$

(2b)${\mu }_Y=\ln \left({\mu }_X\right)-{\displaystyle \frac{1}{2}}{\sigma }_Y^2$

4.1.3. Pearson Type 3

Pearson Type 3 is a versatile, three-parameter model suitable for skewed data; its Gamma distribution-like properties enable the analysis of variables with varying coefficients of variation, making it applicable to precipitation, streamflow, and water quality data.

(3)$f\left(x\right)={\displaystyle \frac{1}{\left|\alpha \right|\mathsf{\Gamma }\left(\beta \right)}}\left[{\left({\displaystyle \frac{x-\xi }{\alpha }}\right)}^{\beta -1}\right]\exp \left[-{\displaystyle \frac{\left(x-\xi \right)}{\alpha }}\right]$

Its parameters are as follows:

(3a)$\beta =4/{\gamma }^2$

(3b)$\alpha ={\sigma }_{\gamma }/2$

(3c)$\xi =\mu -2\sigma /\gamma$

This distribution is prominently employed in hydrology [5].

4.1.4. Log Pearson Type 3

Log Pearson Type 3 is a flexible probability model that combines logarithmic transformation with the Pearson Type 3 distribution; this combination enables the analysis of skewed data with extreme values.

(4)$f\left(x\right)={\displaystyle \frac{1}{\left|\alpha \right|x\mathsf{\Gamma }\left(\beta \right)}}\left[{\left({\displaystyle \frac{\ln \left(x\right)-\xi }{\alpha }}\right)}^{\beta -1}\right]\exp \left[-{\displaystyle \frac{(ln\left(x\right)-\xi )}{\alpha }}\right]$

Phien and Ajirajah [29] assessed the suitability of this distribution for modeling flood and maximum rainfall data, as well as its applicability to annual rainfall and streamflow sequences.

4.1.5. Generalized Extreme Value (GEV)

Generalized Extreme Value (GEV) is a flexible model for extreme events; its cumulative distribution function allows for the analysis of both upper and lower tails, making it suitable for modeling extreme precipitation, flood, and drought events.

(5)$f\left(x\right)={\alpha }^{-1}\exp \left[-\left(1-k\right)y-\exp \left(-y\right)\right],$

(5a) $y=-k^{-1}\log \left\{1-{\displaystyle \frac{k\left(x-\xi \right)}{\alpha }}\right\},k\ne 0$

(5b) $y={\displaystyle \frac{x-\xi }{\alpha }},k=0$

The GEV distribution is extensively endorsed in European countries for its exceptional ability to accurately model flood data [30].

4.1.6. GEV MIN (L-Moments)

A variant of the GEV model, GEV MIN is specifically designed for analyzing extreme minima; its inverse cumulative distribution function enables the modeling of low-frequency events, such as droughts and minimum streamflow.

(6)$f\left(x\right)=\exp \left(-{[1+\xi ({\displaystyle \frac{x-\mu }{\sigma }})}^{-{\displaystyle \frac{1}{\xi }}}\right),for1+\xi \left({\displaystyle \frac{x-\mu }{\sigma }}\right)>0$

4.1.7. GEV MAX (Kappa Specified, L-Moments)

A specialized GEV model for extreme maxima, GEV MAX’s cumulative distribution function is tailored for analyzing high-frequency events, such as floods, heavy precipitation, and maximum streamflow.

(7)$f\left(x\right)=\mathit{exp}\left(-{[1-k({\displaystyle \frac{x-\mu }{\sigma }})}^{{\displaystyle \frac{1}{k}}}\right),for1-k\left({\displaystyle \frac{x-\mu }{\sigma }}\right)>0$

4.1.8. Pareto

Pareto is used to model phenomena where a large proportion of effects are attributable to a small proportion of causes. It is given by the following formula:

(8)$f\left(x,x_m,\alpha \right)=\left\{\begin{array}{c}{\displaystyle \frac{{\alpha x}^{\alpha }}{x^{\alpha +1}}}\\ 0\end{array}\right.,\begin{array}{c}x\ge x_m\\ x<x_m\end{array}$

where $x>0$ is the minimum threshold value. And $\alpha >0$ is the shape parameter (Pareto index).

4.1.9. Gumbell Max

Gumbell Max models extreme (maximum) values of a dataset, and is used in hydrology, meteorology, and extreme data analysis.

(9)$f\left(x,\mu ,\beta \right)={\displaystyle \frac{1}{\beta }}·e^{{\displaystyle \frac{x-\mu }{\beta }}}·e^{{-e}^{-{\displaystyle \frac{x-\mu }{\beta }}}},xϵ\mathbb{R}$

where μ is the location parameter. And β > 0 is the scale parameter.

4.1.10. Galton

The Galton distribution describes variables whose distribution is normal on a logarithmic scale. It is used in particle size analysis, economic growth, etc. It is given by the following equation:

(10)$f\left(x,\mu ,\sigma \right)={\displaystyle \frac{1}{x·\sigma ·\sqrt{2\pi }}}·e^{-{\displaystyle \frac{{\left(lnx-\mu \right)}^2}{{2\sigma }^2}}},x>0$

where μ is the location parameter (on a logarithmic scale). And σ > 0 is the scale parameter (on a logarithmic scale).

4.1.11. Gamma

The Gamma distribution models waiting time and related phenomena by the sum of independent exponential random variables. It is given by the following equation:

(11)$f\left(x,\mu ,\beta \right)={\displaystyle \frac{{\beta }^{\alpha }}{\Gamma \left(\alpha \right)}}·x^{\alpha -1}·e^{-\beta x},x>0$

where σ > 0, is the shape parameter and β > 0 is the scale parameter. And Γ(α) is the Gamma function.

4.1.12. Weibull

The Weibull distribution is common in survival analysis, reliability, and failure modeling. It is given by the following equation:

(12)$f\left(x,\lambda ,k\right)=\left\{\begin{array}{c}{\displaystyle \frac{k}{\lambda }}·{\left({\displaystyle \frac{x}{\lambda }}\right)}^{k-1}·e^{{-\left({\displaystyle \frac{x}{\lambda }}\right)}^k}\\ 0\end{array}\right.,\begin{array}{c}x\ge 0\\ x<0\end{array}$

where λ > 0 is the scale parameter. And k > 0 is the shape parameter.

4.1.13. Exponential

The exponential distribution models the times between events in a Poisson process. It is a continuous probability distribution.

(13)$f\left(x,\lambda \right)=\left\{\begin{array}{c}{\lambda e}^{-\lambda x}\\ 0\end{array}\right.,\begin{array}{c}x\ge 0\\ x<0\end{array}$

where λ > 0 is the event rate.

4.2. Goodness-of-Fit Test

To assess the validity of a specified probability distribution model, goodness-of-fit test statistics are applied. A plethora of normality tests are available, including Empirical Distribution Function (EDF) tests, which quantify the divergence between empirical and theoretical distributions [31]. Prominent EDF tests include the Kolmogorov-Smirnov (K-S), the Anderson-Darling (A-D) test, and Cramer-Von Mises test [32]. The K-S was applied in this study due to its simplicity and computational ease of implementation, coupled with the fact that it does not require the estimation of additional parameters.

Furthermore, RMSE was employed to evaluate model performance and identify the best-fitting model.

4.2.1. Kolmogorov Smirnov (K-S) Test

This test is a non-parametric statistical test used to assess the goodness-of-fit between observed and theoretical distributions. It quantifies the maximum distance between cumulative distribution functions, enabling the detection of significant deviations. It is designed to compare the empirical cumulative frequency $S_n\left(x\right)$ with the cdf of an assumed theoretical distribution $F_x\left(x\right)$. For a sample size n, the values are sorted in a non-decreasing sequence, $X_1<X_2<...<X_n$, and the K-S statistic is applied to each data value in the ascending order.

(14)$S_n\left(x\right)=0;ifX<X_1$

(14a) $={\displaystyle \frac{k}{n}};if{X}_k\le X<X_{k+1}$

(14b) $=1;ifX>X_n$

The K-S test statistic is the maximum difference between $S_n\left(x\right)$ and $F_x\left(x\right)$

(15)$D_n=\max \left|F_x\left(x\right)-S_n\left(x\right)\right|$

(15a) $P\left(D_n\le D_n^{\alpha }\right)=1-\alpha$

Following the next syntax, the critical value is $D_n^{\alpha }$, the significance level is $\alpha$, and $k$ is the rank order of the dataset.

4.2.2. Mean Square Error (MSE)

Mean square error (MSE) is the mean of the squares of the errors between the real and predicted values. It is similar to the RMSE since it measures the accuracy of the model; however, it does not have a square root. In addition, it penalizes large errors more and has the same units as the square data. They follow the following expression.

(16)$MSE={\displaystyle \frac{1}{n}}\sum _{j=1}^n{(x_i-X)}^2$

where $x_i$ are the actual values and X represents the predicted values.

4.2.3. Root Mean Square Error (RMSE)

The lowest RMSE values signify the best-fitting model, yielding the standard deviation of the prediction uncertainty. RMSE assesses the difference between actual and estimated values. The RMSE has the following expression:

(17)$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{j=1}^n{(x_i-X)}^2}$

where $x_i$ is the estimated value and X is the actual value.

4.2.4. R² (Coefficient of Determination)

The Coefficient of Determination measures how well the model explains the variability of the observed data. The equation is presented as follows:

(18)$R^2=1-{\displaystyle \frac{{SS}_{rea}}{{SS}_{tot}}}$

where ${SS}_{rea}$ is the sum of the residual squared errors and ${SS}_{tot}$ is the total sum of squares (total variability).

where the result of this metric can be interpreted as follows:

$R^2=1$—the model explains the data perfectly.

$R^2=0$—the model does not explain it better than the average.

$R^2<0$—the model is worse than using the mean as a prediction.

4.2.5. MAE (Mean Absolute Error)

Mean Absolute Error (MAE) is the average of the absolute values of the errors between the actual and predicted values. This metric measures the error without disproportionately penalizing large errors (unlike MSE and RMSE). It has the same units as the data and is usually more interpretable for data that are not very dispersed. The equation is presented as follows:

(19)$MAE={\displaystyle \frac{1}{n}}\sum _{j=1}^n\left|x_i-X\right|$

4.3. Return Period

The probability of exceedance in a given time is as follows:

(20)$P\left(x\ge x_T\right)={\displaystyle \frac{1}{T}}$

(20a)$T={\displaystyle \frac{1}{1-P(x\le x_T)}}$

The probability of exceedance based on observed data is as follows:

(20b)$P\left(i\right)={\displaystyle \frac{i}{N+1}}$

where $i$ is the position of the observation and $n$ is the number of observations.

4.4. Traditional Approach

In Peru, a traditional approach is used to estimate precipitation at different return periods using statistical distributions, as indicated in the MTC Hydrology manual [33]. It contains these steps:

1. Collection of Precipitation Data:

   Gather historical data on maximum precipitation.

2. Sort Data from Largest to Smallest:

   Organize precipitation data for analysis.

3. Calculate Exceedance Probability:

   Use the formula described in Section 4.3.

4. Statistical Analysis: Selection of Probability Distribution:

   Assess which distribution (Gumbel, Log-Normal, Weibull, etc.) best fits the data.

5. Perform Goodness-of-Fit Tests (Kolmogorov-Smirnov, etc.):

   Apply tests to evaluate how well theoretical distributions fit the empirical data.

6. Evaluate Distribution Fit:

   Does the selected distribution pass the goodness-of-fit test?

   Yes: Proceed to the next step.

   No: Select another distribution and return to step 5.

7. Adjust Parameters of the Selected Distribution (Maximum Likelihood or Method of Moments):

   Adjust the parameters using appropriate methods.

8. Make Precipitation Predictions for Different Return Periods:

   Calculate precipitation values for different return periods using exceedance probabilities.

This approach is described in Figure 6. It has a lineal flow with an iterative step to find the best-fit distribution for each station.

4.5. Random Forest

By integrating multiple decision trees, RF achieves robust predictions through averaging, exhibiting superiority in handling high-dimensional feature spaces and complex data structures, thereby ensuring reliable performance. It represents a machine learning approach that merges the principles of classification and regression trees with bagging techniques, incorporating an added level of randomness [34]. RF allows for the effortless tuning of parameters and requires minimal computational resources [35]. Its effectiveness has been widely established in multiple fields [36]; an example of this is Behrens, et al. [37], who applied it in their analysis of inflation forecasting. It is a supervised learning algorithm that utilizes regression trees as its base learner. Notably, RF can generate multiple trees without pruning, thereby enhancing its predictive capabilities. The training process incorporates two sources of randomness. Firstly, the algorithm employs a bootstrap sampling technique, where each tree is constructed from a random subset of the original data with replacement, allowing for repeated samples. Secondly, RF introduces feature randomness by randomly selecting a subset of candidate predictor variables at each node, and choosing the optimal split based on the most suitable value. The accuracy of the predictions is sensitive to hyperparameter tuning, yet no standardized approach exists for selecting optimal parameters, necessitating a trial-and-error approach [36]. For this study, the Python sklearn library [38] was utilized to create the RF model.

5. Results

Following a thorough evaluation of the data and the application of the statistical methods described in the Methodology section, we have obtained results that provide new insights into an alternative methodology for estimating precipitation return periods. The results are presented below, organized into different sections to facilitate comprehension of the analysis stages.

5.1. Implementation of the Methodology Based on Random Forest

In this study, an RF model was employed to estimate precipitation associated with different return periods, comparing its predictions with those obtained using traditional statistical distributions. The key implementation stages are detailed below:

1. Selection of Input and Output Variables:

   The observed annual maximum precipitation values were ranked from highest to lowest, and their exceedance probabilities were calculated. The exceedance probability was used as the sole predictor variable (input), while precipitation values served as the target variable (output). This approach was based on the direct relationship between exceedance probability and precipitation in frequency analysis, a common methodology in probabilistic hydrology.

2. Dataset Division:

   The dataset was divided into three subsets to ensure robust model evaluation:

   Training (60%): To adjust the model parameters.

   Validation (20%): To optimize hyperparameters through cross-validation.

   Testing (20%): To evaluate the model’s final performance.

   The division was performed randomly using a fixed seed (random_state = 42) to ensure the reproducibility of results.

3. Hyperparameter Optimization:

   Different hyperparameter configurations were explored to maximize model performance. Specifically, these configurations were as follows:

   Number of trees (n_estimators): Rating from 50 to 500, in increments of 50.

   Minimum number of samples to split a node (min_samples_split): Ranging from 2 to 10, in increments of 2. A 5-fold cross-validation was applied to compute the negative mean squared error (MSE) and select the configuration that minimized this error. This process ensured that the model avoided overfitting and generalized well to unseen data.

4. Model Evaluation:

   The final model was trained using the optimal hyperparameters and evaluated on the test set. The evaluation metrics included:

   Mean Squared Error (MSE) and Root Mean Squared Error (RMSE): To quantify the average error.

   Coefficient of Determination (R²): To measure the proportion of explained variance.

   Mean Absolute Error (MAE): To assess the average magnitude of errors without bias.

5. Prediction for Different Return Periods:

   Return periods of 2, 5, 10, 20, 50, and 100 years were defined, and their corresponding exceedance probabilities were calculated. These probabilities were input into the model to obtain precipitation predictions associated with each return period.

6. Visualization of Results:

   The relationship between observed and predicted exceedance probabilities was plotted to visually assess the model’s fit.

The methodology and algorithm were developed by the authors, and it is described in Figure 7, where the flow is lineal in comparison to the traditional approach seen in Section 4.4.

Following the previously described steps, once the data from SENAMHI were collected, they were organized and then used to calculate the probability of exceedance and divide the dataset. Subsequently, a hyperparameter search based on cross-validation yielded the results presented in Table 2.

5.2. Best-Fit Distributions for Each Station and Comparison with Random Forest

The traditional approach involves using the annual maximum precipitation corresponding to each station, followed by ordering all these data from highest to lowest to determine the probability of occurrence for each value. This can be graphed to better understand the occurrence probabilities in relation to precipitation levels. At this point, probability density functions (PDFs) come into play, as they can fit the data and then evaluate the chosen function for a specific value, which in this case is the return period. However, the choice of the function is not arbitrary; it must undergo a goodness-of-fit test. In this study, the Kolmogorov-Smirnov (K-S) test was selected.

Following this line of thought, the innovative approach incorporates the RF algorithm, which, like the traditional approach, starts by using the annual maximum precipitation and the probability of occurrence for each value. The difference lies in the fact that RF learns the patterns in the data and can adapt more effectively than traditional statistical functions.

In this way, the model can learn from the respective training set and improve its predictions. The test set was then subjected to the metrics (MSE, RMSE, R², and MAE) to evaluate accuracy, and the PDFs were similarly evaluated using the same test set that was used for the RF algorithm. For this analysis, the Hydrognomon software was utilized, which is able to determine best-fit distribution and predict precipitations to specific return periods. The results for each station are shown in Table 3. It should be noted that the El Frayle, Caraveli, Cotahuasi, Punta Atico, Sibayo, and Yanaquihua stations had missing values in their annual maximum precipitation dataset, for which the moving averages method was used to fill in the missing data. Table 4 shows the different results of the metrics applied to the best-fit distribution for each station in the Arequipa department and Table 5 shows the results using the methodology based on the RF algorithm.

To better understand the behavior of the data and their distribution, Figure 8 presents the frequency analysis of the distributions that show an acceptable fit to the annual maximum precipitation data. On the other hand, Figure 9 highlights the distribution that offers the best fit for the annual maximum precipitation based on its probability of occurrence.

5.3. Estimating the Return Period Using the Best-Fit Distribution and Random Forest

Once the best-fitting PDFs for each station are identified, the precipitation for each selected return period can be determined. Table 6 presents the return periods (2, 5, 10, 20, 50, and 100 years) along with the corresponding results.

Figure 10 and Figure 11 show the distribution of the predictions obtained from the PDFs listed in Table 6, which in this case represent precipitation values as a function of the return period.

On the other hand, the precipitation results for the previously mentioned return periods are displayed in Table 7.

Finally, the distribution of the values is shown in Figure 12 and Figure 13.

6. Discussion

By focusing on Arequipa, this study fills a significant gap in the hydrological research of key regions of Peru, providing a replicable methodology for other areas facing similar data problems and climate challenges. This strengthens its contribution to sustainable development and resilience to climate change. This commonly applied method includes distributions such as GEV, Pearson 3, and Log Pearson 3, which allow estimates to be made based on available historical data. Achieving national resilience to disasters in the face of hydrological phenomena is essential for countries such as Peru [39,40], where adequately estimating rainfall is a major challenge.

When compared to similar studies worldwide, the methodology presented here aligns with global trends in leveraging advanced modeling techniques for hydrological studies. For instance, Mandal and Choudhury [10] used probability functions to predict extreme rainfall patterns in island environments, while Zhai, Wang, Feng, Xue, Ma, Tian and Sun [11] applied methods to analyze long-term rainfall trends in Asia. In contrast, this study emphasizes the applicability of machine learning techniques, specifically Random Forest (RF), to address challenges unique to mountainous and arid regions like Arequipa.

One of the primary strengths of RF is its ability to handle high-dimensional data, making it ideal for applications with a large number of predictor variables. RF is robust to outliers and noise in the data. However, RF has a tendency to overfit when dealing with noisy or irrelevant features, which can lead to poor generalization performance, and also can be difficult to interpret, especially when dealing with complex models or high-dimensional data. Despite these limitations, this study demonstrates that RF outperforms traditional probability density functions (PDFs) in 61% of stations using MSE, RMSE, and R² metrics, and in 69% when employing MAE, confirming its robustness in most cases.

Moreover, the findings reveal that stations with varying standard deviation values show similar fits between RF and PDFs, suggesting no direct relationship between data variability and model choice. This aligns with studies like those of Si, Shao, Zhao, Wei, Hou and Huang [12], which underscore the importance of adapting modeling approaches to local data characteristics rather than relying solely on statistical assumptions.

The research also shows a notable difference in the behavior of RF and PDFs for long return periods. RF predictions tend to stabilize beyond 50 years due to their reliance on available empirical data, whereas traditional distributions like Gumbel continue to extrapolate values, reflecting more extreme behavior. This observation is consistent with findings in other regions where PDFs remain the preferred choice for high return period estimations. However, RF’s superior performance for short and medium return periods highlights its utility in contexts requiring accurate modeling within observed data ranges, such as planning hydraulic infrastructures and identifying vulnerable areas.

As observed globally, studies analyzing rainfall patterns with probability functions have been conducted for cities, regions, and entire countries. However, this study is among the first to apply RF in Peru, filling a critical gap and providing a foundation for future research. By addressing the unique challenges of Arequipa, including limited meteorological data and climatic variability, this research contributes not only to hydrological resilience in Peru but also to the broader field of hydrological modeling in data-scarce environments.

7. Conclusions

This study aims to provide an innovative approach to the calculation of precipitation for different return periods with a new method to improve accuracy, as well as offer an alternative for regions facing similar cases of data scarcity. Currently, there are cities and regions around the world lacking updated information from their meteorological stations due to various reasons, such as high maintenance costs and low public interest, among others.

Improved precision in precipitation estimates for various return periods has a significant impact on water resource planning and management. Achieving higher accuracy in these estimates enables the more effective and efficient planning of water resources, including allocation for agriculture, industry, and domestic consumption. This, in turn, reduces flood risk and associated damage by identifying high-risk areas and taking preventive measures. Moreover, optimal water supply allocation is ensured through the efficient distribution of available water, meeting the needs of populations and industries. Improved precipitation estimates also yield significant cost savings in the infrastructure, maintenance, and operation of water systems. This integrated approach to resource management is crucial for ensuring the sustainability and resilience of water systems in the face of increasing climate change.

Finally, a future research line involves analyzing daily, monthly, and annual rainfall patterns in Peruvian regions. Along these lines, there are other machine learning algorithms capable of performing regression, as has been conducted in this study (e.g., neural networks). Thus, another research avenue is to explore the calculation using different algorithms and to analyze the differences each one offers.

Author Contributions

Conceptualization, J.A.-V., S.V.-F. and G.A.; methodology, J.A.-V., S.V.-F. and G.A.; software, J.A.-V.; validation, J.A.-V., G.A. and S.V.-F.; formal analysis, J.A.-V. and G.A.; investigation, J.A.-V. and G.A.; resources, J.A.-V., A.J.E.V., J.J.-Q. and E.R.-C.; data curation, J.A.-V.; writing—original draft preparation, J.A.-V., S.V.-F. and J.B.; writing—review and editing, J.A.-V., S.V.-F., A.J.E.V. and J.B.; visualization, J.A.-V., J.J.-Q. and E.R.-C.; supervision, J.A.-V., A.J.E.V. and J.B.; project administration, J.A.-V., A.J.E.V. and J.B.; funding acquisition, A.J.E.V. and J.B. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding author, J.B. The data are not publicly available due to ethical restrictions.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Time series from La Pampilla station to Las Salinas station.

Figure 2. Time series from Andahua station to Chichas station.

Figure 3. Time series from Chivay station to Imata station.

Figure 4. Time series from La Angostura station to Pampa Blanca station.

Figure 5. Time series from Pillones station to Yanaquihua station.

Figure 6. Flow chart of traditional approach.

Figure 7. Flow chart of RF model.

Figure 8. Q-Q plot of the distributions at the Las Salinas station.

Figure 9. Q-Q plot of the L-Moments GEV Min distribution at the Las Salinas station.

Figure 10. Return periods for each station using PDFs.

Figure 11. Return periods for each station using PDFs.

Figure 12. Return periods for each station using RF.

Figure 13. Return periods for each station using RF.

Table 1

RF studies in Peru.

ID	Author	Line of Study	Application	Study Zone
1	Cerna-Cueva, et al. [18]	Agricultural Science	The influence of high Andean grasslands on the annual occurrence of landslides is predicted from estimation models such as RF.	Trujillo
2	Thomas, et al. [19]	Environmental Sciences	Predictive models based on Random Forest are developed to generate national and regional maps of cadmium concentrations in soils and cocoa beans in Peru.	Piura
3	López, et al. [20]	Computing and Meteorology	A more effective machine learning model has been identified to associate dengue cases with climatic variables in the Loreto region, Peru, using epidemiological and meteorological data collected between 2016 and 2022.	Loreto
4	Portuguez-maurtua, et al. [21]	Meteorology	To evaluate different methods to fill gaps in daily precipitation data series in the Malas, Omas, and Cañete river basins (located on the Pacific slope in central Peru), using regression models (RM) and machine learning (ML) techniques.	Lima
5	Fernandez-Palomino, et al. [22]	Hydrometeorology	To present a novel approach to estimate precipitation patterns and generate a new hydrologically corrected daily precipitation dataset, called RAIN4PE, in order to improve the accuracy of hydrological simulation in Peru and Ecuador.	Peru and Ecuador
6	Perez-Suarez, et al. [23]	Agronomy	Develops a machine learning model to predict the appearance of the pest Burkholderia glumae in rice crops in the San Martín region, Peru.	San Martín
7	Sales, et al. [24]	Zoology	To reconstruct the spatial patterns of pre-Columbian human occupation in the tropical Andes using a species distribution model (SDM) that integrates environmental variables and archeological data.	Bolivia, Perú and Ecuador
8	Rollenbeck, et al. [25]	Climatology	To present and evaluate two approaches to estimate quantitative precipitation (QPE) from reflectivity data from an X-band weather radar installed in Peru.	Piura
9	Brill, et al. [26]	Natural Hazards	The objective of the text is to develop and apply advanced statistical and data mining methods to model and assess damages caused by compound natural hazards, such as El Niño events.	Perú
10	Bax and Francesconi [27]	Geography	To investigate how climatic conditions and biophysical characteristics of the landscape influence deforestation processes in the tropical Andes of Peru, and how these variables can be used to improve conservation policies and land use regulation.	Tropical Andes of Peru

Table 2

Hyperparameters of the model.

N°	Station Name	Hyperparameters of Random Forest Model
		n Estimators	Min Samples Split
1	La Pampilla	200	2
2	Chiguata	200	2
3	El Frayle	200	2
4	La Joya	200	2
5	Las Salinas	50	2
6	Andahua	200	2
7	Aplao	200	2
8	Ayo	50	2
9	Caraveli	200	2
10	Chichas	200	2
11	Chivay	150	2
12	Choco	200	2
13	Cotahuasi	200	2
14	Huambo	200	2
15	Imata	50	2
16	La Angostura	400	2
17	Machahuay	300	2
18	Madrigal	200	2
19	Orcopampa	200	2
20	Pampa Blanca	150	2
21	Pillones	200	2
22	Porpera	200	2
23	Pullhay	200	2
24	Salamanca	200	2
25	Sibayo	300	2
26	Yanaquihua	150	2

Table 3

Statistical results and best-fit distribution for each station.

N°	Station Name	Mean	Standard Deviation	Best-Fit Statistic Results
				By K-S Test	D Max
1	La Pampilla	16.7	18.4	Log Pearson III	0.04174
2	Chiguata	21.8	11.4	GEV-Min (L-Moments)	0.05815
3	El Frayle	23.6	8.9	GEV-Max (k specified, L-Moments)	0.06645
4	La Joya	1.8	3	Pearson III	0.10001
5	Las Salinas	19.8	9	GEV-Min (L-Moments)	0.0503
6	Andahua	26.2	10	GEV-Max (kappa specified)	0.07339
7	Aplao	2.6	2.8	Pareto (L-Moments)	0.0598
8	Ayo	13.1	8.7	EV1 Max (Gumbel, L-Moments)	0.08204
9	Caraveli	9.4	7.9	Pareto (L-Moments)	0.03421
10	Chichas	17	7.6	Pareto (L-Moments)	0.07672
11	Chivay	26	8.5	Galton	0.0788
12	Choco	17.2	6.8	GEV-Max (L-Moments)	0.06083
13	Cotahuasi	21.8	9.4	GEV-Max (L-Moments)	0.07198
14	Huambo	24.4	8.5	GEV-Max (L-Moments)	0.07738
15	Imata	30	8.5	EV1 Max (Gumbel, L-Moments)	0.05637
16	La Angostura	36.2	8.5	GEV-Min (L-Moments)	0.06593
17	Machahuay	21.9	10.4	GEV-Max (L-Moments)	0.04968
18	Madrigal	24.7	7.4	GEV-Min (L-Moments)	0.07686
19	Orcopampa	25	11.8	GEV-Max (k specified, L-Moments)	0.05989
20	Pampa Blanca	2.3	2.7	GEV-Max (L-Moments)	0.05141
21	Pillones	25.5	9.1	GEV-Max (L-Moments)	0.05231
22	Porpera	29.8	13.9	Gamma	0.07828
23	Punta Atico	2.4	4.4	EV3-Min (Weibull, L-Moments)	0.14002
24	Salamanca	21.9	7.9	GEV-Max (L-Moments)	0.04083
25	Sibayo	32.2	8.6	EV1-Max (Gumbel, L-Moments)	0.06052
26	Yanaquihua	23.7	15.2	Exponential	0.06192

Table 4

Metric results using best-fit distribution.

N°	Station Name	MSE of Best-Fit Distribution	RMSE of Best-Fit Distribution	R2 of Best-Fit Distribution	MAE of Best-Fit Distribution
1	La Pampilla	1.4496	1.204	0.9825	0.7018
2	Chiguata	1.842	1.3572	0.9868	1.1138
3	El Frayle	1.3014	1.1408	0.9752	0.97
4	La Joya	0.1368	0.3698	0.9506	0.3252
5	Las Salinas	3.1163	1.7653	0.961	0.9508
6	Andahua	2.9286	1.7113	0.9274	1.2372
7	Aplao	0.1784	0.4224	0.9706	0.2584
8	Ayo	1.5126	1.2299	0.9659	0.9102
9	Caraveli	0.2142	0.4628	0.9966	0.3932
10	Chichas	0.3978	0.6307	0.9933	0.5282
11	Chivay	0.562	0.7497	0.9921	0.6781
12	Choco	0.5192	0.7206	0.9864	0.5285
13	Cotahuasi	4.876	2.2082	0.9292	1.5047
14	Huambo	1.4806	1.2168	0.9689	1.0231
15	Imata	1.1266	1.0614	0.9727	0.8105
16	La Angostura	1.7125	1.3086	0.9764	1.01
17	Machahuay	1.5692	1.2527	0.9874	1.0057
18	Madrigal	1.0907	1.0444	0.9801	1.0096
19	Orcopampa	1.1255	1.0609	0.9796	0.856
20	Pampa Blanca	0.2357	0.4855	0.9487	0.2442
21	Pillones	0.4197	0.6478	0.9923	0.4755
22	Porpera	4.2254	2.0556	0.9664	1.59
23	Pullhay	0.1283	0.3581	0.9785	0.2056
24	Salamanca	0.4174	0.6461	0.9929	0.5191
25	Sibayo	1.1997	1.0953	0.9846	0.8404
26	Yanaquihua	2.4553	1.5669	0.9856	0.9666

Table 5

Metric results using Random Forest.

N°	Station Name	MSE of RF	RMSE of RF	R2 of RF	MAE of RF
1	La Pampilla	0.3923	0.6263	0.9953	0.5654
2	Chiguata	0.7425	0.8617	0.9947	0.7008
3	El Frayle	1.9389	1.3924	0.9631	0.9134
4	La Joya	0.0064	0.0798	0.9977	0.0404
5	Las Salinas	1.9984	1.4137	0.975	0.7688
6	Andahua	5.2292	2.2868	0.8704	1.2069
7	Aplao	0.0277	0.1665	0.9954	0.1253
8	Ayo	0.6056	0.7782	0.9864	0.6093
9	Caraveli	0.3803	0.6167	0.9939	0.496
10	Chichas	0.5068	0.7119	0.9915	0.4733
11	Chivay	1.3044	1.1421	0.9817	0.8127
12	Choco	0.5063	0.7115	0.9868	0.5409
13	Cotahuasi	1.5652	1.2511	0.9773	0.7652
14	Huambo	3.9086	1.977	0.918	1.147
15	Imata	0.5459	0.7389	0.9868	0.462
16	La Angostura	0.243	0.4929	0.9967	0.4261
17	Machahuay	1.293	1.1371	0.9896	0.8575
18	Madrigal	0.1428	0.3779	0.9974	0.2717
19	Orcopampa	2.4231	1.5566	0.956	1.0194
20	Pampa Blanca	0.1027	0.3205	0.9777	0.1829
21	Pillones	1.5017	1.2254	0.9724	0.7548
22	Porpera	0.7163	0.8464	0.9943	0.6571
23	Pullhay	0.0196	0.1399	0.9967	0.0621
24	Salamanca	0.7139	0.8449	0.9879	0.6471
25	Sibayo	0.372	0.6099	0.9952	0.5282
26	Yanaquihua	3.3151	1.8207	0.9805	1.0363

Table 6

Return period results with PDFs.

N°	Station Name	Return Period Using the Best-Fit Distribution
		2	5	10	20	50	100
1	La Pampilla	12.1171	23.2693	32.7275	43.3751	59.5553	73.5711
2	Chiguata	20.0623	31.205	37.7953	43.5582	50.3423	55.0198
3	El Frayle	21.4111	28.9544	34.7028	40.8594	49.8822	57.5199
4	La Joya	0.50552	2.44163	4.77694	7.544	11.6142	14.9007
5	Las Salinas	17.5087	26.2854	32.2689	37.9196	45.0286	50.1978
6	Andahua	23.9241	31.9219	38.0166	44.5441	54.1105	62.2085
7	Aplao	1.66002	4.18662	6.20626	8.32425	11.2832	13.6484
8	Ayo	11.7147	18.8981	23.6542	28.2163	34.1215	38.5466
9	Caraveli	7.39489	15.9954	21.2182	25.5447	30.146	32.9401
10	Chichas	17.1382	24.9383	27.4542	28.6782	29.3906	29.6203
11	Chivay	25.5179	32.9247	37.1011	40.7182	44.9803	47.9388
12	Choco	16.3433	22.4535	26.2256	29.6519	33.8238	36.7665
13	Cotahuasi	21.193	29.0618	33.4858	37.2233	41.4202	44.1511
14	Huambo	22.9475	30.5055	35.588	40.5228	46.9981	51.9163
15	Imata	28.6667	35.971	40.8071	45.4459	51.4505	55.95
16	La Angostura	34.5117	42.8009	47.9716	52.6245	58.239	62.1876
17	Machahuay	21.0728	30.5602	36.0145	40.7026	46.0692	49.6289
18	Madrigal	24.1877	31.1045	34.8766	38.035	41.6168	44.0134
19	Orcopampa	22.5584	31.2643	37.8987	45.0041	55.4176	64.2326
20	Pampa Blanca	1.45983	3.34676	5.09711	7.28773	11.1191	14.968
21	Pillones	23.8342	31.5677	36.9413	42.2941	49.5232	55.1717
22	Porpera	27.7076	40.4569	48.4225	55.7071	64.6933	71.1579
23	Punta Atico	0.54632	3.03678	6.29706	10.7611	18.5293	25.8296
24	Salamanca	21.0112	28.1703	32.5173	36.4153	41.0938	44.347
25	Sibayo	30.6979	38.5262	43.7093	48.681	55.1163	59.9387
26	Yanaquihua	19.0273	32.9957	43.5624	54.1291	68.0975	78.6642

Table 7

Return period results with RF.

N°	Station Name	Return Period Using Random Forest
		2	5	10	20	50	100
1	La Pampilla	12.3145	23.3705	29.553	41.6665	91.2295	91.2295
2	Chiguata	20.45	33.6835	39.6505	43.268	46.548	46.548
3	El Frayle	21.6095	29.9575	33.919	40.703	51.5485	51.5485
4	La Joya	0.738	3.015	4.3275	6.2485	13.6365	13.6365
5	Las Salinas	18.402	24.776	33.304	41.164	41.778	41.778
6	Andahua	23.9275	31.1865	36.2735	47.167	60.88	60.88
7	Aplao	1.648	4.6795	6.6325	8.052	11.112	11.112
8	Ayo	11.5	16.688	21.798	36.876	39.684	39.684
9	Caraveli	7.335	16.1985	21.254	24.6105	28.5255	28.5255
10	Chichas	17.7705	23.34	26.431	29.764	31.876	31.876
11	Chivay	25.9007	30.54	36.9193	41.9053	47.0293	47.0293
12	Choco	17.0595	21.2495	25.661	32.033	36.7595	36.7595
13	Cotahuasi	21.874	26.9225	30.7595	35.853	50.5435	50.5435
14	Huambo	23.359	30.9575	35.2655	41.2445	48.6385	48.6385
15	Imata	28.716	34.996	39.502	52.996	55.692	55.692
16	La Angostura	34.427	42.8893	49.9313	53.7325	55.8245	55.8245
17	Machahuay	21.924	29.655	37.622	41.5513	46.377	46.377
18	Madrigal	24.622	32.687	34.576	36.8555	40.1955	40.1955
19	Orcopampa	22.618	29.1315	34.7625	46.621	69.7865	69.7865
20	Pampa Blanca	1.4147	3.4	5.5513	8.5553	11.5173	11.5173
21	Pillones	23.4165	31.5425	35.648	42.119	54.5	54.5
22	Porpera	29.698	38.4095	43.0625	51.138	74.688	74.688
23	Punta Atico	0.806	3.467	6.872	13.211	19.6095	19.6095
24	Salamanca	21.4755	28.717	32.197	35.1875	43.4105	43.4105
25	Sibayo	30.689	39.798	45.3777	48.4993	52.8307	52.8307
26	Yanaquihua	17.854	36.8733	45.5273	57.5333	65.0053	65.0053

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