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Supervised Machine Learning Application for Developing
a Predictive Model of the Monthly Phase of the Pacific
Decadal Oscillation
Indalecio Mendoza Uribe
Instituto Mexicano de Tecnología del Agua,
Mexico
indalecio_mendoza@tlaloc.imta.mx
Abstract. In this work, supervised machine learning was applied, using regres-
sion trees, to develop a predictive model of the monthly phase of the Pacific De-
cadal Oscillation. This oscillation is associated with the alteration of weather pat-
terns, mainly in the North Pacific and southwestern North America. As charac-
teristics, the records of the PDO phase of the 24 months prior to the forecast target
month were used. The predictive model developed presented an acceptable ca-
pacity to estimate the monthly phase of the PDO. This according to the perfor-
mance evaluation statistics corresponding to the Mean Absolute Error, Maximum
Error, Mean Quadratic Error and Pearson's Correlation, which obtained ranges of
[0.55,1.07], [1.58,3.29], [0.55,1.82] and [0.30,0.74] respectively for 20% of test
data for the period 1854-2020.
Keywords: Artificial intelligence, climate, regression trees.
1 Introduction
In climatology, machine learning has great potential, especially in phenomena of long
temporal development, as is the case of the Pacific Decadal Oscillation (PDO). PDO is
mainly characterized by changes in sea surface temperature (SST) in the Pacific Ocean
over 20° north latitude, as well as variation in sea level pressure and wind patterns. The
study of the PDO has gained relevance in recent years due to its association with the
alteration of weather patterns, mainly in the North Pacific and southwestern North
America [1, 2, 3, 4]. Alterations in the climate have significant socioeconomic impacts,
especially in countries that base their development on the management of their natural
resources [5].
In the area of artificial intelligence, various machine learning techniques have been
applied to understand, describe and predict the behavior of natural phenomena. Ovando
et al. [6] developed a model based on neural networks to predict the occurrence of frost
in Argentina, based on meteorological data of temperature, relative humidity, cloud
cover, wind direction and speed. On the other hand, Téllez-Valero et al. [7] developed
a system based on machine learning methods that improves the acquisition of data from
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Research in Computing Science 150(4), 2021pp. 15–26; rec. 2020-12-13; acc. 2021-02-10
natural disasters, the system automatically populates a database of natural disasters with
information extracted from online newspaper news. In addition, Haro-Rivera [8] ap-
plied a decision tree to identify predominant meteorological variables in the province
of Chimborazo, Ecuador. Finally, in this list of examples, Suárez et al. [9] analyzed the
meteorological phenomenon called DANA, which caused serious floods, human losses,
economic and infrastructure damage in the southeast of Spain during the month of Sep-
tember 2019, studying the phenomenon from the perspective of data analysis.
Machine learning is a data analysis technique that gives computers the ability to
learn from experience without relying on a given equation as a model. These algorithms
look for natural patterns in the data that generate knowledge. Algorithms adaptively
improve their performance as the number of samples available for learning increases.
In a general way, we can classify machine learning techniques as supervised and unsu-
pervised.
A supervised learning algorithm takes a set of known data (inputs) and known re-
sponses for this data (outputs) to train a model that can generate reasonable predictions
in response to new data. Supervised learning uses classification and regression tech-
niques to develop predictive models. In comparison, unsupervised learning looks for
hidden patterns or intrinsic structures in the data. Used to infer information from data
sets consisting of input data with no labeled responses. Among the most common un-
supervised learning techniques are neural networks [10], k-means [11], among other.
The objective of this work was to apply supervised machine learning through regres-
sion trees to develop a predictive model of the monthly phase of the Pacific Decadal
Oscillation. As characteristics, the records of the PDO phase of the 24 months prior to
the target month of prognosis were used.
2 Method
The development of the predictive model was carried out by applying three procedures.
First, the historical data set of the monthly value of the PDO was obtained for the period
1854-2020. These data were organized by month and grouped into training and test
data. Second, for each month of the year the regression tree corresponding to the pre-
dictive model was generated with the training data. Third, the monthly predictive mod-
els were applied on the test data sets. The results were evaluated using three continuous
error measurement metrics and one of correlation.
2.1 Dataset
The PDO is a pattern of anomalies of the SST, this fluctuation oscillates between -4
and 4 degrees centigrade, corresponding to the cold and warm phase respectively. The
PDO values indicate the variation of the SST with respect to the historical average. The
data was obtained from National Oceanic and Atmospheric Administration through the
URL https://www.ncdc.noaa.gov/teleconnections/pdo/data.csv. The data set corre-
sponds to the monthly deviation of the SST for the period 1854-2020 (see Fig. 1). For
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each forecast month (label) the values of the previous 24 months were assigned as char-
acteristics. The characteristics and labels for each month of the year were grouped in
separate files to facilitate their processing.
Fig. 1. Monthly anomaly of the Pacific Decadal Oscillation for the period 1854-2020 [12].
Machine learning consists of learning some properties of a data set and then verify-
ing those properties with another data set. A common practice in machine learning is
to evaluate an algorithm by dividing the data into two subsets. The majority set is dom-
inated by training data, from which the algorithm learns some properties. While the
second set of data is called test data, with which the ability of the model to predict
through the learned properties is verified. For this study, the training and test data set
were divided into a proportion of 80 and 20% respectively.
2.2 Generation of the Predictive Model
For each month of the year, the regression tree corresponding to the predictive model
was generated. Each predictive model was trained with 80% corresponding train-
ing data.
Classification and regression trees (CART) were developed by Breiman et al. [13].
Tree models where the target variable can take a finite set of values are called classifi-
cation trees. On the other hand, trees where the target variable can take continuous
values are called regression trees.
Let Y be the response variable and x be the vector with the set of predictor variables,
the problem corresponds to establishing a relationship between Y and x in such a way
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that it is possible to predict Y based on the values of x. Mathematically looking for
probability P(Y | x1, x2, …, xk).
The construction of the tree is done following a recursive binary division approach,
let N be the number of data and Nj the number of cases in class j.
The probability that a case is in class j given that it was located in the terminal node
t, is given by the Eq. 1.
(1)
and comply with:
(2)
Thus, the set of P(j|t) are the relative proportions of the cases in class j at node t [8].
To obtain the optimal tree, evaluate each subdivision among all possible trees, get
the root node and the subsequent ones, the algorithm must measure the predictions
achieved and evaluate them to select the best one. Fig. 2 shows a simplified form of a
regression tree.
Fig. 2. Simplified form of a regression tree.
In this study, machine learning was applied through the Scikit-Learn library of the
Python programming language, which integrates a wide range of machine learning al-
gorithms for supervised and unsupervised problems [14].
Specifically, the tree.DecisionTreeRegressor method was used to create the instance
of the predictive model; train_test_split to divide the training/test data set; mean_ab-
solute_error, mean_squared_error y max_error to measure mean absolute error, mean
square error and maximum error respectively; finally, the function plot_tree was used
to graph the regression trees.
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2.3 Statistical Validation of the Predictive Model
Monthly predictive models were applied on the corresponding test data sets. For the
evaluation of the monthly predictive model of the PDO phase, three continuous error
measurement metrics and Pearson's correlation were used. These metrics are recom-
mended for evaluating forecasts of a deterministic nature. These metrics are de-
scribed below.
The Mean Absolute Error (MAE) measures the magnitude of the errors in a set of
predictions, regardless of their direction [15, 16]. It corresponds to the average of the
absolute differences between the prediction and the observation where all the individual
differences have the same weight (Eq. 3):
(3)
where Pi is the prediction value at position i, Oi is the value observed at position i and
n is the sample size.
The Maximum Error (ME) allows to identify the largest absolute value of the ob-
served error between the prediction and the observation (Eq. 4). It belongs to the set of
objective functions used for the calibration of models [17]:
(4)
The Root Mean Square Root (RMSE) measures the mean magnitude of the error.
Corresponds to the square root of the average of the squared differences between the
prediction and the observation, therefore this measure has been used in the evaluation of
forecasting models [18, 19]. Amplifies and penalizes with greater force those errors of
greater magnitude (Eq. 5):
(5)
Pearson's Correlation, denoted as r (Eq. 6), is a normalized measure widely used to
establish relationships between two continuous quantitative variables [20, 21]. It allows
to show the joint variability and therefore to typify what happens with the data. The
coefficient can score values ranging from -1.0 to 1.0 and is interpreted as follows: val-
ues close to 1.0 indicate that there is a strong association between the variables, that is,
they increase or decrease in the same direction.
On the other hand, values close to -1.0 indicate that there is a strong negative asso-
ciation between the variables, that is, as one variable increases, the other decreases. A
value of 0.0 indicates that there is no correlation or it is a null correlation [22].
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(6)
where is the mean value of the predictions and is the mean value of the observa-
tions.
3 Results
For the creation of the monthly predictive models based on regressive trees, the con-
structor of the DecisionTreeRegressor class was used. Table 1 lists the parameters used
during the creation of the predictive model with which the best results were obtained.
Table 1. Predictive model creation parameters.
Parameter
Value
Description
criterion
mse
Function to measure the quality of the division.
splitter
best
Strategy used to choose the division at each node.
max_depth
None
Maximum depth of the tree. None indicates that nodes are
expanded until all sheets are pure or until all sheets contain
less than min_samples_split samples.
min_samples_split
2
The minimum number of samples required to divide an in-
ternal node.
min_samples_leaf
1
The minimum number of samples required to be in a leaf
node.
max_features
12
The number of features to consider when looking for the
best division.
random_state
5
Controls the randomness of the estimator. To obtain a de-
terministic behavior during the setting random_state must
be set to an integer.
As part of the training, the algorithm identifies the impact on the prognosis of each
of the characteristics. As can be seen in Table 2, in general, with 12 characteristics,
more than 90% importance is obtained in the forecast.
These 12 characteristics are not the same for all months of the year, therefore, in the
training stage, the 24 characteristics are initially considered, but the algorithm is in-
structed to only select the 12 most relevant characteristics. This reduction in dimensions
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allows the algorithm to be optimized by eliminating characteristics that do not contrib-
ute to the forecast.
Table 2. Percentage of importance by characteristics for monthly predictive models.
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Characteristics
1
1
0
1
0
3
8
4
2
0
1
3
1
2
0
1
1
0
0
3
1
0
0
2
4
7
3
1
0
3
1
1
2
1
1
0
3
3
1
4
0
1
2
0
2
0
0
3
5
4
0
0
5
5
2
1
3
2
0
2
5
1
2
0
0
6
0
1
1
0
1
3
0
2
0
2
1
0
7
1
0
0
4
3
2
1
2
0
0
1
0
8
0
2
0
0
2
0
1
1
3
0
1
0
9
2
1
1
1
0
2
2
1
5
0
1
6
10
4
3
2
0
3
1
0
3
2
2
1
3
11
3
0
2
0
0
0
1
2
0
0
2
4
12
0
1
4
0
0
0
1
0
3
2
4
2
13
8
6
0
0
0
0
1
4
0
2
0
1
14
0
1
1
2
1
0
2
0
1
0
1
0
15
6
0
3
1
5
0
1
1
1
2
1
0
16
3
1
2
1
1
0
0
2
3
1
0
0
17
0
3
1
0
1
2
0
0
1
7
5
7
18
1
2
1
0
0
3
0
5
0
1
1
11
19
2
1
1
5
1
0
2
2
3
4
4
2
20
0
3
1
1
2
6
3
3
0
2
1
2
21
5
1
0
5
1
0
3
2
2
6
0
2
22
0
1
1
3
2
2
1
37
7
1
1
0
23
39
49
56
50
51
35
58
14
36
38
41
40
24
19
20
15
23
18
31
15
8
27
18
24
11
Algorithm 1 presents in a simplified way the sequence of steps to divide the data
into the training/test subsets, feed the classifier (predictive model) with the training
data, apply the classifier on the test data, calculate model performance evaluation met-
rics, graphing and data storage. Clarification is made that the algorithm does not detail
the modules of dataReadingMonth() and graphingStorage().
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Algorithm 1: Simplified sequence to generate, train, apply and validate the monthly predic-
tive model
for month in range(0,12):
totalCharacteristics, totalLabels = dataReadingMonth(month)
trainingCharacteristics, testCharacteristics, trainingLabels, testLabels = \
train_test_split(totalCharacteristics, totalLabels,train_size=0.80, \
test_size=0.20, random_state= 5)
# Creation of the instance (object) of type DecisionTreeRegressor (predictive model)
predictiveModel = tree.DecisionTreeRegressor(criterion = 'mse', splitter = 'best', \
max_depth = None, min_samples_split = 2, min_samples_leaf = 1, \
max_features = 12, random_state=5)
# Feed the classifier with the training data (train the predictive model)
predictiveModel.fit(trainingCharacteristics,trainingLabels)
# Apply the predictive model to the test data set
predictions = predictiveModel.predict([testCharacteristics])
predictedLabels = predictions[0]
# Calculate MAE, ME, RMSE and r performance metrics
mae = round(mean_absolute_error(testLabels,predictedLabels),2)
me = round(max_error(testLabels,predictedLabels),2)
rmse = round(mean_squared_error(testLabels,predictedLabels),2)
pearson = sc.pearsonr(testLabels,predictedLabels)
r = round(pearson[0],2)
storageGraph(month,predictiveModel,mae,me,rmse,r)
Figures 3 and 4 show arbitrarily the trees corresponding to the predictive models for
the months of June and December, respectively.
Fig. 3. Regression tree for the month of June. The predictive model was trained with 80% of data
from the period 1854-2020. The strongest fill color indicates the majority class for classification.
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Research in Computing Science 150(4), 2021 ISSN 1870-4069
Fig. 4. Regression tree for the month of December. The predictive model was trained with 80%
of data from the period 1854-2020. The strongest fill color indicates the majority class for clas-
sification.
Monthly predictive models were applied for 20% of test data. Table 3 shows the
results of the four statistical metrics applied by the monthly predictive model. Besides
that, Fig. 5 shows the dispersion diagrams with the comparison between the observed
and predicted data.
Table 3. Result of the statistical metrics of the monthly predictive models.
Target Month
MAE
ME
RMSE
r
January
0.79
2.49
1.03
0.59
February
0.66
2.09
0.70
0.64
March
0.61
2.44
0.66
0.72
April
0.74
1.58
0.74
0.74
May
0.86
2.52
1.13
0.62
June
0.64
1.79
0.62
0.77
July
1.01
2.47
1.44
0.55
August
1.07
3.29
1.82
0.30
September
1.01
2.83
1.58
0.38
October
0.69
1.60
0.68
0.74
November
0.55
1.75
0.55
0.77
December
0.89
3.53
1.31
0.47
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Fig. 5. Monthly dispersion diagrams between observed and predicted values for 20% of test
data for the 1854-2020 period.
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4 Conclusions
Of the 24 characteristics considered, it was identified that characteristic 23 in eleven
months and characteristic 22 in the month of July, predominated as root node in the
trees of the predictive models, that is, these characteristics have a greater impact on
forecasts. In addition, it was distinguished that in 12 characteristics more than 90% of
importance is obtained in the prognosis.
The predictive model developed using machine learning presented an acceptable ca-
pacity to estimate the monthly phase of the PDO. This according to the results of the
performance evaluation statistics MAE, ME, RMSE and r obtained for 20% of test data,
with ranges of [0.55, 1.07], [1.58, 3.29], [0.55, 1.82] y [0.30, 0.74] respectively. There-
fore, it is considered that the predictive model developed can constitute a reference
forecasting tool, but not an exact one.
As future work, it is proposed to continue with the validation and adjustment of the
predictive model for its application in larger time windows, such as for seasonal fore-
cast (3 months), or even annual forecast.
Regarding the functionality of the Scikit-Learn library, this turned out to be docile
to implement and very efficient in its performance. The computational cost required for
the training and testing of the predictive model was of the order of seconds on a per-
sonal computer.
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