Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
https://doi.org/10.1007/s00477-021-02018-9
(0123456789().,-volV)(0123456789().
,- volV)
ORIGINAL PAPER
Artificial Intelligence models for prediction of the tide level in Venice
Francesco Granata1
•
Fabio Di Nunno1
Accepted: 2 April 2021 / Published online: 9 April 2021
The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
Abstract
The city of Venice is an extraordinary architectural, artistic and cultural heritage. Unfortunately, its conservation is
increasingly threatened by particularly significant high tides. Predicting the tide level in Venice, especially the high waters,
is an essential task for the protection of the city and the lagoon. Complex statistical or hydrodynamic models, which require
a large amount of input data, are currently used for this purpose. An effective alternative can be provided by models based
on Artificial Intelligence algorithms. In this study, several different forecasting models were developed and each model
was built in three variants, varying the implemented machine learning algorithm: M5P Regression Tree, Random Forest
and Multilayer Perceptron. Until now, regression tree models had never been used to forecast tide levels. All the proposed
models proved to be able to forecast the tide level in Venice with good accuracy. The M5P algorithm provided the best
performance in most cases. All the models based on M5P were characterized by a coefficient of determination between
0.924 and 0.996, while the Relative Absolute Error was between 5.98 and 26.84%. In addition, good predictions were
achieved by neglecting meteorological factors, even in the case of exceptionally high waters. Finally, satisfactory outcomes
were also obtained with a forecast horizon of several hours, while a further specific comparison showed that the models
based on the considered Machine Learning algorithms are able to outperform the AutoRegressive Integrated Moving
Average models with exogenous input variables in forecasting high water.
Keywords Tide prediction Machine learning M5P Random Forest Neural networks
1 Introduction
Tide prediction is often a matter of prime importance for
the peoples living along the coasts. Tides can strongly
affect navigation, fishing and recreational activities. Tidal
fluctuations have a major influence on coastal ecosystems.
The design of coastal engineering works requires knowledge of the extreme values of tidal oscillations. Additionally, many coastal cities are at risk of being flooded when
extreme high tide events occur and need to be properly
protected.
One of the most famous cases is undoubtedly represented by the city of Venice, which is in the middle of the
largest lagoon (Fig. 1) in the Mediterranean Sea (about 550
& Francesco Granata
f.granata@unicas.it
1
Department of Civil and Mechanical Engineering, University
of Cassino and Southern Lazio, Via G. Di Biasio 43,
03043 Cassino, FR, Italy
km2), located in the upper Adriatic Sea. The lagoon is
connected to the Adriatic Sea by three narrow inlets. Its
complex morphology changes continuously from the
mainland to the sea (Carniello et al. 2009). Life in the
Venetian Lagoon, and especially in the city of Venice, has
always been strongly conditioned by the high tide (Camuffo 1993). The observed tide in Venice is the sum of two
contributions: the astronomical tide, caused by the gravitational attraction of the Moon and the Sun, and the change
in sea level induced by weather conditions (Franco et al.
1982; Fagherazzi et al. 2005). In ordinary conditions, the
meteorological effects are small and the observed water
level is little different from that induced by the astronomical tide. The latter is basically semidiurnal: two
maximum heights and two minimum heights occur within
24 h. On the days of new moon and full moon the highest
tidal fluctuations are observed. The predictability of the
astronomical tide may be altered, even significantly, by
weather factors.
In the case of particular weather conditions, with low
pressure and strong winds from the South-East (Scirocco),
123
2538
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
Fig. 1 The Venice Lagoon, with the locations of the measuring stations
the meteorological effects become relevant: if they are in
phase with a maximum astronomical tide, they can lead to
the phenomenon of high water (‘‘acqua alta’’), conventionally defined as a tide level greater than 80 cm with
reference to the Punta della Salute tide gauge. At this tide
level, the flooding of the lower part of the city begins.
When the sea level exceeds 110 cm large areas begin to be
flooded.
When a strong wind from North-East (Bora) in the
northern Adriatic blows together with Scirocco in middle
Adriatic, the convergence of wind-induced marine currents
may further enhance the phenomenon. An even worse situation occurs in the event of a storm with severe low
pressure on the upper Adriatic and contemporary high
pressure on the lower Adriatic.
When a storm surge occurs on the Adriatic Sea, the
latter behaves like a resonant cavity due to its shape,
originating seiches of progressively decreasing amplitude
(Franco et al. 1982). Of course, this phenomenon also
123
affects the tide height. In recent decades the high waters
have become more frequent, also due to eustatism and
subsidence (Carminati et al. 2003; Tosi et al. 2009).
Moreover, the frequency and magnitude of high tide are
correlated to interdecadal climatic oscillations (Fagherazzi
et al. 2005).
Venice is a unique cultural, artistic and architectural
heritage in the world. Its defense is a matter of significant
national interest in Italy. In order to protect Venice from
high waters, the MOSE system is nearing completion. It
consists of 4 arrays of barriers that will be raised and
isolate the lagoon from the Adriatic Sea, when the tide
height will exceed 110 cm (Umgiesser 2020). Therefore,
the prediction of the tide level in the city of Venice, with a
particular focus on high waters, is a matter of considerable
practical interest, discussed for a long time (Finizio et al.
1972; Zaldivar et al. 2000; Carniello et al. 2005; Wei and
Billings 2006; Zampato et al. 2007). Currently, in order to
predict tide levels, both statistical models are used, such as
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
the BIGSUMDP expert model (Tosoni and Canestrelli
2011), and hydrodynamic models, such as the SHYFEM
model (Umgiesser et al. 2014). These are very complex
models, which require a very large number of input data.
In predicting complex natural phenomena, models
deriving from artificial intelligence studies have become
increasingly popular in recent years (e.g. Kişi 2007; Wang
et al. 2009; Nourani et al. 2014; Najafzadeh et al.
2016, 2018; Granata et al. 2018, 2020; Choubin et al. 2019;
Najafzadeh and Saberi-Movahed 2019; Ghorbani et al.
2019; Di Nunno and Granata 2020; Malik et al. 2020;
Najafzadeh and Oliveto 2020; Saberi-Movahed et al. 2020;
Aghelpour and Varshavian 2021). So far, these models
have been infrequently applied to tide prediction problems.
Deo and Chaudhari (1998) used Artificial Neural Networks
to predict tidal fluctuations in three different locations on
the coast of India. The networks were trained using three
different algorithms: error backpropagation, cascade correlation, and conjugate gradient. The authors found that
continuous prediction of the tidal levels over 15-min and
30-min intervals was good despite complex tidal propagation in the considered areas. Okwuashi and Ndehedehe
(2017) used Support Vector Regression for tide prediction
in Lagos, Nigeria. In particular, they compared the results
obtained with different kernel functions and obtained the
highest accuracy with a polynomial kernel. Imani et al.
(2018) applied Extreme Learning Machine and Relevance
Vector Machine models to forecast tide levels in Chiayi,
Taiwan. They showed that these algorithms outperformed
Support Vector Machines and Radial Basis Function networks in predicting daily mean, maximum and minimum
sea levels. Riazi (2020) proposed a deep neural networkbased model to forecast tide level in different locations of
Queensland, Australia. The model was developed considering 4 input neurons, 3 neurons for astronomical forces
and 1 neuron for geology and geomorphology factors, and
showed good accuracy.
In modeling problems involving the use of machine
learning algorithms, the main challenging aspects are the
selection of the algorithm, the selection of adequately
representative variables, and the availability of suitable data sets.
In this study, several different machine learning-based
models were built for the forecasting of the tide level in
Venice. Models differ in input variables. Three variants of
each model were developed, changing the implemented
algorithm: M5P Regression Tree, Random Forest, and
Multilayer Perceptron, used as a benchmark. Until now,
M5P and Random Forest had never been used to predict
tide levels. These algorithms are generally very accurate
and fast. Furthermore, they are particularly efficient in
dealing with highly non-linear problems. At the same time,
studies focusing on the prediction of high waters in Venice
2539
using Artificial Intelligence algorithms are not available so
far in the literature.
After a comparative analysis of the accuracy of the
different models and algorithms, the study focused in
particular on the ability of the simplest models to predict
the phenomenon of high water, in order to provide useful
operational tools. In addition, in order to further validate
the effectiveness of the above models, a comparison with
the classic AutoRegressive Integrated Moving Average
with exogenous inputs models was proposed.
2 Materials and methods
2.1 M5P algorithm
The M5P algorithm (Quinlan 1992) develops a regression
tree to get predictions. A Regression tree is a decision tree
in which the target variables are real numbers. In a
regression tree the following elements can be identified: a
root node, which contains all the data, internal nodes,
which impose conditions on the input variables, and leaf
nodes, represented by linear regression models on the target values of the variables.
During the development process the input dataset is
recursively divided into sub-domains and a multivariable
linear regression model is built in each of them. At the
initial step of the procedure, the entire dataset is divided
into two subsets, evaluating all the possible binary split on
every field. In the next steps, each partition is divided into
smaller subsets, so as the tree branches out. At each stage,
the procedure selects the splitting in two distinct partitions
that maximizes a function of the Least-Squared Deviation
(LSD), namely the within variance R(t) in the node t:
1 X
RðtÞ ¼
ðyi ym ðtÞÞ2
ð1Þ
NðtÞ i2t
where N(t) is the number of subset units in the node t, yi is
the target variable value for the i-th unit and ym is the mean
of the target variable in the node t. LSD also represents the
impurity of the node. The function to be maximized is
/ðsp ; tÞ ¼ RðtÞ pL RðtL Þ pR RðtR Þ
ð2Þ
in which tL and tR are respectively the left and right nodes
created by the split sp, while pL and pR are the portion units
allocated to the left and right child node. The split sp that
maximizes the value of U(sp, t) is selected.
The algorithm halts if a stopping rule occurs. The
stopping rules may refer to a minimum impurity level, a
minimum variation of the impurity provided by new splits,
a minimum number of elements in each node, or a maximum tree depth.
123
2540
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
A regression tree model may be suffering from overfitting when the tree development is complete. This may
reduce the model forecasting capability. This problem can
be solved by using a pruning technique, which reduces the
size of the tree removing the branches that do not significantly contribute to the predictive ability.
2.2 Random Forest
A Random Forest is an ensemble of simple regression trees
(Breiman 2001), whose predictions are combined to evaluate the final output. Several training subsets of the same
size are randomly selected, with replacement, from the
initial dataset. Then, a regression tree is built for each
subset. These trees will be different and will not lead to the
same results. However, Random Forest algorithm uses a
modified tree learning process (Granata and de Marinis
2017), which introduces an additional step of randomization. While in standard regression trees each node is partitioned using the best splitting among all features (i.e.
input variables), in a random forest each node is divided
considering a subset of the input variables, randomly
chosen at that node. The number of features to be considered for the division of each node is kept constant during
the tree growth process. Each tree is developed as far as
possible. There is no pruning process. Random Forest is
usually robust to outliers and is very stable. In this study,
forests consisting of 100 regression trees were considered.
The preliminary analyzes carried out for the optimization
of the models showed that increasing the number of trees
beyond this value did not lead to an improvement in performance, but only in the calculation times.
2.3 Multilayer Perceptron
A Multilayer Perceptron (MLP) is a class of feedforward
Artificial Neural Network (Ruck et al. 1990), which can
perform regression operations. A MLP contains at least
three layers of nodes: an input layer, a hidden layer and an
output layer. The input layer is made up of a set of neurons
representing the input variables. Each neuron in the hidden
layer processes the values from the previous layer with a
weighted linear summation and a non-linear activation
function. The values from the last hidden layer are transformed into output values by the output layer. The backpropagation algorithm is used for training.
In this study were used neural networks with 1 layer of
hidden neurons, whose number of neurons was equal to
(number of input variables ? 1)/2. Sigmoid was selected
as activation function. The assumed learning rate was 0.3,
while momentum rate for the backpropagation algorithm
was 0.2. These models’ structures proved to be optimal in
the addressed cases.
123
2.4 Input dataset, efficiency criteria,
and implementation
The tide level data were obtained from the measurements
carried out by the tide gauge of Punta della Salute (Fig. 1).
Instead, the weather data were acquired from measurements carried out at the CNR Platform, outside the Venetian Lagoon (Fig. 1). In the first phase of the study, all data
collected, at 30-min intervals, between January 1, 2012 and
December 31, 2015 were considered. The choice of the
dataset was imposed by the simultaneous availability of the
tide and weather data, provided by the Institute for Environmental Protection and Research (ISPRA).
The gravitational attraction of the celestial bodies was
taken into account by introducing the height of the astronomical tide among the input data. The astronomical tide at
the time of interest can be easily calculated, through harmonic analysis, by the equation (Ferla et al. 2007):
AT ¼ A0 þ
N
X
An cosðrn t jn Þ
ð3Þ
n¼1
where An is the amplitude, rn the angular frequency, jn the
phase delay of component n. The values of these parameters can be easily found on the Venice Municipality website (www.comune.venezia.it).
Four different well-known criteria were employed to
assess the effectiveness of the developed models:
•
•
•
•
the coefficient of determination R2,
the Mean Absolute Error (MAE),
the Root Mean Squared Error (RMSE),
the Relative Absolute Error (RAE).
These metrics are defined below:
!
Pm
2
2
i¼1 ðfi yi Þ
R ¼ 1 Pm
2
i¼1 ðya yi Þ
ð4Þ
In which m is the total number of experimental data, fi is
the forecasted value for data point i, and yi is the observed
value for data point i;
Pm
jfi yi j
ð5Þ
MAE ¼ i¼1
m
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm
2
i¼1 ðfi yi Þ
RMSE ¼
ð6Þ
m
Pm
jfi yi j
RAE ¼ Pmi¼1
ð7Þ
j
i¼1 ya yi j
where ya is the averaged value of the observed data. R2 can
be interpreted as the proportion of the variance in the
dependent variable that is predictable from the independent
variable: it is a goodness of fit measure. MAE evaluates the
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
average magnitude of the errors in a set of predictions,
without considering their sign. RMSE is the standard
deviation of the residuals, i.e. the prediction errors. RAE is
a normalization of the total absolute error, which is divided
by the absolute error of a simple predictor: the arithmetic
mean of the experimental values.
The four adopted metrics allow to fully characterize the
efficiency of the models, while alternative metrics generally do not alter the comparison (Granata 2019).
The considered algorithms were executed in MATLAB
environment. The optimization of the models was carried
out by means of a grid search procedure, which changed
the values of the parameters and checked the values of the
above defined metrics, in order to maximize coefficient of
determination and minimize errors. Of the initial dataset,
made up of over 70,000 vectors, 85% was used for model
training and 15% for testing and validation. A preliminary
analysis showed that the results are completely equivalent
to those obtainable with a k-fold cross validation. The raw
data underwent a normalization preprocessing, i.e. all
attributes were rescaled to the range 0 to 1. Normalization
proved useful in improving the efficiency of algorithms
because the input data had significantly varying scales.
3 Results and discussion
The previously described algorithms were used to develop
nineteen different tide level prediction models. They
mainly differ in the input variables. In addition, three
variants were developed for each model, changing the
implemented algorithm. The characteristics of all the
considered models are summarized in Table 1, together
with the values of the metrics referring to the testing phase.
For summary reasons, only the results of the models considered to be more interesting are shown and discussed
below.
Model A is characterized by the more complex structure. It has 28 input variables: the astronomical tide AT, the
wind speed WS, the wind direction WD, the barometric
pressure BP, and the previously observed tide levels, every
hour from 24 h before (Z-24) to 1 h before (Z-1) the time of
the forecast (Table 1). Model G has 9 input variables: the
astronomical tide AT, the weather parameters WS, WD,
and BP, the tide levels observed 24 h before Z-24, 6 h
before Z-6, 3 h before Z-3, 2 h before Z-2, and 1 h before Z1. Model I has 8 input variables and differs from Model G
only because Z-6 is missing among the input variables.
Model N has 7 input variables: AT, WS, WD, BP, Z-24, Z-2,
Z-1. Model Q has 6 input variables and differs from Model
N because Z-24 is missing among the input variables. In the
previous models, the meteorological variables refer to 2 h
before the forecast time. Model T is the simplest, being
2541
characterized by only 3 input variables: AT, Z-2, Z-1. This
model completely neglects weather variables. Model U has
24 input variables: the astronomical tide AT, the meteorological parameters WS, WD, and BP, and the previously
observed tide levels, every hour from 24 h before (Z-24) to
5 h before (Z-5) the time of the forecast.
As an example, Fig. 2 shows the comparisons between
the predicted and measured values of the tide level, for
Model A and the three different algorithms. The comparison for all the test values is shown on the left, while a
comparison in the form of a time series is shown on the
right, for a short period of time characterized by high
waters.
Model A (Fig. 2) led to very accurate results. M5P
(R2 = 0.9958, RAE = 5.98%) proved to be the best performing algorithm of the three considered. RF showed a
significant decrease in performance (R2 = 0.9902, RAE =
9.41%), contrary to expectations. MLP provided results
(R2 = 0.9958, RAE = 6.17%) comparable to those
obtained with M5P. The significant drawback of using
model A lies in the large number of observed sea level
values to be included in the input data.
The forecasting capabilities of Models G and I were
very similar. In addition, the three different algorithms
provided similar results. M5P was a little more accurate
than the other two procedures. All algorithms predicted
high waters with good accuracy, while RF, similarly to
what was shown by model A (Fig. 2c), exhibited a slight
tendency to overestimate low tide levels.
Model N, which differs from Model I in that it does not
include Z-3 among the input variables, was slightly less
accurate than the latter. Again, M5P outperformed both RF
and MLP. In any case, the differences in the performance
of the three algorithms were minimal. Model Q, which
includes among the input variables Z-2 and Z-1, in addition
to the astronomical tide and meteorological factors, showed
a further slight decrease in accuracy. M5P confirmed to be
the most performing algorithm (R2 = 0.9882, RAE =
10.54%). Finally, Model T, based only on theastronomical
tide and the observed values Z-2 and Z-1, also led to good
predictions. M5P algorithm provided again the best outcomes (R2 = 0.9855, RAE = 11.78%). Finally, Model U,
which will be discussed at the end of this section, was the
least accurate, though still with satisfactory results
(R2 = 0.9341, RAE = 24.45% for Random Forest).
It should be noted that, unexpectedly, the M5P algorithm outperformed RF in most cases. This result did not
depend on the number of trees considered in the random
forest.
The developed models showed different levels of
accuracy in different estimation ranges of tide height. For a
deeper understanding of this issue, the entire range of
observed tide values was divided into three subintervals: -
123
2542
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
Table 1 Summary of models and results (N.I.V. = number of input variables)
MODEL
N.I.V
INPUT VARIABLES
ALGORITHM
R2
MAE
[cm]
RMSE
[cm]
RAE
A
28
AT, WS, WD, BP, Z-24, Z-23, Z-22, Z-21, Z-20, Z-19, Z-18, Z-17, Z-16, Z15, Z-14, Z-13, Z-12, Z-11, Z-10, Z-9, Z-8, Z-7, Z-6, Z-5, Z-4, Z-3, Z-2, Z-1
M5P
0.9958
1.298
1.746
5.98%
RF
0.9902
2.016
2.651
9.41%
MLP
0.9958
1.341
1.791
6.17%
M5P
0.9843
2.557
3.391
11.77%
RF
0.9732
3.341
4.366
15.59%
MLP
0.9857
2.399
3.228
11.19%
M5P
RF
0.9952
0.9924
1.337
1.767
1.831
2.324
6.24%
8.25%
MLP
0.9950
1.619
2.119
7.56%
M5P
0.9948
1.405
1.911
6.56%
RF
0.9918
1.832
2.399
8.55%
MLP
0.9944
1.844
2.361
8.61%
M5P
0.9936
1.583
2.104
7.39%
RF
0.9916
1.872
2.435
8.74%
MLP
0.9926
1.76
2.295
8.14%
M5P
0.9724
3.459
4.409
16.15%
RF
0.9736
3.355
4.306
15.66%
MLP
0.9728
3.784
4.753
17.66%
M5P
0.9936
1.608
2.129
7.50%
RF
0.9920
1.823
2.36
8.51%
MLP
0.9928
1.749
2.271
8.16%
M5P
RF
0.9720
0.9720
3.483
3.451
4.436
4.431
16.26%
16.11%
MLP
0.9716
3.679
4.627
17.17%
M5P
0.9928
1.704
2.275
7.95%
RF
0.9902
2.029
2.621
9.47%
MLP
0.9920
1.941
2.523
8.98%
B
27
AT, WS, WD, BP, Z-24, Z-23, Z-22, Z-21, Z-20, Z-19, Z-18, Z-17, Z-16, Z15, Z-14, Z-13, Z-12, Z-11, Z-10, Z-9, Z-8, Z-7, Z-6, Z-5, Z-4, Z-3, Z-2
C
16
AT, WS, WD, BP, Z-23, Z-21, Z-19, Z-17, Z-15, Z-13, Z-11, Z-9, Z-7, Z-5,
Z-3, Z-1
D
12
AT, WS, WD, BP, Z-22, Z-19, Z-16, Z-13, Z-10, Z-7, Z-4, Z-1
E
F
G
10
9
9
AT, WS, WD, BP, Z-21, Z-17, Z-13, Z-9, Z-5, Z-1
AT, WS, WD, BP, Z-24, Z-18, Z-12, Z-6, Z-1
AT, WS, WD, BP, Z-24, Z-6, Z-3, Z-2, Z-1
H
8
AT, WS, WD, BP, Z-24, Z-18, Z-6, Z-1
I
8
AT, WS, WD, BP, Z-24, Z-3, Z-2, Z-1
L
M
N
O
P
Q
R
7
7
7
7
6
6
5
123
AT, WS, WD, BP, Z-24, Z-3, Z-2
AT, WS, WD, BP, Z-24, Z-12, Z-1
AT, WS, WD, BP, Z-24, Z-2, Z-1
AT, WS, WD, BP, Z-3, Z-2, Z-1
AT, WS, WD, BP, Z-24, Z-1
AT, WS, WD, BP, Z-2, Z-1
AT, WS, WD, Z-2, Z-1
M5P
0.9692
3.582
4.681
16.72%
RF
0.9657
3.77
4.907
17.59%
MLP
0.9685
4.051
5.223
18.90%
M5P
0.9677
3.787
4.788
17.68%
RF
0.9679
3.748
4.771
17.49%
MLP
0.9675
4.084
5.166
19.07%
M5P
0.9908
1.971
2.573
9.19%
RF
0.9892
2.123
2.742
9.91%
MLP
0.9902
2.076
2.687
9.60%
M5P
0.9918
1.802
2.419
8.41%
RF
MLP
0.9880
0.9902
2.294
2.076
2.923
2.687
10.61%
9.60%
M5P
0.9661
3.901
4.897
18.21%
RF
0.9649
3.949
4.969
18.44%
MLP
0.9641
4.312
5.379
20.13%
M5P
0.9882
2.277
2.926
10.54%
RF
0.9862
2.456
3.137
11.46%
MLP
0.9872
2.365
3.001
10.94%
M5P
0.9859
2.496
3.179
11.65%
RF
0.9841
2.652
3.369
12.38%
MLP
0.9837
3.043
3.778
14.20%
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
2543
Table 1 (continued)
R2
MODEL
N.I.V
INPUT VARIABLES
ALGORITHM
S
4
AT, BP, Z-2, Z-1
M5P
0.9882
2.261
2.928
10.55%
RF
0.9859
2.483
3.190
11.59%
MLP
0.9866
2.505
3.160
11.69%
M5P
RF
0.9855
0.9809
2.524
2.907
3.213
3.702
11.78%
13.57%
MLP
0.9835
2.980
3.719
13.91%
M5P
0.9242
5.749
7.319
26.84%
RF
0.9341
5.237
6.823
24.45%
MLP
0.9361
5.425
6.992
25.32%
T
3
AT, Z-2, Z-1
U
24
AT, WS, WD, BP, Z-24, Z-23, Z-22, Z-21, Z-20, Z-19, Z-18, Z-17, Z-16, Z15, Z-14, Z-13, Z-12, Z-11, Z-10, Z-9, Z-8, Z-7, Z-6, Z-5
60–0, 0–80, and 80–120 cm. The RAE was assessed in
each subinterval, for each model. The values are shown in
Fig. 3a. All models showed the lowest RAE values in the
range 0–80 cm, and the highest RAE values in the range
80–120 cm. These higher RAE values for high waters was
due to the lower number of data available for models’
training that are in the range 80–120 cm. Of the three
algorithms, M5P showed the lowest RAE value in all
subintervals.
However, the increase of RAE in the range Z [ 80 cm
did not involve a significant reduction in accuracy, as can
be seen from the box-plots of the residuals (i.e. the difference between expected and measured values) reported in
Fig. 3b for the different considered M5P-based models. For
each given model, the residuals were comparable in the
different subintervals. In addition, The distribution of
residuals is rather symmetrical for all models.
Furthermore, Fig. 3b shows that in the case of high
waters, Model T was characterized by errors comparable to
those of most other models. Since Model T does not require
meteorological data as input, it was possible to retrain the
model with a longer time series (January 1, 2009–December 31, 2018) and verify its ability to predict high
waters in a much larger number of cases.
Figure 4 shows on the left the comparison between the
expected and observed values of the high waters that
occurred between 2 July 2017 and 31 December 2018, on
the right the comparison in the form of a time series
between expected and observed values during the exceptional event started on October 29, 2018. The results of the
comparisons are satisfactory. Model T, especially in the
M5P-based variant, proved to be able to adequately predict
even the most severe high water phenomena. In some
cases, RF and MLP slightly underestimated severe events
MAE
[cm]
RMSE
[cm]
RAE
(Fig. 4c and Fig. 4e), while M5P did not show this tendency (Fig. 4a).
As for the October 29, 2018 event, both M5P (Fig. 4b)
and MLP (Fig. 4f) were able to accurately predict all high
tide peaks, while they were not as effective with the first
two low tide peaks. RF, on the other hand, appreciably
underestimated the more severe high tide peak (Fig. 4d).
It is interesting to note that, from a preliminary analysis,
it was found that the results of the different models showed
negligible variations if the weather data referred to 1 h or
3 h before the forecast time, instead of 2 h. This result also
confirms a poor sensitivity of forecast models based on
machine learning to meteorological parameters.
Model L, which was not considered in detail in the
discussion of the results, shows that as the forecast horizon
increases, there is a slight reduction in accuracy. Model U
further confirms this trend. However, Model U responds to
the need to provide reliable predictions even with wider
forecast horizons, in this case 5 h. This forecast horizon
should, for example, enable the procedure for lifting the
MOSE gates to be activated. Even if the accuracy of Model
U is lower than that of the models previously discussed,
while still requiring a large number of input parameters, it
demonstrates how an approach based on regression tree
algorithms, or on MLP neural networks, is adequate to get
good forecasts even several hours in advance.
Figure 5 shows the predictions obtained with the three
Model U variants over the period from 15 to 18 October
2015. The three variants of the model provided good
results, accurately predicting the first peak of 111 cm, even
if they then appreciably overestimated the third and the
fourth peak, which were lower.
The same Fig. 5 shows a comparison with 2 different
AutoRegressive Integrated Moving Average with exogenous inputs (ARIMAX) models, used for the prediction of
123
2544
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
Fig. 2 Model A: comparisons between the predicted and measured values
the same tide event and with the same forecast horizon,
equal to 5 h. Autoregressive integrated moving average
123
models, in different variants, has been widely used in
hydrology for time series predictions (Box and Jenkins
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
2545
Fig. 3 Accuracy assessment in
different subintervals
1968; Jalalkamali et al. 2015; Papacharalampous et al.
2018; Aghelpour et al. 2021). However, a detailed
description of an ARIMAX model is not reported in this
article because it is beyond the scope of this study. It can be
found, for example, in Fan et al. (2009).
The ARIMAX-1 model, which requires both the astronomical tide and all the meteorological parameters indicated above as exogenous variables, led to an
underestimation of the first peak, to an overestimation of
the second and fourth peaks, while correctly predicted the
third and fifth peaks.
The ARIMAX-2 model, which requires only the astronomical tide as exogenous input variable, provided poor
predictions over the entire investigated period. This result
also highlights a greater sensitivity of ARIMAX based
models to meteorological parameters in comparison with
the Machine Learning based-models. Overall, ARIMAX
models appear to be less effective than models based on
Machine Learning algorithms in predicting the wide
oscillations observed during high water phenomena.
A more in-depth comparison between forecasting models based on Machine Learning algorithms and ARIMAX
models, as well as the development of hybrid models based
on both (Moeeni and Bonakdari 2017), could be the subject
of future developments of this research. Further future
developments of this research aim to obtain models characterized by remarkable accuracy even with long-term
forecast horizons.
123
2546
Fig. 4 Prediction of high waters by Model T
123
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
2547
140
RF
M5P
MLP
Experimental data
ARIMAX-1
ARIMAX-2
120
100
Z [cm]
80
60
40
20
0
-20
Fig. 5 Forecast of tide fluctuations with a 5 h advance: comparison between ML-based models and ARIMAX models
4 Conclusions
Authors’ contributions Both authors contributed equally to this
work.
The prediction of the tide level in Venice, especially of the
most severe high waters, is an essential task for the protection of the city and the lagoon.
In this study, several different forecasting models were
developed and each model was built in three variants,
changing the adopted machine learning algorithm: M5P
Regression Tree, Random Forest and Multilayer Perceptron. The main findings of this study can be summarized as
follows:
Funding This research received no external funding.
• all the proposed models, in all their variants, predicted
the tide height in Venice with good accuracy;
• M5P outperformed the other two algorithms in most
cases;
• good forecasts can also be obtained by completely
neglecting the weather features, even in the case of
exceptionally high waters;
• it is possible to develop models based on M5P, RF and
MLP that provide accurate predictions even with
forecast horizons of several hours;
• M5P, Random Forest, and MLP appear to be able to
outperform ARIMAX models in predicting the wide
oscillations observed during high water phenomena.
A constantly updated training dataset will also allow to
take into account changes in sea level due to climate
change and subsidence.
Data availability The data are freely available online on the website:
https://www.venezia.isprambiente.it/index.php?folder_id=20&lang_
id=2
Declarations
Conflicts of interest The authors declare that they have no conflicts of
interest.
References
Aghelpour P, Varshavian V (2020) Evaluation of stochastic and
artificial intelligence models in modeling and predicting of river
daily flow time series. Stoch Environ Res Risk Assess 34:33–50
Aghelpour P, Bahrami-Pichaghchi H, Varshavian V (2021) Hydrological drought forecasting using multi-scalar streamflow
drought index, stochastic models and machine learning
approaches, in northern Iran. Stoch Environ Res Risk Assess
26(4):1–21
Box GEP, Jenkins GM (1968) Some recent advances in forecasting
and control. J R Stat Soc Ser C (Appl Stat) 17(2):91–109
Breiman L (2001) Random forests. Mach Learn 45(1):5–32
Camuffo D (1993) Analysis of the sea surges at Venice from AD 782
to 1990. Theor Appl Climatol 47(1):1–14
Carminati E, Doglioni C, Scrocca D (2003) Apennines subductionrelated subsidence of Venice (Italy). Geophys Res Lett 30(13):
1–4
Carniello L, Defina A, Fagherazzi S, D’Alpaos L (2005) A combined
wind wave–tidal model for the Venice lagoon, Italy. J Geophys
Res Earth Surf 110:1–15
123
2548
Stochastic Environmental Research and Risk Assessment (2021) 35:2537–2548
Carniello L, Defina A, D’Alpaos L (2009) Morphological evolution of
the Venice lagoon: evidence from the past and trend for the
future. J Geophys Res Earth Surf 114:1–10
Choubin B, Mosavi A, Alamdarloo EH, Hosseini FS, Shamshirband
S, Dashtekian K, Ghamisi P (2019) Earth fissure hazard
prediction using machine learning models. Environ Res
179:108770
Deo MC, Chaudhari G (1998) Tide prediction using neural networks.
Comput Aided Civ Infrastruct Eng 13(2):113–120
Di Nunno F, Granata F (2020) Groundwater level prediction in Apulia
region (Southern Italy) using NARX neural network. Environ
Res 190:110062
Fagherazzi S, Fosser G, D’Alpaos L, D’Odorico P (2005) Climatic
oscillations influence the flooding of Venice. Geophys Res Lett
32:1–10
Fan J, Shan R, Cao X (2009) The analysis to Tertiary-industry with
ARIMAX model. J Math Res 1(2):156–163
Ferla M, Cordella M, Michielli L, Rusconi A (2007) Long-term
variations on sea level and tidal regime in the lagoon of Venice.
Estuar Coast Shelf Sci 75(1–2):214–222
Finizio C, Palmieri S, Riccucci A (1972) A numerical model of the
Adriatic for the prediction of high tides at Venice. Q J R
Meteorol Soc 98(415):86–104
Franco P, Jeftic L, Rizzoli PM, Michelato A, Orlic M (1982)
Descriptive model of the Northern Adriatic. Oceanol Acta
5(3):379–389
Ghorbani MA, Deo RC, Karimi V, Kashani MH, Ghorbani S (2019)
Design and implementation of a hybrid MLP-GSA model with
multi-layer perceptron-gravitational search algorithm for
monthly lake water level forecasting. Stoch Environ Res Risk
Assess 33:125–147
Granata F (2019) Evapotranspiration evaluation models based on
machine learning algorithms—a comparative study. Agric Water
Manag 217:303–315
Granata F, de Marinis G (2017) Machine learning methods for
wastewater hydraulics. Flow Meas Instrum 57:1–9
Granata F, Saroli M, de Marinis G, Gargano R (2018) Machine
learning models for spring discharge forecasting. Geofluids
2018:8328167
Granata F, Gargano R, de Marinis G (2020) Artificial intelligence
based approaches to evaluate actual evapotranspiration in
wetlands. Sci Total Environ 703:135653
Imani M, Kao HC, Lan WH, Kuo CY (2018) Daily sea level
prediction at Chiayi coast, Taiwan using extreme learning
machine and relevance vector machine. Glob Planet Change
161:211–221
Jalalkamali A, Moradi M, Moradi N (2015) Application of several
artificial intelligence models and ARIMAX model for forecasting drought using the Standardized Precipitation Index. Int J
Environ Sci Technol 12:1201–1210
Kişi Ö (2007) Streamflow forecasting using different artificial neural
network algorithms. J Hydrol Eng 12(5):532–539
Malik A, Tikhamarine Y, Souag-Gamane D, Kisi O, Pham QB (2020)
Support vector regression optimized by meta-heuristic algorithms for daily streamflow prediction. Stoch Environ Res Risk
Assess 34:1755–1773
Moeeni H, Bonakdari H (2017) Forecasting monthly inflow with
extreme seasonal variation using the hybrid SARIMA-ANN
model. Stoch Environ Res Risk Assess 31:1997–2010
Najafzadeh M, Oliveto G (2020) Riprap incipient motion for
overtopping flows with machine learning models. J Hydroinf
22(4):749–767
123
Najafzadeh M, Saberi-Movahed F (2019) GMDH-GEP to predict free
span expansion rates below pipelines under waves. Mar Georesour Geotechnol 37(3):375–392
Najafzadeh M, Etemad-Shahidi A, Lim SY (2016) Scour prediction in
long contractions using ANFIS and SVM. Ocean Eng
111:128–135
Najafzadeh M, Saberi-Movahed F, Sarkamaryan S (2018) NFGMDH-Based self-organized systems to predict bridge pier
scour depth under debris flow effects. Mar Georesour Geotechnol 36(5):589–602
Nourani V, Baghanam AH, Adamowski J, Kisi O (2014) Applications
of hybrid wavelet–artificial intelligence models in hydrology: a
review. J Hydrol 514:358–377
Okwuashi O, Ndehedehe C (2017) Tide modelling using support
vector machine regression. J Spat Sci 62(1):29–46
Papacharalampous GA, Tyralis H, Koutsoyiannis D (2018) Comparison of stochastic and machine learning methods for multi-step
ahead forecasting of hydrological processes. Stoch Environ Res
Risk Assess 33(2):481–514
Quinlan JR (1992) Learning with continuous classes. In: 5th
Australian joint conference on artificial intelligence, vol 92,
pp 343–348
Riazi A (2020) Accurate tide level estimation: a deep learning
approach. Ocean Eng 198:107013
Ruck DW, Rogers SK, Kabrisky M (1990) Feature selection using a
multilayer perceptron. J Neural Netw Comput 2(2):40–48
Saberi-Movahed F, Najafzadeh M, Mehrpooya A (2020) Receiving
more accurate predictions for longitudinal dispersion coefficients
in water pipelines: training group method of data handling using
extreme learning machine conceptions. Water Resour Manag
34(2):529–561
Tosi L, Rizzetto F, Zecchin M, Brancolini G, Baradello L (2009)
Morphostratigraphic framework of the Venice Lagoon (Italy) by
very shallow water VHRS surveys: evidence of radical changes
triggered by human-induced river diversions. Geophys Res Lett
36:L09406
Tosoni A, Canestrelli P (2011) Il modello stocastico per la previsione
di marea a Venezia. Atti Ist Veneto Sci Lett Arti 169:2010–2011
Umgiesser G (2020) The impact of operating the mobile barriers in
Venice (MOSE) under climate change. J Nat Conserv 54:125783
Umgiesser G, Ferrarin C, Cucco A, De Pascalis F, Bellafiore D,
Ghezzo M, Bajo M (2014) Comparative hydrodynamics of 10
Mediterranean lagoons by means of numerical modeling.
J Geophys Res Oceans 119(4):2212–2226
Wang WC, Chau KW, Cheng CT, Qiu L (2009) A comparison of
performance of several artificial intelligence methods for forecasting monthly discharge time series. J Hydrol
374(3–4):294–306
Wei HL, Billings SA (2006) An efficient nonlinear cardinal B-spline
model for high tide forecasts at the Venice Lagoon. Nonlinear
Process Geophys 13(5):577–584
Zaldivar JM, Gutiérrez E, Galván IM, Strozzi F, Tomasin A (2000)
Forecasting high waters at Venice Lagoon using chaotic time
series analysis and nonlinear neural networks. J Hydroinf
2(1):61–84
Zampato L, Umgiesser G, Zecchetto S (2007) Sea level forecasting in
Venice through high resolution meteorological fields. Estuar
Coast Shelf Sci 75(1–2):223–235
Publisher’s Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.