Time Series Cross-Sequence Prediction
KIRIL KOPARANOV, ELENA ANTONOVA, DANIELA MINKOVSKA, KRASIN GEORGIEV
Department of Programming and Computer Technologies,
Department of Aeronautics,
Technical University of Sofia,
8 Kliment Ohridski blvd., 1000 Sofia,
BULGARIA
Abstract: - In the modern transport industry, vast and diverse information arrays, particularly those including
time series data, are rapidly expanding. This growth presents an opportunity to improve the quality of
forecasting. Researchers and practitioners are continuously developing innovative tools to predict their future
values. The goal of the research is to improve the performance of automated forecasting environments in a
systematic and structured way. This paper investigates the effect of substituting the initial time series with
another of a similar nature, during the training phase of the model’s development. A financial data set and the
Prophet model are employed for this objective. It is observed that the impact on the accuracy of the predicted
future values is promising, albeit not significant. Based on the obtained results, valuable conclusions are drawn,
and recommendations for further improvements are provided. By highlighting the importance of diverse data
incorporation, this research assists in making informed choices and leveraging the full potential of available
information for more precise forecasting outcomes.
Key-Words: - artificial intelligence, automated environments, financial time series, forecasting, machine
learning, model.
Received: November 11, 2023. Revised: May 23, 2024. Accepted: June 23, 2024. Published: July 19, 2024.
1 Introduction
Being inherently stochastic due to various factors,
financial time series make accurate predictions
challenging. With constant advancements in
information technology, especially in the field of
calculation speed and data storage, the debate
regarding the impact of data volume versus data
nature on forecasting accuracy has been largely
settled, leading researchers to focus on refining the
understanding of individual factors’ influence. One
of these factors that can be quantified and analyzed
is financial news, with the use of appropriate tools
for the development of an automated forecasting
environment, [1], [2], [3]. Other authors are
directing their attention towards data preprocessing
techniques and the creation of novel formats and
models to enhance prediction reliability, [4].
Constituting a particular investment research field,
trading signal analysis is the object of interest of
scientists who employ hybrid approaches,
combining technical analysis with machine learning
tools, [5]. Time series forecasting plays a valuable
role in numerous domains, such as water
purification, where it enables the innovation and
expansion of real-time automatic control systems,
resulting in significant energy savings, particularly
due to their swift operations, [6]. Additionally, it is
worth noting that the accumulated expertise from
time series forecasting in the realm of securities
trading can be effectively utilized in the planning
and management of diverse resources, for example,
computing and communication. Particularly, it can
aid in the migration, timely redirection, and
allocation of virtual machines to physical ones, [7].
Within modern society, the demand for stable,
continuous, error-free, and resource-efficient
systems is principally pronounced in the energy
industry. It is logical and expected to exploit the
latest technological achievements in these specific
economic areas. For instance, in wind farms, where
control automation and failure prevention are
crucial, secure, user-friendly, and minimally user-
dependent time series forecasting plays a vital role,
[1], [8]. It also significantly contributes to
addressing the pressing issue of container
throughput at ports, which has arisen because of the
rearrangement of logistics chains brought about by
the COVID-19 pandemic, [9].
The accuracy and dependability of time series
predictions are pivotal concerns that underpin
numerous processes in modern society, [10]. Some
authors have started using components from the
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.131
Kiril Koparanov, Elena Antonova,
Daniela Minkovska, Krasin Georgiev
E-ISSN: 2224-2899
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artificial neural network toolkit in conjunction with
other models to make advancements in addressing
related issues, [10], [11], [12]. Other researchers are
dedicating their efforts to expanding and enhancing
already existing environments for automated
prediction, [2]. Alternatively, they are conducting
experiments to scrutinize outcome precision, [13].
Another research field involves the creation of
hybrid models, often incorporating diverse financial
data of various origins, [12].
The importance of having efficient and user-
friendly time series forecasting methods that do not
require extensive training is underscored. In
response to this need, automated forecasting
environments have emerged in the market,
prompting the requisite for a comprehensive
evaluation of their performance under different data
types. By examining the data from multiple
indicators fed into the forecasting system, this study
aims to assess the results, draw conclusions, and
provide recommendations. Due to the lack of
research demonstrating how the nature of data
affects prediction accuracy, this paper focuses on
revealing the outcomes of replacing the original data
sequence with a distinct, yet similarly structured one
during the model’s training phase. Financial time
series were chosen for this purpose, as their intrinsic
complexity makes future value predictions
challenging. Moreover, they authentically showcase
the model’s accuracy, thus underscoring the
relevance of the approach. Machine learning tools
offer numerous beneficial prospects in the fields of
investment and statistical analysis, [14]. Prophet
was selected as the experimental environment due to
the rapid adoption of automated forecasting
environments in recent years. Its swift establishment
as a trustworthy instrument, [15], [16], [17], along
with its ability to predict time series values for
relatively long future periods, [18], can aid investors
in making well-informed decisions about profitable
investments, [19].
It can be argued that automated forecasting
environments are not viewed as a research tool, but
as one to aid end-user forecasts. The novelty of the
applied approach is that new knowledge is acquired
about the quality of their activity under different
conditions, and this is the way to expand the scope
and improve their functionality. All this would lead
to an increase in the reliability of forecasting with
the use of these environments, as well as in the
accessibility of these products to a wider range of
users.
2 Methodology
The experiment consists of submitting data into the
automatic forecasting system based on Prophet, [1]
to determine the impact of each feature (indicator).
It was assumed that automatic forecasting systems,
such as Prophet, are good enough. In addition, the
input data and the model were restricted to a single-
time series with no additional features. In this
context, the conventional assumption for the
univariate analysis that the past values of a variable
are the best predictor of its future behavior was
challenged.
The used real-world time series data is obtained
from Yahoo Finance - daily quotations for The
Boeing Company. It contains 14,465 records, from
January 2, 1962, to June 19, 2019. Figure 1
illustrates the “Open” price quotations. The input
data is divided into sets for training and testing, set
in a 4:1 ratio, as visualized and labeled in Figure 1
once again.
Table 1 displays the statistical characteristics of
the four distinct features (“Open”, “High”, “Low”,
and “Close”) within the data set. It is evident that
these features exhibit a considerable level of
proximity, given the extensive duration of the time
series, spanning nearly 60 years.
Table 1. Statistical Characteristics of The Input Data
Characteristic
Open
High
Low
Close
mean
42.735
43.187
42.276
42.748
std
66.868
67.551
66.168
66.897
min
0.38272
0.39095
0.38272
0.38272
max
446.01
446.01
440.19
440.62
25%
2.2798
2.3004
2.2551
2.2798
50%
19.250
19.438
19.063
19.25
75%
54.700
55.180
53.900
54.630
Prophet was preferred as a suitable forecasting
platform, as it has had many user installations in
recent years. Also, all Python-based software
solutions and libraries undergo very fast
development, due to the large number of developers
and users involved. The Prophet forecasting model,
shown in Equation (1) is based on an additive
regression model, accounting for trend, seasonality,
and holidays, [7]:
t
h(t) s(t) g(t) y(t)
, (1)
where:
g(t) - non-periodic trend (piecewise linear or
logistic growth curve for modeling non-
periodic changes in time series);
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s(t) - periodic changes (e.g., weekly/yearly
seasonality), modeled using Fourier series
and dummy variables;
h(t) - holiday effects (user-provided) with
irregular schedules;
εt - unusual changes not accommodated by
the model (error).
The forecasting model is treated as a curve-
fitting problem through probabilistic programming.
Therefore, the inherent uncertainty intervals for the
trend component are considered. Additionally, the
model incorporates “changepoints”, allowing the
function parameters to change.
Prophet predictions are univariate. This means
that the future values of a single variable can be
predicted based on the past values of the same
variable. Considering the selected data set, the
conventional univariate prediction involves only
“Close” prices, or only “Open” prices, etc.
In this study, the experiment involved predicting
a long target sequence by training the model on
historical data from the same sequence (stage one)
and from a different sequence (stage two). The
former will also be called independent sequence
prediction and the latter - cross sequence prediction.
The target/feature sequences considered were the
“Open”, “High, “Low”, and “Close” prices. For
example, the model can be trained to predict future
“Open” prices using past “Open” prices (denoted as
“Open/Open”). Similarly, if it predicts future
“High” prices using past “Open” prices, this is
denoted as “High/Open”.
The performance evaluation of the predictions
involves using two metrics - Mean Absolute Error
(MAE) and Mean Squared Error (MSE). Their
equations (2 and 3) are displayed below:
(2)
(3)
The parameters in the formulae above are as
follows:
pt - actual value;
p
t - predicted value;
n - number of samples
The specific target/feature will be denoted in
brackets, e.g. MAE(Open/Open), MAE(Open/High),
MAE(High/Open), etc.
The difference in the performances between the
different approaches for calculating a target is
denoted and calculated as:
MAE_difference(target_i, feature_j) =
MAE(target_i/feature_j) - MAE(target_i/feature_i)
(4)
e.g.
MAE_difference(Open, High) =
MAE(Open/High) - MAE(Open/Open)
A closer examination was performed on
predictions related to the “Open” prices to confirm
the reliability of the outcomes. For this purpose,
data from numerous companies was downloaded
from Yahoo Finance. Data sets containing more
than 5,000 records (similar in size to that of The
Boeing Company) were separated for analysis of
long sequences, while data sets with fewer than
5,000 entries were used to represent short
sequences. Data sets with fewer than 100 records
were disregarded.
For each experiment within the set Ξ, a
comparison of performance metrics is carried out,
and the number of improvements is counted:
(5)
e.g.
All experiments were conducted in Google
Colaboratory, a cloud-based integrated development
environment (IDE). They were performed using:
2.1 The Python Programming Language
(3.10.12)
As a programming language well-known for its
readability and simplicity, Python is easy to learn
and helpful to developers who want to create a wide
range of applications - from web development to
data analysis and machine learning. Its adaptability
and cross-platform compatibility present it as a great
option for creating software that functions smoothly
across various operating systems, [20].
2.2 The NumPy library (1.22.4)
This library offers effective data structures and
functions for handling large arrays and matrices
enabling quick and vectorized operations. It is
widely used for numerical computing and scientific
computing tasks. Moreover, NumPy is an
indispensable part of data analysis, machine
learning, and simulation tasks because it easily
integrates with other libraries and tools in the
scientific Python ecosystem, [21].
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Kiril Koparanov, Elena Antonova,
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Fig. 1: Input Data Set: “Open” Stock Prices of The Boeing Company (USD)
2.3 The Pandas library (1.5.3)
It can handle and process structured data, as it offers
user-friendly data structures and tools for data
analysis.
With built-in features such as filtering, sorting,
aggregating, and merging, tabular data can be
efficiently stored and manipulated using the
DataFrame - the fundamental data structure in
Pandas. It also provides strong indexing and slicing
capabilities for rapid and effective data retrieval,
[22].
2.4 The MatPlotLib library (3.7.1)
With this library, plots of all kinds (line, scatter, bar,
and histogram) can be quickly generated. Thanks to
its wide range of customization choices, users can
alter the colors, labels, axes, titles, and other
elements of their plots, [23].
2.5 The SkLearn library (1.2.2)
Since it is based on NumPy, SciPy, and MatPlotLib
(the libraries mentioned above), it is a strong and
extensive library for machine learning tasks -
handling missing values, selecting features, and
transforming data. While providing a variety of
preprocessing methods, SkLearn also offers others
for model selection and assessment metrics to
evaluate and contrast the performance of various
machine-learning models, [24].
2.6 The Prophet procedure (1.1.4)
This procedure meant to work with time series data
is based on a decomposable model that includes
elements for trend seasonality and holiday effects. It
serves as a flexible and user-friendly interface for
time series forecasting and is appropriate for both
short- and long-term forecasting tasks because it
automatically recognizes and models a variety of
patterns and outliers in the data provided, [25].
2.7 The Jupyter Notebook
As an open-source web application that facilitates
the creation and sharing of documents with live
code explanations and visualizations, it aims to
make the process of implementing logic and
interpreting results faster and more enjoyable. The
supported programming languages include Python
and R. Code is interactively run in the interface and
results are visible right away, [26].
3 Results
The experiment assessment involves utilizing
Prophet for automated time series forecasting on a
data set comprising daily stock quotes for The
Boeing Company. The influence of each feature is
evaluated using Equations (2) and (3). The
experiment was carried out in two stages. Initially,
separate (independent) training and forecasting were
performed for each feature (“Open”, “High, “Low”,
“Close”) for a period of about 3,000 days into the
future. The resulting indicators are presented in
Table 2. In the second stage, the error was
calculated individually for each feature once more
by comparing the predicted prices of the “Open”
feature with the actual ones for “Open”, “High”,
“Low” and “Close” (cross-training), as shown in
Table 3. The performances of both stages are
compared in Figure 2. The differences (4) between
the indicators of both stages are displayed in Table
4. From there, it is observed that there is no
substantial change in prediction accuracy, and the
results remain comparable, often slightly better.
This suggests that the proposed method leads to
some improvement that could substantiate further
study. The comparable performances can be
attributed to the similarity in nature and values of
the input data.
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The most favorable outcome is obtained when
predicting the “Low” price. This reflection is
supported by the results of both performance
measures (Table 2 and Table 3), and it is visually
represented in Figure 2. Finally, the derived patterns
remain consistent across both evaluations.
Table 2. Approximate Mean Absolute Error & Mean
Squared Error for the First Stage
Open/Open
High/High
Low/Low
Close/Close
MAE
54.24
54.85
53.63
54.28
MSE
7707.6
7927.6
7485.4
7718.3
Table 3. Approximate Mean Absolute Error & Mean
Squared Error for The Second Stage (Using “Open”)
Open/Open
High/Open
Low/Open
Close/Open
MAE
54.24
54.88
53.62
54.28
MSE
7707.6
7963.1
7451.9
7719.5
Table 4. Differences between the Results of Both Stages
Open
High
Low
Close
Mean Absolute Error
0
-0.027
0.009
-0.006
Mean Squared Error
0
-35.50
33.53
-1.227
Fig. 2: “Open”, “High”, “Low” & “Close” Price
Prediction Errors for The First Stage (Independent
Training) & The Second Stage (Cross Training)
Subsequently, the focus shifts to the prediction of
solely the “Open” prices. This is achieved by
employing models that have been cross-trained on
sequences, related to the “High”, “Low”, and
“Close” prices. The results are presented in Table 5,
which also includes those of the independent
prediction (“Open/Open”). The absolute change in
the prediction error metric (4), pertaining to the
“Open” prices, is visualized in Figure 3.
Table 5. Approximate Mean Absolute Error &
Mean Squared Error for the Second Stage
(Predicting “Open”)
Open/Open
Open/High
Open/Low
Open/Close
MAE
54.24
54.22
54.26
54.24
MSE
7707.6
7673.7
7742.8
7706.3
To check the consistency of the findings, the
experiment was reiterated using multiple data sets
characterized by varying sequence lengths (5010
short and 2165 long sequences). The total
experiment count and the instances exhibiting
performance improvement (5) are summarized in
Table 6. The number of cases with “Open” price
prediction error improvement after cross-training
with “High”, “Low” and “Close” prices is recorded
in the first three rows respectively. We can see that
the prediction of “Open” prices based on “Low”
prices is better than predictions trained on “Open”
prices in 2815 out of 5010 cases with a lower MSE
for short sequences and in 1129 out of 2165 cases
for long sequences. The number of cases with
“Open” price prediction error improvement after
cross-training with either “High”, “Low”, “Close”
or prices is recorded in the following four rows,
named 0, 1, 2, and 3, based on the number of
features with improvement. The predictions from
either “High”, “Low” or “Close” prices are better in
1797 + 2108 + 550 = 4455 out of 5010 cases for
short sequences and in 870 + 985 + 150 = 2005 out
of 2165 cases for long sequences.
Figure 4 illustrates a graphical representation of
the model’s forecast for the test period (2,893 days
into the future), obtained during the initial phase of
the experiment. The visualization is as indicated:
Blue Line: Prediction;
Black Scatter Plot: Train Data;
Blue Scatter Plot: Test Data;
Light Blue Area: Prediction Deviations.
It is noticeable that the trend towards a general,
permanent, and continuous increase in price has
been captured. In recent decades, however, it has
surged significantly. Consequently, fluctuations
have also increased, which is not satisfactorily
reflected in the forecast. As the forecast duration
extends, the expected widening of the confidence
interval is observed.
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Fig. 3: “Open” Price Prediction Errors for Different Training Sequences (“Open”, “High”, “Low”, “Close”) -
Relative to the “Open” Price Prediction
Table 6. Number of Cases with “Open” Price Prediction Error Improvement by Cross-Training & Number
of Cases Aggregated by Number of Features with Improvement (“High”, “Low”, and/or “Close”)
Short Sequences
Long Sequences
Feature Sequence /
Count
# MAE
Reduction
# MSE
Reduction
# MAE
Reduction
# MSE
Reduction
High
2,226
2,174
1,019
1,023
Low
2,776
2,815
1,131
1,129
Close
2,696
2,674
1,143
1,138
0
528
555
154
160
1
1,817
1,797
876
870
2
2,014
2,108
988
985
3
551
550
147
150
Total
5010
2165
Fig. 4: Example Model Prediction for the Train & Test Periods
4 Conclusion
The conducted experiments and achieved results
represent a preliminary stage in knowledge
expansion and employment of automated
forecasting systems, as well as in considering
various factors on their effectiveness. It was
demonstrated that by training a model on input
sequences that are distinct, yet akin, accurate
forecasting is possible. The improvement, although
not substantial, remains consistent and warrants
additional investigation. In the context of
conventional machine learning, a similar concept
involves using various attributes of the target.
Suggestions can be offered to further diversify the
input data by incorporating disparate characteristics.
Acknowledgement:
The authors would like to thank the Research and
Development Sector at the Technical University of
Sofia for the financial support.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Funding was provided by the Research and
Development Sector at the Technical University of
Sofia.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
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WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2024.21.131
Kiril Koparanov, Elena Antonova,
Daniela Minkovska, Krasin Georgiev
E-ISSN: 2224-2899
1618
Volume 21, 2024
