ABSTRACT
With the advent of modern computing facilities, there is the need to research into new
forecasting techniques that are applicable to natural phenomena characterized by
uncertainties in their problem formulation. A natural phenomenon of particular research
interest is rainfall because of its critical importance to the agricultural economy and food
security in Nigeria. The aim of this thesis is the application of the NeuroFuzzy Soft
Computing tool and time series techniques in the field of forecasting and modeling of
statistical phenomenon using rainfall forecasting of Zaria as a case study. The emerging
NeuroFuzzy technique is a hybrid intelligent system unifying the benefits of the
computational techniques of Fuzzy Logic and Artificial Neural Networks (Neural Nets):
Herein the learning capabilities of Neural Nets and the representation power (knowledge
base) of Fuzzy Logic Systems are optimally exploited. The goal of this research effort is
to quantify the performance of the proposed technique when compared with other
techniques using monthly thirteen-year data on weather variables collected from Nigerian
Meteorological Agency (NiMet), Zaria and Department of Soil Science, Ahmadu Bello
University, Zaria. The developed NeuroFuzzy System is implemented as a ‘pure’ Fuzzy
Logic System using the Fuzzy Technology Language (FTL) as the Rain Forecast Model
(RFM). A fuzzy time series technique is then applied to forecast rainfall so as to have a
basis for comparison with the developed Rain Forecast Model (RFM). The developed
model has been tested extensively relying on the validation data of 2003 to 2005. For the
sake of comparison, existing fuzzy time series method has been applied for the same
period. It is shown that the developed Rain Forecast Model (RFM) performed
significantly better than the fuzzy time series method within the validation period based
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on the root mean square returned by the two methods. Furthermore, the results obtained
from the developed Rain Forecast Model (RFM) and the fuzzy time series methods for
the entire forecast period of 2003 to 2010 are also compared. The results obtained are
presented and discussed from the standpoint of a degree of consistency exhibited by the
two methods and their computational time requirements.
TABLE OF CONTENTS
TITLE PAGE i
DECLARATION ii
CERTIFICATION iii
DEDICATION iv
ACKNOWLEDGEMENTS v
TABLE OF CONTENTS vii
LIST OF FIGURES xi
LIST OF TABLES xv
LIST OF SYMBOLS xvi
ABSTRACT xviii
CHAPTER ONE: INTRODUCTION AND BACKGROUND
1.0 INTRODUCTION 1
1.1 OVERVIEW OF SOFT COMPUTING TECHNIQUES 2
1.2 LITERATURE REVIEW 7
1.3 STATEMENT OF PROBLEM 13
1.4 RESEARCH OBJECTIVES AND METHODOLOGY 19
1.5 THESIS OUTLINE 22
CHAPTER TWO: SOFT COMPUTING TECHNIQUES
2.0 INTRODUCTION 23
2.1 ARTIFICIAL NEURAL NETWORK 23
2.1.1 NEURAL NET MATHEMATICAL MODEL 26
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2.1.2 NEURAL NET CLASSIFICATIONS 28
2.1.3 ACTIVATION FUNCTION 31
2.1.4 NEURAL NET LEARNING: THE ERROR BACK PROPAGATION 34
2:2 FUZZY LOGIC SYSTEM 37
2.2.1 MATHEMATICAL UNCERTAINTY 38
2.2.2 FUZZY SET THEORY 38
2.2.2.1 FUZZY SET OPERATORS AND HEDGES 39
2.2.3 FUZZY LOGIC SYSTEM 40
2.2.4 FUZZY LOGIC SYSTEM LAYER STRUCTURE 43
2.2.4.1 FUZZIFICATION 44
2.2.4.2 FUZZY INFERENCE 45
2.2.4.3 DEFUZZIFICATION 48
2.3 NEURO-FUZZY SYSTEMS 50
2.3.1 NEURAL NETS AND FUZZY LOGIC SYSTEMS INTEGRATION 50
2.3.2 NEURO-FUZZY SYSTEM STRUCTURES 53
2.3.2.1 MAMDANI NEURO-FUZZY SYSTEM 53
2.3.2.2 TAKAGI-SUGENO (TS) NEURO-FUZZY SYSTEM 55
2.3.3 NEURO-FUZZY MODEL: THE MODIFIED nfMod MODEL 57
2.3.3.1 TRIANGULAR-NORMS (T-norm) AND CONORMS (T-conorm or S-norm)
59
2.3.4 LEARNING ALGORITHMS 61
2.3.4.1 HEBBIAN LEARNING RULE 61
2.3.4.2 WINNER TAKES ALL (WTA) LEARNING RULE 62
2.3.4.3 DELTA RULE AND ERROR BACK PROPAGATION (EBP) 62
2.3.5 NEURO-FUZZY LEARNING ALGORITHM 64
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CHAPTER THREE: DEVELOPMENT OF FORECAST MODEL
3.1 DEVELOPMENT OF NEUROFUZZY-BASED FORECAST MODEL 69
3.2 RAIN FORECAST MODEL 87
CHAPTER FOUR: TIME SERIES PREDICTION OF WEATHER VARIABLES
4.1 INTRODUCTION 95
4.2 SIMPLE MOVING AVERAGES 97
4.3 SIMPLE EXPONENTIAL SMOOTHING METHOD 99
4.4 TESTING WITH THE RAIN FORECAST MODEL (RFM) 102
CHAPTER FIVE: FUZZY TIME SERIES FORECASTING
5.1 INTRODUCTION 106
5.2 FUZZY TIME SERIES 107
5.3 FORECASTING METHODOLOGY 108
CHAPTER SIX: RESULTS AND ANALYSIS
6.1 INTRODUCTION 117
6.2 RAIN FORECAST MODEL (RFM) 118
6.3 FORECASTING RAINFALL USING FORECASTED WEATHER VARAIABLES
121
6.4 FORECASTING RAINFALL USING FUZZY TIME SERIES 123
6.5 RAINFALL FORECAST RESULTS 125
6.6 SIGNIFICANCE OF RESULTS OBTAINED FROM INVESTIGATION 127
CHAPTER SEVEN: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK
7.1 INTRODUCTION 130
7.2 MAJOR HIGHLIGHTS 131
7.3 CONSTRAINTS 132
7.4 CONCLUSIONS 133
7.5 SUGGESTIONS FOR FURTHER WORK 134
REFERENCES 137
APPENDIX A1: COMPLETE TRAINING DATA (1993 – 2002) 150
APPENDIX A2: COMPLETE RULE BASE (625 RULES) 152
APPENDIX A3: RULE BASE OF THE RAIN FORECAST MODEL (57 RULES)
167
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APPENDIX A4: COMPLETE PROGRAM LISTING 169
APPENDIX A5: FORECAST RESULTS OF MOVING AVERAGES AND
EXPONENTIAL SMOOTHING METHODS 177
APPENDIX A6: THE COMPLETE FORECASTED RAINFALL DATA OBTAINED
FROM THE RAIN FORECAST MODEL (RFM) 180
APPENDIX A7: FUZZY TIME SERIES DATA 182
CHAPTER ONE
INTRODUCTION AND BACKGROUND
1.0 INTRODUCTION
Soft Computing (or Computational Intelligence) is a field of knowledge in the realm of
Artificial Intelligence that deals with machine incorporations of human expertise. Soft
Computing tools include Fuzzy Logic, Artificial Neural Networks, Neuro-Fuzzy, and
Rough Sets. These tools are generally robust as they can handle imprecise or noisy data,
and non-linear data and are quite useful when the underlying relationships between the
data are not fully understood or impractical to model. Unlike hard computing, which is
conventional computing, that requires a precisely stated analytical model and often a lot
of computation time, the soft computing technique is tolerant of imprecision, uncertainty,
partial truth, and approximation. In essence the role model of the Soft Computing
technique is the human mind.
The application of soft computing to forecasting is the main focus of this research. This
introductory chapter, therefore, presents the motivations for this research; the relevant
literature review of soft computing technique is also presented.
1.1 OVERVIEW OF SOFT COMPUTING TECHNIQUES
Machine intelligent behaviour is determined by the ability to realize machine
incorporations of human expertise, flexibility of the architecture, laws of inference
procedure and the speed of learning. All these form the main constituents of the field of
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knowledge called Computational Intelligence or Soft Computing [1].”Intelligent” in
this case refers to the utilization of engineering techniques that have, to one extent or
another, been borne out of human reasoning, adaptation or learning, biological cognitive
structures or principles of evolution [2].
Soft Computing can be defined as a consortium of methodologies that work
synergistically and provides in one form or another flexible information processing
capability for handling real-life ambiguous situations [3]. When exploited in a synergistic
manner, Soft Computing tools can be used for construction of powerful computationally
intelligent systems [4][5] that could possess human-like expertise within a specific
domain, adapt themselves and learn to do better in changing environments, and explain
how they make decisions [6][7]. Typical Soft Computing tools include Fuzzy Logic,
Artificial Neural Networks, Neuro-Fuzzy and Rough Sets. They offer real advantages
over conventional (hard computing) modeling including the ability to handle large
amounts of dynamic, non-linear or noisy data and they can especially be useful when the
underlying relationships are not fully understood [4][8].
Soft Computing techniques solve complex problems by utilizing the knowledge of an
expert and incorporating the imprecision and uncertainty associated with various aspects
of the problem [3][4]. This knowledge can be obtained by instruction and/or learning
rather than by investigating the problem in details. The objective is getting an acceptable
solution at minimum cost by seeking an approximate solution to an imprecisely or
precisely formulated problem. In traditional hard computing, the main issues are
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precision, certainty and rigorous computations while in contrast, in soft computing the
principal notion is that precision and certainty carry a cost. As such the challenge is to
exploit the tolerance for imprecision by devising methods of computation that lead to an
acceptable solution at minimal cost [3]. And because many contemporary problems such
as pattern information processing and classification (handwriting, speech, objects,
images, etc.) and forecasting do not lend themselves to precise solutions, soft computing
tools are becoming useful considering the following properties:
i) Ability to learn from experimental data; and
ii) Power of generalization, which is derived from approximating or interpolating
to produce outputs from previously unseen inputs by using outputs from
previous learned inputs.
If a network is trained on a data set and the error is e1 and then a new data set
is applied that produces error e2, 1 2 e e is then a measure of the generalizing
ability of the network [9]
Computation, reasoning and decision-making should exploit (wherever possible) the
tolerance for imprecision, uncertainty, approximate reasoning and partial truths for
obtaining low cost solutions. This is possible with Soft Computing taking the following
features into consideration:
i) Parallel Processing: The mathematical power of machine intelligence is
normally attributed to the neural-like system architecture used and the faulttolerance
arising from the massively interconnected structure [1]. Neural Nets,
Fuzzy Logic and other hybrid Neuro-Fuzzy Systems are dynamic, parallel
23
processing systems that can estimate a function without any mathematical model
and learn from experience with sample data [3].
ii) Fuzzy Levels: Instead of “0” and “1” digital levels, Soft Computing systems use
fuzzy/continuous levels. This ensures that much more information is passed
through the system [1].
iii) Survivability: Soft Computing systems can survive in the presence of faults. As
such, they may work correctly if they are partially damaged [1] or are operating in
an environment of uncertainty and imprecision. This is responsible for Soft
Computing systems being able to understand distorted speech, decipher sloppy
handwriting, recognize and classify images, summarize text and more generally
making rational decisions in an environment of uncertainty and imprecision [3].
Fuzzy Logic is therefore a technology capable of modeling vagueness, handling
uncertainty and supporting human-type-reasoning [3]. It allows representation of human
decision and evaluation processes in algorithmic form (IF <situation> THEN <action>.
Thus, if the desired performance of a technical system can be described for certain
distinctive cases by rules, Fuzzy Logic can effectively put the knowledge into a solution
[10].
A Fuzzy Logic System is developed in essentially three (3) steps, as shown in Figure 1.1:
Figure 1.1: Fuzzy Logic Block Diagram
FUZZIFICATION FUZZY
INFERENCE
DEFUZZIFICATION
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i) FUZZIFICATION is using Fuzzy Sets to translate real values into linguistic
variable values and using Membership Functions to graphically represent the
variables.
ii) FUZZY INFERENCE is evaluating fuzzy ‘IF <situation> – THEN <action>’
rules that define relationships between linguistic variables. The IF part is called
the ‘Antecedent’ while the THEN part is called the ‘Consequent’
iii) DEFUZZIFICATION is obtaining crisp output values from results of fuzzy
inference
Fuzzy Logic Systems are designed using linguistic variables (first described by Lofti
Zadeh in 1965). A linguistic variable can belong to more than one set in the universal
Fuzzy Set of variables according to its membership degree (from 0 to 1). This is a major
departure from conventional Boolean logic with only two states: High or Low. The fuzzy
rule base is developed in a manner that ‘nearly’ reflects a human decision base for a
similar situation.
Neural Net is an information processing technique that is inspired by the way biological
nervous systems, such as the brain, process information. It is, therefore, a system
composed of many simple processing elements (neurons) operating in parallel whose
function is determined by network structure (e.g. feedforward) and connection (synapses)
strength [11]. The Neural Net, just like the Fuzzy Logic System, also has three basic
layers: INPUT LAYER, HIDDEN LAYER and OUTPUT LAYER as in Figure 1.2.
25
Input Layer Hidden Layer Output Layer
Figure 1.2: A Simple Neural Network
Developing a Neural Net solution means teaching the net desired behaviour using sample
data sets of inputs and corresponding outputs. This is called the LEARNING PHASE.
The Neural Net is created ‘dumb’ and it is the LEARNING ALGORITHM that modifies
the individual neurons of the net and their connection weights in such a way that reflects
the desired behaviour. After learning, the Neural Net enters into a WORKING PHASE,
where its behaviour is now deterministic [10].
The basic idea of Neuro-Fuzzy approach is to set up a complete fuzzy rule base not by
entering the rules but by presenting a set of training samples and selecting the set of rules
that represent the samples. This is important because in many real world applications,
knowledge that describes desired system behaviour is contained in data sets. It then
means that the “IF <situation> THEN <action>” rules of a Fuzzy System must be derived
from the data set manually, which can be difficult and time consuming especially with
large data set. This is where a Neural Net presents a solution since it can train itself from
data [10][12].
26
Fuzzy Logic Systems can be used for knowledge representation in Neural Systems
because FUZZIFICATION can represent INPUT LAYER, FUZZY RULES can represent
HIDDEN LAYER and DEFUZZIFICATION can represent OUTPUT LAYER. However,
in each of these mechanisms, parameters must be found that can be used as weights in the
Neural Net training. In a fuzzification (input) layer, each neuron represents an input
Membership Function of the antecedent of a fuzzy rule. In a fuzzy inference (hidden)
layer, fuzzy rules are fired and the value at the end of each rule represents the initial
weight of the rule, and will be adjusted to its appropriate level at the end of the training.
In the defuzzification (output) layer, each neuron represents a consequent proposition
[3].
1.2 LITERATURE REVIEW
Forecasting is the process of generating information for the possible future development
of a process from data about its past and its present whereas modeling is the process of
finding global underlying structures, models and formulae that can explain the behaviour
of the process in the long run and that can be used for long term prediction as well as
understanding the past [25].
Methodologies, basically statistical, have and still are being used for forecasting. These
include:
i) Time-Series, which is a method that uses a set of historical values at equally
spaced time intervals to predict an outcome.
27
ii) Regression Analysis, which is a technique in which one random variable is
predicted from the value of another random variable and/or several other
random variables.
iii) Group Method for Data Handling (GMDH), which is a technique that is
able to construct regressive mathematical model of high order that can accept
a great number of variables. It divides the set of observations into ‘Training’
and ‘Checking’ sets.
iv) Moving Average, which is a technique used to forecast values based on a
weighted average of past values [26].
These approaches are ill defined to represent vague input data and human judgment. They
also have the major shortcoming of the difficulty in deriving mathematical models that
approximate highly non-linear behaviour. The NeuroFuzzy methodology, as a Soft
Computing tool, has the advantages of improving the adaptability of the forecasting
system to sudden changes and the fact that it can simulate arbitrarily non-linear systems
[13][27].
One of the most important functions inherent in human thinking is the ability to learn
how to predict as no action is performed without predicting, in one way or the other the
results of the action. The simplest method of predicting the future is based on the
assumption that the future will be like the recent past and present [28]. The existence of
past data constitutes observable time series and time-series forecasting assumes that a
time series is a combination of a pattern and some random error. The goal is to separate
28
the pattern from the error by understanding the pattern’s trend (its long-term increase or
decrease), and its seasonality (the change caused by seasonal factors) [29].
The Neuro-Fuzzy technique as an evolving Soft Computing technique is being used to
develop prediction models that are relatively more accurate and more robust than
previous statistical time-series based models. This is because these models are able to
handle noisy, vague and non-linear situations amongst others. The average forecasting
error of the fuzzy time series approach is also smaller than that of most statistical timeseries
based models. Its main advantage is the fact that human experience and knowledge
can be applied to the forecasting procedure.
Wang et al [13] used a Neuro-Fuzzy methodology that combined Neural Nets trained on
historical data with PROTREN (a Fuzzy Logic algorithm that detects the onset of
transients with extremely high efficiency and reliability) for nodal load forecasting.
PROTREN extracts various features from the on-line signals and processes them in a
fuzzy way to acquire the trend information that is needed by the Neural Nets for load
forecasting. The methodology improves the adaptability of the system to sudden changes
or special events that may influence the load by temporarily distorting the general pattern
and thus making the load signal highly unpredictable.
Abraham et al [6][7] used an Evolving Fuzzy Neural Network (EFuNN) Neuro-Fuzzy
methodology implementing a Mamdani type Fuzzy Inference System (FIS) to develop a
monthly rainfall time series prediction model for Kerala, India. Based on information
29
from the previous four years, the network would predict the amount of rain in each month
of the fifth year. In order to achieve good generalization properties, three-month
information centred over the predicted month of the fifth year in each of the previous four
years was used. The codes were executed using MATLAB and C++.
Ahmad et al [14] used a Neural Net with Error Back-Propagation learning algorithm to
forecast weather. The model developed is able to learn the pattern of rainfall in order to
produce a precise forecasting result. The Neural Net technique is applied to the hourly
local weather of Johor, Malaysia with 77-78% forecast accuracies reported. The
reliability of the Neural Net approach is constrained by the fact that its operation cannot
be explained in addition to the heavy computational requirements of the Error Back-
Propagation algorithm.
Lotfi [15] used the learning Fuzzy Inference System (FIS) to forecast maximum daily
electricity load based on previous data available for electricity load and average daily
temperature. The problem was tackled in two different stages by creating two different
models. The first model will predict the temperature and the second model uses the
predicted temperature to forecast the maximum electricity load. Initial fuzzy rules are
generated and then the numerical data from the Eastern Slovakian Electricity Corporation
are used to learn the parameters of the learned Fuzzy Inference System,
Cuevas et al [16] used the Adaptive Neuro-Fuzzy Inference System (ANFIS) Takagi-
Sugeno (TS) Neuro-Fuzzy model to predict target movements with minimum error for a
30
real time visual tracking system. Real time visual tracking is a complicated problem due
to the dynamics of the objects involved in the process. The prediction model developed
helps to reduce the delay’s (caused by motors and mechanisms used for the camera’s
movements being significantly slow) effects in the control for visual tracking.
Kooths [17] used a Neuro-Fuzzy system called the Neuro-Fuzzy Expectation Generator
(NFEG) connected to a business cycle simulation model using MAKROMAT-nfx
software and Neuro-Fuzzy Expectations-editor (NFE-editor) in order to predict rate of
inflation based on the current unemployment rate and the observable money growth rate
for expectations modeling in macroeconomic theory. The software allows for analyzing
how the Neuro-Fuzzy Expectation Generator (NFEG) interacts with the economic system
when the system is exposed to shocks.
Lee et al [18] used a Similar Adaptive Neuro-Fuzzy Inference System (SANFIS)
Mamdani Neuro-Fuzzy model to qualitatively evaluate a candidate’s next moves in a 19 x
19 Go board game. This is a game for two players, who alternately play a single stone
onto the 19 x 19 intersection (361 points), with the aim of surrounding more territory than
the opponent.
This work, which aims to develop a rainfall forecast model for Zaria, as a case study, uses
a modified neuro-fuzzy model (nfMod) based on Mamdani type Neuro-Fuzzy System,
the fuzzyTECH Professional Edition V5.54h Neuro-Fuzzy module, Fuzzy Technology
Language (FTL) and time series prediction techniques. According to Gorzalczany et al
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[19], the neuro-fuzzy (nfMod) model is one of the least sensitive to removal of weakest
rules from the rule base (using such techniques as -cut (alpha-cut)) and is also better
able to generalize from the learned rules than other such models like Adaptive Neuro-
Fuzzy Inference System (ANFIS), Neuro-Fuzzy Identification (NFIDENT), etc. The
neuro-fuzzy (nfMod) model is characterized by its high transparency and interpretability
when compared from the other types of models.
A Time Series is a sequence of observations which are ordered in time and according to
Tomé et al [20] and Easton et al [21], time series prediction is a problem with wide
range of applications, including Economics, Finance, Meteorology, Energy Systems
Planning, Population or Traffic prediction amongst others. Typical statistical time series
techniques include the smoothing techniques of Moving Average and Exponential
Smoothing. According to Easton et al [21] and Roberts [22], a moving average is a form
of average which has been adjusted to allow for seasonal or cyclical components of a
time series while exponential smoothing is used to reduce irregularities in the time series
data thus providing a means of predicting future values of the time series.
Time series prediction is basically a modeling problem, which can better be solved using
Soft Computing techniques. This is because it is possible that the underlying relationship
between the data may not be known [17]. The first step in the solution is establishing a
non-linear mapping between inputs and outputs, after which the model can be used to
predict future values based on past and present observations [30].
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The fuzzy time series technique used to forecast the rainfall is based on the technique
adopted by Abbasov et al [23] in forecasting the annual population growth of
Azerbaijan. Lee et al [24] also used a similar fuzzy time series model with the Box-
Jenkins time series technique to evaluate the Taipei’s monthly unemployment rate series.
1.3 STATEMENT OF THE PROBLEM
Weather is very difficult to predict despite the data and information that may be available.
This is because weather can change very quickly (e.g. if the wind changes direction
slightly, the rain may pass over and fall elsewhere) which means the forecast can quickly
be out of date [31]. In general, climate and rainfall in particular, are highly non-linear
phenomena in nature [32][7] exhibiting what is known as the “Butterfly Effect”. Edward
Lorenz (1961) first explained this butterfly effect whilst at Massachusetts Institute of
Technology (MIT) [7} as “Small variations of the initial condition of a dynamical system
may produce large variations in the long term behavior of the system”. The phrase refers
to the idea that a butterfly’s wings might create tiny changes in the atmosphere that
ultimately cause a tornado to appear (or prevent a tornado from appearing). The flapping
wing represents a small change in the initial condition of the system, which causes a
chain of events leading to large-scale phenomena.
As a result of this effect, while some regions of the world (e.g. Africa) are noticing
systematic decrease in annual rainfall, others notice increase in flooding and severe
storms (El-Nino, Hurricane Ivan, Tropical storm Jeanne, Hurricane Katrina etc in the
Caribbean and America). Typical reported cases include:
33
i) El Nińo warming in the Central Eastern Equatorial Pacific tends to cause
droughts in Indonesia and Australia and floods in California.
ii) Above normal Sea Surface Temperatures (SSTs) off Angola tend to produce
wetter conditions in Brazil’s Nordeste [32]
Soft Computing techniques are well suited for developing the prediction model as a result
of the non-linear nature of rainfall. The idea of using a Neuro-Fuzzy approach came
both, from Neural Nets’ abilities to learn and generalize from sets of historical patterns,
and the complexities of the problem (unpredictability of rainfall), which is more suitable
to be modeled using fuzzy techniques.
Rain is one of nature’s greatest gifts and is very vital to the agricultural economies and
food security of developing countries like Nigeria and also sustenance of the
environment. In Nigeria, rainfall determines the zonal pattern of crops and the seasonal
activities of farmers. It is then important that rainfall trends be identified as any
unforeseen deviations can cause unwanted disruptions. This has assumed an even greater
importance due to the threats posed by global warming and greenhouse effect [6].
Rainfall is governed by the interaction of the moist tropical maritime air mass and the
dry, cool tropical continental air mass. Air mass is defined as a large uniform (with
respect to temperature and water vapour) body of air within the atmosphere. Rainfall is
characterized by its extreme variability, both of intensity and duration with a temporal
and spatial pattern. The characteristic and intensity of the prevailing weather conditions
are determined by the surface location of the moisture boundary zone separating the two
34
air masses. The boundary is known as Inter Tropical Divergence (ITD). It is the location
of a place in relation to the position of the Inter Tropical divergence (ITD) that
determines its weather situation [33].
Zaria is located within the Sudan Savannah zone lying on a plateau of about 670.56m
above sea level and on latitude 118/N and longitude 741/E. It lies within a region with
distinct dry and wet seasons, which are influenced by two distinct air masses. One from
the North is dry and continental in origin and the other from over the Atlantic in the
South is moist, cool and equatorial maritime. It is the decline in air pressure over land
that shifts convection towards the land bringing in maritime moisture, rainfall and
cloudiness [33]. The wet season occurs in the high sun period and is dominated by the
South-West winds coming in around April or May and lasting till around October. About
65% of the rains occur between July and September. The dry season lasts between
November and March and is practically rainless. The months of December to February,
the harmattan season, are usually cold and dry due to the influence of the continental air
mass (dry-dusty wind) from the desert regions of North Africa [33].
Using weather variables (Wind Direction, Wind Speed and Relative Humidity) collected
from the Nigeria Meteorological Agency (NiMet), Zaria and the Meteorological Unit of
the Department of Soil Science, Ahmadu Bello University, Zaria on a monthly average
basis over a thirteen- (13) year period (1993 – 2005), a rainfall prediction model is to be
developed. Data from 1993 to 2002 is used as the training set whilst the data from 2003
to 2005 is used as the validation data. However, since the exact relationships between
35
these variables are not known, data driven methods are more suitable in developing the
prediction model. These methods perform a kind of function fitting by using multiple
parameters on the existing information in order to predict the possible relationships in the
near future. The non-linear relationships between rainfall and the weather variables used
in this work are shown in the scatter diagrams of Figures 1.3 – 1.5 whilst their trends for
the period 1993 – 2002 are shown in Figure 1.6. The scatter diagram, which is a graphical
representation of the pairs of data, is a tool for making assessments about the
relationships between random variables.
0
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REL_HUM
Figure 1.3: Plot of Relative Humidity v. Amount of Rainfall
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WIND_DIR
Figure 1.4: Plot of Wind Direction v. Amount of Rainfall
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Figure 1.5: Plot of Wind Speed v. Amount of Rainfall
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Rainfall and Weather Variables
REL_HUM
WIND_DIR
WIND_SPD
RAINFALL
Figure 1.6: Rainfall and Weather Variables Trend For 1993 – 2002.
The Soft Computing model is trained on rainfall data corresponding to a certain period in
the past (training set) and cross validation of prediction made by the network over some
other period (validation set)[28]. One of such methods is Neuro-Fuzzy and the design
issues with respect to Neuro-Fuzzy solution development include the following:
i) Number and types of input and output Membership Functions: How many
Membership Functions can model a decision properly?
ii) Number of Rules
iii) Performance Function
iv) Optimization Method
v) Data Partitioning
vi) Number of Epochs [34][35].
The Neuro-Fuzzy Module provides training methods for supervised learning [36]. The
standard method employed is a combination of Error Back Propagation and Competitive
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Learning (Winner Neurons). After a system output is computed by forward propagation,
an output error is identified by comparing the system output with the given sample output
data [37][38]. The error is then used to determine the fuzzy rule most suited for
influencing system behaviour. Using the selected Learning Rate, the plausibility of the
fuzzy rule is modified before the subsequent sample data set is read [10].
1.4 RESEARCH OBJECTIVES AND METHODOLOGY
The objective of the research is applying the Soft Computing technique of Neuro-Fuzzy
to develop a model to forecast (or predict) future values of an event, in this case the
amount of rainfall, on a monthly average basis, using Zaria as a case study:
The fuzzyTECH Professional Edition V5.54h Neuro-Fuzzy Module is used to
carry out data training.
The forecast model, Rain Forecast Model (RFM), is developed using the Fuzzy
Technology Language (FTL).
Time series techniques are used to predict the weather variables that are used with
the forecast model to predict the rainfall.
The Neuro-Fuzzy Module uses an iterative computation [10] to find the Fuzzy System
that best matches all given samples. In most cases, the samples must be trained several
times in order to reach a satisfactory result.
The methodology adopted in data training and in developing the Neuro-Fuzzy System
using the fuzzyTECH Neuro-Fuzzy module involves the following:
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a) Obtain training sample data.
b) Cluster the sample data (if necessary). The data obtained may have to be
preprocessed to remove redundant data and to resolve conflicts in the data.
c) Create an empty Fuzzy Logic System i.e. the rule set will have all rules with DoS
(Degree of Support) = 0. This means the rule set is COMPLETE but FALSE;
this is required as Neuro-Fuzzy training can only start with an existing rule set. A
DoS value gives the weight for each value to be used in the rule aggregation step
of fuzzy inference. The value is between 0 and 1.
d) Collection of expert knowledge about the process and entering all existing
knowledge (if any) in the solution.
e) Selection of the components of the Fuzzy Logic System to be trained. All or
specific components can be opened for learning.
f) Configuring the Neuro-Fuzzy module by specifying the LEARNING METHOD
and setting parameters for it. The learning method can be any of RealMethod,
RandomMethod, Batch_Learn and Batch_Random.
g) Train with the sample data to learn parameters of the Fuzzy System.
h) Evaluate system performance and validation of results. This is accomplished by
testing trained system with sample test data. This will help in minimizing the
occurrence of “over training”.
i) Manual optimization using an interactive approach with the aid of the watch
window to eliminate functionally redundant or unnecessary rules..
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The methodology adopted in developing the forecast solution using fuzzy time series
involves the following:
i) Definition of universal set U containing the interval between the minimum
and maximum variations in the rainfall amounts for the month under
consideration.
ii) Division of universal set U into equal-length intervals containing variation
values corresponding to different rainfall amounts for the month under
consideration.
iii) Determining the set of fuzzy sets or the respective values of the linguistic
values.
iv) Fuzzifying the input data that is, converting numerical values into fuzzy
values.
v) Selection of parameter (w > 1) corresponding to the time period prior to the
concerned month and calculation of the relationships matrix.
vi) Defuzzifying obtained results, that is, conversion into quantitative values.
The overall methodology adopted in carrying out this investigation involves the
following:
1) Data training and development of the Neuro-Fuzzy System using the
fuzzyTECH Neuro-Fuzzy module, as described above.
2) Implementation as a pure Fuzzy Logic System using the Fuzzy Technology
Language (FTL) resulting in the development of the Rain Forecast Model
(RFM).
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3) Validation of the Rain Forecast Model (RFM) using the validation data.
4) The weather variables, Relative Humidity, Wind Direction and Wind Speed
are forecasted using the time series techniques of Simple Moving Average and
Exponential Smoothing. The forecasted values are used to forecast for the
rainfall using the Rain Forecast Model (RFM) developed.
5) Obtaining rainfall forecast values using fuzzy time series technique, which are
compared with the values obtained from the Rain Forecast Model (RFM).
1.5 THESIS OUTLINE
This work is divided into seven chapters. Chapter One is the introductory chapter where
the objective of the research is defined and the methodology to be applied explained.
Chapter Two describes the theoretical background for the Soft Computing techniques of
Neural Nets, Fuzzy Logic System and Neuro-Fuzzy Systems. The Rain Forecast model
(RFM) is developed in Chapter Three, while Chapter Four describes the method used in
forecasting the weather variables used as inputs for the developed model. The fuzzy time
series method to be used for comparison purposes is described in Chapter Five while the
obtained results are analyzed and compared in Chapter Six. The conclusion is in Chapter
Seven. References and Appendices are provided at the end of the thesis.
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