Traffic Flow Forecasting Using Grey Neural Network Model
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Transcript of Traffic Flow Forecasting Using Grey Neural Network Model
Traffic flow forecasting
ASEMINAR REPORT
ON
Traffic Flow Forecasting Using Grey Neural Network Model
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ABSTRACT
In this Report, a kind of Grey Neural Network (abbreviates GNN) is
proposed which combines grey system theory with neural network, that is,
the GNN model has been built by adding a grey layer before neural input
layer and a white layer after neural output layer. Gray neural network can
elaborate advantages of both grey model and neural network, and enhance
further precision of forecasting. The GNN model is employed to forecast a
real vehicle traffic flow of JINGSHI highway with favor precision and result,
which is firstly applied GNN to traffic flow forecasting. Evaluation method
has been used for comparing the performance of forecasting techniques.
The experiments show that the GNN model is outperformed GM model and
neural network model, and traffic flow forecasting based on GNN is of
validity and Feasibility. In this study, we consider an application of grey
system theory to the time series data forecasting problem, called grey
forecasting, where grey implies incomplete or uncertain, and grey system
describes a system lacking information about structure messages, operation
mechanism and or behavior documents. In case of bad data lacking
information, grey forecasting method is known to be effective in time series
data analysis. We present the design of grey forecasting model, and
compare it with other methods.
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CONTENTS
Pg .no
CHAPTER 1: INTRODUCTION
1.1 General………………………………………………………………..2
1.2 Importance of Traffic Forecasting in Highway Sector…………………2
1.3 Need and Strategy of Forecasting……………………………………...2
1.4 Experiences in Traffic Forecasting…………………………………….3
1.5 Traffic Flow Forecasting Models……………………………………....4
CHAPTER 2: ARTIFICIAL NEURAL NETWORKS (ANN)
2.1 What is NN?.......................................................................................................5
2.2 History of ANN……………………………………………………….5
2.3 Why use NN…………………………………………………………..7
2.4 Biological Inspiration …………………………………………………7
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2.5 Application of ANN…………………………………………………..9
2.6 ANN Model and Architecture
2.6.1 Neuron Model………………………………………………….11
2.6.2 Network architecture…………………………………………...16
CHAPTER 3: GREY SYSTEM THEORY AND TIME SERIES ANALYSIS
3.1 Back Ground of Grey System Theory………………………………...20
3.2 Fundamental concepts of GST and its main contents………………....21
3.3 Grey Time Series Analysis…………………………………………….23
3.4 Grey Forecasting Model………………………………………………23
CHAPTER4: GREY NEURAL NETWORKS
4.1 Construction of Grey Neural network Model………………………...28
4.2 Experiment Result and Comparison of GNN, GM (1, 1) &NN ……..30
CHAPTER5: Conclusions.....………………………………………………………34
References………………………………………………………………………….35
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List of figures pg.no
1. Schematic diagram of biological neurons 7
2. Single Input Neuron 12
3. Hard Limit Transfer Function 13
4. Linear Transfer Function 14
5. Log Sigmoid Transfer Function 14
6. Multi Input Neuron 16
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7. Neuron with R inputs with Abbrivated notations 17
8. Layers of neurons 17
9. Topology of Feed Forward Neural Network 19
10. Three layer network 20
11. The construction of grey neural network model 29
List of tables pg.no
1. Transfer Functions 15
2. The Results obtained from three Forecasting Models and Compares 34
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CHAPTER 1
INTRODUCTION
Traffic flow forecasting is significant to traffic programming, traffic guide,
traffic controlling, traffic management, traffic security, etc. It has become
an emphasis question for discussion in traffic engineering domain and one
kernel study in Intelligent Transportation System. Grey system theory and
neural networks have been successfully used to predict traffic Grey system
theory utilizes accumulated generating data instead of original data to build
forecasting model, which makes raw data stochastic weak, or reduces noise
influence in a certain extent, therefore, intrinsic regularity of data can be
searched easily, and model can be built with relatively little data. Neural
network has been a primary nonlinear forecasting method because of its
ability of self-learning, nonlinear map and parallel distributed manipulation.
Traffic system is a complicated system with rather great stochastic,
traffic flow possess characteristic of great time-dependent and nonlinear. If
combine grey system theory with neural networks to build GNN (Grey
Neural Network), we can exploit sufficiently the characteristic of grey
system model requiring less data and feature of nonlinear map of neural
network, and develop both advantages, thus raise predicting precision much
more. In this paper, a kind of forecasting model combining grey system
theory with neural networks is proposed, which adds a grey layer before
neural input layer and a white layer after neural output layer. The GNN
model is firstly applied to forecast a real vehicle traffic flow of JINGSHI
highway with favorable precision and prediction result. Evaluation methods
are used for comparing the performance of forecasting techniques, which
show that the GNN model is outperformed GM model and neural network.
The experiment shows that this kind information manipulation and
forecasting method based on GNN is of validity and feasibility.
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1.1 GENERAL
TRAFFIC FORECASTING
Traffic flow forecasting is significant in traffic programming, traffic guide,
traffic controlling, traffic management, traffic security etc. It has become an
emphasis question for discussion in traffic engineering domain and in
intelligent transportation system. Forecasting of data is a key element of
management decision making. It becomes all the more important when
decision involves huge investments.
1.2 IMPORTANCE OF TRAFFIC FLOW FORECASTING IN HIGHWAY
SECTOR:
Transportation is a basic infrastructural facility for the economical, social,
cultural and administrative development country. It has been recognized
that the sustainable development of an area is dependent on the type and
quantum of the transportation infrastructure linking the various centers of
human population, employment, economic growth and market centers.
Fast depleting financial and other resources and over increasing travel
needs call for careful planning and optimum resource utilization in the road
sector. as all the decisions regarding planning, construction and
maintenance of road sector are based on estimates of the traffic for the
design period, it is necessary to cut down the dependence on the chance
while forecasting the traffic. over estimation of traffic will result in more
than necessary capital being tied up in a fewer projects, thus preventing
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other potential projects being taken up. where as under estimation of the
traffic will result in premature failure of the pavement structure, causing
heavy financial losses increased maintenance costs.
1.3 NEED AND STRATEGY OF FORECASTING
Existence in an environment governed by time requires allocation of
available time among competing resources in some optimal manner. This is
accomplished by making forecasts of future activities and taking the proper
actions as suggested by these forecasts. The time series underlying the
process to be forecasts is bound to be influenced by many casual factors.
Some forcing the time series up while conflicting factors act to force the
series down nevertheless it is essential to make forecasts in order to
effectively adjust budget and resources.
Forecasts by extending the patterns revealed by smoothing techniques, is a
very speculative procedure. It must be assumed to start with that past is a
mirror of the future the past trends and cycles will continue in the
future .this is seldom the case ,in the end ,mathematical forecasting
procedures and judgments must work hand in hand .thus one must not only
smooth the data and try to extend the signal components in to the future
but also predict the impact of unknown factors such as political events,
research and inventions, new land use development ,changes in the present
land use ,vehicle use ,change of behavior of vehicle user etc in connections
with traffic volumes. The subjective evaluations must, in turn, be used to
conditions the forecast obtained from the mathematical forecasting model.
1.4 EXPERIENCES IN TRAFFIC FORECSTING:
Numbers of methods are available for forecasting ranging from the simplest
methods such as the one using the most recent observations as to forecast
to highly complex approaches like econometric system of simultaneous
equations. However, the methods for generating forecasts can be broadly
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classified as qualitative, depending upon the extent to which mathematical
and statistical methods are used.
Quantitative methods, market research methods, panel consensus,
historical analogy, visionary forecasts etc, involve subjective estimation
through the experts opinion from a panel of forecasts.
Hence such forecast may differ from panel to panel or expert to expert.
Sometimes the divergence in opinion among the expert is so extensive that
it becomes hard to imagine any substantial could be placed in the results.
On the other hand substantial forecasting procedures explicitly define how
the forecast is determined .the logic is clearly stated and the operations are
mathematical. The methods involve examination of historical data to
determine the underlying process generating the variable and assuming
that the process is stable; use this knowledge to extrapolate the process
into the future. The two basic types of these models are time series models
and casual models.
Casual models exploit the relationship between the time series of interest
and one or more other time series data of casual variables. Knowing the
future values o the casual variables, one can use the model to forecast the
dependent variable. But the future value of casual variable may itself be
obtained by forecasting it either by casual models or time series models.
Hence this method is complex to operate. Some of the casual models are
regression analysis, econometric models, input-output models; anticipation
surveys etc.Time series models use only the time history of the variable
being forecasted in order to develop a model for predicting future values.
The selection of appropriate forecasting methods is influenced by the
following factors such as ,
1. Form of forecasts required.
2. Forecasts horizon, period and interval.
3. Data availability.
4. Accuracy required.
5. Behavior of process being forecast.
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6. Cost of development.
7. Ease of pattern.
8. Management comprehension and cooperation.
1.5 TRAFFIC FLOW FORECASTING MODELS:
Several types of mathematical models currently exist and are used to
forecast the traffic flow.
These models range from simple regression to complicated transition
probability method. On the other hand grey forecasting model and neural
networks, fuzzy logic have been applied in traffic flow forecasting to certain
extent.
Development of traffic forecasting models has been an active area in the
last couple of decades, which constitute a key component of management
decision making. The traffic forecasting model, when considered as a
system with inputs of historical and current data and outputs of future data,
behaves in a nonlinear fashion and varies with time of day. Traffic data are
found to change abruptly during the transition times of entering or leaving
rush hours. Accurate and real time models are needed to approximate the
nonlinear time variant functions between system inputs and outputs from a
continuous stream of training data.
There has been a steady increase in both rural and urban freeway traffic in
recent years resulting in congestion in many freeway systems. Accurate and
timely forecasting of traffic flow is of paramount importance for effective
management of traffic congestion and decision making. The basic types of
forecasting models are given below.
1. Time series models.
2. Local regression models.
3. Kalman filters theory.
4. Neural network approach.
4. Markov chin model.
5. Fuzzy neural approach.
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CHAPTER 2
ARTIFICIAL NEURAL NETWORKS
2.1 WHAT IS NEURAL NETWORK?
An Artificial Neural Network (ANN) is an information processing paradigm
that is inspired by the way biological nervous systems, such as the brain,
process information. The key element of this paradigm is the novel
structure of the information processing system. It is composed of a large
number of highly interconnected processing elements (neurons) working in
unison to solve specific problems. ANNs, like people, learn by example. An
ANN is configured for a specific application, such as pattern recognition or
data classification, through a learning process. Learning in biological
systems involves adjustments to the synaptic connections that exist between
the neurons. This is true of ANNs as well.
2.2 HISTORY OF ANN:
The history of artificial neural networks is filled with colorful, creative
individuals from many different fields, many of whom struggled for decades
to develop concepts that we now take for granted. This history has been
documented by various authors. One particularly interesting book is Neuro
computing: Foundations of Research by John Anderson and Edward
The history of neural networks has progressed through both conceptual
innovations and implementation developments. These advancements,
however, seem to have occurred in fits and starts rather than by steady
evolution.
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Some of the background work for the field of neural networks occurred in
the late 19th and early 20th centuries. This consisted primarily of
interdisciplinary work in physics, psychology and neurophysiology by such
scientists as Hermann von Helmholtz, Ernst Mach and Ivan Pavlov. This
early work emphasized general theories of learning, vision, conditioning,
etc.,and did not include specific mathematical models of neuron operation.
The modern view of neural networks began in the 1940s with the work of
Warren McCulloch and Walter Pitts [McPi43], who showed that networks of
artificial neurons could, in principle, compute
any arithmetic or logical function. Their work is often acknowledged as the
origin of the neural network field.
McCulloch and Pitts were followed by Donald Hebb , who proposed that
classical conditioning (as discovered by Pavlov) is present because of the
properties of individual neurons. He proposed a mechanism for learning in
biological neurons.
The first practical application of artificial neural networks came in the late
1950s, with the invention of the perception network and associated learning
rule by Frank Rosenblatt. Rosenblatt and his colleagues built a perception
network and demonstrated its ability to perform pattern recognition. This
early success generated a great deal of interest in neural network research.
Unfortunately, it was later shown that the basic perception network could
solve only a limited class of problems unfortunately, both Rosenblatt’s and
Windrows networks suffered from the same inherent limitations, However,
they were not able to successfully modify their learning algorithms to train
the more complex networks. During the 1980s both of these impediments
were overcome, and research in neural networks increased dramatically.
New personal computers and workstations, which rapidly grew in
capability, became widely available.
In addition, important new concepts were introduced. The second key
development of the 1980s was the back propagation algorithm for training
multilayer perception networks, which was discovered independently by
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several different researchers. The most influential publication of the back
propagation algorithm was by David Rumelhart and James McClelland
[RuMc86].
These new developments reinvigorated the field of neural networks. In the
last ten years, thousands of papers have been written, and neural networks
have found many applications. The field is buzzing with new theoretical and
practical work. Many of the advances in neural networks have had to do
with new concepts, such as innovative architectures and training rules. Just
as important has been the availability of powerful new computers on which
to test these new concepts.
Neural networks will not only have their day but will have a permanent
place, not as a solution to every problem, but as a tool to be used in
appropriate situations. In addition, remember that we still know very little
about how the brain works. The most important advances in neural
networks almost certainly lie in the future.
2.3 WHY USE NEURAL NETWORKS?
Neural networks, with their remarkable ability to derive meaning from
complicated or imprecise data, can be used to extract patterns and detect
trends that are too complex to be noticed by either humans or other
computer techniques. A trained neural network can be thought of as an
"expert" in the category of information it has been given to analyse. This
expert can then be used to provide projections given new situations of
interest and answer "what if" questions.
Other advantages include: Adaptive learning: An ability to learn how to do
tasks based on the data given for training or initial experience.
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1. Self-Organization: An ANN can create its own organization or
representation of the information it receives during learning time.
2. Real Time Operation: ANN computations may be carried out in
parallel, and special hardware devices are being designed and
manufactured which take advantage of this capability.
Fault Tolerance via Redundant Information Coding: Partial destruction of a
network leads to the corresponding degradation of performance. However,
some network capabilities may be retained even with major network
damage
2.4Biological Inspiration:
An Artificial Neural Network (ANN) is an information processing paradigm
that is inspired by the way biological nervous systems, such as the brain,
process information.
The brain consists of a large number (approximately 1011) of highly
connected elements (approximately 104connections per element) called
neurons. For our purposes these neurons have three principal components:
the dendrites, the cell body and the axon. The dendrites are tree-like
receptive networks of nerve fibers that carry electrical signals into the cell
body. The cell body effectively sums and thresholds these incoming signals.
The axon is a single long fiber that carries the signal from the cell body out
to other neurons. The point of contact between an axon of one cell and a
dendrite of another cell is called a synapse. It is the arrangement of neurons
and the strengths of the individual synapses, determined by a complex
chemical process that establishes the function of the neural network. Figure
1 is a simplified schematic diagram of two biological neurons.
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Fig:1 Schematic diagram of biological neurons
Artificial neural networks do not approach the complexity of the brain.
There are, however, two key similarities between biological and artificial
neural networks. First, the building blocks of both networks are simple
computational devices (although artificial neurons are much simpler than
biological neurons) that are highly interconnected. Second, the connections
between neurons determine the function of the network.
It is worth noting that even though biological neurons are very slow when
compared to electrical circuits (10-3 s compared to 10-9 s), the brain is able
to perform many tasks much faster than any conventional computer. This is
in part because of the massively parallel structure of biological neural
networks; all of the neurons are operating at the same time. Artificial neural
networks share this parallel structure
.
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2.5 APPLICATIONS
The applications are expanding because neural networks are good at
solving problems, not just in engineering, science and mathematics, but in
medicine, business, finance and literature as well.
Their application to a wide variety of problems in many fields makes them
very attractive. Also, faster computers and faster algorithms have made it
possible to use neural networks to solve complex industrial problems that
formerly required too much computation.
Neural networks have been applied in many fields A list of some
applications mentioned in the literature follows
Aerospace
High performance aircraft autopilots, flight path simulations, aircraft
control systems, autopilot enhancements, aircraft component simulations,
aircraft component fault detectors
Automotive
Automobile automatic guidance systems, warranty activity analyzers
Banking
Check and other document readers, credit application evaluators
Defense
Weapon steering, target tracking, object discrimination, facial recognition,
new kinds of sensors, sonar, radar and image signal processing including
data compression, feature extraction and noise suppression, signal/image
identification
Electronics
Code sequence prediction, integrated circuit chip layout, process control,
chip failure analysis, machine vision, voice synthesis, nonlinear modeling
Entertainment
Animation, special effects, market forecasting
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Financial
Real estate appraisal, loan advisor, mortgage screening, corporate bond
rating, credit line use analysis, portfolio trading program, corporate
financial analysis, currency price prediction
Insurance
Policy application evaluation, product optimization
Manufacturing
Manufacturing process control, product design and analysis, process and
machine diagnosis, real-time particle identification, visual quality inspection
systems, beer testing, welding quality analysis, paper quality prediction,
computer chip quality analysis, analysis of grinding operations, chemical
product design analysis, machine maintenance analysis, project bidding,
planning and management, dynamic modeling of chemical process systems
Medical
Breast cancer cell analysis, EEG and ECG analysis, prosthesis design,
optimization of transplant times, hospital expense reduction, hospital
quality improvement, and emergency room test advisement
Robotics
Trajectory control, forklift robot, manipulator controllers, vision systems
Speech
Speech recognition, speech compression, vowel classification, text to
speech synthesis
Securities
Market analysis, automatic bond rating, and stock trading advisory systems
Telecommunications
Image and data compression, automated information services, real-time
translation of spoken language, customer payment processing systems
Transportation
Truck brake diagnosis systems, vehicle scheduling, routing systems
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2.6 ANN MODEL AND ARCHITECTURE
2.6.1 NEURON MODEL
Single-Input Neuron:
A single-input neuron is shown in Figure 2. The scalar input is multiplied by
the scalar weight to form , one of the terms that is sent to the summer. The
other input,, is multiplied by a bias and then passed to the summer. The
summer output, often referred to as the net input , goes into a transfer
function , which produces the scalar neuron output . (Some authors use the
term activation function rather than transfer function and offset rather than
bias.)
The weight corresponds to the strength of a synapse, the cell body is
represented by the summation and the transfer function, and the neuron
output represents the signal on the axon.
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Fig:2 SINGLE INPUT NEURON
The neuron output is calculated as a= f (wp+b)
If, for instance w = 3 p = 2 and ,b=-1.5 then a= f(3(2)– 1.5)= f(4.5) .
The actual output depends on the particular transfer function that is
chosen. The bias is much like a weight, except that it has a constant input of
1.However, if you do not want to have a bias in a particular neuron, it can
be omitted.
Note that w and b are both adjustable scalar parameters of the neuron.
Typically the transfer function is chosen by the designer and then the
parameters w and b will be adjusted by some learning rule so that the
neuron input/output relationship meets some specific goal.
Transfer Functions:
The transfer function in Figure 2 may be a linear or a nonlinear function of .
A particular transfer function is chosen to satisfy some specification of the
problem that the neuron is attempting to solve.
A variety of transfer functions have been included and Three of the most
commonly used functions are discussed below.
Hard Limit Transfer Function:
The hard limit transfer function, shown on the left side of Figure 3 , sets the
output of the neuron to 0 if the function argument is less than 0, or 1 if its
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argument is greater than or equal to 0. We will use this function to create
neurons that classify inputs into two distinct categories.
Fig:3 HARD LIMIT TRANSFER FUNCTION
The graph on the right side of Figure 3 illustrates the input/output
characteristic of a single-input neuron that uses a hard limit transfer
function. Here we can see the effect of the weight and the bias. Note that an
icon for the hard limit transfer function is shown between the two figures.
Such icons will replace the general in network diagrams to show the
particular transfer function that is being used.
Linear Transfer Function
The output of a linear transfer function is equal to its input:
a = n
As illustrated in Figure 4.
Neurons with this transfer function are used in the ADALINE networks
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Fig:4 LINEAR TRANSFER FUNCTION
Log Sigmoid Transfer Function:
FIG:5 LOG SIGMOID TRANSFER FUNCTION
This transfer function takes the input (which may have any value between
plus and minus infinity) and squashes the output into the range 0 to 1,
according to the expression.
The log-sigmoid transfer function is commonly used in multilayer networks
that are trained using the back propagation algorithm, in part because this
function is differentiable.
Most of the transfer functions are summarized in Table1.
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TABLE 1 TRANSFER FUNCTIONS
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Multiple-Input Neuron:
Weight matrix:
Typically, a neuron has more than one input. A neuron with R inputs is
shown in Figure 6. The individual inputs p1,p2,p3….. are each weighted by
corresponding elements w1,w2,w3…..of the weight matrix W .
FIG:6 MULTI INPUT NEURON
The neuron has a bias , which is summed with the weighted inputs to form
the net input :
This expression can be written in matrix form:
Now the neuron output can be written as
Weight indices:
We have adopted a particular convention in assigning the indices of the
elements of the weight matrix. The first index indicates the particular
neuron destination for that weight. The second index indicates the source of
the signal fed to the neuron. Thus, the indices w 1,2 in say that this weight
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represents the connection to the first (and only) neuron from the second
source.
We would like to draw networks with several neurons, each having several
inputs. Further, we would like to have more than one layer of neurons. You
can imagine how complex such a network might appear if all the lines were
drawn. It would take a lot of ink, could hardly be read, and the mass of
detail might obscure the main features. Thus, we will use an abbreviated
notation.
A multiple-input neuron using this notation is shown in Figure7.
FIG:7 Neuron with R inputs with Abbrivated
notations.
2.6.2 NETWORK ARCHITECTURES
Commonly one neuron, even with many inputs, may not be sufficient. We
might need five or ten, operating in parallel, in what we will call a layer.
A Layer of Neurons
A single-layer network of S neurons is shown in Figure8. Note that each of
the R inputs is connected to each of the neurons and that the weight matrix
now has S rows.
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FIG:8 Layers of neurons
The layer includes the weight matrix, the summers, the bias vector b , the
transfer function boxes and the output vector a. Some authors refer to the
inputs as another layer, but we will not do that here Each element of the
input vector p is connected to each neuron through the weight matrix W.
Each neuron has a bias bi, a summer, a transfer function f and an output
ai .Taken together, the outputs form the output vector a. It is common for
the number of inputs to a layer to be different from the number of neurons
(i.e. )..
The input vector elements enter the network through the weight matrix W:
As noted previously, the row indices of the elements of matrix W indicate
the destination neuron associated with that weight, while the column
indices indicate the source of the input for that weight. Thus, the indices in
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w3,2 say that this weight represents the connection to the third neuron from
the second source.
A layer whose output is the network output is called an output layer. The
other layers are called hidden
Layers. It is shown in fig 9:
FIG:9 Topology of Feed Forward Neural Network
Multiple layers:
Now consider a network with several layers. Each layer has its own weight
matrix W, its own bias vector b, a net input vector n and an output vector a .
We need to introduce some additional notation to distinguish between these
layers. We will use superscripts to identify the layers. Specifically, we
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append the number of the layer as a superscript to the names for each of
these variables. Thus, the weight matrix for the first layer is written as W1,
and the weight matrix for the second layer is written as W2 as shown in fig
10.
FIG:10 Three layer network
As shown, there are R inputs, S1 neurons in the first layer, S2 neurons in the
second layer, etc. As noted, different layers can have different numbers of
neurons.
The outputs of layers one and two are the inputs for layers two and three.
Thus layer 2 can be viewed as a one-layer network with R = S1 inputs, S=S2
neurons, and an S1xS2 weight matrix. The input to layer 2 is a1, and the
output is a2.
How to Pick Architecture
Problem specifications help define the network in the following ways:
1. Number of network inputs = number of problem inputs
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2. Number of neurons in output layer = number of problem outputs
3. Output layer transfer function choice at least partly determined by
Problem specification of the output
CHAPTER 3
3.1 THE BACKGROUND OF GST
Based on widespread divisions in activities of scientific research, the highly
synthetic tendency has brought forward many cross-disciplinary research
activities possessing significant methodological meanings. The systems
science has revealed more profoundly and essentially some important
internal relations among the subjects, who have deeply promoted the
integrative progress of modem science and technology. With the help of
these newly emerging fields of study, many complicated problems,
unsolvable before, can be resolved successfully and much deeper
understandings about the nature have been brought forward. These cross
disciplinary theories include, to say a few, the systems theory, information
theory and cybernetics, which were formulated during the end of the 1940s,
the theory of dissipative structures, synergetic and fractals, which started
to be known during the end of the 1960s and the beginning of 197Os, the
ultra circular theory and general systems theory, which have been more
maturing after late 1970s.
In a systems research, due to noises from both inside and outside of the
system of our concern and the limitation of our cognitive level, the
information people obtain is always uncertain and limited in scope. With the
development of science and technology and the progress of the social
society, people’s understanding about the uncertainties of various systems
is much more profound than ever before, and the study on uncertainties is
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also more in-depth. During the later half of 20* century, in the field of
systems science and engineering, a variety of systems theories and
methodologies on uncertainty had been emerging constantly. For instance,
Professor L.A. Zaden established fuzzy mathematics in the 1960s, Professor
J. L.Deng pioneered a difficult and fruitful research on grey systems theory,
Professor 2. Pawlark initiated rough sets theory in the 198Os, and Professor
Wang gnang-yuan contributed a great deal in the area of unascertained
mathematics. All these theories mentioned above are significant
achievements in the research on unascertained systems, and provided the
needed theories and methodologies for describing and dealing with
numerous unascertained information from different aspects.
3.2 FUNDEMENTAL CONCEPT OF GST AND ITS MAIN CONTENTS
In the year of 1980, grey systems theory was brought forward by Professor
Deng Ju-long from China. It was a new theory and method applicable to the
study of unascertained problems with few data and or poor information.
Grey systems theory works on unascertained systems with partially known
and partially unknown information by drawing out valuable information by
generating and developing the partially known information. It can describe
correctly and monitor effectively the systemic operational behavior.
Many systems, such as social, economic, agricultural, industrial, ecological
biological systems, are named based on the fields and ranges where the
research subjects belong to . In contrary, the name of grey systems is
chosen based on the colors of the subjects under investigation. For example,
in control theory, the darkness of colors has been commonly used to
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indicate the degree of clarity of information. One of the most well accepted
representations is the so-called “black box”, which stands for an object with
its internal relations or structure totally unknown to the investigator .Here,
we will use ‘Mack” to represent unknown information, “white” for
completely known information ,and “grey” for those information which are
partially known and partially unknown. Accordingly, we will name the
systems with completely unknown information as black systems, and the
systems with partially known and partially unknown information as grey
systems, respectively.
In our daily social economic and scientific research activities, we often face
situations of incomplete information. For example, in some studies of
agriculture, even though all the information, related to the area which is
planted, the quality of seeds, fertilizers, irrigation, et al., is completely
known, it is still difficult to estimate the production quantity and the
consequent annual income due to various unknowns or vague information
related to labor quality, the level of technology employed, natural
environment, weather conditions, et al. As for the case of insects control, we
might have known very well the relationship between the special kind of
insect and its Natural enemies. But it might still be difficult for us to achieve
the desirable certainty due to the reason that we do not have enough
information regarding the relationship between the insects of our concern
and the baits, its natural enemies and the baits, one natural enemy and
other natural enemies, one kind of insect and other kinds of insects, et al.
For each adjustment of a price system in our economy, the decision makers
often face the difficulty of not knowing the definite information on the effect
of the price change on consumers, on the prices of goods, et al. All liquid
pressure systems are difficult to control due to some immeasurable
quantities. Electricity systems are hard to observe because of the stochastic
parameters of the voltage and currents, which is caused by not having
enough knowledge on motion and parameters. In a general social or
economic system, it is difficult to analyze the effect of the input on the
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output for the reasons that there do not exist clear differences between the
“interior” and the “exterior”, the system self and its environment, and that
the boundary of the system may be sometime easy to tell or on other
occasions difficult to clarify. In stochastic works, a same economic variable
could be seen as endogenous by some scholars and external by some other
scholars. The appearance of such a phenomenon is due to the lack of
Modeling information, or the reason that an appropriate systems model
has not been found, or the fact that the right observation and control
variables have not been employed.
Having been developed for more than 20 years, grey Systems theory has
already built up the framework of a new discipline. Its main contents
include: a theory system based on hazy integration, an analysis system
depending on space of grey incidence, a modeling system with GM as its
vital part, a methodological system on the foundation of grey sequence
generation, and a technological system Constructed mainly by systems
analysis, modeling, forecasting, decision, controlling and optimization. Hazy
Integration, grey algebraic system, grey equations and grey matrix are the
foundation of grey systems theory, and there are still many problems worth
further studying in order to perfect itself. Grey systems analysis consists of
mainly grey incidence analysis, grey clustering and grey statistical
evaluation, et al. The generation of grey sequence relies on functions of
sequence operators including buffer operator (weakening operator,
strengthening operator), average generation operator, stepwise ratio
generation operator, inverse accumulating generation operator and
accumulating generation operator, et al. Grey systems modeling is famished
based on the thought of five-step-modeling. And hidden laws are found
through the generation of grey numbers or functions of sequence operators.
The new promise of using discrete data sequence to construct continuous
dynamical differential equations is achieved by interchanging grey
difference equations with grey differential equations. Grey prediction is a
quantitative prediction based on GM (1,l). According to the effectiveness
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and characteristics, grey predictions can be classified as following six
classes: (I) Serial predictions; (2) Interval predictions; (3) Disaster
predictions; (4) Seasonal disaster predictions; (5) Stock-market-like
predictions; and (6) Systems predictions. Grey decision making includes :
(1)Grey target decision makings; (2) Grey incidence decision-making, (3)
Grey statistics, (4) Grey clustering decision making, (5) Grey situation
decision making, and (6) Grey stratified decision making. The main contents
of grey control cover the control problems of grey systems of intrinsic
characteristics and controls based on grey systems methodology, such as
grey incidence control, control of GM (1,l) prediction, et al. And grey linear
programming, grey nonlinear programming, grey integer programming,
grey dynamic programming and et al. are all involved in grey optimal
technology.
3.3 GREY TIME SERIES ANALYSIS
Grey system theory was proposed in early 1980s, as a tool for considering
systems with uncertainties in extensive applications .Using the concept of
black box, if the characteristics of a system is known, we call the system
white, on the contrary if the characteristics of a system is unknown, w-e call
the system black. While grey system is defined between the two as a system
which is partially known, e.g., - law of movement or characteristics of the
system is partially understood, or - factors used in the system description
are not well defined or uncertain, or * relations among factors are not
known.
In grey system theory, the cases with incomplete information are treated by
using grey factors, grey numbers and grey relations, which describe
uncertainties, give numerical forms of grey factors and deal with the
incomplete relations respectively.
In system analysis or modeling, object data are generally collected under
various conditions. The data may contain errors from noises and other
unknown factors. Grey system theory is to bring a grey system close to a
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white system, or to estimate rough characteristics of the system based on
the known incomplete information. Grey system theory is concerned with
the mathematics for grey numbers, which has been proposed useful for
studies in fields of liner planning, forecasting decision making, system
control, etc., .In this study, we present the forecasting method based on
grey system theory (grey forecasting) for time series data analysis, in a
comparison with conventional techniques.
3.4 GREY FORECASTING MODEL
In grey forecasting, the forecasting models are based on generating
operations to the time series data sequence. For example, AGO
(accumulated generating operation), an iterative addition to the time
series data, has been proposed as one of generating operations. AGO is
defined as follows.
Suppose x(0) is an original discrete n the dimensional sequence with
elements x0(k)
K=1, 2… n, i.e
x (0) ={x(0) (1),…..x(0) (n)}………………(1)
Then AGO is defined as
x(0) ={x(0) (1),…..x(0) (n)}
Where
Similarly, AGO to time series x(r-1) is given as
Where
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In fact,x(r) can be accordingly viwed as a result of the approximated
exponential law to x(r-1) .
As an example consider an intial time series x(0) ,eg;x(0)
={1,2,1.5,3},according to definition of eqn(2) the AGOs can be obtained as
AGO x(0) : x(1) = {1,3,4.5,7.5}
AGO x(1) : x(2) = {1,4,8.5,16}…….
In the design of a forecasting model using grey system theory, the model is
called a grey model (GM). The grey models are given by grey differential
equations, which are groups of abnormal differential equations with
variations in behavior parameters, or grey difference equations which are
groups of abnormal difference equations with variations in structure, rather
than the first-order differential equations or the difference equations in
conventional cases.
Basically, the model in grey forecasting is GM (1, l), which is built based on
AGO to a time series sequence only. Here, GM (n, m) denotes a GM
including nth-order differential or n difference equations with m variables.
Suppose x(0) is an original discrete n th dimensional sequence, and x(1) is the
AGO on x(0) ,i.e.;
x(0) = {x(0) (1),x(0) (2),…..,x(0) (n)},
x(1) = {x(1) (1),x(1) (2)……,x(1) (n)} = AGO x(0) .
The forecasting model GM (1, l) is described by using following equation:
and x(1) (k) is a group of real numbers, which is determined as if and only
if x(1) (k) is relative to α(1) (x(1)(k))
Compared with the form of normal first-order differential equation, i.e.,
(dx (t)/dt)+ax (t) =b, a, b: constants. (4)
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The difference (1) (x (1) (k)) is corresponding to dx (t)/dt, and so is x (1) (k) to
x (t). x(1) (k) is called the background value of (1) (x (1) (k)).i.e;the value of (1)
(x (1) (k)) depends only on that of x(1) (k).
That is the forecasting model GM (1, I) by eq.(3-a)is based on a
difference equation concerning
Since α(1) (x(1)(k))= x(0) (k),k=1,2,3…….n from eqn (3-b).the difference can be
rewritten as
Where a is called the development of GM, and b is called the grey input.
The fifth eqn will be satisfied when, if and only if
When k=2, 3….n.
Where z (1) (k) is the mean of x (1) (k) defined as
Where k=2, 3….n
Under the demand for parallel shooting, eq (5) can therefore be
transformed to
Where k=2, 3…n.
Where a, b are determined to minimize the least square error on
x (1) (k)-(b-a Z(1) (k)) where
k=2,3……..n.
i.e. Min: ET E
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Where
Based on identification algorithm [2], optimal a and b are given by
For the given discrete n th-dimensional sequence x(0),the forecasting model
is then determined,with a and b shown by eqn(7).and the sequence x^ (1) (k)
is given by
Where k=1, 2…
(8)
Is said to be the response of the GM (1, 1)
Accordingly the following sequence
Is said to be the GM (1,1) sequence of the AGO and
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Is called the GM(1,1) sequence while the sequence
Is called the forecasting sequence of GM (1, 1)
There by, the grey forecasting for a given time series data sequence x={x
(0), x (1)…….x (n)} is to determined the correspondent forecasting
sequence of GM (1, 1) by eq (11)
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CHAPTER 4
4.1 THE CONSTRUCTION OF GREY NEURAL NETWORK
The grey system theory has been initially presented by Deng . The grey
system puts each stochastic variable as a grey quantity or a grey procedure
that changes within a given range or a certain time period. It does not rely
on statistical method to deal with the grey quantity; instead, it uses grey
generating method to deal with these disorderly and unsystematic raw data
and then changes them into a time series data with regularity. In this way,
the stochastic degree of the grey quantity is reduced, and it is easy for some
functions to characterize the grey quantity. Grey Neural Network model has
been built according to above ideology. GNN model has three basic parts: a
grey layer, a general neural network (such as back propagation), and a
white layer. The grey layer before neural input nodes has accumulated
generating operation (AGO) to initial input data, then these new data
generated by the accumulated generating operation are feed into the
network, at last, the white layer after neural output nodes inverses
accumulated generation to the output data of the network Therefore, the
prediction value we need is obtained. The construction of GNN model is
shown in fig.1
Fig11: the construction of grey neural network
model
Neural network design includes determining network structure, the number
of layers and the number of neurons in every layer. Generally, the neural
network adopts neural network back propagation with three layers, and the
NN learning algorithm is error back propagation. Let the number of input
nodes be n, the number of hidden nodes ism, and the number of output
neurons is one for one step prediction. We often use n*m*l to describe the
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NN frame. The number of input nodes, i.e., the value of n can be determined
by the grey relational analysis, that is, taking into account the relationships
existed between several known traffic flow and the prediction value. The
value of m can be determined by thorough tests. The GNN model
mechanism is described in the following. Suppose the neural network in the
GNN model has n input nodes, the original data x(0) with n+l entries taken
as training sample is
where x(0) (i) is the time series data at time i . Based on the initial sequence
x(0) , a new sequence x(1) with n+l entries is generated by the, accumulated
generating operation,
where x("(kJ is derived as follows:
Let
Where the pair [z, y] constitutes one train sample for neural network back
propagation model, z is input data and y is output data. Get a vector with
n+l elements at one time from initial time series data in turn, if the length
of initial time series data is N, we can obtain N-n train samples to train NN.
When the GNN is successfully trained, it can be used to predict traffic flow.
The forecast is estimated through one operation of the inverse of the
accumulated generating operation. The prediction value of x (0) (n + 1) can
be written as follows
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where x^(1) (n+I) is output value of the Neural Network in GNN model, x^(0)
(n + 1) is output value of the white layer in GNN model, it is prediction
value of x(0) (n+1) at time n+l. Besides the most common method
accumulated generating operation, the grey generating operation done to
raw data also includes multipoint-moving-average, opening the n power or
takes the logarithmic transformation to raw data. The original data has
been preprocessed by grey generating operation before feeding into a
neural network the unknown system can be easily characterized by then on
linear function of neural network. Thus, the training time of the network
can be shortening, so, while the prediction precision advanced, the
convergent process also can be speeded up.
4.2 EXPERIMENT RESULTS
The time series data of traffic volume in the period of a day from 5: OO to
17:00 at XUSHUI west in JlNGSHI highway have been used as test data
sets, there are 72traffic flow data regarding a small car as a unit, and the
sampling interval between two adjacent data is 10 minutes .Three
forecasting model, i.e., the Grey Neural Network model, the GM (1,l) grey
model, the Neural Network model are used to forecast this same traffic
flow. The time series data from no.1 to no.62 are used as known data (i.e.,
in-sample data) to forecast the last 10 data from no.63 to no.72 (i.e., out-of-
sample data). The difference between the actual and the forecast are used
to evaluate the accuracy of the forecasting models. Four criteria, i.e., the
mean root of squares error (MRSE), the mean absolute percentage error
(MAPE), the maximum absolute percentage error(MAXAPE), the minimum
absolute percentage error(MINAF'E) are used to compare the performance
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of the GNN model against other two models, i.e., the grey forecasting model
GM (1,l) and the neural network model
The Forecasting Results of GM (1,l) Grey Model
The GM (1,l) grey model has been built using a time series data with 10
data, in order to ensure the forecasting accuracy of GM (1,l J grey model,
the equal dimension GM1276(],I) is applied, that is, after predict one traffic
flow data, add 3.3a new dam to the sequence at the end, meanwhile take
out the oldest datum from the head of the sequence, then, rebuilt the
GM(I,I) grey model to forecast the next traffic flow data In this way, the new
superseding the old, forecasting one by one, all need prediction results can
be obtained. Start from the 53th data to build grey model and then forecast
one data, iterate 10 times, then the last 10 traffic flow forecasting results
can be estimated, as shown in table 1.
The Forecasting Results of Neural Network Model:
Use the neural network back propagation model to build the traffic flow
forecasting model, where the choice of input nodes is derived from the grey
relation. According to the grey relation analysis,[x(l), x(2), x(3),x(4)] is taken
as input data, x(5)is taken as forecast data, and the neural network is
selected as 4*4*1. Take 62 data from no.] to no.62 for train data network,
the train data is preprocessed within the range [OJ] by standardization in
order to ensure the neural network train procedure convergent. Take
iteration as 15000, learning rate as 0.01, learning goal as sum of square
error 0.1. Set the initial neurons connection weighs as stochastic real
number belonging to[-1,1].
The neurons connection weighs and bias of a success trained neural
network are as follows:
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Use this neural network model to forecast the last l0 traffic flow data, the forecasting results are showed in table
The Forecasting Results of GNN Model:
Apply GNN model to forecasting traffic flow. The raw data goes through one
operation of the accumulated generating operation done by the grey layer,
the forecast is estimated through one operation of the inverse of
accumulated generating operation done by the white layer The neural
network in GNN model has 3,layers, from the result of the grey relation, the
number of input nodes is 4,and the number of neurons in hidden layer is
also defined as 4 by try.
Take 62 data from no. 1 to no. 62 for train data sets to train neural network,
the train data is preprocessed within the range [0,1] in Order to ensure the
neural network train procedure convergent. Take iteration as 15000
learning rate as 0.01, learning goal as Sum of Square error 0.1. set the
initial neurons connection weighs as stochastic real number belonging to [-
1,1].
The neurons connection weighs and bias of a success trained neural network in the GNN model are as follows:
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Use this grey neural network model to forecast the last l0 traffic flow data,
the forecasting results are also showed in table 1.
Evaluations and Comparisons:
In this section, the performance of the previous models in forecasting the
traffic flow in highway is reported. The measurement criteria include
MRSE, MAPE, MAXAPE, and MINAPE. The out-of-sample error in table 1
indicates that the grey neural network model is outperformed the GM (1,l)
model and the neural network model. The MAPE of GM(1,l) grey model and
the neural network model are 15.4884 and 13.0101, respectively, while the
grey neural network model is the lowest at 11.5597. The MRSE of the GNN
model is 4.7544, which is better than the GM (1,l) model and is the same as
the neural network model. The GNN model still has the lowest MAXAPE.
The MINAPE of the GNN model is better than the neural network model
and is the same as the GM (1,l) model.
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Table2: the results obtained from three forecasting models and
compares
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CHAPTER 5
CONCLUSION
The comparison of three forecasting models, i.e., the grey neural network
model, the neural network model and the GM (1,l) grey model demonstrates
that the grey neural network model is outperformed the GM (1,l) model and
the neural network model. If some improvement measure done to the GNN
model, such as to choose different neural network type, to add neurons in
hidden layer, to add learning time, or to choose representative samples
training neural network, the prediction accuracy would enhance further,
and the GNN model would be more practical.
In brief, the grey neural network model exploits sufficiently the
characteristic of the preprocessed data handled by the grey operation with
stochastic reduced and regularity raised and the nonlinear map feature of
neural network, makes the convergent process of the network fast, and
while advances the prediction precision. Therefore, the GNN model is a
novel practical method with rather high accuracy
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