traffic flow forecasting using grey neural network model

Traffic flow forecasting

ASEMINAR REPORT

ON

Traffic Flow Forecasting Using Grey Neural Network Model

Transportation division


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.



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



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



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



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



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.



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



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



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.



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.



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.



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



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.



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.



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

.



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



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



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.



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



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



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.



TABLE 1 TRANSFER FUNCTIONS



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



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.



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



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



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



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



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



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



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



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



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



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)



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



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



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)



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



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



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



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:



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:



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.



Table2: the results obtained from three forecasting models and

compares



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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