Vehicle Headway Distribution Models on Two-Lane Two-Way Undivided RoadsAM Publications
The time headway between vehicles is an important flow characteristic that affects the safety, level of service, driver behavior, and capacity of a transportation system. The present study attempted to identify suitable probability distribution models for vehicle headways on 2-lane 2-way undivided (2/2 UD) road sections. Data was collected from three locations in the city of Semarang: Abdulrahman Saleh St. (Loc. 1), Taman Siswa St. (Loc. 2) and Lampersari St. (Loc.3). The vehicle headways were grouped into one-second interval. Three mathematical distributions were proposed: random (negative-exponential), normal, and composite, with vehicle headway as variable. The Kolmogorov-Smirnov test was used for testing the goodness of fit. Traffic flows at the selected locations were considered low, with traffic volume ranged between 400 to 670 vehicles per hour per lane. The traffic volume on Loc.1 was 484 vehicles per hour, that on Loc. 2 was 405 vehicles per hour, and that on Loc. 3 was 666 vehicles per hour. Random distribution showed good fit at all locations under study with 95% confidence level. Normal distribution showed good fit at Loc. 1 and Loc. 2, whereas composite distribution fit only at Loc. 1. It was suggested that random distribution is to be used as an input in generating traffic in traffic analysis at highway sections where traffic volume are under 500 vehicles per hour.
Vehicle Headway Distribution Models on Two-Lane Two-Way Undivided RoadsAM Publications
The time headway between vehicles is an important flow characteristic that affects the safety, level of service, driver behavior, and capacity of a transportation system. The present study attempted to identify suitable probability distribution models for vehicle headways on 2-lane 2-way undivided (2/2 UD) road sections. Data was collected from three locations in the city of Semarang: Abdulrahman Saleh St. (Loc. 1), Taman Siswa St. (Loc. 2) and Lampersari St. (Loc.3). The vehicle headways were grouped into one-second interval. Three mathematical distributions were proposed: random (negative-exponential), normal, and composite, with vehicle headway as variable. The Kolmogorov-Smirnov test was used for testing the goodness of fit. Traffic flows at the selected locations were considered low, with traffic volume ranged between 400 to 670 vehicles per hour per lane. The traffic volume on Loc.1 was 484 vehicles per hour, that on Loc. 2 was 405 vehicles per hour, and that on Loc. 3 was 666 vehicles per hour. Random distribution showed good fit at all locations under study with 95% confidence level. Normal distribution showed good fit at Loc. 1 and Loc. 2, whereas composite distribution fit only at Loc. 1. It was suggested that random distribution is to be used as an input in generating traffic in traffic analysis at highway sections where traffic volume are under 500 vehicles per hour.
Rank adjustment strategies for Dynamic PageRank : REPORTSubhajit Sahu
This is my report on:
Rank adjustment strategies for Dynamic PageRank (v1).
While doing research work under Prof. Dip Banerjee, Prof. Kishore Kothapalli.
Abstract — To avoid calculating ranks of vertices in a dynamic graph from scratch for every snapshot, the ones computed in the previous snapshot of the graph can be used, with adjustment. Four different rank adjustment strategies for dynamic PageRank are studied here. These include zero-fill, 1/N-fill, scaled zero-fill, and scaled 1/N-fill. Results indicate that the scaled 1/N-fill strategy requires the least number of iterations, on average. As long as the graph has no affected dead ends (including dead ends in the previous snapshot), unaffected vertices can be skipped with this adjustment strategy.
Index terms — PageRank algorithm, Dynamic graph, Rank adjustment, Initial ranks.
Slides of my seminar on optimal transport and its applications in machine learning, image processing and mechanistic modelling.
Github with code: https://github.com/MichielStock/Teaching/tree/master/Optimal_transport
This paper presents a geometric approach to the coordinatization of a measured space called the Map
Maker’s algorithm. The measured space is defined by a distance matrix for sites which are reordered and
mapped to points in a two-dimensional Euclidean space. The algorithm is tested on distance matrices
created from 2D random point sets and the resulting coordinatizations compared with the original point
sets for confirmation. Tolerance levels are set to deal with the cumulative numerical errors in the
processing of the algorithm. The final point sets are found to be the same apart from translations,
reflections and rotations as expected. The algorithm also serves as a method for projecting higher
dimensional data to 2D.
REDUCING TIMED AUTOMATA: A NEW APPROACHijistjournal
Today model checking is the most useful verification method for real time systems, so there is a serious need for improving its efficiency with respect to both time and resources. In this paper we present a new approach for reducing timed automata. In fact regions of a region automaton are aggregated according to a coarse equivalence class partitioning based on traces. We will show that the proposed algorithm terminates and preserves original timed automaton. Proposed algorithms are implemented by model transformation with Atom3 tool.
It is head to head analysis to study the impact of multi short operation segments with a direct flight operation. A case study of SAH-CAI-SAH and SAH-ADE-CAI-ADE-SAH, here we study the impact of introducing Aden sector with the direct flight of SAH-CAI
Hope to enjoy
Rank adjustment strategies for Dynamic PageRank : REPORTSubhajit Sahu
This is my report on:
Rank adjustment strategies for Dynamic PageRank (v1).
While doing research work under Prof. Dip Banerjee, Prof. Kishore Kothapalli.
Abstract — To avoid calculating ranks of vertices in a dynamic graph from scratch for every snapshot, the ones computed in the previous snapshot of the graph can be used, with adjustment. Four different rank adjustment strategies for dynamic PageRank are studied here. These include zero-fill, 1/N-fill, scaled zero-fill, and scaled 1/N-fill. Results indicate that the scaled 1/N-fill strategy requires the least number of iterations, on average. As long as the graph has no affected dead ends (including dead ends in the previous snapshot), unaffected vertices can be skipped with this adjustment strategy.
Index terms — PageRank algorithm, Dynamic graph, Rank adjustment, Initial ranks.
Slides of my seminar on optimal transport and its applications in machine learning, image processing and mechanistic modelling.
Github with code: https://github.com/MichielStock/Teaching/tree/master/Optimal_transport
This paper presents a geometric approach to the coordinatization of a measured space called the Map
Maker’s algorithm. The measured space is defined by a distance matrix for sites which are reordered and
mapped to points in a two-dimensional Euclidean space. The algorithm is tested on distance matrices
created from 2D random point sets and the resulting coordinatizations compared with the original point
sets for confirmation. Tolerance levels are set to deal with the cumulative numerical errors in the
processing of the algorithm. The final point sets are found to be the same apart from translations,
reflections and rotations as expected. The algorithm also serves as a method for projecting higher
dimensional data to 2D.
REDUCING TIMED AUTOMATA: A NEW APPROACHijistjournal
Today model checking is the most useful verification method for real time systems, so there is a serious need for improving its efficiency with respect to both time and resources. In this paper we present a new approach for reducing timed automata. In fact regions of a region automaton are aggregated according to a coarse equivalence class partitioning based on traces. We will show that the proposed algorithm terminates and preserves original timed automaton. Proposed algorithms are implemented by model transformation with Atom3 tool.
It is head to head analysis to study the impact of multi short operation segments with a direct flight operation. A case study of SAH-CAI-SAH and SAH-ADE-CAI-ADE-SAH, here we study the impact of introducing Aden sector with the direct flight of SAH-CAI
Hope to enjoy
Interventions required to meet business objectives - from Forecasting Methods,
Forecast Accuracy / Error Reduction,
Integrate – Sales Forecast / Production to undertaking a CPFR
Physique et Chimie de la Terre Physics and Chemistry of the .docxLacieKlineeb
Physique et Chimie de la Terre / Physics and Chemistry of the Earth 2022 / 2023
Homework
Physics of the Earth
Deadline : 10th of november
The Herglotz-Wiechert method and
Earth’s mantle seismic velocities profiles
The goal of this problem is to build a model of the P and S wave velocity profiles in the Mantle,
from travel times tables build from observations. To do this, we will use the Herglotz-Wiechert method,
a method developed by Gustav Herglotz and Emil Wiechert at the beginning of the twentieth century.
We consider a seismic ray going from point S to point A, as depicted on figure 1. We denote by ∆
the angular distance of travel (i.e. the angle ŜCA), and by T (∆) the travel time of the seismic wave
as a function of angular distance. We recall that in spherical geometry the ray parameter is defined as
p = r sin i(r)
V (r) , (1)
and is constant along a given ray. Here r is the distance from the center of the Earth, i(r) is the
incidence angle (i.e. the angle between the ray and the vertical direction at a given r), and V (r) is the
wave velocity. We denote by R = 6371 km the radius of the Earth.
∆
d∆
R
p
p + dp
i
A
A’B
S
C
rb
Figure 1 – Two rays coming from the same source S with infinitesimally different ray parameters p and p + dp. Their
angular distances of travel are ∆ and ∆+ d∆, and their travel-times are T and T + dT . The line going through points A
and B is perpendicular to both rays.
1 Constant velocity model
Let us first assume that the wave velocity V does not vary with depth.
1. Draw on a figure the ray going from a source S to a point A of the surface, without any reflexion.
This ray could represent either the P or S phase.
2. Find the expression of the travel time T along this ray as a function of ∆.
3. Find the expression of the incidence angle i of the ray at point A as a function of the epicentral
distance ∆, and then show that the ray parameter is given by
p = R
V
cos
∆
2
. (2)
1/3
Physique et Chimie de la Terre / Physics and Chemistry of the Earth 2022 / 2023
2 Linking p to T and ∆
We now turn to a more realistic model and allow for radial variations of the waves velocities.
4. By considering two rays coming from the same source with infinitesimally different ray parameters
p and p + dp, and travel times T and T + dT (Figure 1a), demonstrate that
p = dT
d∆
. (3)
Hints : (1) Since the two rays are very close, the arcs connecting A to A’, A to B, and B to A’
can be approximated as straight lines. (2) Show first that the segment AB is part of a wavefront.
What does it imply for the times of arrivals at points A and B ?
5. Check that the expressions of p and T found for the constant velocity model are consistent with
eq. (3).
3 Travel time curves and estimate of the p(∆) curves
You will find on Chamillo a file containing travel time tables obtained from the global Earth’s seis-
mological model ak135 (either a text file, AK135tables.txt, or an Excel spreadsheet, AK135tables.xlsx).
The f.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The Geometric Characteristics of the Linear Features in Close Range Photogram...IJERD Editor
The accuracy of photogrammetry can be increased with better instruments, careful geometric
characteristics of the system, more observations and rigorous adjustment. The main objective of this research is
to develop a new mathematical model of two types of linear features (straight line, spline curve) in addition to
relating linear features in object space to the image space using the Direct Linear Transformation (DLT). The
second main objective of the present paper is to study of some geometric characteristics of the system, when the
linear features are used in close range photogrammetric reduction processes. In this research, the accuracy
improvement has been evaluated by adopting certain assessment criteria, this will be performed by computing
the positional discrepancies between the photogrammetrically calculated object space coordinates of some check
object points, with the original check points of the test field, in terms of their respective RMS errors values. In
addition, the resulting least squares estimated covariance matrices of the check object point's space coordinates.
To perform the above purposes, some experiments are performed with synthetic images. The obtained results
showed significant improvements in the positional accuracy of close range photogrammetry, when starting node,
end nodes, and interior node on straight line and spline curve are increased with certain specifications regarding
the location and magnitude of each type of them.
A Novel Technique in Software Engineering for Building Scalable Large Paralle...Eswar Publications
Parallel processing is the only alternative for meeting computational demand of scientific and technological advancement. Yet first few parallelized versions of a large application code- in the present case-a meteorological Global Circulation Model- are not usually optimal or efficient. Large size and complexity of the code cause making changes for efficient parallelization and further validation difficult. The paper presents some novel techniques to enable change of parallelization strategy keeping the correctness of the code under control throughout the modification.
Air Cargo Forecast 2023 for Aviation IndustryMohammed Awad
Air Cargo Forecast addresses the trends in terms of figures based on the IATA air cargo newsletter. It looks that Air Cargo is the only business unit in aviation that has not been impacted by Covid 19
BCG Matrix Analysis for Airlines for period Dec 2019Mohammed Awad
The BCG Matrix provides a visual representation that aids decision-makers in understanding the relative position of each business unit within the overall portfolio. It helps in resource allocation, strategic planning, and identifying areas that require attention or investment. Remember that the implementation may vary based on the specific dynamics and challenges each airline faces.
Forecasting is an important tool for planning, so without planning, there will be a disaster that may happen, i.e what will happen in case of Force Major
For study new destinations, Gravity Model frequently use, which is a tough approach that may not work sometime due to many economic factors in the world and war crises.
Since we use small size of data, the model is not fit properly , but the approach is clearly defined.
Marsa Alam Airport should be supported by Airport Hotel,
Marsa Alam Tourism team, Egypt Air, and Egyptian Civil Aviation Authorities should talk the same language to setup an effective tourist package that will attractive the tourist.
Rush Hour Analysis,
Every city has its own pattern of traffic congestion that may reflects by a rush hour. Here we use leveling approach, that define by two factors - Rotational and Displacement. the idea is to define the right repeated cycle with minimum errors, and define the Rush Hour !
Has Airline Forecasting changed forever?Mohammed Awad
it is concern forecasting in Aviation,
two case studies are addressded, i.e global world, and Delta Airline, which shows the impact of Covid19, and its recovery in the future.
Back To Norms,
study about defining the seasonality patterned of Dubai Airport
Using Forecasting for both short term and long term - keeping the out come results.
The Organisation for Economic Co-operation and Development (OECD) is an international organisation that works to build better policies for better lives. their goal is to shape policies that foster prosperity, equality, opportunity and well-being for all. they draw on 60 years of experience and insights to better prepare the world of tomorrow.
Back to norms, is an article that address the event of Black Swan, which is a very hard situation for airline industry, but it shows that there is some light at the end of the tunnel. that may recover soon.
Fare Mapping Analysis,
It is the best way to establish a foundation for Fare, It gives a complete picture for the condition of the flight for sector SHJ-AMM using the input of G9 airline (Al Arabia). in terms of CASK, Aircraft capacity and Fares.
Back To Norms,
It is addressing the impact of Covid19, and when to recover, both IATA are used Pax and Cargo, then we used Dlephi method to reflect the judgment of expert people to reflect their opinion in real life.
This research is up todate - we used IATA information
Is Low Cost Carrier Profitable, this time we hold differents senarios by varing load factors and fares, the network is huge and large 128 sectors. hope to enjoy
Enterprise Excellence is Inclusive Excellence.pdfKaiNexus
Enterprise excellence and inclusive excellence are closely linked, and real-world challenges have shown that both are essential to the success of any organization. To achieve enterprise excellence, organizations must focus on improving their operations and processes while creating an inclusive environment that engages everyone. In this interactive session, the facilitator will highlight commonly established business practices and how they limit our ability to engage everyone every day. More importantly, though, participants will likely gain increased awareness of what we can do differently to maximize enterprise excellence through deliberate inclusion.
What is Enterprise Excellence?
Enterprise Excellence is a holistic approach that's aimed at achieving world-class performance across all aspects of the organization.
What might I learn?
A way to engage all in creating Inclusive Excellence. Lessons from the US military and their parallels to the story of Harry Potter. How belt systems and CI teams can destroy inclusive practices. How leadership language invites people to the party. There are three things leaders can do to engage everyone every day: maximizing psychological safety to create environments where folks learn, contribute, and challenge the status quo.
Who might benefit? Anyone and everyone leading folks from the shop floor to top floor.
Dr. William Harvey is a seasoned Operations Leader with extensive experience in chemical processing, manufacturing, and operations management. At Michelman, he currently oversees multiple sites, leading teams in strategic planning and coaching/practicing continuous improvement. William is set to start his eighth year of teaching at the University of Cincinnati where he teaches marketing, finance, and management. William holds various certifications in change management, quality, leadership, operational excellence, team building, and DiSC, among others.
3.0 Project 2_ Developing My Brand Identity Kit.pptxtanyjahb
A personal brand exploration presentation summarizes an individual's unique qualities and goals, covering strengths, values, passions, and target audience. It helps individuals understand what makes them stand out, their desired image, and how they aim to achieve it.
What are the main advantages of using HR recruiter services.pdfHumanResourceDimensi1
HR recruiter services offer top talents to companies according to their specific needs. They handle all recruitment tasks from job posting to onboarding and help companies concentrate on their business growth. With their expertise and years of experience, they streamline the hiring process and save time and resources for the company.
Kseniya Leshchenko: Shared development support service model as the way to ma...Lviv Startup Club
Kseniya Leshchenko: Shared development support service model as the way to make small projects with small budgets profitable for the company (UA)
Kyiv PMDay 2024 Summer
Website – www.pmday.org
Youtube – https://www.youtube.com/startuplviv
FB – https://www.facebook.com/pmdayconference
Discover the innovative and creative projects that highlight my journey throu...dylandmeas
Discover the innovative and creative projects that highlight my journey through Full Sail University. Below, you’ll find a collection of my work showcasing my skills and expertise in digital marketing, event planning, and media production.
Attending a job Interview for B1 and B2 Englsih learnersErika906060
It is a sample of an interview for a business english class for pre-intermediate and intermediate english students with emphasis on the speking ability.
Affordable Stationery Printing Services in Jaipur | Navpack n PrintNavpack & Print
Looking for professional printing services in Jaipur? Navpack n Print offers high-quality and affordable stationery printing for all your business needs. Stand out with custom stationery designs and fast turnaround times. Contact us today for a quote!
"𝑩𝑬𝑮𝑼𝑵 𝑾𝑰𝑻𝑯 𝑻𝑱 𝑰𝑺 𝑯𝑨𝑳𝑭 𝑫𝑶𝑵𝑬"
𝐓𝐉 𝐂𝐨𝐦𝐬 (𝐓𝐉 𝐂𝐨𝐦𝐦𝐮𝐧𝐢𝐜𝐚𝐭𝐢𝐨𝐧𝐬) is a professional event agency that includes experts in the event-organizing market in Vietnam, Korea, and ASEAN countries. We provide unlimited types of events from Music concerts, Fan meetings, and Culture festivals to Corporate events, Internal company events, Golf tournaments, MICE events, and Exhibitions.
𝐓𝐉 𝐂𝐨𝐦𝐬 provides unlimited package services including such as Event organizing, Event planning, Event production, Manpower, PR marketing, Design 2D/3D, VIP protocols, Interpreter agency, etc.
Sports events - Golf competitions/billiards competitions/company sports events: dynamic and challenging
⭐ 𝐅𝐞𝐚𝐭𝐮𝐫𝐞𝐝 𝐩𝐫𝐨𝐣𝐞𝐜𝐭𝐬:
➢ 2024 BAEKHYUN [Lonsdaleite] IN HO CHI MINH
➢ SUPER JUNIOR-L.S.S. THE SHOW : Th3ee Guys in HO CHI MINH
➢FreenBecky 1st Fan Meeting in Vietnam
➢CHILDREN ART EXHIBITION 2024: BEYOND BARRIERS
➢ WOW K-Music Festival 2023
➢ Winner [CROSS] Tour in HCM
➢ Super Show 9 in HCM with Super Junior
➢ HCMC - Gyeongsangbuk-do Culture and Tourism Festival
➢ Korean Vietnam Partnership - Fair with LG
➢ Korean President visits Samsung Electronics R&D Center
➢ Vietnam Food Expo with Lotte Wellfood
"𝐄𝐯𝐞𝐫𝐲 𝐞𝐯𝐞𝐧𝐭 𝐢𝐬 𝐚 𝐬𝐭𝐨𝐫𝐲, 𝐚 𝐬𝐩𝐞𝐜𝐢𝐚𝐥 𝐣𝐨𝐮𝐫𝐧𝐞𝐲. 𝐖𝐞 𝐚𝐥𝐰𝐚𝐲𝐬 𝐛𝐞𝐥𝐢𝐞𝐯𝐞 𝐭𝐡𝐚𝐭 𝐬𝐡𝐨𝐫𝐭𝐥𝐲 𝐲𝐨𝐮 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐚 𝐩𝐚𝐫𝐭 𝐨𝐟 𝐨𝐮𝐫 𝐬𝐭𝐨𝐫𝐢𝐞𝐬."
Improving profitability for small businessBen Wann
In this comprehensive presentation, we will explore strategies and practical tips for enhancing profitability in small businesses. Tailored to meet the unique challenges faced by small enterprises, this session covers various aspects that directly impact the bottom line. Attendees will learn how to optimize operational efficiency, manage expenses, and increase revenue through innovative marketing and customer engagement techniques.
Digital Transformation and IT Strategy Toolkit and TemplatesAurelien Domont, MBA
This Digital Transformation and IT Strategy Toolkit was created by ex-McKinsey, Deloitte and BCG Management Consultants, after more than 5,000 hours of work. It is considered the world's best & most comprehensive Digital Transformation and IT Strategy Toolkit. It includes all the Frameworks, Best Practices & Templates required to successfully undertake the Digital Transformation of your organization and define a robust IT Strategy.
Editable Toolkit to help you reuse our content: 700 Powerpoint slides | 35 Excel sheets | 84 minutes of Video training
This PowerPoint presentation is only a small preview of our Toolkits. For more details, visit www.domontconsulting.com
What is the TDS Return Filing Due Date for FY 2024-25.pdfseoforlegalpillers
It is crucial for the taxpayers to understand about the TDS Return Filing Due Date, so that they can fulfill your TDS obligations efficiently. Taxpayers can avoid penalties by sticking to the deadlines and by accurate filing of TDS. Timely filing of TDS will make sure about the availability of tax credits. You can also seek the professional guidance of experts like Legal Pillers for timely filing of the TDS Return.
Personal Brand Statement:
As an Army veteran dedicated to lifelong learning, I bring a disciplined, strategic mindset to my pursuits. I am constantly expanding my knowledge to innovate and lead effectively. My journey is driven by a commitment to excellence, and to make a meaningful impact in the world.
1. A Multiplicative Time Series Model For
Air Transport Demand Forecasting
Presented By
Mohammed Salem Awad
Consultant
YEMEN
2. A Multiplicative Time Series Model For
Air Transport Demand Forecasting
Jairam Singh1 A. A. Bashaswan2 Mohd Salem Awad3
Abstract:
The air travel demand is influenced by a variety of factors which renders the number
of passengers to vary as a seasonality fluctuating non-stationary time series. For non-
seasonal series it is possible to obtain a parsimonious representation in the form of
ARMA model but seasonality makes the model cumbersome. In the present paper a
multiplicative model of the type (p,d,q)lx(P,D,Q)s has been used to represent a non-
stationary time series displaying seasonality at an interval of 'S' observations. The data
of the International Passengers from the records of Yemenia. Yemen Airways have
been used to estimate the parameters of this model. The model gives a fairly good
forecast values.
1. Introduction:
There are several modes of transportations for passengers and freight and air
transport is a part of a larger product. Air travel is not an end in it self. Air travel
demand is dependent on the demand for the other product of leisure and business.
Airline product is characterized by the passengers and the freight mix strategy. A seat
in the aircraft is very much like another. Similarly the freight product is also
homogeneous but the role airfreight tends to be underestimated although it amounts to
one quarter of the RTKs output and one tenth of the total revenue. Therefore, the
determinants of air travel demand are many such as personal income, cost of air
travel, convenience and speed level of trade, population distribution and changes and
the of economic activity. The institutional factors such as festivals working practice
and school holidays cause the seasonal fluctuation in the demand.
Figure 1. Shows a time series of the total of international airline passenger's quoted by
Yemenia / Aden Center [10] given in Appendix A.
1
Prof. of Industrial Management, Department of Mechanical Engineering, Faculty of Engineering,
Aden University.
2
Associate Prof. of Mechanical Engineering, Faculty of Engineering, Aden University
3
Mechanical Engineer, YEMENIA Yemen Airways.
3.
4.
5. This series exhibits a periodic behavior with period S = 12 months, in the late summer
months a secondary peak occurs in the spring.
One of the deficiencies in the analysis of a time series has been the confusion of
fitting a series and forecasting it. A common method of analyzing a time series is to
decompose it arbitrarily into three components – a "trend", "seasonal component", and
a "random component". The trend can be fitted by a polynomial and the seasonal
components by a Fourier series. A forecast can then be made by projecting these fitted
functions. Such methods can give an extremely misleading results in cases where a
part of the time series may look to be quadratic some times due to some of the random
deviates which is taken to be the characteristics of the demand data, if they are fitted
to it.
Another model involving sines and cosines may be used to present seasonal variations
which may be written as
Zt
j1
π j Zt j a t ψ a
j 1
j t j at …………………………. ( 1 )
With suitable values of the coefficients π j and ψ j , is entirely adequate to describe
many seasonal time series analysis. The problem is to choose a suitable system of
parsimonious parameterization for such models involving a mixture of sines and
cosines and polynomial terms to allow the changes in the level of series.
For non-seasonal series, it is usually possible to obtain a useful and
parsimonious representation in the form of ARMA model as
Φ (B) Zt Θ (B) a t …………………………………………. ( 2 )
Where B is backward shift operator
Moreover, the generalized autoregressive operators (B) determines the eventual
forecast function which is the solution of the difference equation
(B) Zt ( l ) 0
(1 Φ1 B - Φ2 B2 - Φ 3B3 .......... p Bb ) ZT (l) 0 ………………. ( 3 )
Φ
6. Where B is understood to operate on l . In representing a seasonal behavior we shall
want the forecast function to trace out a periodic pattern. If (B) represents a
forecast a forecast function which is a sine wave with a twelve month period, adaptive
with phase and amplitude, will satisfy the difference equation.
(1 - 3 B B 2 ) Z t ( l ) 0
However, it is not sure that the periodic behavior is truly represented. It might need
many sine – cosine components
If we give some thought as to what happens when we try to induce stationary by
differencing d times and we write
(B) (B) (1 - B) d
Which is equivalent to setting d roots of equation
(B) 0 , equal to unity. When such a presentation proved adequate 1 - B
was used as a simplifying operators to convert a non-stationary series into a stationary
series. The fundamental fact that a seasonal time series will have observations similar
to each other after a certain intervals which is called a period. Therefore, the operation
B s Z t will represent an observation before s interval i.e
B s Z t Z t -s
and then the series which exhibits seasonally will be Z t , Z t -s , Z t -2s , Z t -3s ,......
This series may also be expected to be non-stationary, therefore, a simplifying
operator , Z t (1 - B s )Z t Z t Z t -s might be useful to make it stationary [2,3,4].
2. The Multiplicative Model
The seasonal effect implies that an observation for a particular month, say April is
related to the observations for previous Aprils. Suppose the t th observation Z t is the
7. month of April. We might be able to link this observation Z t , with observations in
previous April by a model of the form
(B s ) S Z t (B S ) t …………………………………….. (4)
D
Where S=12, S 1 B S and (B s ) , (B S ) , are polynomial in BS of degree P and
Q, respectively and satisfying the stationary and invertibility criteria. Similarly a
model of the form [5,6,7,8]
(B s ) S Z t -1 (B S ) t -1 …………………………..………( 5 )
D
,might be used to link the current behavior for March with the previous March
observations, and so on, for each of the twelve months.
Now the error components t , t -1 , t - 2 ,........would not in general is uncorrelated. For
example, the total airline passengers in April 1990, while related to April totals,
would also relate to totals in March 1990, February of 1990 and January of 1990, etc.
Thus, we would expect that t , in eq. (4) would be related to t -1 , in eq.(5) and so on.
A second model may be introduced to take care of such relationships,
(B) d t q (B) a t ……..………………………………… ( 6 )
,where at is a white noise process and (B) and q (B) , are the polynomials of degree
p and q respectively, and 1 1 B .
Substituting Eq. (6) in (4), we get
p (B) (B S ) d S Z t q (B)(B S ) a t ……………………….. ( 7 )
D
,where for this particular example, S=12. The resulting multiplicative process will
be said to be the order of (p,d,q)x(P,D,Q)S
8. 3. Choice of Transformation of the Data from Yemen Airways
( YEMENIA Aden Center )
It is particularly true for seasonal model that the weighted averages of the
previous data values, which comprise the forecasts, may extend far back into the
series; care is therefore needed in choosing a transformation in terms of which a
parsimonious linear model will closely apply over a sufficient stretch of the series. A
data based transformation may help determine in what metric the amplitude of the
seasonal components is roughly independent of the level of the series. Let us assume
that some power transformation, z = x for ≠ 0 , z = ln x for = 0, may be
needed to make the model (7) appropriate. The approach of Box and Cox [11] may be
followed and the maximum likelihood value obtained by fitting the model to x() = (x
- 1)/ x-1 for various values of which results in the smallest residual sum of
squares S , In this expression x is the geometric mean of the series. It was shown by
Box and Jenkins [9] by the airline data, the maximum likelihood value is thus close to
= 0 confirming for this particular example, the appropriateness of the logarithmic
transformation.
The monthly totals of the passengers in international travel shows a seasonal
behavior with period S = 12. The data are shown in Table 1. , which represents the
logarithms of the airline data.
Table 1. Natural Logarithmic of Monthly
Passengers Total in international Air Travel
By Yemenia, Aden Center (Using Aden as a hub).
9. 4. Representation of the Airline Data by
Multiplicative by (p,d,q) x ( P,D,Q)12 Model
The arrangement of Table 1. emphasizes the fact that in periodic data there are two
main intervals which are important. We expect relationships to occur:
(a) between the observations of the same month in the successive years
(b) between the observations in the successive months in a particular years.
Identification of Multiplicative Model
A tentative identification of time series model is done by analysis of historical
data. Usually at least 50 observations are required to achieve satisfactory results. The
primary tool used in this analysis is the autocorrelation function.
Consider the time Z1, Z2 , ………………..ZN
The theoretical autocorrelation function is
EZ t μ Z t -κ μ
ρκ , κ 0,1,2, .......... K
, (8)
σ2
z
Where σ 2 is the variance of the series. The quantity ρ κ , is called autocorrelation
z
at lag k. Obviously ρ 0 1. The theoretical autocorrelation function is never known
with a certainty, and must be estimated. Satisfactory estimate of ρ κ is the sample
autocorrelation function
Z
N -k
1
N-k
t
Z Z t -κ Z
t 1
ρκ N
, ………………….………….. (9)
Z
1 2
t Z
N t 1
N
For useful results, we would usually compute the first k , autocorrelations
4
As a supplemental aid the partial autocorrelation function often proves useful. We
shall define partial autocorrelation coefficient kk as the kth , coefficients in an
autoregressive process of order k. It can be shown ( Ref. 9 chapter 3 ) that the partial
autocorrelation coefficients satisfy the following Yule-Walker equations.
10. ρ j k1 ρ j-1 k2 ρ j- 2 k3 ρ j- 3 .......... kk ρ j-k
... ……….……… (10)
j 1,2, ..........
k
ˆ
This partial autocorrelation coefficient may be estimated by substituting ρ j , for
ρ j , in Eq. (10), Yielding
ρ j k1 ρ j-1 k2 ρ j- 2 k3 ρ j- 3 .......... kk ρ j- k
ˆ ˆ ˆ ˆ ... ˆ ………… ( 11 )
j 1,2, ..........
k
And solving the resulting equation for k 1,2, ......K to obtain 11 , 22 , ....,kk
, the sample partial correlation function.
From the estimated autocorrelation function, which can be conveniently
exhibited by a graph, a tentative model autocorrelation function patterns. These
patterns are discussed in reference [9].
A sample autocorrelation and partial autocorrelations function of non-stationary
time series die down extremely slowly from a value of one. If this type of behavior is
exhibited, the usual approach is to compute the autocorrelation and the partial
autocorrelation functions for the first difference of the series. If these functions
behave according to the characteristics of a stationary series. Then one degree of
differencing is necessary to achieve stationary.
In case of Yemenia ( Aden Center) data after first differencing, the
autocorrelations for all lags beyond the first is zero ( see table 2 ). Therefore, an
IMA(0,1,1) model is appropriate. This contains no seasonal components.
Suppose, we want to use this model to link the data 12 months apart then the model
would be
12 Z t (1 - B 12 ) t ……………………………….. (12)
Further, we want to employ a similar model using a linear filter to link the data only
one month apart. This gives a model
t Z t (1 - B) a t ……………………..………… (13)
Where and will have different values.
11. Then on combining expression (12) and (13), we would obtained the seasonal
multiplicative model
12 Z t (1 - θB) (1 - B 12 ) a t ……………………………..…………….. ( 14)
of the order (0,1,1)x(0,1,1)12 . The model written explicitly is
Z t - Z t 1 - Z t -12 Z t -13 a t - θa t -1 θa t -13 ………………….. (15)
The invertibility region of this model is defined by
1 θ 1 and 1 1
From the table 2, we see that the autocorrelation function does not die down rapidly
and it can be concluded from this that the logged data of the time series is non-
stationary. Therefore, some degree of differencing will be necessary to produce
stationary. The first difference of time series is taken and its autocorrelation function
is calculated as shown in table 2. It appears that the simple differencing reduces the
auto correlations in general but a heavy periodic components remains, this is evident
particularly at large lags. Sample differencing with respect to period 12 results in
correlations which are firstly persistently positive and then persistently negative. By
contrast the differencing 12 Z markedly reduces the correlation coefficient
throughout.
12. Table 2. Estimated Autocorrelations of Various
Differencing of the Logged Airlines Data.
Autocorrelations
ˆ ˆ
Estimation: Iterative Calculation of least squares estimates θ , and .
An iterative linearization technique may be used in straight forward situation to
supply the least squares estimates and their standard approximates errors. For the
present examples we can write approximately
a t,0 θ θ 0 χ 0 χ at
1,t 2,t
Where
a a
χ 1,t - χ 2,t -
θ θ 0Θ 0 Θ θ 0Θ 0
,and where θ 0 , and Θ 0 are guessed values and a t,0 at θ0Θ0 , The derivative are
mostly easily computed numerically [9].
For (p,d,q) x ( P,D,Q)12 model the preliminary estimates of autocorrelations
functions would be
13. -θ -Θ
ρ1 ρ 12
1 θ 2 , and 1 Θ 2 ………………. (16)
On substitution the sample estimates
r1 0.337 , and r12 0.189 in (16) [9], we obtain
2 2
0.337(1 θ ) θ and 0.189(1 Θ ) Θ .
2 2
Or …… θ 2.967 θ 1 0 , and Θ 5.291 Θ 1 0
ˆ ˆ
From these a rough estimates of θ 0.3876 , and Θ 0.1962 ……
4.2.1. Iterative Estimation of θ , and Θ
χ 1,t , can be written as
χ1,t at ω, β1,0, .....,β.....βk,0 at ω, β1,0, .....,β1,0 δ1 .....βk,0 δ1 ….. (17)
Consider the fitting of the airline data to (0,1,1)x(0,1,1) 12 process
ω t 12 Z t (1 - θB)(1 - ΘB 12 ) a t
The beginning of the calculation is shown in table (3) for the estimated value of
ˆ ˆ
θ 0.3876 , and Θ 0.1962 The back – forecasted values of [at] were actually
obtained using the expression as worked out in Appendix A.
e t 12 Z t θe t-1 Θe t -12 θΘe t -13
The value of [at] can be obtained by
a t 12 Z t θa t-1 Θa t-12 θΘa t -13
ˆ ˆ
χ 1,t , can be evaluated by eq. (17) giving an increments in θ and Θ till S(,)
becomes the minimum. After many iterations improves to 0.3225 and , 0.3712.
14. 5. Forecasting:
Forecasting are best computed directly from the difference equation it self. Thus using
the seasonal model (15) for forecasting at a lead time , and origin t is given by
Z t Z t 1 Z t 12 Z t 13 a t θa t 1 Θa t 12 θΘa t 13 ……..(18)
After setting = 0.3225, and = 0.3712 , the minimum means squared errors
forecast at lead time , and origin t is given by
Z t Z t 1 Z t 12 Z t 13 a t θa t 1 Θa t 12 θΘa t 13 …….
ˆ (19)
Here we refer to
Z t EZ t θ, Θ, Z t , Z t ,......... .......... .... ……... (20)
As the conditional expectation of Z t , taken at origin t, In this expression the
parameters are supposed exactly known and knowledgeable of series Z t , Z t 1 , is
supported to the extend into the remote past. This practical application depends upon
The facts that.
a) Invertible models fitted to the actual data usually yield forecasts which depends
appreciably only on the recent values of the series.
b) The forecasts are insensitive to small changes in parameter values such as are
introduced by estimation errors.
Now
Z t 1 j 0
Z t 1
………………. (21)
Z j
ˆ j 0
t
a t 1 j 0
a t 1
…………….. (22)
0 j 0
15. Thus to obtain the forecasts, we simply replace unknown ZS, by forecast, and
unknown a S, by zeros.
The known aS , are of course the one step ahead forecast errors already computed,
ˆ
that is, a t Z t Z t -1 …… (1)
For example, to obtain the three months ahead forecast, we have
Z t 3 Z t 2 Z t 9 Z t -10 a t 3 0.3225 a t 2 0.3712 a t 9 0.1197a t 10
Taking conditional expectation at origin,
Z t 3 Z t 2 Z t 9 Z t -10 0.3712a t 9 0.1197a t 10
ˆ ˆ
That is
Z t 3 Z t 2 Z t 9 Z t -10 0.3712 Z t -9 Z t -10 (1) 0.1197 Z t -10 Z t -11 (1) …
ˆ ˆ ˆ ˆ
Hence
Z t 3 Z t 2 0.6288 Z t 9 0.8803 Z t -10 0.3712 Z t -10 (1) 0.1197 Z t -11 (1)
ˆ ˆ ˆ ˆ (1)
This expresses the forecasts in terms of ZS and previous forecast of ZS
16. Conclusions:
It is obvious from the forecast values shown in Fig. 1., that the simple model
containing only two parameters faithfully reproduces the seasonal pattern and
supplies excellent forecasts. It is to be remembered that like all predications obtained
from the general pattern linear stochastic model, the forecast function is adaptive.
When the seasonal pattern changes, these will be appropriately projected into the
forecast. Of course, a forecast for a lead time of 36 may necessarily contains a fairly
larger error. However, in practice, an initially remote forecast will be continually
updated and as the lead shortens, greater accuracy will be possible.
The model presented here is robust to moderate changes in the values of values of the
parameters. Thus, if = 0.35 , and = 0.4 , instead of 0.3225 and 0.37, the forecast
would not be greatly affected. This is true for the forecasts made several steps ahead
e.g 12 months. This has been seen by studying the sum of squares surfaces of
modifying the values of the parameters by one step ahead forecasts.
References:
1- Brown, R.G. "Smoothing, Forecasting and Predication of Discrete Time Series",
Prentice Hall, New Jersey, 1962.
2- Box, G.E.P. and Jenkins G.M.. "Some Statistical Aspects of Adaptive
Optimization and Control", Jour. Royal stat. Soc. B24,297,1962.
3- Box, G.E.P. and Jenkins G.M.. "Further Contribution to Adaptive Quality:
Control Simultaneous estimation of Dynamos ; non zero costs", Bull Ins. Stat.
34th Seminars, 1963.
4- Box, G.E.P. and Jenkins G.M.. "Mathematical Models for Adaptive Control and
Optimization", A.I. Ch. E.-J. Chem. E. Symposium Series, 4, 61, 1965.
5- Box, G.E.P. and Jenkins G.M.. "Models for Forecasting Seasonal and Non
Seasonal Time Series" , Advanced Seminar On Spectral Analysis of Time Series,
ed B. Harris, 271, John Wiley, New York, 1967.
6- Box, G.E.P. and Jenkins G.M.. "Some Recent Advances in Forecasting and
Control, I". Applied Statistics, 17,91,1968.
7- Box, G.E.P. and Jenkins G.M.. "The Time Series Analysis Forecasting and
Control", Holden.-Day Singapore 1976.
8- Yule, G. U. , "On the Method of Investigating Periodicity in Disturbed Series
with Special Reference to Walfers Sunspot Number". Phil. Trans A226, 267,
1927.
9- Daniels , H.E. " Approximate Distribution of Serial Correlation Coefficients"
Biometrica, 43, 169, 1956.
10- Yemenia: Records of International Passengers, 1996.
11- Box, G.E.P. and Cox D.R., "An Analysis of Transformation". Jour. Of Rayal
Stat. Soc. B26, 211. 1964.
17. APPENDIX A :
Data International Airline Passenger quoted by Yemenia / Aden Centre
18. APPENDIX B :
CALCULATION OF THE UNCONDITIONAL SUM OF SQUARES FOR
THE MODEL ωt 12 Zt (1 - θB) (1 - B12 ) at
With ωt 12 Zt , the model ( 0,1,1) x (0,1,1)12 may be written in either the
forward or backward form.
ωt (1 - θB) (1 - B12 ) at
Or
ωt (1 - θF ) (1 - F12 ) et
And where μ E ω t is assumed to be zero. Hence we can write
e t ω t θ e t 1 e t 12 θΘe t 13 ……………… B1
a t ω t θ a t 1 a t 12 θΘa t 13 ……………… B2
Where ω t ω t for t = 1, 2, ………, n and is the back forecast of ω t for t 0
There are N = 48 observations in the airline series. Accordingly, in Table, these are
designated as Z12 , Z11 , Z10 , Z9 ,……….. Z35 . The ω ' s obtained by differencing
from the series ω 1 , ω 2 , ω 3 , ,......... ω n , where n = 35 we shall start calculation e 35
.......
by setting unknowns e ' s equal to zero.
Using (B.1) we get
e 35 ω 35 θ 0 0 θΘ 0
e 34 ω 34 θ e 35 0 θΘ 0
e 33 ω 33 θ e 34 e 35 θΘ 0
………….
…………..
e 4 ω 4 θ e 2 e 3 θΘ e 14
19. Table B1. Computation at
..........ω -12 , then B.2 is used to calculate a t
We can calculate ω 0 , ω -1 , ..........
Since each a t is function of previously occurring ω ' ' and a -1 0 , j > 12
Now
24
Sθ, Θ a
t 12
t
2
The next iteration would start with e ' s starting of the iteration using forecast values
......... ,from a ' s already calculated.
ω n1 , ω n 2 , ω n 3 ..........
Table 3. Numerical Calculation of Derivative from Airline Data
t Z a t,0 a t θ
0
a t θmδ a a δ
t θ t θδ
-12 8.1715 -0.0545 -0.0545 -0.0545 0
-11 7.9596 0.0995 0.0783 0.0778 0.0545
-1
0
1
35 9.8182 0.7071 0.6377 0.6361 0.1560
20. Using the whole series, the iteration can be proceeded.
a χ
n
t,0 1t
t 0
θ - θ0 n
χ
t 0
1t
2
Then changing the value θ 0 θ and repeating the same procedure, the values of
θ and can be found out which minimizes
n
Sθ, Θ a
t0
t0
2