Kardi Teknomo
Kardi Teknomo Kardi Teknomo Kardi Teknomo
   
 
  Research
  Publications
  Tutorials
  Resume
  Resources
  Contact

Visit Tutorials below:
Adaptive Learning from Histogram
Adjacency matrix
Analytic Hierarchy Process (AHP)
Analysis of Algorithm
ArcGIS tutorial
Arithmetic Mean
Aroon Oscillator
Bayes Theorem
Bootstrap Sampling
Bray Curtis Distance
Break Even Point
Chebyshev Distance
City Block Distance
Conditional Probability
Complex Number
Continued Fraction
CryptArithmetic
Data Analysis from Questionnaire
Data Revival from Statistics
Decimal to Rational
Decision tree
Difference equations
Digital Root
Discriminant analysis
Divisibility
Eigen Value using Excel
Euclidean Distance
Euler Integration
Euler Number
Excel Iteration
Excel Macro
Excel Tutorial
Expectation Maximization (EM) Algorithm
Factorial Function
Feasibility Study
Financial Analysis
Financial Education
Gaussian Mixture Model
Generalized Inverse
Generalized Mean
Geometric Mean
Ginger Bread Man and Chaos
Graph Theory
Growth Model
Hamming Distance
Harmonic Mean
Hierarchical Clustering
Independent Events
Incident matrix
Jaccard Coefficient
Kernel basis function
Kernel Regression
k-Means clustering
K Nearest Neighbor
LAN Connections Switch
Learning from data
Lehmer Mean
Linear Algebra
Logarithm Rules
Mahalanobis Distance
MapReduce
Market Basket Analysis
Mean Absolute Deviation
Mean and Average
Mean, median, mode
Minkowski Distance
Minkowski Mean
Monte Carlo Simulation
Multi Agent System
Maximum Likelihood
Multicriteria decision making
Mutivariate Distance
Neural Network
Newton Raphson
Non-Linear Transformation
Normalization Index
Normalized Rank
Ordinary Differential Equation
Page Rank
Palindrome
PI
Power rules
Prime Factor
Prime Number
Q Learning
Quadratic Function
Queueing Theory
Rank Reversal
Recursive Statistics
Regression Model
Reinforcement Learning
Root of Polynomial
Runge-Kutta
Scenario Analysis
Sierpinski gasket
Sieve of Erastosthenes
Similarity and Distance
Solving System Equation
Standard deviation
String Distance
Summation Tricks
Support Vector Machines
System dynamic
Time Average
Tower of Hanoi
Variance
Vedic Square
Visual Basic (VB) tutorial
What If Analysis

 

Weaving Section Numerical Example

By Kardi Teknomo, PhD.

< Previous | Index | Next >

Suppose we have a simple weaving section consisting of 8 nodes and 8 links.  For simplicity, without losing generality, we consider only 9 agents and generate 9 ordinal trajectories to be put into the network. The trajectories consist of 4 types in this simple network, but we repeat some of the trajectories. The adjacency matrix, the network and trajectories are given in the following figure. Zeroes in the matrix are deleted to enhance the pattern of non-zero entries. The non-zero entries are located on the upper diagonal to indicate that it is acyclic directed graph. The upper triangular property, however, is very much dependent on the labeling of the nodes. For example, if we swapped the labeling of the nodes ‘a’ and ‘h’, then the non-zero entries will not be restricted to the upper triangular, even though the graph is still a directed acyclic graph. See Buckley &Harary (1990) for proof that acyclic directed graph can be labeled as upper triangular of the adjacency matrix.

 

Weaving SectionWeaving SectionWeaving Section

From the network structure, the path matrix isgiven below as computed using a simple breadth-first search approach using the adjacency matrix as input. Again, the zeroes are deleted from the matrices to enhance the view. Infinite elements indicate there exists no path between the two nodes. The maximum non-infinity entries represent the diameter of the network (which is 3 in this case). Binarized path matrix is the matrix that results from transforming all non-zero and non-infinity elements into one.

Weaving SectionWeaving Section

 

On the network structure, we can also define external matrix as the difference between path matrix and adjacency matrix. In the binarized external matrix, all non-zero and non-infinity elements of external matrix are also transformed to one

Weaving SectionWeaving Section

Now we discuss how the network is utilized by putting ordinal graph trajectories into the network. Matrix-set (with tilde) is a matrix in which entries are sets of trajectories. The matrix Weaving Sectionrepresents the set of trajectories passing through a given direct link. The matrix and its associated figure are shown below where the set of trajectory ids are placed on each link. For instance, link ac is used by trajectories id {1, 3, 5, 7, 9}.

Weaving SectionWeaving Section

When we count the number of elements of sets of trajectory ids on each link, we get the flow matrix. Associated with the flow matrix is the flow pattern on each link as shown in the figure below. The number on each link is the explicit number of trajectories passing through that particular link. We also put the number visually as the weight of the arrow as flow pattern.

Weaving SectionWeaving Section

A trajectory passes through a sequence of nodes from a source to a sink. Any ordered two nodes of the node sequence of a trajectory can be seen as origin and destination nodes. For instance, trajectory #1 of a->c->e->g will produce origin destination pair nodes of {a->c, a->e, a->g, c->e, c->g, e->g}. Putting these OD pair nodes into matrix produces origin-destination-set matrix Weaving Section. The entries of this matrix are set of trajectory ID numbers of the OD pair nodes. Equation (14) formalizes the formulas to obtain the origin-destination-set matrix through the intersection between two sets of trajectories: a trajectories that goes out of a node origin and a set of trajectories that goes in to a node destination.

Weaving Section

The tilde notation represents a matrix whose elements are sets instead of scalar values. If we count the number of elements in each set in the matrix, we get the OD matrix

Weaving Section

SinceWeaving Section, then we haveWeaving Section. Clearly, we can see that Weaving Section.  Considering only sources and sinks as in traditional OD matrix, we take partial element of the matrix Weaving Sectionand the associated OD produces graphs as shown below.

Weaving SectionWeaving Section

Now we exclude matrix set

Weaving Section

Then we get the count of the elements in each set in the matrix to obtain

Weaving Section

The Hadamard product of the binarized eternal matrix with the OD matrix will also produce the desire line matrix because there is no alternative route flow. The desire line pattern visualization is also shown below.

Weaving SectionWeaving Section

 

< Previous | Index | Next >

 

 

Share and save this tutorial
Add to: Del.icio.us  Add to: Digg  Add to: StumbleUpon   Add to: Reddit   Add to: Slashdot   Add to: Technorati   Add to: Netscape   Add to: Newsvine   Add to: Mr. Wong Add to: Webnews Add to: Folkd Add to: Yigg Add to: Linkarena Add to: Simpy Add to: Furl Add to: Yahoo Add to: Google Add to: Blinklist Add to: Blogmarks Add to: Diigo Add to: Blinkbits Add to: Ma.Gnolia Information

 

These tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi. (2013) Relationship between Generalized Origin Destination and Flow Matrix – A Tutorial
http://people.revoledu.com/kardi/research/trajectory/od/

 

 
© 2007 Kardi Teknomo. All Rights Reserved.
Designed by CNV Media