Random Walk 1-D and 2-D
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Random Walk in 1-D and 2-D Andri Rahmadhani Department of Physics ITB – KAIST Internship Program Summer 2012
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Week 1...Presentation about random Walk in 1-D and 2-D as basic theory of echonphysics
Transcript of Random Walk 1-D and 2-D
Random Walk in 1-D and 2-D
Andri RahmadhaniDepartment of Physics ITB – KAIST Internship Program
Summer 2012
Basic Theory Method & Algorithm Results References
Outline
What is random walk? Algorithm that represents trajectory of
random steps The direction of random walk cannot be
predicted from the past actions.
Basic Theory
Example : Flipped coin problem
Basic Theory
Figure 1. One of possible random walks position after 5 flips
Example : Time-Series Graph (1-D)
Basic Theory
Figure 2. Time-Series random walk with N (steps) = 1000
and equal probability 0.5
Example : Square Lattice (2-D)
Basic Theory
Figure 3. Two dimensional random walk in square
lattice
The mathematics of 1-D Random Walk◦ Sum/Position after n steps: ; ◦ Expected Value:
◦ Expected translation distance after n steps:
Basic Theory
Basic Theory The mathematics of 2-D Random Walk
◦ Sum/Position after n steps: ; ◦ Absolute Value:
◦ Expected Value:
◦ Expected translation distance after n steps:
Simulation using Matlab Algorithm of Random Walk (1-D & 2-D)
1) Set the number of steps, trials, random variable, and probability of random variable
2) Generate random number and moves3) Repeat step 2 for n times which already set up in step 14) Plot sample graph5) Calculate mean/expectation value6) Increase number of steps and repeat step 2 until
reaching maximum number of steps7) Plot distance graph and print the results
Method & Algorithm
Random Walk in 1-D
Results
Figure 4. Value and distance for 10 steps with 10 trials
0 1 2 3 4 5 6 7 8 9 10-8
-6
-4
-2
0
2
4
Steps (N)
Val
ue
1st trial 2nd trial 3rd trial 8th trial 9th trial 10th trial
1 2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Random Walk in 1-D
Results
Figure 5. Value and distance for 100 steps with 10 trials
0 10 20 30 40 50 60 70 80 90 100-15
-10
-5
0
5
10
15
Steps (N)
Val
ue
1st trial 2nd trial 3rd trial 8th trial 9th trial 10th trial0 10 20 30 40 50 60 70 80 90 100
0
2
4
6
8
10
12
14
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Random Walk in 1-D
Results
Figure 6. Value and distance for 10 steps with 10000 trials
0 1 2 3 4 5 6 7 8 9 10-6
-5
-4
-3
-2
-1
0
1
2
3
Steps (N)
Val
ue
1st trial 2nd trial 3rd trial 9998th trial 9999th trial 10000th trial
1 2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Random Walk in 1-D
Results
0 10 20 30 40 50 60 70 80 90 100-20
-15
-10
-5
0
5
Steps (N)
Val
ue
1st trial 2nd trial 3rd trial 9998th trial 9999th trial 10000th trial
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10
11
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Figure 7. Value and distance for 100 steps with 10000 trials
Random Walk in 1-D◦ Curve Fitting :
Results
10 steps
10 trials
100 steps
10 trials
10 steps
100 trials
100 steps
100 trials
10 steps
1000 trials
100 steps
1000 trials
10 steps
10000 trials
100 steps
10000 trials
a 1.007 0.9786 0.8612 0.9993 1.006 0.9943 1.003 1
b 0.1199 0.08314 0.1896 0.08167 -0.003658 0.03825 -0.003868 0.006539
R2 0.8451 0.7447 0.9554 0.9651 0.9976 0.9946 0.9997 0.9995
Table 1. Curve fitting property for
Random Walk in 2-D
Results
Figure 8. Value and distance for 10 steps with 10 trials
-4 -3 -2 -1 0 1 2 3 4 5-2
-1.5
-1
-0.5
0
0.5
1
x
y
1st trial End 2nd trial End 9th trial End 10th trial End
1 2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
4
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Random Walk in 2-D
Results
Figure 9. Value and distance for 100 steps with 10 trials
-10 -5 0 5 10 15-15
-10
-5
0
5
10
15
x
y
1st trial End 2nd trial End 9th trial End 10th trial End
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
14
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Random Walk in 2-D
Results
Figure 10. Value and distance for 10 steps with 10000 trials
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
x
y
1st trial End 2nd trial End 9999th trial End 10000th trial End1 2 3 4 5 6 7 8 9 10
1
1.5
2
2.5
3
3.5
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
Random Walk in 2-D
Results
Figure 11. Value and distance for 100 steps with 10000 trials
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
Number of steps (N)
Dis
tanc
e <
|Sn|
>
Distance Curve Fitting N1/2
-6 -4 -2 0 2 4 6-10
-5
0
5
10
15
20
x
y
1st trial End 2nd trial End 9999th trial End 10000th trial End
Random Walk in 2-D◦ Curve Fitting :
Results
10 steps
10 trials
100 steps
10 trials
10 steps
100 trials
100 steps
100 trials
10 steps
1000 trials
100 steps
1000 trials
10 steps
10000 trials
100 steps
10000 trialsa 1.168 1.002 1.01 1.019 1.004 1.002 0.9961 1
b -0.3342 -0.1072 0.003024 -0.1196 0.007713 -0.0141 0.009183 -0.002094
R2 0.8919 0.7645 0.9865 0.9782 0.9975 0.9975 0.9998 0.9998
Table 2. Curve fitting property for
Root-mean-square of random walk final position or the distance is equal with root-square of number of random walk steps.
Valid for large number of random walks trials and equal probability.
Conclusion
Pearson, K. (1905). The problem of the Random Walk. Nature. 72, 294.
Van Kampen N. G., Stochastic Processes in Physics and Chemistry, revised and enlarged edition (North-Holland, Amsterdam) 1992.
McCrea, W. H. and Whipple, F. J. W. "Random Paths in Two and Three Dimensions." Proc. Roy. Soc. Edinburgh 60, 281-298, 1940.
References
Thank youEnd of slide
