Random Walk Simulation

Perform a bootstrap analysis to study how the distance travelled grows with time for a two-dimensional random walk

We will simulate a random walk in two dimensions, and in particular will find numerical estimates for the distance traveled as a function of time. First let's warm up by looking at walks of fixed length.

Simulate a symmetric random walk of length 10^3 in 2D:

In[1]:=

SeedRandom[1234];dataRW=Transpose@RandomFunction[RandomWalkProcess[0.5],{0,10^3},2]["ValueList"];

Visualize the walk:

In[2]:=

Graphics[Line[dataRW],AspectRatioAutomatic]

Out[2]=

Simulate 1000 random walks of fixed length 10^3:

In[3]:=

data2d=Table[Transpose@RandomFunction[RandomWalkProcess[0.5],{0,10^3},2]["ValueList"],1000];

View the first steps of some of the paths:

In[4]:=

data2d〚1;;5,1;;10〛//Column

Out[4]=

{{0,0},{-1,-1},{-2,0},{-3,1},{-4,0},{-5,1},{-6,0},{-7,1},{-8,2},{-9,1}}

{{0,0},{1,1},{0,2},{-1,1},{-2,0},{-1,1},{0,0},{1,-1},{2,-2},{1,-1}}

{{0,0},{1,-1},{0,0},{-1,-1},{0,0},{1,-1},{2,0},{3,1},{4,0},{3,-1}}

{{0,0},{1,1},{0,2},{1,1},{0,2},{1,3},{0,4},{1,3},{2,2},{1,3}}

{{0,0},{1,1},{2,2},{3,1},{4,2},{3,1},{2,2},{3,1},{2,0},{3,1}}

Find the ending points for fixed

and plot a histogram:

In[5]:=

finalPts=data2d〚All,-1〛;Histogram3D[finalPts]

Out[5]=

Compute the average distance travelled in this collection of walks:

In[6]:=

avgTravel=Mean[Function[{x,y},N@Sqrt[

]]@@@finalPts]

Out[6]=

39.2459

Now let's study how the net distance travelled varies with walk length. It is known that for large

, the net distance covered is proportional to the square root of

. Let's see if we can confirm this numerically.

Write a function to calculate the distance traveled as a function of the walk-length,

, averaged over

independent walks:

In[7]:=

Clear[d]d[t_,n_]:=Module[{data,finalPts},data=Table[Transpose@RandomFunction[RandomWalkProcess[0.5],{0,t},2]["ValueList"],n];finalPts=data〚All,-1〛;Mean[Function[{x,y},N@Sqrt[

]]@@@finalPts]]

Compute the distance travelled for a range of

values, averaging over 100 walks:

In[8]:=

numWalks=100;

In[9]:=

tVals={100,300,600,1000,2000,3000};

In[10]:=

dVals=d[#,numWalks]&/@tVals//EchoTiming

⌚

0.32699

Out[10]=

{12.3828,21.0842,31.106,39.6968,61.5684,68.7477}

Attempt to fit these data to a power law:

In[11]:=

fit=FindFit[Transpose[{tVals,dVals}],at^b,{a,b},t]

Out[11]=

{a1.18845,b0.510933}

Plot the data points, along with the best fit power law:

In[12]:=

Show[{Plot[at^b/.fit,{t,1,3500}],ListPlot[Transpose[{tVals,dVals}]]}]

Out[12]=

The above analysis tells us nothing about the sizes of statistical errors on the data points, or on the resultant fit. To get these, let's apply a bootstrap analysis.

Generate an ensemble of 1000 data samples:

In[13]:=

numBoots=1000;

In[14]:=

dValsEnsemble=Table[d[#,numWalks],numBoots]&/@tVals;

Generate 1000 sets of resampled data:

In[15]:=

dValsBoot=(RandomChoice[#,numBoots]&/@dValsEnsemble);

Compute the mean and standard deviation of distance travelled for these samples:

In[16]:=

dVals2=Mean/@dValsBoot

Out[16]=

{12.5314,21.626,30.8598,39.5926,55.961,68.494}

In[17]:=

dErrs=1/Sqrt[numWalks]StandardDeviation/@dValsBoot

Out[17]=

{0.0643032,0.109354,0.155531,0.210671,0.27833,0.352747}

Fit the mean distance travelled as a function of time to a power law:

In[18]:=

fit2=FindFit[Transpose[{tVals,dVals2}],at^b,{a,b},t]

Out[18]=

{a1.26433,b0.498654}

Plot the data points along with the best fit power-law function. Note that the displayed error bars are magnified 10 times actual size for visibility:

In[19]:=

Show[{Plot[at^b/.fit2,{t,1,3500}],ListPlot[Transpose[{tVals,Around@@@Transpose[{dVals2,10dErrs}]}],PlotMarkersAutomatic]},PlotLabel"Power law function with magnified error bars"]

Out[19]=

To get the error bars in the fit parameters, we fit each of the resampled ensembles:

In[20]:=

fits=FindFit[Transpose[{tVals,#}],at^b,{a,b},t]&/@Transpose[dValsBoot];//Quiet

In[21]:=

fits〚1;;10〛

Compute the mean and standard deviation of the exponent across the bootstrap ensemble, showing that the distance travelled as a function of step count varies according to a square-root law, as expected:

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Contributed by: Paco Jain (Wolfram Research)

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Random Walk Simulation

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Random Walk Simulation

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