Brownian Motion - Revision history

Bradley Monk: /* A Correlated Trajectory */

2025-11-06T06:52:03Z

A Correlated Trajectory

← Older revision		Revision as of 23:52, 5 November 2025
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	==A Correlated Trajectory==		==A Correlated Trajectory==

	[[~~Image~~:~~FancyStatistics_09~~.png\|right\|thumb\|300px]]		[[File:Brownian Motion Cross Correlation.png\|right\|thumb\|300px]]

	<p>So what does the autocorrelation sequence look like for a trajectory that is not generated from independent samples? In the following example, the displacement at each time interval depends in part on the displacement in the last interval plus a random number. The autocorrelation is not nearly as sharp as the one generated by independent random samples.</p>		<p>So what does the autocorrelation sequence look like for a trajectory that is not generated from independent samples? In the following example, the displacement at each time interval depends in part on the displacement in the last interval plus a random number. The autocorrelation is not nearly as sharp as the one generated by independent random samples.</p>
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	title('Autocorrelation');		title('Autocorrelation');
	</pre>		</pre>




	=Additional Resources=		=Additional Resources=

Bradley Monk: /* MATLAB Tricks */

2025-11-06T06:47:09Z

MATLAB Tricks

← Older revision		Revision as of 23:47, 5 November 2025
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	==MATLAB Tricks==		==MATLAB Tricks==

	[[~~Image~~:~~FancyStatistics_08~~.png\|right\|thumb\|300px]]		[[File:Xcorr Acorr.png\|right\|thumb\|300px]]

	<p>Matlab can generate (and plot) an amazing amount of data. For example, the <tt>xcorr</tt> function can compute the auto and cross correlations of a large set of data values. If A is an MxN matrix, <tt>xcorr(A)</tt> returns a size 2*M-1 x N^2 matrix whose columns contain the cross-correlation sequences for all combinations of the columns of A. The football shape of the data is a result of the way that the data is windowed when computing the correlation.</p>		<p>Matlab can generate (and plot) an amazing amount of data. For example, the <tt>xcorr</tt> function can compute the auto and cross correlations of a large set of data values. If A is an MxN matrix, <tt>xcorr(A)</tt> returns a size 2*M-1 x N^2 matrix whose columns contain the cross-correlation sequences for all combinations of the columns of A. The football shape of the data is a result of the way that the data is windowed when computing the correlation.</p>
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	title('All Possible Auto and Cross Correlations in the x Dimension');		title('All Possible Auto and Cross Correlations in the x Dimension');
	</pre>		</pre>





	==A Correlated Trajectory==		==A Correlated Trajectory==

Bradley Monk: /* Cross Correlation */

2025-11-06T06:39:00Z

Cross Correlation

← Older revision		Revision as of 23:39, 5 November 2025
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	==Cross Correlation==		==Cross Correlation==

	[[~~Image~~:~~FancyStatistics_07~~.png\|right\|thumb\|300px]]		[[File:Displacement Cross Correlation.png\|right\|thumb\|300px]]

	<p>With two vector arguments, <tt>xcorr(x,y)</tt> returns a cross correlation matrix. The cross correlation can be used to test the relationship (or lack thereof) between one particle's trajectory and another's. If two particles move independently, the cross correlations should be very small. The code below computes the cross correlation between the x displacements of particle simulations number 1 and 2.</p>		<p>With two vector arguments, <tt>xcorr(x,y)</tt> returns a cross correlation matrix. The cross correlation can be used to test the relationship (or lack thereof) between one particle's trajectory and another's. If two particles move independently, the cross correlations should be very small. The code below computes the cross correlation between the x displacements of particle simulations number 1 and 2.</p>
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	Title('Particle 1, 2 x-axis Displacement Cross Correlation');		Title('Particle 1, 2 x-axis Displacement Cross Correlation');
	</pre>		</pre>





	==MATLAB Tricks==		==MATLAB Tricks==

Bradley Monk: /* Additional Resources */

2024-06-23T10:12:30Z

Additional Resources

← Older revision		Revision as of 03:12, 23 June 2024
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	*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]		*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]
	*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active Diffusion Positions in the Nucleus]		*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active Diffusion Positions in the Nucleus]
			*[[Stochastic Diffusion]]




	~~==See Also==~~

	*[[Stochastic Diffusion]]
	*[[Category:Diffusion]]		*[[Category:Diffusion]]
	*[[Category:Neurobiology]]		*[[Category:Neurobiology]]

Bradley Monk: /* Additional Resources */

2024-06-23T10:11:45Z

Additional Resources

← Older revision		Revision as of 03:11, 23 June 2024
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	=Additional Resources=		=Additional Resources=
	<br>		<br>

	*[http://mathworld.wolfram.com/RandomWalk2-Dimensional.html Wolfram Alpha - Random Walk]		*[http://mathworld.wolfram.com/RandomWalk2-Dimensional.html Wolfram Alpha - Random Walk]
	*[http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd.html Democritus - MSD Theory]		*[http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd.html Democritus - MSD Theory]
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	*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]		*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]
	*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active Diffusion Positions in the Nucleus]		*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active Diffusion Positions in the Nucleus]




	==See Also==		==See Also==

Bradley Monk: /* Additional Resources */

2024-06-23T10:11:23Z

Additional Resources

← Older revision		Revision as of 03:11, 23 June 2024
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	==Additional Resources==		=Additional Resources=
			<br>

	*[http://mathworld.wolfram.com/RandomWalk2-Dimensional.html ~~wolfram alpha~~]		*[http://mathworld.wolfram.com/RandomWalk2-Dimensional.html Wolfram Alpha - Random Walk]
	*[http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd.html Democritus]		*[http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd.html Democritus - MSD Theory]
	*[http://www.mathworks.com/matlabcentral/fileexchange/40692-mean-square-displacement-analysis-of-particles-trajectories ~~Matlab toolbox~~]		*[http://www.mathworks.com/matlabcentral/fileexchange/40692-mean-square-displacement-analysis-of-particles-trajectories MSD MATLAB Toolbox]
	*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]		*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]
	*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active ~~diffusion positions~~ the ~~nucleus in mouse oocytes~~]		*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active Diffusion Positions in the Nucleus]

	==See Also==		==See Also==

Bradley Monk: /* notes & articles */

2024-06-23T10:09:36Z

notes & articles

← Older revision		Revision as of 03:09, 23 June 2024
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	==~~notes & articles~~==		==Additional Resources==

	~~Here are several good resources for explaining and calculating MSD:~~
	*[http://mathworld.wolfram.com/RandomWalk2-Dimensional.html wolfram alpha]		*[http://mathworld.wolfram.com/RandomWalk2-Dimensional.html wolfram alpha]
	*[http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd.html Democritus]		*[http://www.compsoc.man.ac.uk/~lucky/Democritus/Theory/msd.html Democritus]
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	*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]		*[http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3055791/ Mean Square Displacement Analysis of Single-Particle Trajectories with Localization Error: Brownian Motion in Isotropic Medium]
	*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active diffusion positions the nucleus in mouse oocytes]		*[http://www.nature.com/ncb/journal/v17/n4/full/ncb3131.html Active diffusion positions the nucleus in mouse oocytes]



	==See Also==		==See Also==

Bradley Monk: /* 100 Years of BIO Lab Data in 1 Second */

2024-06-23T10:08:46Z

100 Years of BIO Lab Data in 1 Second

← Older revision		Revision as of 03:08, 23 June 2024
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	~~==100 Years of BIO Lab Data in 1 Second==~~

	~~[[Image:FancyStatistics_05.png\|right\|thumb\|300px]]~~

	<p>So does the uncertainty of the squared and summed behave as expected? The following plot is a simulation of 5000 data sets - 50 at each value of N just as above. (This is what a plot of all the data from students doing the BIO lab for the next hundred years might look like. Some lazy groups decided to use very low values of N.)</p>

	~~<pre>~~
	~~clf;~~
	~~hold on;~~

	~~for i= 1:100~~
	~~for j = 1:50~~
	~~dx = randn(1,i);~~
	~~dy = randn(1,i);~~
	~~m = mean( dx .^ 2 + dy .^ 2 );~~
	~~plot(i,m,'x');~~
	~~end~~
	~~end~~

	~~plot(1:100, 2 + 2./sqrt(1:100), 'k', 'LineWidth', 2); % plot upper error bar in dark black~~
	~~plot(1:100, 2 - 2./sqrt(1:100), 'k', 'LineWidth', 2); % plot lower error bar in dark black~~

	~~hold off;~~
	~~xlabel('Population Size (N)');~~
	~~ylabel('Population Mean');~~
	~~title('Population Mean of Chi Squared (2 DOF) Distributed Random Variable versus Population Size');~~
	~~</pre>~~





	~~===Planning Your Time in the Lab===~~

	Sampling error sets a lower bound on the uncertainty in your estimate of the diffusion coefficient from experimental data. To plan your time in the lab, it will be important to understand how much data you should take for each experimental condition. Because the particles go out of focus and drift away from of the observation area, it can be difficult to make movies longer than about 5 seconds. At a frame rate of 10 frames per second, how many movies would you have to make to achieve a final uncertainty of 10%? 1%? .1%? What level of uncertainty will you try for in the lab?

Bradley Monk: /* Mean Squared Displacement (MSD) */

2024-06-23T10:07:24Z

Mean Squared Displacement (MSD)

← Older revision		Revision as of 03:07, 23 June 2024
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	=== Mean Squared Displacement (MSD)===		=== Mean Squared Displacement (MSD)===
			<br>

	::<math>		::<math>
	MSD = { 2d \cdot D}		MSD = { 2d \cdot D}
	</math>		</math>


	:: <big><code>d</code></big> : dimensions		:: <big><code>d</code></big> : dimensions
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	<big><code>MSD (in units: '''µm²''')</code></big> is the mean square displacement of a particle over some time period.		<big><code>MSD (in units: '''µm²''')</code></big> is the mean square displacement of a particle over some time period.



	=== Standard deviation (k) of the normal distribution of step lengths ===		=== Standard deviation (k) of the normal distribution of step lengths ===

Bradley Monk: /* Fancy Statistics and Plots */

2024-06-23T10:06:16Z

Fancy Statistics and Plots

← Older revision		Revision as of 03:06, 23 June 2024
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	=~~==Fancy~~ Statistics and Plots===		=Motion Statistics and Plots=

	<p>This section looks at the statistical properties of the simulated data in more detail. In particular, it discusses:</p>		<p>This section looks at the statistical properties of the simulated data in more detail. In particular, it discusses:</p>