My Math Forum  

Go Back   My Math Forum > Science Forums > Physics

Physics Physics Forum


Reply
 
LinkBack Thread Tools Display Modes
September 8th, 2016, 12:33 AM   #1
Newbie
 
Joined: Aug 2016
From: Germany

Posts: 5
Thanks: 0

Question Taking the binary logarithm - error propagation?

Hi,

How do I calculate the error (in my case represented by the standard deviation) of a set of data which are converted to their binary logarithm?

I have for example 10 numerical values for which I can calculate the standard deviation. After converting these 10 values to their binary logarithm, I can either calculate the new standard deviation based on these new 10 values as I did before or I can use the rules of the Gaussian error propagation for calculating the new error (standard deviation). The formula for the latter is shown in the attachment.

I get different results using one or the other method. Which way of calculating the standard deviation is correct (and why)?
Attached Images
File Type: jpg Error propagation for binary logarithm.jpg (3.8 KB, 18 views)
Biochemist is offline  
 
September 8th, 2016, 02:23 PM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,590
Thanks: 1434

to make sure I understand this before looking in detail at it.

You have 10 samples from an underlying Gaussian distribution $X_n \sim N(\mu_X, \sigma_X)$

You can estimate $\sigma$ the usual way to come up with the sample variance $\hat{\sigma}_X$

Now let a new random variable

$Y \sim \log_2(X)$

You can process these samples to obtain

$Y_n = \log_2(X_n)$

and you want to determine how to calculate $\hat{\sigma}_Y$, presumably from the $Y_n$

Is this correct?
romsek is offline  
September 9th, 2016, 01:29 AM   #3
Newbie
 
Joined: Aug 2016
From: Germany

Posts: 5
Thanks: 0

Yes, this is correct. This is what I want to do.

Actually, I can think of two ways how to do it but I don't know which is the right one.
Biochemist is offline  
September 9th, 2016, 03:45 PM   #4
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,590
Thanks: 1434

since the underlying distribution is Gaussian, what do you plan to do for negative values who's binary logarithm is undefined?
romsek is offline  
September 9th, 2016, 04:54 PM   #5
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,590
Thanks: 1434

I found this portion of a pdf file that explains what you need to do pretty well.

You still have the truncation problem.
Attached Images
File Type: jpg clipboard.jpg (91.2 KB, 4 views)
romsek is offline  
September 13th, 2016, 12:12 AM   #6
Newbie
 
Joined: Aug 2016
From: Germany

Posts: 5
Thanks: 0

Quote:
Originally Posted by romsek View Post
since the underlying distribution is Gaussian, what do you plan to do for negative values who's binary logarithm is undefined?
Acutally, due to the nature of the data there are no negative values.
Biochemist is offline  
September 13th, 2016, 12:13 AM   #7
Newbie
 
Joined: Aug 2016
From: Germany

Posts: 5
Thanks: 0

Quote:
Originally Posted by romsek View Post
I found this portion of a pdf file that explains what you need to do pretty well.

You still have the truncation problem.
Thanks. What do you mean by truncation problem?
Biochemist is offline  
September 13th, 2016, 05:06 AM   #8
Newbie
 
Joined: Aug 2016
From: Germany

Posts: 5
Thanks: 0

I found this page here where it is recommended not to use error propagation rules at all if the individual values can be calculated, i.e. if the replicated data be transformed directly.

https://books.google.de/books?id=aIq...arithm&f=false

So, it seems like I don't have to bother with the Gaussian error propagation here.
Biochemist is offline  
Reply

  My Math Forum > Science Forums > Physics

Tags
binary, error, logarithm, propagation, taking



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Error propagation question GreenLamp Advanced Statistics 2 October 23rd, 2015 11:42 AM
Solving equation without taking logarithm mhacker064 Elementary Math 1 April 15th, 2014 08:06 AM
propagation error Kinroh Physics 1 November 3rd, 2013 06:22 PM
Error propagation from a distribution Shadly Advanced Statistics 1 April 5th, 2011 02:02 PM
propagation coefficient - Complex Numbers ferret Complex Analysis 2 February 23rd, 2011 01:36 PM





Copyright © 2019 My Math Forum. All rights reserved.