Lysario - by Panagiotis Chatzichrisafis

Export data from Scilab for PSTricks plots

Panagiotis — Sun, 25 Jan 2015 13:36:28 +0000

It is possible to load data from a file for plots with PSTricks in TeX. According to the PSTricks manual all that is needed is a file with a list of coordinate pairs delimited by curly braces , parentheses, commas.
One easy solution is to use the write command from scilab. Define the following function:


function pstrickswrite2d(filename,data_matrix)
//Author: Panagiotis Chatzichrisafis
//Date: 25.01.2015
//Description: 
//The function pstrickswrite2d(filename,data_matrix) has two inputs: 
//: The path and the filename which shall be used to write 
//            formated output for PSTricks 
//: The matrix containing the data to be written into the file provided by
//                argument.
//The output written to the  file has a format which is suitable for 
//the TeX PSTricks package fileplot or dataplot commands.  
    _num_decimal_places_=5; 
    file_desc=file('open',filename,'unknown','formatted');
    _max_num_ = max(abs(data_matrix));
    _num_integer_digits_ = ceil(log10(_max_num_)); 
    _num_format_(1) ='F';
    // add number of integer digits and decimal digits plus 2 for sign information
    _num_format_(2) = string(_num_integer_digits_+_num_decimal_places_+2); 
    _num_format_(3) ='.'; 
    _num_format_(4) = string(_num_decimal_places_);
    _num_format_ =strcat(_num_format_);
    _format_(1)   ='( ''{'' '   ;
    _format_(2)   = _num_format_; 
    _format_(3)   = ', '','' '  ;
    _format_(4)   = _num_format_; 
    _format_(5)   = ' ''}'' )'  ;
    _format_=strcat(_format_);
    write(file_desc,data_matrix,_format_); 
    file('close',file_desc);
endfunction

and load it into the scilab workspace.

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 94 exercise 4.2

Panagiotis — Tue, 24 Jun 2014 18:54:07 +0000

In [1, p. 94 exercise 4.2] we are asked to consider the estimator

(1)

where

(2)

for the process of Problem 4.1. We are informed that this estimator may be viewed as an averaged periodogram. In this point of view the data record is sectioned into blocks (in this case, of length 1) and the periodograms for each block are averaged. We are asked to find the mean and variance of and compare the result to that obtained in [2].
Solution: We note that according to the boundaries of problem 4.1 ( for which the process is white Gaussian with zero mean) we obtain the fact that the sum is distributed according to a distribution [3, p. 682] with mean and variance . The average periodogram is obtained by and thus the mean and the average is obtained by

Thus taking the average periodogram of one sample section has no statistical advantage for the result of the estimator.

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 94 exercise 4.1

Panagiotis — Sat, 01 Feb 2014 16:40:19 +0000

In problem [1, p. 94 exercise 4.1] we will show that the periodogram is an inconsistent estimator by examining the estimator at , or

(1)

If is real white Gaussian noise process with PSD

(2)

we are asked to find the mean an variance of . We are asked if the variance converge to zero as . The hint provided within the exercise is to note that

(3)

where the quantiiy inside the parenthesis is Solution: Because is distributed according to a normal distribution the squared variable is distributed according to a with mean and variance . From the previous relations we obtain the mean and the variance of the periodogram as

We see that while the mean converges to the true power spectral density, the variance does not converge to zero. Thus the estimator is inconsistent. QED.

Installing OpenCV 2.4.7 on MacOS Snow Leopard and get it work with macam drivers

Panagiotis — Fri, 31 Jan 2014 18:57:32 +0000

I recently wanted to update my OpenCV installation and try to find out why my usb camera – using the macam drivers – was not working with the libraries installed on my system. So this time i had the intention to dig into the code of OpenCV release. That meant using the easy way – mac ports – to install the OpenCV libraries was not an option. The latest OpenCV software can be downloaded at the OpenCV homepage. I downloaded OpenCV version 2.4.7 and extracted it within a local folder i created solely for OpenCV. The package comes with a cmake file (CMakeLists.txt) for creating different packages and setting up the build environment for OpenCV. cmake is open source software and is it’s purpose can be described as cross plattform build generator.That means it is used in order to provide environments for different plattforms that can be used to create a build for some project. Several tutorials and webinars may be found on the homepage of the developing company kitware. I strongly recommend the webinar “Introduction to CMake Course” in order to get an understanding of the cmake process. cmake uses special generator commands that allow you to deploy the project for different purposes. You may like to create make files for Unix for example, or prepare the project to be used with XCode. The actual available generators may be displayed (as of cmake version 2.8.3) by typing into the commandline the command cmake –help. Here’s the excerpt of my terminal output to the command:

[...]

Generators

The following generators are available on this platform:
 Unix Makefiles                 = Generates standard UNIX makefiles.
 Xcode                          = Generate Xcode project files.
 CodeBlocks - Unix Makefiles    = Generates CodeBlocks project files.
 Eclipse CDT4 - Unix Makefiles  = Generates Eclipse CDT 4.0 project files.
 KDevelop3                      = Generates KDevelop 3 project files.
 KDevelop3 - Unix Makefiles     = Generates KDevelop 3 project files.

[...]

In my case i wanted to create Xcode projects out of the source directory. For this purpose i created a new path into the directory where i downloaded and extracted the opencv software (let’s call this place

mypath_to_opencv

where

mypath_to_opencv/opencv-2.4.7

is the path to the opencv-2.4.7 sources). Start your terminal and type:

mkdir xcode
cd xcode
cmake -G Xcode mypath_to_opencv/opencv-2.4.7

With the last command cmake is told to generate Xcode projects out of the files at the location were opencv-2.4.7 was extracted. The Xcode projects will be generated into the path where we called cmake. In the above mentioned case this path is mypath_to_opencv/xcode. If you only want to use the libraries for using the OpenCV API you may like to read Tilo Mitras article about the procedure, while i still suggest that the easiest approach is using macports. Now let’s go back to our Xcode generated build. If you look through your build tree you ‘ll find several Xcode Proejects. The top level of your build tree (xcode) includes for example mypath_to_opencv/xcode/OpenCV.xcodeproj . The samples directory, in case the sample build has been enabled in ccmake, has the Xcode project file mypath_to_opencv/xcode/samples/c/c_samples.xcodeproj In my case the standard cmake configuration had activated the OCL dependency which caused during the build in my systems problem because i have not installed any OCL driver on my system. I had to reconfigure the system and deactivate OCL by the command (yes the command below is ccmake (not cmake) a commandline configuration tool for cmake)

ccmake -G Xcode /opencv-2.4.7

and switching OCL off cmake variable WITH_OPENCL. So now i was ready for experimentation. The error message i got when building and running OpenCV XCode examples that were accessing a camera was “QTKit didn’t find any attached Video Input Devices !”. The same error was reported in a post by karlphillip at stackoverflow. Unfortunately i couldn’t fix my problem with the solution provided there. Debugging didn’t provide any clues why this was happening and somehow i came across a post by Andrew Janke on stackoverflow which gave me the right momentum. The Macam version that i am using (macam-cvs-build-2009-09-25) could be build without any changes for 32 bit system while the 64 bit build failed. OpenCV (2.4.7) on the other hand is generated with 32/64 bit architecture support when running the previous mentioned cmake commands. It is possible to reconfigure the Xcode project by changing the architecture option in ccmake and regenerate the xcode build directory. Trying to built OpenCV with 32 bit support failed on my system due to incompatible libraries that were installed on my machine. The following libraries were only for 64 bit architecture available:

libdc1394.dylib
libavcodec.dylib (part of ffmpeg)
libavformat.dylib (part of ffmpeg)
libavutil.dylib (part of ffmpeg)
libswscale.dylib (part of ffmpeg)
libbz2.dylib

My problem now was, when i build examples i.e. example_video_homography that the 64 bit libraries were used with the 32 bit macam driver. So i tried to compile everything with a 32 bit architecture. Using macports i adapted the opt/local/etc/macports/macports.config file to make builds for 32-bit architecture build_arch i386 I reinstalled the above mentioned libraries. Voila ! Happy end, my good old USB camera is now working with the OpenCV libraries !

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 62 exercise 3.19

Panagiotis — Fri, 17 May 2013 21:09:39 +0000

In [1, p. 62 exercise 3.19] we are asked to find for the multiple sinusoidal process

the ensemble ACF and the temporal ACF as , where the ‘s are all uniformly distributed random variables on and independent of each other. We are also asked to determine if this random process is autocorrelation ergodic.
Solution: We note that the joint p.d.f of the uniformly random variables is given by , with domain we can proceed to calculate the ensemble autocorrelation function which is defined as:

As in exercise [2] we can use the trigonometric relation [3, p. 810] :

(1)

(, ) to further simplify the expression for the ensemble autocorrelation function:





			(2)

Having obtained the ensemble autocorrelation function we can proceed to obtain the temporal autocorrelation function, which we hope will be a good approximation to the ensemble autocorrelation function . By definition the temporal autocorrelation function is given by





			(3)

The second sum of the previous relation can be simplified by one of the derived formulas of [2], for :

Thus the temporal autocorrelation function may be expressed, as long as as:



			(4)

In order to simplify the relation further, especially the first sum, we will again use (1), with and , and thus :



			(5)

We see that both distinct parts are of the form , and thus it remains to simplify this relation. Again using a trigonometric formula [3, p. 810]:

(6)

with and we can simplify the relation as:

Furthermore it was shown in [2] that , for , so we can further simplify the previous relation by:


			(7)

Thus finally we can simplify (5) by:



			(8)

So finally the temporal autocorrelation (4) can be reduced using (8) to the following formula:




			(9)

Rearranging the terms on the right we see that the temporal autocorrelation function equals the ensemble autocorrelation function with an additional error , which we have derived for the the case when :

At once we see again that when that the error goes to zero . Thus the random process of the sum of sinusoids is also autocorrelation ergodic as long as . It is also easy to recognize, by using the argumentation of the previous exercise [2] , that this is not true if does not hold. In this case parts of the the error of equation (3) are proportional to as was already observed in [2] . QED.

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 62 exercise 3.18

Panagiotis — Fri, 24 Aug 2012 15:37:26 +0000

In [1, p. 62 exercise 3.18] we are asked to find the temporal autocorrelation function for the real sinusoidal random process of Problem [1, p. 62 exercise 3.16]:

as . As a second step we are asked to determine if the random process autocorrelation ergodic.
Solution: The sinusoidal process of Problem [1, p. 62 exercise 3.16] is given by , and the autocorrelation function was found to be equal to (see solution of exercise 3.16 [2]) . The temporal autocorrelation is equal to :

The previous relation can be furthermore simplified by the property of trigonometric functions [3, p. 810] :

(1)

and thus the autocorrelation can be written as:

Again it is possible to further simplify the previous relation, this time by using the following trigonometric formula [3, p. 810]:

(2)

Because the sinus function is odd the symmetric sum equals zero. Generally the same is not true for the cosine sum. For for example the sum equals and thus even for the temporal autocorrelation function will be in error to the true autocorrelation function. In general it is true that . The error to the true autocorrelation function is equal to , and thus the process is in general not autocorrelation ergodic, because as we have shown for e.g. the temporal autocorrelation doesn’t even converge for large . That is:

While it is true that there are frequencies for which the temporal autocorrelation doesn’t converge, there are also cases when the process is autocorrelation ergodic. To see this we have to further simplify the relation for the temporal autocorrelation. The sum can be further simplified, by using , and observing that the resulting sum is a geometric progression ([3, p. 7] ) with :

Using this identity the error to the true autocorrelation function can be rewritten as:

Thus by only imposing that the process is autocorrelation ergodic.

Scilab Startup Configuration on MacOsX

Panagiotis — Sat, 05 May 2012 14:20:52 +0000

Sometimes it is useful to have several scripts (with standard definitions) loaded immediately into scilab.
On MacOsX Scilab executes at startup the first lines of the file .scilab. The file has to be placed (if it does already exist) under the directory:


/home/User" />/.Scilab/scilab-version" />

If the file isn’t there just create one and call your scripts with the definitions to be loaded into the scilab workspace.

For example you might want to load definitions of physical constants into the workspace that are defined within ‘physical_constants.sce’. For this purpose add a line like the following into the .scilab file:

exec('Users/User" />/Development/scilab/Base/physical_constants.sce');

Book Review: Logicomix – An epic search for the truth

Panagiotis — Thu, 26 Apr 2012 20:45:44 +0000

Last year i got a very different mathematical book as a gift. A comic novel from Doxiadis, Papadimitriou and Papadatos: Logicomix. It is easy to read and provides some insight of the mathematical quest in the area of logic of the last century. You’ll meet people like Bertrand Russel, David Hilbert, Gödel and many more outstanding persons. Some work results of the main actors are presented in that way that is accessible for a wide audience – according to the rules used in popular science media. So even mathematical illiterate persons may enjoy the comic and due to the very nice illustrations i think even kids could enjoy it. The “showdown” is Gödel’s incompleteness theorem and i enjoyed the way the result was presented. The authors admit at the end of the book that the storyline is only loosely based on historical facts but actually this doesn’t reduce the value of the novel. Reading this comic was a nice way to relax during a cold winter’s day. Give it a try !

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 62 exercise 3.17

Panagiotis — Thu, 19 Apr 2012 20:43:42 +0000

In [1, p. 62 exercise 3.17] we are asked to verify that the variance of the sample mean estimator for the mean of a real WSS random process

is given by [1, eq. (3.60), p. 58]. For the case when is real white noise we are asked to what the variance expression does reduce to. A hint that is given is to use the relationship from [1, eq. (3.64), p. 59].
Solution: Let’s reproduce the corresponding equations cited in the problem statement, so [1, eq. (3.60), p. 58] is given by

while [1, eq. (3.64), p. 59] is given by:

First we note that the mean of the sample mean converges to the true mean of the WSS random process :

From this we can derive the variance of the as:

The squared sample mean can be written as:

From the previous relation we can derive that the mean squared sample mean is given by:


			(1)


			(2)

In the previous relations the factors and denote that for the second summand, while for the third summand degenerates to zero. The three double sums can be further simplified:

(3)

The second double sum can be written by:







			(4)

while similar to the previous derivation the third double sum can be simplified to:







			(5)

Using (3),(4) and (5) in (2) we derive the relationship:




			(6)

Observe that we could have replaced in (1) the sum by its equivalent form and applied relation [1, eq. (3.64), p. 59] of the hint. Then we would have come to the same result. In the approach taken in this solution we also derived the relation provided by the hint. Now relating the autocovariance to the autocorrelation function:

and replacing the autocorrelation in (6) by we obtain the variance of the sample mean as:



			(7)

Noting that [2, p. 125]: we can simplify the second part of the previous equation by:




			(8)

So finally we can rewrite the variance of the sample (7) mean as :



			(9)

which is the relation of the sample mean used in [1, eq. (3.60), p. 58]. QED.

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 62 exercise 3.16

Panagiotis — Fri, 13 Apr 2012 17:58:22 +0000

In [1, p. 62 exercise 3.16] we are asked to show that the random process

(1)

, where is uniformly distributed on , is WSS by finding its mean and ACF. Using the same assumptions we are asked to repeat the exercise for a single complex sinusoid

(2)

Solution: The mean of the random process is given by:

and is thus independent of the sampling variable. Let the p.d.f of be then the ACF of the random process is given by




			(3)

From the trigonometric formula [2, p. 810 rel 21.2-12]:

(4)

by setting and we obtain because :

Thus it is shown that the random process is WSS. Repeating the same for a single complex sinusoid we obtain for the mean that it is equal to zero and independent of :




			(5)

The autocorrelation of the complex sinusoid is obtained by:

which again is independent of and thus the complex sinusoid is also wide sense stationary. We see that in this case, the real part of the complex sinusoid equals the double of the autocorrelation of the real sinusoid.