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  #11  
Old 12-20-2017, 11:45 AM
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Fourier methods use complex numbers, but to the best of my knowledge they don't specifically rely on complex random variables.

I found a CAS note that describes complex random variables and some potential actuarial applications. I don't know how mainstream those ideas are or will be, but odds are good that someone will find it useful.
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Therefore, it is likely that complex random variables and vectors will find their way
into actuarial science. But it will take years, even decades, and technology and education will have to
prepare for it
I should take a look at the imaginary component of our loss reserves.
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  #12  
Old 12-20-2017, 01:24 PM
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Originally Posted by ultrafilter View Post
Fourier methods use complex numbers, but to the best of my knowledge they don't specifically rely on complex random variables.

I found a CAS note that describes complex random variables and some potential actuarial applications. I don't know how mainstream those ideas are or will be, but odds are good that someone will find it useful.
That study note is kind of dumb. 40 pages of complex analysis followed by three pages that indicate the author has not actually looked at the fields where complex analysis actually gets used.

If you combine signal processing methods used for audio and images with statistical inference, odds are you’re going to work with a complex random variable at some point. The question here was “why does this machine learning book discuss complex random variables?” A plausible answer is “because audio and image recognition are popular applications of machine learning.” I don’t own the book and I’m trying to entertain myself by seeing if I can predict its contents.

That’s a separate question from “do actuaries need to know complex random variables to do machine learning?” The answer is probably not unless they are bringing in a specific method from signal processing and even then, they could probably adapt the method into some form of time series analysis using real numbers for parameters.

However, for learning the concepts, image and audio processing may be better to study since there is an abundance of data where concrete and unequivocal results can be demonstrated.
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Old 12-20-2017, 03:38 PM
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I should take a look at the imaginary component of our loss reserves.
Isn't that the definition of IBNR???
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  #14  
Old 12-21-2017, 08:55 PM
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My review of Chapter 1: http://ymmathstat.blogspot.com/2017/...s-machine.html
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Old 12-24-2017, 12:24 PM
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This is actually all covered in undergrad engineering although these topics would be covered in separate courses.
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Old 12-24-2017, 01:14 PM
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This is actually all covered in undergrad engineering although these topics would be covered in separate courses.
I came to a similar conclusion after talking with a former Physics Ph.D. student. I've bought a physics book as a result of this (Mathematical Methods in the Physical Sciences by Boas) and will be going through that text alongside this study.

Math degree programs, I've realized, go through topics too slowly for applications.
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  #17  
Old 12-24-2017, 01:23 PM
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So for a periodic process, the fourier transform constitutes sufficient statistics of the process. The Nyquist sampling theorem gives conditions for a sequence of discrete samples to be sufficient to reconstruct a continuous periodic function.

The ergodic hypothesis is the premise that a sample of a cross-section of a system yields sufficient staistics of the behavior of the system over time. Observations of a population over a short interval can be used as the basis of estimating probabilities for individuals over a longer interval.

There is a related but slightly different notion of ergodicity as the property that a sufficiently large time slice of an ergodic process gives you sufficient statistics for the process. This is conceptually similar to reconstructing a periodic function from a finite discrete sample. In this sense, both periodic processes and processes that are ergodic in this sense are characterized by frequencies or distributions of events. These are assumptions that events have rates of occurrence which can be estimated from observations.

Information theory is based on a transformation of problems involving probabilities to problems involving discrete codes. Under this correspondence, maximum likelihood estimates are mapped to representations that minimize the quantity of data used to describe the system.

These are just ideas you’ve seen in statistics wearing other hats.
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Old 12-29-2017, 06:00 PM
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Quote:
Originally Posted by ultrafilter View Post
Fourier methods use complex numbers, but to the best of my knowledge they don't specifically rely on complex random variables.

I found a CAS note that describes complex random variables and some potential actuarial applications. I don't know how mainstream those ideas are or will be, but odds are good that someone will find it useful.
See spectral analysis in time series. Very standard stuff.
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  #19  
Old 12-29-2017, 07:59 PM
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Originally Posted by clarinetist View Post
I came to a similar conclusion after talking with a former Physics Ph.D. student. I've bought a physics book as a result of this (Mathematical Methods in the Physical Sciences by Boas) and will be going through that text alongside this study.

Math degree programs, I've realized, go through topics too slowly for applications.
No. You just hate waiting and appreciating the beauty and ideas of mathematics.
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  #20  
Old 12-29-2017, 09:42 PM
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Quote:
Originally Posted by clarinetist View Post
I came to a similar conclusion after talking with a former Physics Ph.D. student. I've bought a physics book as a result of this (Mathematical Methods in the Physical Sciences by Boas) and will be going through that text alongside this study.

Math degree programs, I've realized, go through topics too slowly for applications.
why are you buying that book? You need to stop buying books and start printing free lecture notes. FFT and fourier analysis are covered in a standard math curriculum.
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