# AS3: Fourier techniques and statistics

Posted September 26th, 2012 by mpags

The course looks at two fundamental tools of modern experimental science: Fourier analysis and statistics.

Statistics allows us to address questions such as: How precisely do my data constrain the parameters of my model? Which of two or more models of a dataset is preferred? Is the preferred model singled out above all others? How large a sample of objects do I need to observe on how large a telescope to distinguish two rival theories? Is a putative detection real? Statistical analysis is a basic part of experimental design when one is noise limited. The application of statistics has been transformed by the digital computing revolution and it is now common practice to apply methods that may require hundreds of thousands of independent model fits to explore the range of models supported by data. Statistics will be introduced starting with probability and probability distributions, and then looking at estimation, likelihood and maximum likelihood, the analysis of variance (ANOVA), model fitting, and Monte Carlo and Markov Chain Monte Carlo methods.

Most will be familiar with Fourier transforms as a means of extracting frequency components from a time series. They are also used extensively in image processing where spatial data can be converted into a representation in terms of spatial frequencies. Such transformations enable many physical problems to be solved more straightforwardly. For example, how long did it take for the Earth to cool from molten rock, or how well can a telescope resolve an image? Is this set of data correlated with that set? The course will start from the basic mathematics of Fourier methods and their computational implementation, notably via the Fast Fourier Transform (FFT). Technical issues such as aliasing and leakage will be discussed. This will lead on to the Fourier approach to understanding noise in data, looking at power spectral densities and their relation to random noise. The final lecture will look at some noise sources that set fundamental limits to the accuracy of measurement.

As well as paper-and-pen based exercises, there will be computational exercises implemented using Matlab, Maple and Python.

## Module title: Fourier techniques and statistics (AS3)

### Module convenors:

Tom Marsh (Warwick) and Clive Speake (Birmingham)The course looks at two fundamental tools of modern experimental science: Fourier analysis and statistics.

Statistics allows us to address questions such as: How precisely do my data constrain the parameters of my model? Which of two or more models of a dataset is preferred? Is the preferred model singled out above all others? How large a sample of objects do I need to observe on how large a telescope to distinguish two rival theories? Is a putative detection real? Statistical analysis is a basic part of experimental design when one is noise limited. The application of statistics has been transformed by the digital computing revolution and it is now common practice to apply methods that may require hundreds of thousands of independent model fits to explore the range of models supported by data. Statistics will be introduced starting with probability and probability distributions, and then looking at estimation, likelihood and maximum likelihood, the analysis of variance (ANOVA), model fitting, and Monte Carlo and Markov Chain Monte Carlo methods.

Most will be familiar with Fourier transforms as a means of extracting frequency components from a time series. They are also used extensively in image processing where spatial data can be converted into a representation in terms of spatial frequencies. Such transformations enable many physical problems to be solved more straightforwardly. For example, how long did it take for the Earth to cool from molten rock, or how well can a telescope resolve an image? Is this set of data correlated with that set? The course will start from the basic mathematics of Fourier methods and their computational implementation, notably via the Fast Fourier Transform (FFT). Technical issues such as aliasing and leakage will be discussed. This will lead on to the Fourier approach to understanding noise in data, looking at power spectral densities and their relation to random noise. The final lecture will look at some noise sources that set fundamental limits to the accuracy of measurement.

As well as paper-and-pen based exercises, there will be computational exercises implemented using Matlab, Maple and Python.

Duration:

2hrsAcademic year:

2012-2013Starts:

11/01/2013 - 11:00