mhjensen
3016 lines
122 KiB
Plaintext
3016 lines
122 KiB
Plaintext
translating doconce text in book.do.txt to ipynb
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*** error: could not open the file src/plot_Hudson.py used in
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@@@CODE src/plot_Hudson.py
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translating doconce text in book.do.txt to ipynb
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copy complete file src/plot_Hudson.py (format: pypro)
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copy complete file src/Hudson_Bay.py (format: pypro)
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*** error: figure file "fig/Hudson_Bay_sim" does not exist!
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translating doconce text in book.do.txt to ipynb
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copy complete file src/plot_Hudson.py (format: pypro)
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copy complete file src/Hudson_Bay.py (format: pypro)
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figure file fig/Hudson_Bay_data:
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can use fig/Hudson_Bay_data.png for format ipynb
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figure file fig/Hudson_Bay_sim:
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can use fig/Hudson_Bay_sim.png for format ipynb
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collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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output in book.ipynb
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*** error: file has a mako construction ${\bf \hat{J}'
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but seemingly no definition in <%...%>'
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(it is not a command-line given mako variable either).
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However, if this is a variable in a Makefile or Bash script
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run with --no_mako - and you cannot use mako and Makefile or Bash variables
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in the same document!
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*** error: file has a mako construction ${\bf \hat{J}'
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but seemingly no definition in <%...%>'
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(it is not a command-line given mako variable either).
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However, if this is a variable in a Makefile or Bash script
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run with --no_mako - and you cannot use mako and Makefile or Bash variables
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in the same document!
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translating doconce text in book.do.txt to ipynb
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*** error: found multiple labels:
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eq:def_covariance eq:autocorrelation_time eq:error_estimate_corr_time
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translating doconce text in book.do.txt to ipynb
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copy complete file src/plot_Hudson.py (format: pypro)
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copy complete file src/Hudson_Bay.py (format: pypro)
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figure file fig/Hudson_Bay_data:
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can use fig/Hudson_Bay_data.png for format ipynb
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figure file fig/Hudson_Bay_sim:
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can use fig/Hudson_Bay_sim.png for format ipynb
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collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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output in book.ipynb
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translating doconce text in book.do.txt to ipynb
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copy complete file src/plot_Hudson.py (format: pypro)
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copy complete file src/Hudson_Bay.py (format: pypro)
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figure file fig/Hudson_Bay_data:
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can use fig/Hudson_Bay_data.png for format ipynb
|
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figure file fig/Hudson_Bay_sim:
|
|
can use fig/Hudson_Bay_sim.png for format ipynb
|
|
collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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output in book.ipynb
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translating doconce text in book.do.txt to ipynb
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ERROR: 2 !bblock do not match 4 !eblock directives
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Two !eblock after each other!
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!eblock
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===== Numpy and arrays =====
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"Numpy":"http://www.numpy.org/" provides an easy way to handle arrays in Python. The standard way to import this library is as
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!bc pycod
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import numpy as np
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!ec
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Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,
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!bc pycod
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n = 10
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x = np.random.normal(size=n)
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print(x)
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!ec
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We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.
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Another alternative is to declare a vector as follows
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!bc pycod
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import numpy as np
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x = np.array([1, 2, 3])
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print(x)
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!ec
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Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++
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start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as
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!bc pycod
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import numpy as np
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x = np.log(np.array([4, 7, 8]))
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print(x)
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!ec
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In the last example we used Numpy's unary function $np.log$. This function is
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highly tuned to compute array elements since the code is vectorized
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and does not require looping. We normaly recommend that you use the
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Numpy intrinsic functions instead of the corresponding _log_ function
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from Python's _math_ module. The looping is done explicitely by the
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_np.log_ function. The alternative, and slower way to compute the
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logarithms of a vector would be to write
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!bc pycod
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import numpy as np
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from math import log
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x = np.array([4, 7, 8])
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for i in range(0, len(x)):
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x[i] = log(x[i])
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print(x)
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!ec
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We note that our code is much longer already and we need to import the _log_ function from the _math_ module.
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The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the _automatic_ keyword in C++). To change this we could define our array elements to be double precision numbers as
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!bc pycod
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import numpy as np
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x = np.log(np.array([4, 7, 8], dtype = np.float64))
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print(x)
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!ec
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or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
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!bc pycod
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import numpy as np
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x = np.log(np.array([4.0, 7.0, 8.0])
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print(x)
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!ec
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To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the _itemsize_ functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as
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!bc pycod
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import numpy as np
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x = np.log(np.array([4.0, 7.0, 8.0])
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print(x.itemsize)
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!ec
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===== Matrices in Python =====
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Having defined vectors, we are now ready to try out matrices. We can
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define a $3 \times 3 $ real matrix $\hat{A}$ as (recall that we user
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lowercase letters for vectors and uppercase letters for matrices)
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!bc pycod
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import numpy as np
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A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
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print(A)
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!ec
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If we use the _shape_ function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as
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!bc pycod
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import numpy as np
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A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
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# print the first column, row-major order and elements start with 0
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print(A[:,0])
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!ec
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We can continue this was by printing out other columns or rows. The example here prints out the second column
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!bc pycod
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import numpy as np
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A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
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# print the first column, row-major order and elements start with 0
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print(A[1,:])
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!ec
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Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the "Numpy website for more details":"http://www.numpy.org/". Useful functions when defining a matrix are the _np.zeros_ function which declares a matrix of a given dimension and sets all elements to zero
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!bc pycod
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import numpy as np
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n = 10
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# define a matrix of dimension 10 x 10 and set all elements to zero
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A = np.zeros( (n, n) )
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print(A)
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!ec
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or initializing all elements to
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!bc pycod
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import numpy as np
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n = 10
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# define a matrix of dimension 10 x 10 and set all elements to one
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A = np.ones( (n, n) )
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print(A)
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!ec
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or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
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!bc pycod
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import numpy as np
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n = 10
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# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
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A = np.random.rand(n, n)
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print(A)
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!ec
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As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.
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As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors
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$\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as
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!bt
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\[
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\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
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\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
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\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
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\end{bmatrix},
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\]
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!et
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where for example
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!bt
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\[
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\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
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\]
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!et
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The Numpy function _np.cov_ calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values.
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The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $3\times n$ matrix $\hat{W}$
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!bt
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\[
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\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
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x_1 & y_1 & z_1 \\
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x_2 & y_2 & z_2 \\
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\dots & \dots & \dots \\
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x_{n-2} & y_{n-2} & z_{n-2} \\
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x_{n-1} & y_{n-1} & z_{n-1}
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\end{bmatrix},
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\]
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!et
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which in turn is converted into into the $3\times 3$ covariance matrix
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$\hat{\Sigma}$ via the Numpy function _np.cov()_. We note that we can also calculate
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the mean value of each set of samples $\hat{x}$ etc using the Numpy
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function _np.mean(x)_. We can also extract the eigenvalues of the
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covariance matrix through the _np.linalg.eig()_ function.
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!bc pycod
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# Importing various packages
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import numpy as np
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n = 100
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x = np.random.normal(size=n)
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print(np.mean(x))
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y = 4+3*x+np.random.normal(size=n)
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print(np.mean(y))
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z = x**3+np.random.normal(size=n)
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print(np.mean(z))
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W = np.vstack((x, y, z))
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Sigma = np.cov(W)
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print(Sigma)
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Eigvals, Eigvecs = np.linalg.eig(Sigma)
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print(Eigvals)
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!ec
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!bc pycod
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import numpy as np
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import matplotlib.pyplot as plt
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from scipy import sparse
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eye = np.eye(4)
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print(eye)
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sparse_mtx = sparse.csr_matrix(eye)
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print(sparse_mtx)
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x = np.linspace(-10,10,100)
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y = np.sin(x)
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plt.plot(x,y,marker='x')
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plt.show()
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!ec
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===== Meet the Pandas =====
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FIGURE: [fig/pandas.jpg, width=600 frac=0.8]
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Another useful Python package is
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"pandas":"https://pandas.pydata.org/", which is an open source library
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providing high-performance, easy-to-use data structures and data
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analysis tools for Python. _pandas_ stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
|
|
_pandas_ has two major classes, the _DataFrame_ class with two-dimensional data objects and tabular data organized in columns and the class _Series_ with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below.
|
|
_pandas_ allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.
|
|
|
|
The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of _pandas_, in particular in connection with classification of data.
|
|
|
|
!bc pycod
|
|
import pandas as pd
|
|
from IPython.display import display
|
|
data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
|
|
'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
|
|
'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
|
|
'Date of Birth T.A.': [2968, 2890, 2931, 2980]
|
|
}
|
|
data_pandas = pd.DataFrame(data)
|
|
display(data_pandas)
|
|
!ec
|
|
|
|
In the above we have imported _pandas_ with the shorthand _pd_, the latter has become the standard way we import _pandas_. We make then a list of various variables
|
|
and reorganize the aboves lists into a _DataFrame_ and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.
|
|
Displaying these results, we see that the indices are given by the default numbers from zero to three.
|
|
_pandas_ is extremely flexible and we can easily change the above indices by defining a new type of indexing as
|
|
!bc pycod
|
|
data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
|
|
display(data_pandas)
|
|
!ec
|
|
Thereafter we display the content of the row which begins with the index _Aragorn_
|
|
!bc pycod
|
|
display(data_pandas.loc['Aragorn'])
|
|
!ec
|
|
|
|
We can easily append data to this, for example
|
|
!bc pycod
|
|
new_hobbit = {'First Name': ["Peregrin"],
|
|
'Last Name': ["Took"],
|
|
'Place of birth': ["Shire"],
|
|
'Date of Birth T.A.': [2990]
|
|
}
|
|
data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
|
|
display(data_pandas)
|
|
!ec
|
|
|
|
|
|
Here are other examples where we use the _DataFrame_ functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
|
|
of dimensionality $10\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.
|
|
!bc pycod
|
|
import numpy as np
|
|
import pandas as pd
|
|
from IPython.display import display
|
|
np.random.seed(100)
|
|
# setting up a 10 x 5 matrix
|
|
rows = 10
|
|
cols = 5
|
|
a = np.random.randn(rows,cols)
|
|
df = pd.DataFrame(a)
|
|
display(df)
|
|
print(df.mean())
|
|
print(df.std())
|
|
display(df**2)
|
|
!ec
|
|
|
|
Thereafter we can select specific columns only and plot final results
|
|
!bc pycod
|
|
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
|
|
df.index = np.arange(10)
|
|
|
|
display(df)
|
|
print(df['Second'].mean() )
|
|
print(df.info())
|
|
print(df.describe())
|
|
|
|
from pylab import plt, mpl
|
|
plt.style.use('seaborn')
|
|
mpl.rcParams['font.family'] = 'serif'
|
|
|
|
df.cumsum().plot(lw=2.0, figsize=(10,6))
|
|
plt.show()
|
|
|
|
|
|
df.plot.bar(figsize=(10,6), rot=15)
|
|
plt.show()
|
|
!ec
|
|
We can produce a $4\times 4$ matrix
|
|
!bc pycod
|
|
b = np.arange(16).reshape((4,4))
|
|
print(b)
|
|
df1 = pd.DataFrame(b)
|
|
print(df1)
|
|
!ec
|
|
and many other operations.
|
|
|
|
The _Series_ class is another important class included in
|
|
_pandas_. You can view it as a specialization of _DataFrame_ but where
|
|
we have just a single column of data. It shares many of the same features as _DataFrame. As with _DataFrame_,
|
|
most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.
|
|
As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.
|
|
For multidimensional arrays, we recommend strongly "xarray":"http://xarray.pydata.org/en/stable/". _xarray_ has much of the same flexibility as _pandas_, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both _pandas_ and _xarray_.
|
|
|
|
|
|
|
|
===== Reading Data and fitting =====
|
|
|
|
In order to study various Machine Learning algorithms, we need to
|
|
access data. Acccessing data is an essential step in all machine
|
|
learning algorithms. In particular, setting up the so-called _design
|
|
matrix_ (to be defined below) is often the first element we need in
|
|
order to perform our calculations. To set up the design matrix means
|
|
reading (and later, when the calculations are done, writing) data
|
|
in various formats, The formats span from reading files from disk,
|
|
loading data from databases and interacting with online sources
|
|
like web application programming interfaces (APIs).
|
|
|
|
In handling various input formats, as discussed above, we will mainly stay with _pandas_,
|
|
a Python package which allows us, in a seamless and painless way, to
|
|
deal with a multitude of formats, from standard _csv_ (comma separated
|
|
values) files, via _excel_, _html_ to _hdf5_ formats. With _pandas_
|
|
and the _DataFrame_ and _Series_ functionalities we are able to convert text data
|
|
into the calculational formats we need for a specific algorithm. And our code is going to be
|
|
pretty close the basic mathematical expressions.
|
|
|
|
Our first data set is going to be a classic from nuclear physics, namely all
|
|
available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set.
|
|
|
|
We will show some of the
|
|
strengths of packages like _Scikit-Learn_ in fitting nuclear binding energies to
|
|
specific functions using linear regression first. Then, as a teaser, we will show you how
|
|
you can easily implement other algorithms like decision trees and random forests and neural networks.
|
|
|
|
But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,
|
|
(don't be offended) fitting straight lines!
|
|
|
|
|
|
=== Simple linear regression model using _scikit-learn_ ===
|
|
|
|
We start with perhaps our simplest possible example, using _Scikit-Learn_ to perform linear regression analysis on a data set produced by us.
|
|
|
|
What follows is a simple Python code where we have defined a function
|
|
$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries.
|
|
The numbers in the vector $\hat{x}$ are given
|
|
by random numbers generated with a uniform distribution with entries
|
|
$x_i \in [0,1]$ (more about probability distribution functions
|
|
later). These values are then used to define a function $y(x)$
|
|
(tabulated again as a vector) with a linear dependence on $x$ plus a
|
|
random noise added via the normal distribution.
|
|
|
|
|
|
The Numpy functions are imported used the _import numpy as np_
|
|
statement and the random number generator for the uniform distribution
|
|
is called using the function _np.random.rand()_, where we specificy
|
|
that we want $100$ random variables. Using Numpy we define
|
|
automatically an array with the specified number of elements, $100$ in
|
|
our case. With the Numpy function _randn()_ we can compute random
|
|
numbers with the normal distribution (mean value $\mu$ equal to zero and
|
|
variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear
|
|
dependence as function of $x$
|
|
|
|
!bt
|
|
\[
|
|
y = 2x+N(0,1),
|
|
\]
|
|
!et
|
|
|
|
where $N(0,1)$ represents random numbers generated by the normal
|
|
distribution. From _Scikit-Learn_ we import then the
|
|
_LinearRegression_ functionality and make a prediction $\tilde{y} =
|
|
\alpha + \beta x$ using the function _fit(x,y)_. We call the set of
|
|
data $(\hat{x},\hat{y})$ for our training data. The Python package
|
|
_scikit-learn_ has also a functionality which extracts the above
|
|
fitting parameters $\alpha$ and $\beta$ (see below). Later we will
|
|
distinguish between training data and test data.
|
|
|
|
For plotting we use the Python package
|
|
"matplotlib":"https://matplotlib.org/" which produces publication
|
|
quality figures. Feel free to explore the extensive
|
|
"gallery":"https://matplotlib.org/gallery/index.html" of examples. In
|
|
this example we plot our original values of $x$ and $y$ as well as the
|
|
prediction _ypredict_ ($\tilde{y}$), which attempts at fitting our
|
|
data with a straight line.
|
|
|
|
The Python code follows here.
|
|
!bc pycod
|
|
# Importing various packages
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from sklearn.linear_model import LinearRegression
|
|
|
|
x = np.random.rand(100,1)
|
|
y = 2*x+np.random.randn(100,1)
|
|
linreg = LinearRegression()
|
|
linreg.fit(x,y)
|
|
xnew = np.array([[0],[1]])
|
|
ypredict = linreg.predict(xnew)
|
|
|
|
plt.plot(xnew, ypredict, "r-")
|
|
plt.plot(x, y ,'ro')
|
|
plt.axis([0,1.0,0, 5.0])
|
|
plt.xlabel(r'$x$')
|
|
plt.ylabel(r'$y$')
|
|
plt.title(r'Simple Linear Regression')
|
|
plt.show()
|
|
!ec
|
|
|
|
This example serves several aims. It allows us to demonstrate several
|
|
aspects of data analysis and later machine learning algorithms. The
|
|
immediate visualization shows that our linear fit is not
|
|
impressive. It goes through the data points, but there are many
|
|
outliers which are not reproduced by our linear regression. We could
|
|
now play around with this small program and change for example the
|
|
factor in front of $x$ and the normal distribution. Try to change the
|
|
function $y$ to
|
|
|
|
!bt
|
|
\[
|
|
y = 10x+0.01 \times N(0,1),
|
|
\]
|
|
!et
|
|
|
|
where $x$ is defined as before. Does the fit look better? Indeed, by
|
|
reducing the role of the noise given by the normal distribution we see immediately that
|
|
our linear prediction seemingly reproduces better the training
|
|
set. However, this testing 'by the eye' is obviouly not satisfactory in the
|
|
long run. Here we have only defined the training data and our model, and
|
|
have not discussed a more rigorous approach to the _cost_ function.
|
|
|
|
We need more rigorous criteria in defining whether we have succeeded or
|
|
not in modeling our training data. You will be surprised to see that
|
|
many scientists seldomly venture beyond this 'by the eye' approach. A
|
|
standard approach for the *cost* function is the so-called $\chi^2$
|
|
function (a variant of the mean-squared error (MSE))
|
|
|
|
!bt
|
|
\[ \chi^2 = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
|
|
\]
|
|
!et
|
|
|
|
where $\sigma_i^2$ is the variance (to be defined later) of the entry
|
|
$y_i$. We may not know the explicit value of $\sigma_i^2$, it serves
|
|
however the aim of scaling the equations and make the cost function
|
|
dimensionless.
|
|
|
|
Minimizing the cost function is a central aspect of
|
|
our discussions to come. Finding its minima as function of the model
|
|
parameters ($\alpha$ and $\beta$ in our case) will be a recurring
|
|
theme in these series of lectures. Essentially all machine learning
|
|
algorithms we will discuss center around the minimization of the
|
|
chosen cost function. This depends in turn on our specific
|
|
model for describing the data, a typical situation in supervised
|
|
learning. Automatizing the search for the minima of the cost function is a
|
|
central ingredient in all algorithms. Typical methods which are
|
|
employed are various variants of _gradient_ methods. These will be
|
|
discussed in more detail later. Again, you'll be surprised to hear that
|
|
many practitioners minimize the above function ''by the eye', popularly dubbed as
|
|
'chi by the eye'. That is, change a parameter and see (visually and numerically) that
|
|
the $\chi^2$ function becomes smaller.
|
|
|
|
There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define
|
|
the relative error (why would we prefer the MSE instead of the relative error?) as
|
|
|
|
!bt
|
|
\[
|
|
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
|
|
\]
|
|
!et
|
|
We can modify easily the above Python code and plot the relative error instead
|
|
!bc pycod
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from sklearn.linear_model import LinearRegression
|
|
|
|
x = np.random.rand(100,1)
|
|
y = 5*x+0.01*np.random.randn(100,1)
|
|
linreg = LinearRegression()
|
|
linreg.fit(x,y)
|
|
ypredict = linreg.predict(x)
|
|
|
|
plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
|
|
plt.axis([0,1.0,0.0, 0.5])
|
|
plt.xlabel(r'$x$')
|
|
plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
|
|
plt.title(r'Relative error')
|
|
plt.show()
|
|
!ec
|
|
|
|
Depending on the parameter in front of the normal distribution, we may
|
|
have a small or larger relative error. Try to play around with
|
|
different training data sets and study (graphically) the value of the
|
|
relative error.
|
|
|
|
As mentioned above, _Scikit-Learn_ has an impressive functionality.
|
|
We can for example extract the values of $\alpha$ and $\beta$ and
|
|
their error estimates, or the variance and standard deviation and many
|
|
other properties from the statistical data analysis.
|
|
|
|
Here we show an
|
|
example of the functionality of _Scikit-Learn_.
|
|
!bc pycod
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from sklearn.linear_model import LinearRegression
|
|
from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
|
|
|
|
x = np.random.rand(100,1)
|
|
y = 2.0+ 5*x+0.5*np.random.randn(100,1)
|
|
linreg = LinearRegression()
|
|
linreg.fit(x,y)
|
|
ypredict = linreg.predict(x)
|
|
print('The intercept alpha: \n', linreg.intercept_)
|
|
print('Coefficient beta : \n', linreg.coef_)
|
|
# The mean squared error
|
|
print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
|
|
# Explained variance score: 1 is perfect prediction
|
|
print('Variance score: %.2f' % r2_score(y, ypredict))
|
|
# Mean squared log error
|
|
print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
|
|
# Mean absolute error
|
|
print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
|
|
plt.plot(x, ypredict, "r-")
|
|
plt.plot(x, y ,'ro')
|
|
plt.axis([0.0,1.0,1.5, 7.0])
|
|
plt.xlabel(r'$x$')
|
|
plt.ylabel(r'$y$')
|
|
plt.title(r'Linear Regression fit ')
|
|
plt.show()
|
|
|
|
!ec
|
|
The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields
|
|
$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
|
|
!bt
|
|
\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
|
\]
|
|
!et
|
|
|
|
The smaller the value, the better the fit. Ideally we would like to
|
|
have an MSE equal zero. The attentive reader has probably recognized
|
|
this function as being similar to the $\chi^2$ function defined above.
|
|
|
|
The _r2score_ function computes $R^2$, the coefficient of
|
|
determination. It provides a measure of how well future samples are
|
|
likely to be predicted by the model. Best possible score is 1.0 and it
|
|
can be negative (because the model can be arbitrarily worse). A
|
|
constant model that always predicts the expected value of $\hat{y}$,
|
|
disregarding the input features, would get a $R^2$ score of $0.0$.
|
|
|
|
If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
|
|
!bt
|
|
\[
|
|
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
|
|
\]
|
|
!et
|
|
where we have defined the mean value of $\hat{y}$ as
|
|
!bt
|
|
\[
|
|
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
|
\]
|
|
!et
|
|
Another quantity taht we will meet again in our discussions of regression analysis is
|
|
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.
|
|
The MAE is defined as follows
|
|
!bt
|
|
\[
|
|
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
|
|
\]
|
|
!et
|
|
Finally we present the
|
|
squared logarithmic (quadratic) error
|
|
!bt
|
|
\[
|
|
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
|
|
\]
|
|
!et
|
|
|
|
where $\log_e (x)$ stands for the natural logarithm of $x$. This error
|
|
estimate is best to use when targets having exponential growth, such
|
|
as population counts, average sales of a commodity over a span of
|
|
years etc.
|
|
|
|
We will discuss in more
|
|
detail these and other functions in the various lectures. We conclude this part with another example. Instead of
|
|
a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn.
|
|
|
|
!bc pycod
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
|
|
import random
|
|
from sklearn.linear_model import Ridge
|
|
from sklearn.preprocessing import PolynomialFeatures
|
|
from sklearn.pipeline import make_pipeline
|
|
from sklearn.linear_model import LinearRegression
|
|
|
|
x=np.linspace(0.02,0.98,200)
|
|
noise = np.asarray(random.sample((range(200)),200))
|
|
y=x**3*noise
|
|
yn=x**3*100
|
|
poly3 = PolynomialFeatures(degree=3)
|
|
X = poly3.fit_transform(x[:,np.newaxis])
|
|
clf3 = LinearRegression()
|
|
clf3.fit(X,y)
|
|
|
|
Xplot=poly3.fit_transform(x[:,np.newaxis])
|
|
poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
|
|
plt.plot(x,yn, color='red', label="True Cubic")
|
|
plt.scatter(x, y, label='Data', color='orange', s=15)
|
|
plt.legend()
|
|
plt.show()
|
|
|
|
def error(a):
|
|
for i in y:
|
|
err=(y-yn)/yn
|
|
return abs(np.sum(err))/len(err)
|
|
|
|
print (error(y))
|
|
!ec
|
|
|
|
|
|
|
|
|
|
=== To our real data: nuclear binding energies. Brief reminder on masses and binding energies ===
|
|
|
|
Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding
|
|
energies. A basic quantity which can be measured for the ground
|
|
states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with
|
|
atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).
|
|
|
|
Atomic masses are usually tabulated in terms of the mass excess defined by
|
|
!bt
|
|
\[
|
|
\Delta M(N, Z) = M(N, Z) - uA,
|
|
\]
|
|
!et
|
|
where $u$ is the Atomic Mass Unit
|
|
!bt
|
|
\[
|
|
u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
|
|
\]
|
|
!et
|
|
The nucleon masses are
|
|
!bt
|
|
\[
|
|
m_p = 1.00727646693(9)u,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
|
|
\]
|
|
!et
|
|
|
|
In the "2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu":"http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf"
|
|
there are data on masses and decays of 3437 nuclei.
|
|
|
|
The nuclear binding energy is defined as the energy required to break
|
|
up a given nucleus into its constituent parts of $N$ neutrons and $Z$
|
|
protons. In terms of the atomic masses $M(N, Z)$ the binding energy is
|
|
defined by
|
|
|
|
|
|
!bt
|
|
\[
|
|
BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
|
|
\]
|
|
!et
|
|
where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.
|
|
In terms of the mass excess the binding energy is given by
|
|
!bt
|
|
\[
|
|
BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
|
|
\]
|
|
!et
|
|
where $\Delta_H c^2 = 7.2890$ MeV and $\Delta_n c^2 = 8.0713$ MeV.
|
|
|
|
|
|
A popular and physically intuitive model which can be used to parametrize
|
|
the experimental binding energies as function of $A$, is the so-called
|
|
_liquid drop model_. The ansatz is based on the following expression
|
|
|
|
!bt
|
|
\[
|
|
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
|
|
\]
|
|
!et
|
|
|
|
where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit
|
|
to the experimental data.
|
|
|
|
|
|
|
|
|
|
To arrive at the above expression we have assumed that we can make the following assumptions:
|
|
|
|
* There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.
|
|
|
|
* There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
|
|
|
|
|
|
* There is a Coulomb energy term $a_3\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding.
|
|
|
|
* There is an asymmetry term $a_4\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.
|
|
|
|
We could also add a so-called pairing term, which is a correction term that
|
|
arises from the tendency of proton pairs and neutron pairs to
|
|
occur. An even number of particles is more stable than an odd number.
|
|
|
|
|
|
=== Organizing our data ===
|
|
|
|
Let us start with reading and organizing our data.
|
|
We start with the compilation of masses and binding energies from 2016.
|
|
After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.
|
|
|
|
|
|
We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of _scikit-learn_.
|
|
!bc pycod
|
|
# Common imports
|
|
import numpy as np
|
|
import pandas as pd
|
|
import matplotlib.pyplot as plt
|
|
import sklearn.linear_model as skl
|
|
from sklearn.model_selection import train_test_split
|
|
from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
|
|
import os
|
|
|
|
# Where to save the figures and data files
|
|
PROJECT_ROOT_DIR = "Results"
|
|
FIGURE_ID = "Results/FigureFiles"
|
|
DATA_ID = "DataFiles/"
|
|
|
|
if not os.path.exists(PROJECT_ROOT_DIR):
|
|
os.mkdir(PROJECT_ROOT_DIR)
|
|
|
|
if not os.path.exists(FIGURE_ID):
|
|
os.makedirs(FIGURE_ID)
|
|
|
|
if not os.path.exists(DATA_ID):
|
|
os.makedirs(DATA_ID)
|
|
|
|
def image_path(fig_id):
|
|
return os.path.join(FIGURE_ID, fig_id)
|
|
|
|
def data_path(dat_id):
|
|
return os.path.join(DATA_ID, dat_id)
|
|
|
|
def save_fig(fig_id):
|
|
plt.savefig(image_path(fig_id) + ".png", format='png')
|
|
|
|
infile = open(data_path("MassEval2016.dat"),'r')
|
|
!ec
|
|
|
|
|
|
Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various _matplotlib_ commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.
|
|
!bc pycod
|
|
from pylab import plt, mpl
|
|
plt.style.use('seaborn')
|
|
mpl.rcParams['font.family'] = 'serif'
|
|
|
|
def MakePlot(x,y, styles, labels, axlabels):
|
|
plt.figure(figsize=(10,6))
|
|
for i in range(len(x)):
|
|
plt.plot(x[i], y[i], styles[i], label = labels[i])
|
|
plt.xlabel(axlabels[0])
|
|
plt.ylabel(axlabels[1])
|
|
plt.legend(loc=0)
|
|
!ec
|
|
|
|
Our next step is to read the data on experimental binding energies and
|
|
reorganize them as functions of the mass number $A$, the number of
|
|
protons $Z$ and neutrons $N$ using _pandas_. Before we do this it is
|
|
always useful (unless you have a binary file or other types of compressed
|
|
data) to actually open the file and simply take a look at it!
|
|
|
|
|
|
In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with _pandas_. The file begins with some basic format information.
|
|
!bc pycod
|
|
"""
|
|
This is taken from the data file of the mass 2016 evaluation.
|
|
All files are 3436 lines long with 124 character per line.
|
|
Headers are 39 lines long.
|
|
col 1 : Fortran character control: 1 = page feed 0 = line feed
|
|
format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5
|
|
These formats are reflected in the pandas widths variable below, see the statement
|
|
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
|
|
Pandas has also a variable header, with length 39 in this case.
|
|
"""
|
|
!ec
|
|
|
|
The data we are interested in are in columns 2, 3, 4 and 11, giving us
|
|
the number of neutrons, protons, mass numbers and binding energies,
|
|
respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will
|
|
covert them into the _pandas_ DataFrame structure.
|
|
|
|
!bc pycod
|
|
# Read the experimental data with Pandas
|
|
Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
|
|
names=('N', 'Z', 'A', 'Element', 'Ebinding'),
|
|
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
|
|
header=39,
|
|
index_col=False)
|
|
|
|
# Extrapolated values are indicated by '#' in place of the decimal place, so
|
|
# the Ebinding column won't be numeric. Coerce to float and drop these entries.
|
|
Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
|
|
Masses = Masses.dropna()
|
|
# Convert from keV to MeV.
|
|
Masses['Ebinding'] /= 1000
|
|
|
|
# Group the DataFrame by nucleon number, A.
|
|
Masses = Masses.groupby('A')
|
|
# Find the rows of the grouped DataFrame with the maximum binding energy.
|
|
Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
|
|
!ec
|
|
|
|
We have now read in the data, grouped them according to the variables we are interested in.
|
|
We see how easy it is to reorganize the data using _pandas_. If we
|
|
were to do these operations in C/C++ or Fortran, we would have had to
|
|
write various functions/subroutines which perform the above
|
|
reorganizations for us. Having reorganized the data, we can now start
|
|
to make some simple fits using both the functionalities in _numpy_ and
|
|
_Scikit-Learn_ afterwards.
|
|
|
|
Now we define five variables which contain
|
|
the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves.
|
|
!bc pycod
|
|
A = Masses['A']
|
|
Z = Masses['Z']
|
|
N = Masses['N']
|
|
Element = Masses['Element']
|
|
Energies = Masses['Ebinding']
|
|
print(Masses)
|
|
!ec
|
|
The next step, and we will define this mathematically later, is to set up the so-called _design matrix_. We will throughout call this matrix $\bm{X}$.
|
|
It has dimensionality $p\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit.
|
|
!bc pycod
|
|
# Now we set up the design matrix X
|
|
X = np.zeros((len(A),5))
|
|
X[:,0] = 1
|
|
X[:,1] = A
|
|
X[:,2] = A**(2.0/3.0)
|
|
X[:,3] = A**(-1.0/3.0)
|
|
X[:,4] = A**(-1.0)
|
|
!ec
|
|
With _scikitlearn_ we are now ready to use linear regression and fit our data.
|
|
!bc pycod
|
|
clf = skl.LinearRegression().fit(X, Energies)
|
|
fity = clf.predict(X)
|
|
!ec
|
|
Pretty simple!
|
|
Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.
|
|
!bc pycod
|
|
# The mean squared error
|
|
print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
|
|
# Explained variance score: 1 is perfect prediction
|
|
print('Variance score: %.2f' % r2_score(Energies, fity))
|
|
# Mean absolute error
|
|
print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
|
|
print(clf.coef_, clf.intercept_)
|
|
|
|
Masses['Eapprox'] = fity
|
|
# Generate a plot comparing the experimental with the fitted values values.
|
|
fig, ax = plt.subplots()
|
|
ax.set_xlabel(r'$A = N + Z$')
|
|
ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
|
|
ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
|
|
label='Ame2016')
|
|
ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
|
|
label='Fit')
|
|
ax.legend()
|
|
save_fig("Masses2016")
|
|
plt.show()
|
|
!ec
|
|
|
|
|
|
=== Seeing the wood for the trees ===
|
|
|
|
As a teaser, let us now see how we can do this with decision trees using _scikit-learn_. Later we will switch to so-called _random forests_!
|
|
|
|
|
|
!bc pycod
|
|
|
|
#Decision Tree Regression
|
|
from sklearn.tree import DecisionTreeRegressor
|
|
regr_1=DecisionTreeRegressor(max_depth=5)
|
|
regr_2=DecisionTreeRegressor(max_depth=7)
|
|
regr_3=DecisionTreeRegressor(max_depth=9)
|
|
regr_1.fit(X, Energies)
|
|
regr_2.fit(X, Energies)
|
|
regr_3.fit(X, Energies)
|
|
|
|
|
|
y_1 = regr_1.predict(X)
|
|
y_2 = regr_2.predict(X)
|
|
y_3=regr_3.predict(X)
|
|
Masses['Eapprox'] = y_3
|
|
# Plot the results
|
|
plt.figure()
|
|
plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
|
|
plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
|
|
plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
|
|
plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
|
|
|
|
plt.xlabel("$A$")
|
|
plt.ylabel("$E$[MeV]")
|
|
plt.title("Decision Tree Regression")
|
|
plt.legend()
|
|
save_fig("Masses2016Trees")
|
|
plt.show()
|
|
print(Masses)
|
|
print(np.mean( (Energies-y_1)**2))
|
|
!ec
|
|
|
|
|
|
=== And what about using neural networks? ===
|
|
The _seaborn_ package allows us to visualize data in an efficient way. Note that we use _scikit-learn_'s multi-layer perceptron (or feed forward neural network)
|
|
functionality.
|
|
!bc pycod
|
|
from sklearn.neural_network import MLPRegressor
|
|
from sklearn.metrics import accuracy_score
|
|
import seaborn as sns
|
|
|
|
X_train = X
|
|
Y_train = Energies
|
|
n_hidden_neurons = 100
|
|
epochs = 100
|
|
# store models for later use
|
|
eta_vals = np.logspace(-5, 1, 7)
|
|
lmbd_vals = np.logspace(-5, 1, 7)
|
|
# store the models for later use
|
|
DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
|
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
|
sns.set()
|
|
for i, eta in enumerate(eta_vals):
|
|
for j, lmbd in enumerate(lmbd_vals):
|
|
dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
|
|
alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
|
|
dnn.fit(X_train, Y_train)
|
|
DNN_scikit[i][j] = dnn
|
|
train_accuracy[i][j] = dnn.score(X_train, Y_train)
|
|
|
|
fig, ax = plt.subplots(figsize = (10, 10))
|
|
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
|
ax.set_title("Training Accuracy")
|
|
ax.set_ylabel("$\eta$")
|
|
ax.set_xlabel("$\lambda$")
|
|
plt.show()
|
|
|
|
|
|
|
|
!ec
|
|
|
|
|
|
|
|
|
|
|
|
|
|
===== A first summary =====
|
|
|
|
The aim behind these introductory words was to present to you various
|
|
Python libraries and their functionalities, in particular libraries like
|
|
_numpy_, _pandas_, _xarray_ and _matplotlib_ and other that make our life much easier
|
|
in handling various data sets and visualizing data.
|
|
|
|
Furthermore,
|
|
_Scikit-Learn_ allows us with few lines of code to implement popular
|
|
Machine Learning algorithms for supervised learning. Later we will meet _Tensorflow_, a powerful library for deep learning.
|
|
Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
======= Why Linear Regression (aka Ordinary Least Squares and family) =======
|
|
|
|
Fitting a continuous function with linear parameterization in terms of the parameters $\bm{\beta}$.
|
|
* Method of choice for fitting a continuous function!
|
|
* Gives an excellent introduction to central Machine Learning features with _understandable pedagogical_ links to other methods like _Neural Networks_, _Support Vector Machines_ etc
|
|
* Analytical expression for the fitting parameters $\bm{\beta}$
|
|
* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more
|
|
* Analytical relation with probabilistic interpretations
|
|
* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics
|
|
* Easy to code! And links well with classification problems and logistic regression and neural networks
|
|
* Allows for _easy_ hands-on understanding of gradient descent methods
|
|
* and many more features
|
|
|
|
For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
|
|
Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended.
|
|
|
|
|
|
=== Regression analysis, overarching aims ===
|
|
|
|
Regression modeling deals with the description of the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\bm{x} =[x_0, x_1,\dots, x_{n-1}]^T$.
|
|
The first variable is called the _dependent_, the _outcome_ or the _response_ variable while the set of variables $\bm{x}$ is called the independent variable, or the predictor variable or the explanatory variable.
|
|
|
|
A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$, that is the conditional distribution for $\bm{y}$ with a given $\bm{x}$. The estimation of $p(\bm{y}\vert \bm{x})$ is made using a data set with
|
|
* $n$ cases $i = 0, 1, 2, \dots, n-1$
|
|
* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \dots, n-1$
|
|
* $p$ so-called explanatory (independent or predictor) variables $\bm{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]$ with $i = 0, 1, 2, \dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples.
|
|
The goal of the regression analysis is to extract/exploit relationship between $\bm{y}$ and $\bm{X}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
|
|
|
|
|
|
Consider an experiment in which $p$ characteristics of $n$ samples are
|
|
measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix
|
|
$\mathbf{X}$.
|
|
|
|
The matrix $\mathbf{X}$ is called the *design
|
|
matrix*. Additional information of the samples is available in the
|
|
form of $\bm{y}$ (also as above). The variable $\bm{y}$ is
|
|
generally referred to as the *response variable*. The aim of
|
|
regression analysis is to explain $\bm{y}$ in terms of
|
|
$\bm{X}$ through a functional relationship like $y_i =
|
|
f(\mathbf{X}_{i,\ast})$. When no prior knowledge on the form of
|
|
$f(\cdot)$ is available, it is common to assume a linear relationship
|
|
between $\bm{X}$ and $\bm{y}$. This assumption gives rise to
|
|
the *linear regression model* where $\bm{\beta} = [\beta_0, \ldots,
|
|
\beta_{p-1}]^{T}$ are the *regression parameters*.
|
|
|
|
Linear regression gives us a set of analytical equations for the parameters $\beta_j$.
|
|
|
|
|
|
=== Examples ===
|
|
|
|
In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\bm{y}$,
|
|
consider the model we discussed for describing nuclear binding energies.
|
|
|
|
There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.
|
|
Assuming
|
|
!bt
|
|
\[
|
|
BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},
|
|
\]
|
|
!et
|
|
we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms.
|
|
This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a
|
|
$p\times n$ matrix $\bm{X}$.
|
|
|
|
Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the
|
|
so-called "credit card default data from Taiwan":"https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub". The data set contains data on $n=30000$ credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are $24$ such predictors or attributes leading to a design matrix of dimensionality $24 \times 30000$
|
|
|
|
|
|
===== General linear models =====
|
|
|
|
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\bm{y}=[y_0,y_1,\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\bm{x}=[x_0,x_1,\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function.
|
|
|
|
Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is
|
|
!bt
|
|
\[
|
|
y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i,
|
|
\]
|
|
!et
|
|
where $\epsilon_i$ is the error in our approximation.
|
|
|
|
|
|
For every set of values $y_i,x_i$ we have thus the corresponding set of equations
|
|
!bt
|
|
\begin{align*}
|
|
y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\
|
|
y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\
|
|
y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
!et
|
|
|
|
|
|
Defining the vectors
|
|
!bt
|
|
\[
|
|
\bm{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
\bm{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
\bm{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T,
|
|
\]
|
|
!et
|
|
and the design matrix
|
|
!bt
|
|
\[
|
|
\bm{X}=
|
|
\begin{bmatrix}
|
|
1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\
|
|
1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\
|
|
1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\
|
|
\dots& \dots &\dots& \dots & \dots &\dots\\
|
|
1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\
|
|
\end{bmatrix}
|
|
\]
|
|
!et
|
|
we can rewrite our equations as
|
|
!bt
|
|
\[
|
|
\bm{y} = \bm{X}\bm{\beta}+\bm{\epsilon}.
|
|
\]
|
|
!et
|
|
The above design matrix is called a "Vandermonde matrix":"https://en.wikipedia.org/wiki/Vandermonde_matrix".
|
|
|
|
|
|
|
|
|
|
===== Generalizing the fitting procedure as a linear algebra problem =====
|
|
|
|
We are obviously not limited to the above polynomial expansions. We
|
|
could replace the various powers of $x$ with elements of Fourier
|
|
series or instead of $x_i^j$ we could have $\cos{(j x_i)}$ or $\sin{(j
|
|
x_i)}$, or time series or other orthogonal functions. For every set
|
|
of values $y_i,x_i$ we can then generalize the equations to
|
|
|
|
!bt
|
|
\begin{align*}
|
|
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
|
|
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
|
|
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\
|
|
\dots & \dots \\
|
|
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
!et
|
|
|
|
_Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!_
|
|
|
|
We redefine in turn the matrix $\bm{X}$ as
|
|
!bt
|
|
\[
|
|
\bm{X}=
|
|
\begin{bmatrix}
|
|
x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\
|
|
x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\
|
|
x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\
|
|
\dots& \dots &\dots& \dots & \dots &\dots\\
|
|
x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\
|
|
\end{bmatrix}
|
|
\]
|
|
!et
|
|
and without loss of generality we rewrite again our equations as
|
|
!bt
|
|
\[
|
|
\bm{y} = \bm{X}\bm{\beta}+\bm{\epsilon}.
|
|
\]
|
|
!et
|
|
The left-hand side of this equation is kwown. Our error vector $\bm{\epsilon}$ and the parameter vector $\bm{\beta}$ are our unknow quantities. How can we obtain the optimal set of $\beta_i$ values?
|
|
|
|
We have defined the matrix $\bm{X}$ via the equations
|
|
!bt
|
|
\begin{align*}
|
|
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
|
|
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
|
|
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\
|
|
\dots & \dots \\
|
|
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
!et
|
|
|
|
As we noted above, we stayed with a system with the design matrix
|
|
$\bm{X}\in {\mathbb{R}}^{n\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define
|
|
our matrix as $\bm{X}\in {\mathbb{R}}^{n\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements.
|
|
|
|
|
|
===== Our model for the nuclear binding energies =====
|
|
|
|
In our introductory notes we looked at the so-called "liguid drop model":"https://en.wikipedia.org/wiki/Semi-empirical_mass_formula". Let us remind ourselves about what we did by looking at the code.
|
|
|
|
We restate the parts of the code we are most interested in.
|
|
!bc pycod
|
|
# Common imports
|
|
import numpy as np
|
|
import pandas as pd
|
|
import matplotlib.pyplot as plt
|
|
from IPython.display import display
|
|
import os
|
|
|
|
# Where to save the figures and data files
|
|
PROJECT_ROOT_DIR = "Results"
|
|
FIGURE_ID = "Results/FigureFiles"
|
|
DATA_ID = "DataFiles/"
|
|
|
|
if not os.path.exists(PROJECT_ROOT_DIR):
|
|
os.mkdir(PROJECT_ROOT_DIR)
|
|
|
|
if not os.path.exists(FIGURE_ID):
|
|
os.makedirs(FIGURE_ID)
|
|
|
|
if not os.path.exists(DATA_ID):
|
|
os.makedirs(DATA_ID)
|
|
|
|
def image_path(fig_id):
|
|
return os.path.join(FIGURE_ID, fig_id)
|
|
|
|
def data_path(dat_id):
|
|
return os.path.join(DATA_ID, dat_id)
|
|
|
|
def save_fig(fig_id):
|
|
plt.savefig(image_path(fig_id) + ".png", format='png')
|
|
|
|
infile = open(data_path("MassEval2016.dat"),'r')
|
|
|
|
|
|
# Read the experimental data with Pandas
|
|
Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
|
|
names=('N', 'Z', 'A', 'Element', 'Ebinding'),
|
|
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
|
|
header=39,
|
|
index_col=False)
|
|
|
|
# Extrapolated values are indicated by '#' in place of the decimal place, so
|
|
# the Ebinding column won't be numeric. Coerce to float and drop these entries.
|
|
Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
|
|
Masses = Masses.dropna()
|
|
# Convert from keV to MeV.
|
|
Masses['Ebinding'] /= 1000
|
|
|
|
# Group the DataFrame by nucleon number, A.
|
|
Masses = Masses.groupby('A')
|
|
# Find the rows of the grouped DataFrame with the maximum binding energy.
|
|
Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
|
|
A = Masses['A']
|
|
Z = Masses['Z']
|
|
N = Masses['N']
|
|
Element = Masses['Element']
|
|
Energies = Masses['Ebinding']
|
|
|
|
# Now we set up the design matrix X
|
|
X = np.zeros((len(A),5))
|
|
X[:,0] = 1
|
|
X[:,1] = A
|
|
X[:,2] = A**(2.0/3.0)
|
|
X[:,3] = A**(-1.0/3.0)
|
|
X[:,4] = A**(-1.0)
|
|
# Then nice printout using pandas
|
|
DesignMatrix = pd.DataFrame(X)
|
|
DesignMatrix.index = A
|
|
DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
|
|
display(DesignMatrix)
|
|
!ec
|
|
|
|
With $\bm{\beta}\in {\mathbb{R}}^{p\times 1}$, it means that we will hereafter write our equations for the approximation as
|
|
!bt
|
|
\[
|
|
\bm{\tilde{y}}= \bm{X}\bm{\beta},
|
|
\]
|
|
!et
|
|
throughout these lectures.
|
|
|
|
|
|
|
|
With the above we use the design matrix to define the approximation $\bm{\tilde{y}}$ via the unknown quantity $\bm{\beta}$ as
|
|
!bt
|
|
\[
|
|
\bm{\tilde{y}}= \bm{X}\bm{\beta},
|
|
\]
|
|
!et
|
|
and in order to find the optimal parameters $\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\tilde{y}_i$, namely
|
|
!bt
|
|
\[
|
|
C(\bm{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\bm{y}-\bm{\tilde{y}}\right)^T\left(\bm{y}-\bm{\tilde{y}}\right)\right\},
|
|
\]
|
|
!et
|
|
or using the matrix $\bm{X}$ and in a more compact matrix-vector notation as
|
|
!bt
|
|
\[
|
|
C(\bm{\beta})=\frac{1}{n}\left\{\left(\bm{y}-\bm{X}^T\bm{\beta}\right)^T\left(\bm{y}-\bm{X}^T\bm{\beta}\right)\right\}.
|
|
\]
|
|
!et
|
|
This function is one possible way to define the so-called cost function.
|
|
|
|
|
|
|
|
It is also common to define
|
|
the function $Q$ as
|
|
|
|
!bt
|
|
\[
|
|
C(\bm{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2,
|
|
\]
|
|
!et
|
|
since when taking the first derivative with respect to the unknown parameters $\beta$, the factor of $2$ cancels out.
|
|
!eblock
|
|
translating doconce text in book.do.txt to ipynb
|
|
ERROR: 2 !bblock do not match 4 !eblock directives
|
|
|
|
|
|
Two !eblock after each other!
|
|
|
|
!eblock
|
|
|
|
|
|
===== Numpy and arrays =====
|
|
"Numpy":"http://www.numpy.org/" provides an easy way to handle arrays in Python. The standard way to import this library is as
|
|
|
|
!bc pycod
|
|
import numpy as np
|
|
!ec
|
|
Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,
|
|
!bc pycod
|
|
n = 10
|
|
x = np.random.normal(size=n)
|
|
print(x)
|
|
!ec
|
|
We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.
|
|
Another alternative is to declare a vector as follows
|
|
!bc pycod
|
|
import numpy as np
|
|
x = np.array([1, 2, 3])
|
|
print(x)
|
|
!ec
|
|
Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++
|
|
start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as
|
|
!bc pycod
|
|
import numpy as np
|
|
x = np.log(np.array([4, 7, 8]))
|
|
print(x)
|
|
!ec
|
|
|
|
In the last example we used Numpy's unary function $np.log$. This function is
|
|
highly tuned to compute array elements since the code is vectorized
|
|
and does not require looping. We normaly recommend that you use the
|
|
Numpy intrinsic functions instead of the corresponding _log_ function
|
|
from Python's _math_ module. The looping is done explicitely by the
|
|
_np.log_ function. The alternative, and slower way to compute the
|
|
logarithms of a vector would be to write
|
|
|
|
!bc pycod
|
|
import numpy as np
|
|
from math import log
|
|
x = np.array([4, 7, 8])
|
|
for i in range(0, len(x)):
|
|
x[i] = log(x[i])
|
|
print(x)
|
|
!ec
|
|
We note that our code is much longer already and we need to import the _log_ function from the _math_ module.
|
|
The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the _automatic_ keyword in C++). To change this we could define our array elements to be double precision numbers as
|
|
!bc pycod
|
|
import numpy as np
|
|
x = np.log(np.array([4, 7, 8], dtype = np.float64))
|
|
print(x)
|
|
!ec
|
|
or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
|
|
!bc pycod
|
|
import numpy as np
|
|
x = np.log(np.array([4.0, 7.0, 8.0])
|
|
print(x)
|
|
!ec
|
|
To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the _itemsize_ functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as
|
|
!bc pycod
|
|
import numpy as np
|
|
x = np.log(np.array([4.0, 7.0, 8.0])
|
|
print(x.itemsize)
|
|
!ec
|
|
|
|
|
|
===== Matrices in Python =====
|
|
|
|
Having defined vectors, we are now ready to try out matrices. We can
|
|
define a $3 \times 3 $ real matrix $\hat{A}$ as (recall that we user
|
|
lowercase letters for vectors and uppercase letters for matrices)
|
|
|
|
!bc pycod
|
|
import numpy as np
|
|
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
|
|
print(A)
|
|
!ec
|
|
If we use the _shape_ function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as
|
|
!bc pycod
|
|
import numpy as np
|
|
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
|
|
# print the first column, row-major order and elements start with 0
|
|
print(A[:,0])
|
|
!ec
|
|
We can continue this was by printing out other columns or rows. The example here prints out the second column
|
|
!bc pycod
|
|
import numpy as np
|
|
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
|
|
# print the first column, row-major order and elements start with 0
|
|
print(A[1,:])
|
|
!ec
|
|
Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the "Numpy website for more details":"http://www.numpy.org/". Useful functions when defining a matrix are the _np.zeros_ function which declares a matrix of a given dimension and sets all elements to zero
|
|
!bc pycod
|
|
import numpy as np
|
|
n = 10
|
|
# define a matrix of dimension 10 x 10 and set all elements to zero
|
|
A = np.zeros( (n, n) )
|
|
print(A)
|
|
!ec
|
|
or initializing all elements to
|
|
!bc pycod
|
|
import numpy as np
|
|
n = 10
|
|
# define a matrix of dimension 10 x 10 and set all elements to one
|
|
A = np.ones( (n, n) )
|
|
print(A)
|
|
!ec
|
|
or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
|
|
!bc pycod
|
|
import numpy as np
|
|
n = 10
|
|
# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
|
|
A = np.random.rand(n, n)
|
|
print(A)
|
|
!ec
|
|
|
|
As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.
|
|
As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors
|
|
$\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as
|
|
!bt
|
|
\[
|
|
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
|
|
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
|
|
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
|
|
\end{bmatrix},
|
|
\]
|
|
!et
|
|
where for example
|
|
!bt
|
|
\[
|
|
\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
|
|
\]
|
|
!et
|
|
The Numpy function _np.cov_ calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values.
|
|
The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $3\times n$ matrix $\hat{W}$
|
|
!bt
|
|
\[
|
|
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
|
|
x_1 & y_1 & z_1 \\
|
|
x_2 & y_2 & z_2 \\
|
|
\dots & \dots & \dots \\
|
|
x_{n-2} & y_{n-2} & z_{n-2} \\
|
|
x_{n-1} & y_{n-1} & z_{n-1}
|
|
\end{bmatrix},
|
|
\]
|
|
!et
|
|
|
|
which in turn is converted into into the $3\times 3$ covariance matrix
|
|
$\hat{\Sigma}$ via the Numpy function _np.cov()_. We note that we can also calculate
|
|
the mean value of each set of samples $\hat{x}$ etc using the Numpy
|
|
function _np.mean(x)_. We can also extract the eigenvalues of the
|
|
covariance matrix through the _np.linalg.eig()_ function.
|
|
|
|
!bc pycod
|
|
# Importing various packages
|
|
import numpy as np
|
|
|
|
n = 100
|
|
x = np.random.normal(size=n)
|
|
print(np.mean(x))
|
|
y = 4+3*x+np.random.normal(size=n)
|
|
print(np.mean(y))
|
|
z = x**3+np.random.normal(size=n)
|
|
print(np.mean(z))
|
|
W = np.vstack((x, y, z))
|
|
Sigma = np.cov(W)
|
|
print(Sigma)
|
|
Eigvals, Eigvecs = np.linalg.eig(Sigma)
|
|
print(Eigvals)
|
|
!ec
|
|
|
|
|
|
!bc pycod
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from scipy import sparse
|
|
eye = np.eye(4)
|
|
print(eye)
|
|
sparse_mtx = sparse.csr_matrix(eye)
|
|
print(sparse_mtx)
|
|
x = np.linspace(-10,10,100)
|
|
y = np.sin(x)
|
|
plt.plot(x,y,marker='x')
|
|
plt.show()
|
|
!ec
|
|
|
|
|
|
===== Meet the Pandas =====
|
|
|
|
|
|
FIGURE: [fig/pandas.jpg, width=600 frac=0.8]
|
|
|
|
Another useful Python package is
|
|
"pandas":"https://pandas.pydata.org/", which is an open source library
|
|
providing high-performance, easy-to-use data structures and data
|
|
analysis tools for Python. _pandas_ stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
|
|
_pandas_ has two major classes, the _DataFrame_ class with two-dimensional data objects and tabular data organized in columns and the class _Series_ with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below.
|
|
_pandas_ allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.
|
|
|
|
The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of _pandas_, in particular in connection with classification of data.
|
|
|
|
!bc pycod
|
|
import pandas as pd
|
|
from IPython.display import display
|
|
data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
|
|
'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
|
|
'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
|
|
'Date of Birth T.A.': [2968, 2890, 2931, 2980]
|
|
}
|
|
data_pandas = pd.DataFrame(data)
|
|
display(data_pandas)
|
|
!ec
|
|
|
|
In the above we have imported _pandas_ with the shorthand _pd_, the latter has become the standard way we import _pandas_. We make then a list of various variables
|
|
and reorganize the aboves lists into a _DataFrame_ and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.
|
|
Displaying these results, we see that the indices are given by the default numbers from zero to three.
|
|
_pandas_ is extremely flexible and we can easily change the above indices by defining a new type of indexing as
|
|
!bc pycod
|
|
data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
|
|
display(data_pandas)
|
|
!ec
|
|
Thereafter we display the content of the row which begins with the index _Aragorn_
|
|
!bc pycod
|
|
display(data_pandas.loc['Aragorn'])
|
|
!ec
|
|
|
|
We can easily append data to this, for example
|
|
!bc pycod
|
|
new_hobbit = {'First Name': ["Peregrin"],
|
|
'Last Name': ["Took"],
|
|
'Place of birth': ["Shire"],
|
|
'Date of Birth T.A.': [2990]
|
|
}
|
|
data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
|
|
display(data_pandas)
|
|
!ec
|
|
|
|
|
|
Here are other examples where we use the _DataFrame_ functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
|
|
of dimensionality $10\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.
|
|
!bc pycod
|
|
import numpy as np
|
|
import pandas as pd
|
|
from IPython.display import display
|
|
np.random.seed(100)
|
|
# setting up a 10 x 5 matrix
|
|
rows = 10
|
|
cols = 5
|
|
a = np.random.randn(rows,cols)
|
|
df = pd.DataFrame(a)
|
|
display(df)
|
|
print(df.mean())
|
|
print(df.std())
|
|
display(df**2)
|
|
!ec
|
|
|
|
Thereafter we can select specific columns only and plot final results
|
|
!bc pycod
|
|
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
|
|
df.index = np.arange(10)
|
|
|
|
display(df)
|
|
print(df['Second'].mean() )
|
|
print(df.info())
|
|
print(df.describe())
|
|
|
|
from pylab import plt, mpl
|
|
plt.style.use('seaborn')
|
|
mpl.rcParams['font.family'] = 'serif'
|
|
|
|
df.cumsum().plot(lw=2.0, figsize=(10,6))
|
|
plt.show()
|
|
|
|
|
|
df.plot.bar(figsize=(10,6), rot=15)
|
|
plt.show()
|
|
!ec
|
|
We can produce a $4\times 4$ matrix
|
|
!bc pycod
|
|
b = np.arange(16).reshape((4,4))
|
|
print(b)
|
|
df1 = pd.DataFrame(b)
|
|
print(df1)
|
|
!ec
|
|
and many other operations.
|
|
|
|
The _Series_ class is another important class included in
|
|
_pandas_. You can view it as a specialization of _DataFrame_ but where
|
|
we have just a single column of data. It shares many of the same features as _DataFrame. As with _DataFrame_,
|
|
most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.
|
|
As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.
|
|
For multidimensional arrays, we recommend strongly "xarray":"http://xarray.pydata.org/en/stable/". _xarray_ has much of the same flexibility as _pandas_, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both _pandas_ and _xarray_.
|
|
|
|
|
|
|
|
===== Reading Data and fitting =====
|
|
|
|
In order to study various Machine Learning algorithms, we need to
|
|
access data. Acccessing data is an essential step in all machine
|
|
learning algorithms. In particular, setting up the so-called _design
|
|
matrix_ (to be defined below) is often the first element we need in
|
|
order to perform our calculations. To set up the design matrix means
|
|
reading (and later, when the calculations are done, writing) data
|
|
in various formats, The formats span from reading files from disk,
|
|
loading data from databases and interacting with online sources
|
|
like web application programming interfaces (APIs).
|
|
|
|
In handling various input formats, as discussed above, we will mainly stay with _pandas_,
|
|
a Python package which allows us, in a seamless and painless way, to
|
|
deal with a multitude of formats, from standard _csv_ (comma separated
|
|
values) files, via _excel_, _html_ to _hdf5_ formats. With _pandas_
|
|
and the _DataFrame_ and _Series_ functionalities we are able to convert text data
|
|
into the calculational formats we need for a specific algorithm. And our code is going to be
|
|
pretty close the basic mathematical expressions.
|
|
|
|
Our first data set is going to be a classic from nuclear physics, namely all
|
|
available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set.
|
|
|
|
We will show some of the
|
|
strengths of packages like _Scikit-Learn_ in fitting nuclear binding energies to
|
|
specific functions using linear regression first. Then, as a teaser, we will show you how
|
|
you can easily implement other algorithms like decision trees and random forests and neural networks.
|
|
|
|
But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,
|
|
(don't be offended) fitting straight lines!
|
|
|
|
|
|
=== Simple linear regression model using _scikit-learn_ ===
|
|
|
|
We start with perhaps our simplest possible example, using _Scikit-Learn_ to perform linear regression analysis on a data set produced by us.
|
|
|
|
What follows is a simple Python code where we have defined a function
|
|
$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries.
|
|
The numbers in the vector $\hat{x}$ are given
|
|
by random numbers generated with a uniform distribution with entries
|
|
$x_i \in [0,1]$ (more about probability distribution functions
|
|
later). These values are then used to define a function $y(x)$
|
|
(tabulated again as a vector) with a linear dependence on $x$ plus a
|
|
random noise added via the normal distribution.
|
|
|
|
|
|
The Numpy functions are imported used the _import numpy as np_
|
|
statement and the random number generator for the uniform distribution
|
|
is called using the function _np.random.rand()_, where we specificy
|
|
that we want $100$ random variables. Using Numpy we define
|
|
automatically an array with the specified number of elements, $100$ in
|
|
our case. With the Numpy function _randn()_ we can compute random
|
|
numbers with the normal distribution (mean value $\mu$ equal to zero and
|
|
variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear
|
|
dependence as function of $x$
|
|
|
|
!bt
|
|
\[
|
|
y = 2x+N(0,1),
|
|
\]
|
|
!et
|
|
|
|
where $N(0,1)$ represents random numbers generated by the normal
|
|
distribution. From _Scikit-Learn_ we import then the
|
|
_LinearRegression_ functionality and make a prediction $\tilde{y} =
|
|
\alpha + \beta x$ using the function _fit(x,y)_. We call the set of
|
|
data $(\hat{x},\hat{y})$ for our training data. The Python package
|
|
_scikit-learn_ has also a functionality which extracts the above
|
|
fitting parameters $\alpha$ and $\beta$ (see below). Later we will
|
|
distinguish between training data and test data.
|
|
|
|
For plotting we use the Python package
|
|
"matplotlib":"https://matplotlib.org/" which produces publication
|
|
quality figures. Feel free to explore the extensive
|
|
"gallery":"https://matplotlib.org/gallery/index.html" of examples. In
|
|
this example we plot our original values of $x$ and $y$ as well as the
|
|
prediction _ypredict_ ($\tilde{y}$), which attempts at fitting our
|
|
data with a straight line.
|
|
|
|
The Python code follows here.
|
|
!bc pycod
|
|
# Importing various packages
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from sklearn.linear_model import LinearRegression
|
|
|
|
x = np.random.rand(100,1)
|
|
y = 2*x+np.random.randn(100,1)
|
|
linreg = LinearRegression()
|
|
linreg.fit(x,y)
|
|
xnew = np.array([[0],[1]])
|
|
ypredict = linreg.predict(xnew)
|
|
|
|
plt.plot(xnew, ypredict, "r-")
|
|
plt.plot(x, y ,'ro')
|
|
plt.axis([0,1.0,0, 5.0])
|
|
plt.xlabel(r'$x$')
|
|
plt.ylabel(r'$y$')
|
|
plt.title(r'Simple Linear Regression')
|
|
plt.show()
|
|
!ec
|
|
|
|
This example serves several aims. It allows us to demonstrate several
|
|
aspects of data analysis and later machine learning algorithms. The
|
|
immediate visualization shows that our linear fit is not
|
|
impressive. It goes through the data points, but there are many
|
|
outliers which are not reproduced by our linear regression. We could
|
|
now play around with this small program and change for example the
|
|
factor in front of $x$ and the normal distribution. Try to change the
|
|
function $y$ to
|
|
|
|
!bt
|
|
\[
|
|
y = 10x+0.01 \times N(0,1),
|
|
\]
|
|
!et
|
|
|
|
where $x$ is defined as before. Does the fit look better? Indeed, by
|
|
reducing the role of the noise given by the normal distribution we see immediately that
|
|
our linear prediction seemingly reproduces better the training
|
|
set. However, this testing 'by the eye' is obviouly not satisfactory in the
|
|
long run. Here we have only defined the training data and our model, and
|
|
have not discussed a more rigorous approach to the _cost_ function.
|
|
|
|
We need more rigorous criteria in defining whether we have succeeded or
|
|
not in modeling our training data. You will be surprised to see that
|
|
many scientists seldomly venture beyond this 'by the eye' approach. A
|
|
standard approach for the *cost* function is the so-called $\chi^2$
|
|
function (a variant of the mean-squared error (MSE))
|
|
|
|
!bt
|
|
\[ \chi^2 = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
|
|
\]
|
|
!et
|
|
|
|
where $\sigma_i^2$ is the variance (to be defined later) of the entry
|
|
$y_i$. We may not know the explicit value of $\sigma_i^2$, it serves
|
|
however the aim of scaling the equations and make the cost function
|
|
dimensionless.
|
|
|
|
Minimizing the cost function is a central aspect of
|
|
our discussions to come. Finding its minima as function of the model
|
|
parameters ($\alpha$ and $\beta$ in our case) will be a recurring
|
|
theme in these series of lectures. Essentially all machine learning
|
|
algorithms we will discuss center around the minimization of the
|
|
chosen cost function. This depends in turn on our specific
|
|
model for describing the data, a typical situation in supervised
|
|
learning. Automatizing the search for the minima of the cost function is a
|
|
central ingredient in all algorithms. Typical methods which are
|
|
employed are various variants of _gradient_ methods. These will be
|
|
discussed in more detail later. Again, you'll be surprised to hear that
|
|
many practitioners minimize the above function ''by the eye', popularly dubbed as
|
|
'chi by the eye'. That is, change a parameter and see (visually and numerically) that
|
|
the $\chi^2$ function becomes smaller.
|
|
|
|
There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define
|
|
the relative error (why would we prefer the MSE instead of the relative error?) as
|
|
|
|
!bt
|
|
\[
|
|
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
|
|
\]
|
|
!et
|
|
We can modify easily the above Python code and plot the relative error instead
|
|
!bc pycod
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from sklearn.linear_model import LinearRegression
|
|
|
|
x = np.random.rand(100,1)
|
|
y = 5*x+0.01*np.random.randn(100,1)
|
|
linreg = LinearRegression()
|
|
linreg.fit(x,y)
|
|
ypredict = linreg.predict(x)
|
|
|
|
plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
|
|
plt.axis([0,1.0,0.0, 0.5])
|
|
plt.xlabel(r'$x$')
|
|
plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
|
|
plt.title(r'Relative error')
|
|
plt.show()
|
|
!ec
|
|
|
|
Depending on the parameter in front of the normal distribution, we may
|
|
have a small or larger relative error. Try to play around with
|
|
different training data sets and study (graphically) the value of the
|
|
relative error.
|
|
|
|
As mentioned above, _Scikit-Learn_ has an impressive functionality.
|
|
We can for example extract the values of $\alpha$ and $\beta$ and
|
|
their error estimates, or the variance and standard deviation and many
|
|
other properties from the statistical data analysis.
|
|
|
|
Here we show an
|
|
example of the functionality of _Scikit-Learn_.
|
|
!bc pycod
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from sklearn.linear_model import LinearRegression
|
|
from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
|
|
|
|
x = np.random.rand(100,1)
|
|
y = 2.0+ 5*x+0.5*np.random.randn(100,1)
|
|
linreg = LinearRegression()
|
|
linreg.fit(x,y)
|
|
ypredict = linreg.predict(x)
|
|
print('The intercept alpha: \n', linreg.intercept_)
|
|
print('Coefficient beta : \n', linreg.coef_)
|
|
# The mean squared error
|
|
print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
|
|
# Explained variance score: 1 is perfect prediction
|
|
print('Variance score: %.2f' % r2_score(y, ypredict))
|
|
# Mean squared log error
|
|
print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
|
|
# Mean absolute error
|
|
print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
|
|
plt.plot(x, ypredict, "r-")
|
|
plt.plot(x, y ,'ro')
|
|
plt.axis([0.0,1.0,1.5, 7.0])
|
|
plt.xlabel(r'$x$')
|
|
plt.ylabel(r'$y$')
|
|
plt.title(r'Linear Regression fit ')
|
|
plt.show()
|
|
|
|
!ec
|
|
The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields
|
|
$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
|
|
!bt
|
|
\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
|
\]
|
|
!et
|
|
|
|
The smaller the value, the better the fit. Ideally we would like to
|
|
have an MSE equal zero. The attentive reader has probably recognized
|
|
this function as being similar to the $\chi^2$ function defined above.
|
|
|
|
The _r2score_ function computes $R^2$, the coefficient of
|
|
determination. It provides a measure of how well future samples are
|
|
likely to be predicted by the model. Best possible score is 1.0 and it
|
|
can be negative (because the model can be arbitrarily worse). A
|
|
constant model that always predicts the expected value of $\hat{y}$,
|
|
disregarding the input features, would get a $R^2$ score of $0.0$.
|
|
|
|
If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
|
|
!bt
|
|
\[
|
|
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
|
|
\]
|
|
!et
|
|
where we have defined the mean value of $\hat{y}$ as
|
|
!bt
|
|
\[
|
|
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
|
\]
|
|
!et
|
|
Another quantity taht we will meet again in our discussions of regression analysis is
|
|
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.
|
|
The MAE is defined as follows
|
|
!bt
|
|
\[
|
|
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
|
|
\]
|
|
!et
|
|
Finally we present the
|
|
squared logarithmic (quadratic) error
|
|
!bt
|
|
\[
|
|
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
|
|
\]
|
|
!et
|
|
|
|
where $\log_e (x)$ stands for the natural logarithm of $x$. This error
|
|
estimate is best to use when targets having exponential growth, such
|
|
as population counts, average sales of a commodity over a span of
|
|
years etc.
|
|
|
|
We will discuss in more
|
|
detail these and other functions in the various lectures. We conclude this part with another example. Instead of
|
|
a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn.
|
|
|
|
!bc pycod
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
|
|
import random
|
|
from sklearn.linear_model import Ridge
|
|
from sklearn.preprocessing import PolynomialFeatures
|
|
from sklearn.pipeline import make_pipeline
|
|
from sklearn.linear_model import LinearRegression
|
|
|
|
x=np.linspace(0.02,0.98,200)
|
|
noise = np.asarray(random.sample((range(200)),200))
|
|
y=x**3*noise
|
|
yn=x**3*100
|
|
poly3 = PolynomialFeatures(degree=3)
|
|
X = poly3.fit_transform(x[:,np.newaxis])
|
|
clf3 = LinearRegression()
|
|
clf3.fit(X,y)
|
|
|
|
Xplot=poly3.fit_transform(x[:,np.newaxis])
|
|
poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
|
|
plt.plot(x,yn, color='red', label="True Cubic")
|
|
plt.scatter(x, y, label='Data', color='orange', s=15)
|
|
plt.legend()
|
|
plt.show()
|
|
|
|
def error(a):
|
|
for i in y:
|
|
err=(y-yn)/yn
|
|
return abs(np.sum(err))/len(err)
|
|
|
|
print (error(y))
|
|
!ec
|
|
|
|
|
|
|
|
|
|
=== To our real data: nuclear binding energies. Brief reminder on masses and binding energies ===
|
|
|
|
Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding
|
|
energies. A basic quantity which can be measured for the ground
|
|
states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with
|
|
atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).
|
|
|
|
Atomic masses are usually tabulated in terms of the mass excess defined by
|
|
!bt
|
|
\[
|
|
\Delta M(N, Z) = M(N, Z) - uA,
|
|
\]
|
|
!et
|
|
where $u$ is the Atomic Mass Unit
|
|
!bt
|
|
\[
|
|
u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
|
|
\]
|
|
!et
|
|
The nucleon masses are
|
|
!bt
|
|
\[
|
|
m_p = 1.00727646693(9)u,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
|
|
\]
|
|
!et
|
|
|
|
In the "2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu":"http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf"
|
|
there are data on masses and decays of 3437 nuclei.
|
|
|
|
The nuclear binding energy is defined as the energy required to break
|
|
up a given nucleus into its constituent parts of $N$ neutrons and $Z$
|
|
protons. In terms of the atomic masses $M(N, Z)$ the binding energy is
|
|
defined by
|
|
|
|
|
|
!bt
|
|
\[
|
|
BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
|
|
\]
|
|
!et
|
|
where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.
|
|
In terms of the mass excess the binding energy is given by
|
|
!bt
|
|
\[
|
|
BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
|
|
\]
|
|
!et
|
|
where $\Delta_H c^2 = 7.2890$ MeV and $\Delta_n c^2 = 8.0713$ MeV.
|
|
|
|
|
|
A popular and physically intuitive model which can be used to parametrize
|
|
the experimental binding energies as function of $A$, is the so-called
|
|
_liquid drop model_. The ansatz is based on the following expression
|
|
|
|
!bt
|
|
\[
|
|
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
|
|
\]
|
|
!et
|
|
|
|
where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit
|
|
to the experimental data.
|
|
|
|
|
|
|
|
|
|
To arrive at the above expression we have assumed that we can make the following assumptions:
|
|
|
|
* There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.
|
|
|
|
* There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
|
|
|
|
|
|
* There is a Coulomb energy term $a_3\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding.
|
|
|
|
* There is an asymmetry term $a_4\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.
|
|
|
|
We could also add a so-called pairing term, which is a correction term that
|
|
arises from the tendency of proton pairs and neutron pairs to
|
|
occur. An even number of particles is more stable than an odd number.
|
|
|
|
|
|
=== Organizing our data ===
|
|
|
|
Let us start with reading and organizing our data.
|
|
We start with the compilation of masses and binding energies from 2016.
|
|
After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.
|
|
|
|
|
|
We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of _scikit-learn_.
|
|
!bc pycod
|
|
# Common imports
|
|
import numpy as np
|
|
import pandas as pd
|
|
import matplotlib.pyplot as plt
|
|
import sklearn.linear_model as skl
|
|
from sklearn.model_selection import train_test_split
|
|
from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
|
|
import os
|
|
|
|
# Where to save the figures and data files
|
|
PROJECT_ROOT_DIR = "Results"
|
|
FIGURE_ID = "Results/FigureFiles"
|
|
DATA_ID = "DataFiles/"
|
|
|
|
if not os.path.exists(PROJECT_ROOT_DIR):
|
|
os.mkdir(PROJECT_ROOT_DIR)
|
|
|
|
if not os.path.exists(FIGURE_ID):
|
|
os.makedirs(FIGURE_ID)
|
|
|
|
if not os.path.exists(DATA_ID):
|
|
os.makedirs(DATA_ID)
|
|
|
|
def image_path(fig_id):
|
|
return os.path.join(FIGURE_ID, fig_id)
|
|
|
|
def data_path(dat_id):
|
|
return os.path.join(DATA_ID, dat_id)
|
|
|
|
def save_fig(fig_id):
|
|
plt.savefig(image_path(fig_id) + ".png", format='png')
|
|
|
|
infile = open(data_path("MassEval2016.dat"),'r')
|
|
!ec
|
|
|
|
|
|
Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various _matplotlib_ commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.
|
|
!bc pycod
|
|
from pylab import plt, mpl
|
|
plt.style.use('seaborn')
|
|
mpl.rcParams['font.family'] = 'serif'
|
|
|
|
def MakePlot(x,y, styles, labels, axlabels):
|
|
plt.figure(figsize=(10,6))
|
|
for i in range(len(x)):
|
|
plt.plot(x[i], y[i], styles[i], label = labels[i])
|
|
plt.xlabel(axlabels[0])
|
|
plt.ylabel(axlabels[1])
|
|
plt.legend(loc=0)
|
|
!ec
|
|
|
|
Our next step is to read the data on experimental binding energies and
|
|
reorganize them as functions of the mass number $A$, the number of
|
|
protons $Z$ and neutrons $N$ using _pandas_. Before we do this it is
|
|
always useful (unless you have a binary file or other types of compressed
|
|
data) to actually open the file and simply take a look at it!
|
|
|
|
|
|
In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with _pandas_. The file begins with some basic format information.
|
|
!bc pycod
|
|
"""
|
|
This is taken from the data file of the mass 2016 evaluation.
|
|
All files are 3436 lines long with 124 character per line.
|
|
Headers are 39 lines long.
|
|
col 1 : Fortran character control: 1 = page feed 0 = line feed
|
|
format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5
|
|
These formats are reflected in the pandas widths variable below, see the statement
|
|
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
|
|
Pandas has also a variable header, with length 39 in this case.
|
|
"""
|
|
!ec
|
|
|
|
The data we are interested in are in columns 2, 3, 4 and 11, giving us
|
|
the number of neutrons, protons, mass numbers and binding energies,
|
|
respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will
|
|
covert them into the _pandas_ DataFrame structure.
|
|
|
|
!bc pycod
|
|
# Read the experimental data with Pandas
|
|
Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
|
|
names=('N', 'Z', 'A', 'Element', 'Ebinding'),
|
|
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
|
|
header=39,
|
|
index_col=False)
|
|
|
|
# Extrapolated values are indicated by '#' in place of the decimal place, so
|
|
# the Ebinding column won't be numeric. Coerce to float and drop these entries.
|
|
Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
|
|
Masses = Masses.dropna()
|
|
# Convert from keV to MeV.
|
|
Masses['Ebinding'] /= 1000
|
|
|
|
# Group the DataFrame by nucleon number, A.
|
|
Masses = Masses.groupby('A')
|
|
# Find the rows of the grouped DataFrame with the maximum binding energy.
|
|
Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
|
|
!ec
|
|
|
|
We have now read in the data, grouped them according to the variables we are interested in.
|
|
We see how easy it is to reorganize the data using _pandas_. If we
|
|
were to do these operations in C/C++ or Fortran, we would have had to
|
|
write various functions/subroutines which perform the above
|
|
reorganizations for us. Having reorganized the data, we can now start
|
|
to make some simple fits using both the functionalities in _numpy_ and
|
|
_Scikit-Learn_ afterwards.
|
|
|
|
Now we define five variables which contain
|
|
the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves.
|
|
!bc pycod
|
|
A = Masses['A']
|
|
Z = Masses['Z']
|
|
N = Masses['N']
|
|
Element = Masses['Element']
|
|
Energies = Masses['Ebinding']
|
|
print(Masses)
|
|
!ec
|
|
The next step, and we will define this mathematically later, is to set up the so-called _design matrix_. We will throughout call this matrix $\bm{X}$.
|
|
It has dimensionality $p\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit.
|
|
!bc pycod
|
|
# Now we set up the design matrix X
|
|
X = np.zeros((len(A),5))
|
|
X[:,0] = 1
|
|
X[:,1] = A
|
|
X[:,2] = A**(2.0/3.0)
|
|
X[:,3] = A**(-1.0/3.0)
|
|
X[:,4] = A**(-1.0)
|
|
!ec
|
|
With _scikitlearn_ we are now ready to use linear regression and fit our data.
|
|
!bc pycod
|
|
clf = skl.LinearRegression().fit(X, Energies)
|
|
fity = clf.predict(X)
|
|
!ec
|
|
Pretty simple!
|
|
Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.
|
|
!bc pycod
|
|
# The mean squared error
|
|
print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
|
|
# Explained variance score: 1 is perfect prediction
|
|
print('Variance score: %.2f' % r2_score(Energies, fity))
|
|
# Mean absolute error
|
|
print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
|
|
print(clf.coef_, clf.intercept_)
|
|
|
|
Masses['Eapprox'] = fity
|
|
# Generate a plot comparing the experimental with the fitted values values.
|
|
fig, ax = plt.subplots()
|
|
ax.set_xlabel(r'$A = N + Z$')
|
|
ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
|
|
ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
|
|
label='Ame2016')
|
|
ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
|
|
label='Fit')
|
|
ax.legend()
|
|
save_fig("Masses2016")
|
|
plt.show()
|
|
!ec
|
|
|
|
|
|
=== Seeing the wood for the trees ===
|
|
|
|
As a teaser, let us now see how we can do this with decision trees using _scikit-learn_. Later we will switch to so-called _random forests_!
|
|
|
|
|
|
!bc pycod
|
|
|
|
#Decision Tree Regression
|
|
from sklearn.tree import DecisionTreeRegressor
|
|
regr_1=DecisionTreeRegressor(max_depth=5)
|
|
regr_2=DecisionTreeRegressor(max_depth=7)
|
|
regr_3=DecisionTreeRegressor(max_depth=9)
|
|
regr_1.fit(X, Energies)
|
|
regr_2.fit(X, Energies)
|
|
regr_3.fit(X, Energies)
|
|
|
|
|
|
y_1 = regr_1.predict(X)
|
|
y_2 = regr_2.predict(X)
|
|
y_3=regr_3.predict(X)
|
|
Masses['Eapprox'] = y_3
|
|
# Plot the results
|
|
plt.figure()
|
|
plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
|
|
plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
|
|
plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
|
|
plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
|
|
|
|
plt.xlabel("$A$")
|
|
plt.ylabel("$E$[MeV]")
|
|
plt.title("Decision Tree Regression")
|
|
plt.legend()
|
|
save_fig("Masses2016Trees")
|
|
plt.show()
|
|
print(Masses)
|
|
print(np.mean( (Energies-y_1)**2))
|
|
!ec
|
|
|
|
|
|
=== And what about using neural networks? ===
|
|
The _seaborn_ package allows us to visualize data in an efficient way. Note that we use _scikit-learn_'s multi-layer perceptron (or feed forward neural network)
|
|
functionality.
|
|
!bc pycod
|
|
from sklearn.neural_network import MLPRegressor
|
|
from sklearn.metrics import accuracy_score
|
|
import seaborn as sns
|
|
|
|
X_train = X
|
|
Y_train = Energies
|
|
n_hidden_neurons = 100
|
|
epochs = 100
|
|
# store models for later use
|
|
eta_vals = np.logspace(-5, 1, 7)
|
|
lmbd_vals = np.logspace(-5, 1, 7)
|
|
# store the models for later use
|
|
DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
|
|
train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
|
|
sns.set()
|
|
for i, eta in enumerate(eta_vals):
|
|
for j, lmbd in enumerate(lmbd_vals):
|
|
dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
|
|
alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
|
|
dnn.fit(X_train, Y_train)
|
|
DNN_scikit[i][j] = dnn
|
|
train_accuracy[i][j] = dnn.score(X_train, Y_train)
|
|
|
|
fig, ax = plt.subplots(figsize = (10, 10))
|
|
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
|
|
ax.set_title("Training Accuracy")
|
|
ax.set_ylabel("$\eta$")
|
|
ax.set_xlabel("$\lambda$")
|
|
plt.show()
|
|
|
|
|
|
|
|
!ec
|
|
|
|
|
|
|
|
|
|
|
|
|
|
===== A first summary =====
|
|
|
|
The aim behind these introductory words was to present to you various
|
|
Python libraries and their functionalities, in particular libraries like
|
|
_numpy_, _pandas_, _xarray_ and _matplotlib_ and other that make our life much easier
|
|
in handling various data sets and visualizing data.
|
|
|
|
Furthermore,
|
|
_Scikit-Learn_ allows us with few lines of code to implement popular
|
|
Machine Learning algorithms for supervised learning. Later we will meet _Tensorflow_, a powerful library for deep learning.
|
|
Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
======= Why Linear Regression (aka Ordinary Least Squares and family) =======
|
|
|
|
Fitting a continuous function with linear parameterization in terms of the parameters $\bm{\beta}$.
|
|
* Method of choice for fitting a continuous function!
|
|
* Gives an excellent introduction to central Machine Learning features with _understandable pedagogical_ links to other methods like _Neural Networks_, _Support Vector Machines_ etc
|
|
* Analytical expression for the fitting parameters $\bm{\beta}$
|
|
* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more
|
|
* Analytical relation with probabilistic interpretations
|
|
* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics
|
|
* Easy to code! And links well with classification problems and logistic regression and neural networks
|
|
* Allows for _easy_ hands-on understanding of gradient descent methods
|
|
* and many more features
|
|
|
|
For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
|
|
Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended.
|
|
|
|
|
|
=== Regression analysis, overarching aims ===
|
|
|
|
Regression modeling deals with the description of the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\bm{x} =[x_0, x_1,\dots, x_{n-1}]^T$.
|
|
The first variable is called the _dependent_, the _outcome_ or the _response_ variable while the set of variables $\bm{x}$ is called the independent variable, or the predictor variable or the explanatory variable.
|
|
|
|
A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$, that is the conditional distribution for $\bm{y}$ with a given $\bm{x}$. The estimation of $p(\bm{y}\vert \bm{x})$ is made using a data set with
|
|
* $n$ cases $i = 0, 1, 2, \dots, n-1$
|
|
* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \dots, n-1$
|
|
* $p$ so-called explanatory (independent or predictor) variables $\bm{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]$ with $i = 0, 1, 2, \dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples.
|
|
The goal of the regression analysis is to extract/exploit relationship between $\bm{y}$ and $\bm{X}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
|
|
|
|
|
|
Consider an experiment in which $p$ characteristics of $n$ samples are
|
|
measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix
|
|
$\mathbf{X}$.
|
|
|
|
The matrix $\mathbf{X}$ is called the *design
|
|
matrix*. Additional information of the samples is available in the
|
|
form of $\bm{y}$ (also as above). The variable $\bm{y}$ is
|
|
generally referred to as the *response variable*. The aim of
|
|
regression analysis is to explain $\bm{y}$ in terms of
|
|
$\bm{X}$ through a functional relationship like $y_i =
|
|
f(\mathbf{X}_{i,\ast})$. When no prior knowledge on the form of
|
|
$f(\cdot)$ is available, it is common to assume a linear relationship
|
|
between $\bm{X}$ and $\bm{y}$. This assumption gives rise to
|
|
the *linear regression model* where $\bm{\beta} = [\beta_0, \ldots,
|
|
\beta_{p-1}]^{T}$ are the *regression parameters*.
|
|
|
|
Linear regression gives us a set of analytical equations for the parameters $\beta_j$.
|
|
|
|
|
|
=== Examples ===
|
|
|
|
In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\bm{y}$,
|
|
consider the model we discussed for describing nuclear binding energies.
|
|
|
|
There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.
|
|
Assuming
|
|
!bt
|
|
\[
|
|
BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},
|
|
\]
|
|
!et
|
|
we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms.
|
|
This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a
|
|
$p\times n$ matrix $\bm{X}$.
|
|
|
|
Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the
|
|
so-called "credit card default data from Taiwan":"https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub". The data set contains data on $n=30000$ credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are $24$ such predictors or attributes leading to a design matrix of dimensionality $24 \times 30000$
|
|
|
|
|
|
===== General linear models =====
|
|
|
|
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\bm{y}=[y_0,y_1,\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\bm{x}=[x_0,x_1,\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function.
|
|
|
|
Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is
|
|
!bt
|
|
\[
|
|
y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i,
|
|
\]
|
|
!et
|
|
where $\epsilon_i$ is the error in our approximation.
|
|
|
|
|
|
For every set of values $y_i,x_i$ we have thus the corresponding set of equations
|
|
!bt
|
|
\begin{align*}
|
|
y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\
|
|
y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\
|
|
y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
!et
|
|
|
|
|
|
Defining the vectors
|
|
!bt
|
|
\[
|
|
\bm{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
\bm{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
\bm{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T,
|
|
\]
|
|
!et
|
|
and the design matrix
|
|
!bt
|
|
\[
|
|
\bm{X}=
|
|
\begin{bmatrix}
|
|
1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\
|
|
1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\
|
|
1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\
|
|
\dots& \dots &\dots& \dots & \dots &\dots\\
|
|
1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\
|
|
\end{bmatrix}
|
|
\]
|
|
!et
|
|
we can rewrite our equations as
|
|
!bt
|
|
\[
|
|
\bm{y} = \bm{X}\bm{\beta}+\bm{\epsilon}.
|
|
\]
|
|
!et
|
|
The above design matrix is called a "Vandermonde matrix":"https://en.wikipedia.org/wiki/Vandermonde_matrix".
|
|
|
|
|
|
|
|
|
|
===== Generalizing the fitting procedure as a linear algebra problem =====
|
|
|
|
We are obviously not limited to the above polynomial expansions. We
|
|
could replace the various powers of $x$ with elements of Fourier
|
|
series or instead of $x_i^j$ we could have $\cos{(j x_i)}$ or $\sin{(j
|
|
x_i)}$, or time series or other orthogonal functions. For every set
|
|
of values $y_i,x_i$ we can then generalize the equations to
|
|
|
|
!bt
|
|
\begin{align*}
|
|
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
|
|
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
|
|
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\
|
|
\dots & \dots \\
|
|
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
!et
|
|
|
|
_Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!_
|
|
|
|
We redefine in turn the matrix $\bm{X}$ as
|
|
!bt
|
|
\[
|
|
\bm{X}=
|
|
\begin{bmatrix}
|
|
x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\
|
|
x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\
|
|
x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\
|
|
\dots& \dots &\dots& \dots & \dots &\dots\\
|
|
x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\
|
|
\end{bmatrix}
|
|
\]
|
|
!et
|
|
and without loss of generality we rewrite again our equations as
|
|
!bt
|
|
\[
|
|
\bm{y} = \bm{X}\bm{\beta}+\bm{\epsilon}.
|
|
\]
|
|
!et
|
|
The left-hand side of this equation is kwown. Our error vector $\bm{\epsilon}$ and the parameter vector $\bm{\beta}$ are our unknow quantities. How can we obtain the optimal set of $\beta_i$ values?
|
|
|
|
We have defined the matrix $\bm{X}$ via the equations
|
|
!bt
|
|
\begin{align*}
|
|
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
|
|
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
|
|
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\
|
|
\dots & \dots \\
|
|
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
!et
|
|
|
|
As we noted above, we stayed with a system with the design matrix
|
|
$\bm{X}\in {\mathbb{R}}^{n\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define
|
|
our matrix as $\bm{X}\in {\mathbb{R}}^{n\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements.
|
|
|
|
|
|
===== Our model for the nuclear binding energies =====
|
|
|
|
In our introductory notes we looked at the so-called "liguid drop model":"https://en.wikipedia.org/wiki/Semi-empirical_mass_formula". Let us remind ourselves about what we did by looking at the code.
|
|
|
|
We restate the parts of the code we are most interested in.
|
|
!bc pycod
|
|
# Common imports
|
|
import numpy as np
|
|
import pandas as pd
|
|
import matplotlib.pyplot as plt
|
|
from IPython.display import display
|
|
import os
|
|
|
|
# Where to save the figures and data files
|
|
PROJECT_ROOT_DIR = "Results"
|
|
FIGURE_ID = "Results/FigureFiles"
|
|
DATA_ID = "DataFiles/"
|
|
|
|
if not os.path.exists(PROJECT_ROOT_DIR):
|
|
os.mkdir(PROJECT_ROOT_DIR)
|
|
|
|
if not os.path.exists(FIGURE_ID):
|
|
os.makedirs(FIGURE_ID)
|
|
|
|
if not os.path.exists(DATA_ID):
|
|
os.makedirs(DATA_ID)
|
|
|
|
def image_path(fig_id):
|
|
return os.path.join(FIGURE_ID, fig_id)
|
|
|
|
def data_path(dat_id):
|
|
return os.path.join(DATA_ID, dat_id)
|
|
|
|
def save_fig(fig_id):
|
|
plt.savefig(image_path(fig_id) + ".png", format='png')
|
|
|
|
infile = open(data_path("MassEval2016.dat"),'r')
|
|
|
|
|
|
# Read the experimental data with Pandas
|
|
Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
|
|
names=('N', 'Z', 'A', 'Element', 'Ebinding'),
|
|
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
|
|
header=39,
|
|
index_col=False)
|
|
|
|
# Extrapolated values are indicated by '#' in place of the decimal place, so
|
|
# the Ebinding column won't be numeric. Coerce to float and drop these entries.
|
|
Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
|
|
Masses = Masses.dropna()
|
|
# Convert from keV to MeV.
|
|
Masses['Ebinding'] /= 1000
|
|
|
|
# Group the DataFrame by nucleon number, A.
|
|
Masses = Masses.groupby('A')
|
|
# Find the rows of the grouped DataFrame with the maximum binding energy.
|
|
Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
|
|
A = Masses['A']
|
|
Z = Masses['Z']
|
|
N = Masses['N']
|
|
Element = Masses['Element']
|
|
Energies = Masses['Ebinding']
|
|
|
|
# Now we set up the design matrix X
|
|
X = np.zeros((len(A),5))
|
|
X[:,0] = 1
|
|
X[:,1] = A
|
|
X[:,2] = A**(2.0/3.0)
|
|
X[:,3] = A**(-1.0/3.0)
|
|
X[:,4] = A**(-1.0)
|
|
# Then nice printout using pandas
|
|
DesignMatrix = pd.DataFrame(X)
|
|
DesignMatrix.index = A
|
|
DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
|
|
display(DesignMatrix)
|
|
!ec
|
|
|
|
With $\bm{\beta}\in {\mathbb{R}}^{p\times 1}$, it means that we will hereafter write our equations for the approximation as
|
|
!bt
|
|
\[
|
|
\bm{\tilde{y}}= \bm{X}\bm{\beta},
|
|
\]
|
|
!et
|
|
throughout these lectures.
|
|
|
|
|
|
|
|
With the above we use the design matrix to define the approximation $\bm{\tilde{y}}$ via the unknown quantity $\bm{\beta}$ as
|
|
!bt
|
|
\[
|
|
\bm{\tilde{y}}= \bm{X}\bm{\beta},
|
|
\]
|
|
!et
|
|
and in order to find the optimal parameters $\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\tilde{y}_i$, namely
|
|
!bt
|
|
\[
|
|
C(\bm{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\bm{y}-\bm{\tilde{y}}\right)^T\left(\bm{y}-\bm{\tilde{y}}\right)\right\},
|
|
\]
|
|
!et
|
|
or using the matrix $\bm{X}$ and in a more compact matrix-vector notation as
|
|
!bt
|
|
\[
|
|
C(\bm{\beta})=\frac{1}{n}\left\{\left(\bm{y}-\bm{X}^T\bm{\beta}\right)^T\left(\bm{y}-\bm{X}^T\bm{\beta}\right)\right\}.
|
|
\]
|
|
!et
|
|
This function is one possible way to define the so-called cost function.
|
|
|
|
|
|
|
|
It is also common to define
|
|
the function $Q$ as
|
|
|
|
!bt
|
|
\[
|
|
C(\bm{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2,
|
|
\]
|
|
!et
|
|
since when taking the first derivative with respect to the unknown parameters $\beta$, the factor of $2$ cancels out.
|
|
!eblock
|
|
translating doconce text in book.do.txt to ipynb
|
|
*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
|
|
*** error: figure file "fig/pandas.jpg" does not exist!
|
|
translating doconce text in book.do.txt to ipynb
|
|
*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
|
|
collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
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|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
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environments in math environments.
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|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
|
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
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|
|
|
*** warning: latex envir \begin{eqnarray*} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
output in book.ipynb
|
|
*** error: file has a mako construction ${\bf \hat{J}'
|
|
but seemingly no definition in <%...%>'
|
|
(it is not a command-line given mako variable either).
|
|
However, if this is a variable in a Makefile or Bash script
|
|
run with --no_mako - and you cannot use mako and Makefile or Bash variables
|
|
in the same document!
|
|
|
|
*** error: file has a mako construction ${\bm{J}'
|
|
but seemingly no definition in <%...%>'
|
|
(it is not a command-line given mako variable either).
|
|
However, if this is a variable in a Makefile or Bash script
|
|
run with --no_mako - and you cannot use mako and Makefile or Bash variables
|
|
in the same document!
|
|
|
|
translating doconce text in book.do.txt to ipynb
|
|
*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
|
|
collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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|
Stick to \[ ... \], equation, equation*, align, or align*
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|
environments in math environments.
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|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
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Stick to \[ ... \], equation, equation*, align, or align*
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|
environments in math environments.
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|
|
|
*** warning: latex envir \begin{eqnarray*} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
output in book.ipynb
|
|
translating doconce text in book.do.txt to ipynb
|
|
*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
|
|
collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
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Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
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Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
|
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environments in math environments.
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
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Stick to \[ ... \], equation, equation*, align, or align*
|
|
environments in math environments.
|
|
|
|
*** warning: latex envir \begin{eqnarray*} does not work well in Markdown.
|
|
Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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|
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
|
|
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
|
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Stick to \[ ... \], equation, equation*, align, or align*
|
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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output in book.ipynb
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translating doconce text in book.do.txt to latex
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figure fig/pandas.jpg must have extension(s) .eps, .ps
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*** warning: need to convert from fig/pandas.jpg to fig/pandas.eps
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using ImageMagick's convert program, but the result will
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be loss of quality. Generate a proper fig/pandas.eps file (if possible).
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....image conversion: convert fig/pandas.jpg fig/pandas.eps
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output in book.p.tex
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translating doconce text in book.do.txt to ipynb
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*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
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collected all required additional files in ipynb-book-src.tar.gz which must be distributed with the notebook
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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*** warning: latex envir \begin{cases} does not work well in Markdown.
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Stick to \[ ... \], equation, equation*, align, or align*
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environments in math environments.
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Failed to remove ans_at_end environment
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Failed to remove sol_at_end environment
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output in book.ipynb
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