log-normal.lyx

#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\begin_preamble

\usepackage{amsthm}\usepackage{epsfig}\usepackage{psfrag}\usepackage{lineno}

\bibliographystyle{apalike}

%\setlength{\evensidemargin}{0in} \setlength{\oddsidemargin}{0in}
%\setlength{\topmargin}{0.0in} \setlength{\textwidth}{6.5in}
%\setlength{\textheight}{9in} \setlength{\topskip}{0in}
%\setlength{\headheight}{0in} \setlength{\headsep}{0in}
\end_preamble
\use_default_options false
\begin_modules
knitr
\end_modules
\maintain_unincluded_children false
\language english
\language_package none
\inputencoding auto
\fontencoding default
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100

\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize 12
\spacing single
\use_hyperref false
\papersize default
\use_geometry true
\use_amsmath 2
\use_esint 1
\use_mhchem 0
\use_mathdots 0
\cite_engine natbib_authoryear
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\use_refstyle 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Title
Log-normal
\end_layout

\begin_layout Author
Jeff Laake
\end_layout

\begin_layout Standard
The log-normal distribution can be a bit confusing.
 Even the name is confusing because it is the log of a log-normal that is
 normally distributed.
 Let 
\shape italic
y
\shape default
 be log-normally distributed, then 
\shape italic

\begin_inset Formula $x=\ln(y)$
\end_inset

 
\shape default
is normally distributed.
 Likewise, if x is normally distributed then 
\shape italic

\begin_inset Formula $y=e^{x}$
\end_inset


\shape default
 is log-normally distributed.
 I find it helps if you use the same symbol for the mean and standard deviation
 for each distribution with a subscript.
 Let 
\begin_inset Formula $\mu_{x}$
\end_inset

 and 
\begin_inset Formula $\sigma_{x}$
\end_inset

 be the mean and standard deviation of the normal distribution and 
\begin_inset Formula $\mu_{y}$
\end_inset

 and 
\begin_inset Formula $\sigma_{y}$
\end_inset

 be the mean and standard deviation of the log-normal distribution.
 The following are the key relationships: 1) 
\begin_inset Formula $\mu_{y}=e^{\mu_{x}+\sigma_{x}^{2}/2}$
\end_inset

 and 2) 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none

\begin_inset Formula $CV(y)=\nicefrac{\sigma_{y}}{\mu_{y}}=\sqrt{e^{\sigma_{x}^{2}}-1}$
\end_inset

 which leads to 
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit

\begin_inset Formula $\sigma_{x}^{2}=\ln\left[1+CV^{2}(y)\right]$
\end_inset

.
 
\end_layout

\begin_layout Standard
So for your example, we are assuming that 
\begin_inset Formula $D=e^{\mu_{x}}$
\end_inset

, so 
\begin_inset Formula $\mu_{x}=\ln(D)=\ln(2)=$
\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Sexpr{round(log(2),3)}
\end_layout

\end_inset

 and 
\begin_inset Formula $\sigma_{x}=\sqrt{\ln\left[1+CV^{2}(y)\right]}=\sqrt{\ln\left[1+0.8^{2}\right]}$
\end_inset

=
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
Sexpr{round(sqrt(log(1+.8^2)),4)}
\end_layout

\end_inset

.
 
\end_layout

\begin_layout Standard
Simulation always helps.
 Below I simulate 1E7 random variates working from the log-normal to the
 normal because the former is more intuitive in terms of specifying parameters
 and trust me, it is easy to get confused.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<fig=TRUE>>=
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

# parameters
\end_layout

\begin_layout Plain Layout

muy=10000
\end_layout

\begin_layout Plain Layout

cvy=0.3
\end_layout

\begin_layout Plain Layout

sigmay=cvy*muy
\end_layout

\begin_layout Plain Layout

sigmax=sqrt(log(1+cvy^2))
\end_layout

\begin_layout Plain Layout

mux=log(muy)-sigmax^2/2
\end_layout

\begin_layout Plain Layout

mediany=exp(mux)
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

# get random values
\end_layout

\begin_layout Plain Layout

x=rnorm(1e7,mux,sigmax)
\end_layout

\begin_layout Plain Layout

y=exp(x)
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

# plot
\end_layout

\begin_layout Plain Layout

par(mfcol=c(1,2))
\end_layout

\begin_layout Plain Layout

hist(x)
\end_layout

\begin_layout Plain Layout

hist(y)
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

# relative differences - should be small  
\end_layout

\begin_layout Plain Layout

(mean(x)-mux)/mux
\end_layout

\begin_layout Plain Layout

(sqrt(var(x))-sigmax)/sigmax
\end_layout

\begin_layout Plain Layout

(mean(y)-muy)/muy
\end_layout

\begin_layout Plain Layout

(median(y)-mediany)/exp(mux)
\end_layout

\begin_layout Plain Layout

(sqrt(var(y))-sigmay)/sigmay
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\end_body
\end_document