teaching/scientificComputing
Archived
3
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

fixed some index entries

Browse Source
This commit is contained in:
Jan Benda
2019-12-06 19:29:20 +01:00
parent 2a2e02b37e
commit 983ca0daea
6 changed files with 32 additions and 27 deletions
Download Patch File Download Diff File Expand all files Collapse all files

9
likelihood/lecture/likelihood.tex
View File

@@ -6,8 +6,9 @@
A common problem in statistics is to estimate from a probability
distribution one or more parameters $\theta$ that best describe the
data $x_1, x_2, \ldots x_n$. \enterm{Maximum likelihood estimators}
(\enterm[mle|see{Maximum likelihood estimators}]{mle},
data $x_1, x_2, \ldots x_n$. \enterm[maximum likelihood
estimator]{Maximum likelihood estimators} (\enterm[mle|see{maximum
likelihood estimator}]{mle},
\determ{Maximum-Likelihood-Sch\"atzer}) choose the parameters such
that they maximize the likelihood of the data $x_1, x_2, \ldots x_n$
to originate from the distribution.
@@ -228,7 +229,7 @@ maximized respectively.
\begin{figure}[t]
\includegraphics[width=1\textwidth]{mlepropline}
\titlecaption{\label{mleproplinefig} Maximum likelihood estimation
of the slope of line through the origin.}{The data (blue and
of the slope of a line through the origin.}{The data (blue and
left histogram) originate from a straight line $y=mx$ trough the origin
(red). The maximum-likelihood estimation of the slope $m$ of the
regression line (orange), \eqnref{mleslope}, is close to the true
@@ -257,6 +258,7 @@ respect to $\theta$ and equate it to zero:
This is an analytical expression for the estimation of the slope
$\theta$ of the regression line (\figref{mleproplinefig}).
\subsection{Linear and non-linear fits}
A gradient descent, as we have done in the previous chapter, is not
necessary for fitting the slope of a straight line, because the slope
can be directly computed via \eqnref{mleslope}. More generally, this
@@ -275,6 +277,7 @@ exponential decay
Such cases require numerical solutions for the optimization of the
cost function, e.g. the gradient descent \matlabfun{lsqcurvefit()}.
\subsection{Relation between slope and correlation coefficient}
Let us have a closer look on \eqnref{mleslope}. If the standard
deviation of the data $\sigma_i$ is the same for each data point,
i.e. $\sigma_i = \sigma_j \; \forall \; i, j$, the standard deviation drops