In order to do conventional least-squares fitting to you need to determine .

Step response of NIH 3T3 fibroblasts to ionomycin

For instance, the figure above shows the fluorescent intensity of some 3T3 cells to ionomycin.   The traces are normalized to the initial values.  Notice that the cells don’t recover to the levels at which they began.  I wanted to know the time constant of the decay on the tail of the trace.

I solved this problem by performing the fit inside an iterative minimum-finding function.  First I identified the point of fastest decline and selected that as the point.  Then I iterated to find the value for which minimized the norm of the error:

for best-fit A and .

y0_fit = fminbnd( @(y0) ...
                 norm( y-y0 - exp( polyval( polyfit( x, log(y-y0), 1), x))) ,...
                 -10*y(1), y(1) );

This, along with other analysis:

For a population of cells (Showing mean and standard deviation for each test point. numbers in parentheses indicate number of cells in the sample.):

TeX source

Figure [above] shows how hazards, risks, and costs are considered together in an alternatives analysis.  A successful alternatives analysis should develop and evaluate several different approaches for solving the problem at hand.  The analysis should make conspicuous consideration of hazards, the sources of hazards, and opportunities to avoid hazards, thus ensuring a precautionary footing.  The risks presented by unavoidable hazards should be analyzed using conventional methods.  Decision analytic tools should be used to compare the resulting options and weight the relevant criteria.  The results of the analysis should be reported to the public in a manner that enables informed decision making by consumers while encouraging responsible innovation by industry.  The process should be ongoing and the analysis should be improved continuously.

TeX source

Since large-scale manufacturing processes are highly integrative, with each stage depending on the output from the prior stage, the functional requirements of a single manufacturing stage can be extremely specific and large changes can be disruptive to the manufacturing chain.  An analyst considering a single manufacturing process (limited scope) may have limited flexibility in selecting alternatives.  On the other hand, an analyst considering an entire product system (broad scope) may be able to envision a much wider array of alternatives. Figure [above] shows a generic product life cycle and compares the minimal scope for an alternatives analysis to the maximal scope, which is the entire product system.

I’m quite fond of PSTricks, a LaTeX package which allows the author to make use of PostScript’s extensive computational and graphical capabilities from within a TeX document.  My project for this week was to come up with a way to give these figures to my advisor, who works exclusively in MS Word. So what I needed was a way to export TeX figures to graphics files. Here’s what I came up with.

Essentially, I create a file which includes the \pspicture environment and all the graphics commands, which I call somepicture.fig. I then \input{somepicture.fig} into my main document when I want the picture to appear. When I then want to convert the picture into a standalone graphic, I run a script called figuretops.sh(link to file) which performs the conversion by making use of the very neat “file here” notation in bash:

cat >> filename.tex <<EOF
\documentclass{article}
\usepackage{pstricks-add}
\usepackage{calc}

\begin{document}
\thispagestyle{empty}
\color{white}
\fbox{
\color{black}
\input{}
}
\end{document}
EOF

When I run LaTeX on this file, it gets to the blank \input{} directive and asks the user for the filename. I solve this by echoing the filename into the LaTeX invocation in the bash script. I then clean up temporary files and use

convert -density 300 somefigure.eps somefigure.png

to convert the eps image to a 300-DPI image.

Here’s a source figure, the bash script, and an image it produced:

Here’s another (naively crafted: TeX source) but more interesting figure: