 # Can matplotlib draw normal curves

Hi Rich,

I'm CC-ing my reply to the matplotlib-users list, so other people can
contribute to and benefit from the discussion.

> John, Last evening I read through the user guide and
> tutorial for matplotlib. Although I've done my scientific
> plotting using Gri (a better gnuplot) for years, and more
> recently with PSTricks, the philosophy and syntax of
> matplotlib is quite easy to grasp. Congratulations on a
> masterful job!

Thanks!

> the types of curves I need to plot. A sample is attached. I
> need to be able to draw overlapping normal curves and S- and
> Z-curves at the ends of the range of x values. Can I do this
> with matplotlib, particularly by specifying end points, the
> x value where y=1.0, and the y value of the point of
> inflection?

> Plotting triangular and trapezoidal versions are easy as
> the connecting lines are straight. Doing the normal curves
> (complete and either half) is a challenge.

matplotlib's general approach is that you compute the x and y vertices
of the curves you want to plot and then pass it off to plot. So you
could write a helper function to generate these points given your
parameters of interest and then plot them. Eg for a normal pdf,
matplotlib.mlab provides such a function

from matplotlib.mlab import normpdf
import matplotlib.numerix as nx
import pylab as p

x = nx.arange(-4, 4, 0.01)
y = normpdf(x, 0, 1) # unit normal
p.plot(x,y, color='red', lw=2)
p.show()

Of course, some curves do not have closed form expressions and are not
amenable for such treatment. Some of the matplotlib backends have the
capability to draw arbitrary paths with splines (cubic and quartic)
but this functionality hasn't been exposed to the user yet. If you
need this, let me know and I can provide an interface in an upcoming
release.

I had not heard of "S curves" and "Z curves", but a little bit of
googling [1,2] suggests that the S curve is a sigmoid and the Z curve
is simply 1-sigmoid. If this is what you are looking for, there are
many simple forms for sigmoids: eg, the hill, boltzman, and arc
tangent functions. Here is an example of the boltzman function

import matplotlib.numerix as nx
import pylab as p

def boltzman(x, xmid, tau):
"""
evaluate the boltzman function with midpoint xmin and time constant tau
over x
"""
return 1. / (1. + nx.exp(-x-xmid)/tau)

x = nx.arange(-6, 6, .01)
S = boltzman(x, 0, 1)
Z = 1-boltzman(x, 0.5, 1)
p.plot(x, S, x, Z, color='red', lw=2)
p.show()

little arithmetic you can write a helper function that takes the
midpoint and saturation point as arguments and computes the relevant
parameters and points.

question that you may want to fill the area below the intersection of
the S and Z curves, which you can do with the magic of numerix and the
fill function

import matplotlib.numerix as nx
import pylab as p

def boltzman(x, xmid, tau):
"""
evaluate the boltzman function with midpoint xmin and time constant tau
over x
"""
return 1. / (1. + nx.exp(-x-xmid)/tau)

def fill_below_intersection(x, S, Z):
"""
fill the region below the intersection of S and Z
"""

#find the intersection point
ind = nx.nonzero( nx.absolute(S-Z)==min(nx.absolute(S-Z)))

# compute a new curve which we will fill below
Y = nx.zeros(S.shape, typecode=nx.Float)
Y[:ind] = S[:ind] # Y is S up to the intersection
Y[ind:] = Z[ind:] # and Z beyond it
p.fill(x, Y, facecolor='blue', alpha=0.5)

x = nx.arange(-6, 6, .01)
S = boltzman(x, 0, 1)
Z = 1-boltzman(x, 0.5, 1)
p.plot(x, S, x, Z, color='red', lw=2)
fill_below_intersection(x, S, Z)
p.show()

As these examples illustrate, matplotlib doesn't come with helper
functions for all the kinds of curves people want to plot, but
along with numerix and python, provides the basic tools to enable you
to build them yourself.

Hope this helps!
JDH

I'm CC-ing my reply to the matplotlib-users list, so other people can
contribute to and benefit from the discussion.

John,

That's fine. I didn't want to post the pdf attachment to the list and I
knew that you would have the greatest insight.

matplotlib's general approach is that you compute the x and y vertices of
the curves you want to plot and then pass it off to plot. So you could
write a helper function to generate these points given your parameters of
interest and then plot them. Eg for a normal pdf, matplotlib.mlab provides
such a function

from matplotlib.mlab import normpdf
import matplotlib.numerix as nx
import pylab as p

x = nx.arange(-4, 4, 0.01)
y = normpdf(x, 0, 1) # unit normal
p.plot(x,y, color='red', lw=2)
p.show()

That's a great example, thanks again. I also need to research the formula
for these curves (d'oh!) because my need is not only to draw them but to
calculate the y value for a given x value, and vice-versa.

Of course, some curves do not have closed form expressions and are not
amenable for such treatment. Some of the matplotlib backends have the
capability to draw arbitrary paths with splines (cubic and quartic) but
this functionality hasn't been exposed to the user yet. If you need this,
let me know and I can provide an interface in an upcoming release.

Splines are great graphically but not for generating plots from model
output.

I had not heard of "S curves" and "Z curves", but a little bit of googling
[1,2] suggests that the S curve is a sigmoid and the Z curve is simply
1-sigmoid. If this is what you are looking for, there are many simple forms
for sigmoids: eg, the hill, boltzman, and arc tangent functions. Here is an
example of the boltzman function

Each specialty reassigns words to suit their own jargon: normal curve (math
and statistics), bell curve (social sciences), Gaussian curve (physics) are
all synonyms. The left and right halves (start at y=1.0, decrease
sinusoidally to y=0.0) looks like an uppercase 'Z' if you use your
imagination, while the right half looks like an uppercase 'S'. import matplotlib.numerix as nx
import pylab as p

def boltzman(x, xmid, tau):
"""
evaluate the boltzman function with midpoint xmin and time constant tau
over x
"""
return 1. / (1. + nx.exp(-x-xmid)/tau)

x = nx.arange(-6, 6, .01)
S = boltzman(x, 0, 1)
Z = 1-boltzman(x, 0.5, 1)
p.plot(x, S, x, Z, color='red', lw=2)
p.show()

Great! More research to define the curves I need and to understand what
values I need to feed into the algorithm, and I'll be there.

little arithmetic you can write a helper function that takes the
midpoint and saturation point as arguments and computes the relevant
parameters and points.

I suspected it was possible, but I hadn't seen the functions in the docs;
I'll look at the NumPy docs again. (I need the eig function from to calculate
the prinicpal eigenvector from a symmetrical matrix, but there's no plotting
involved here.)

question that you may want to fill the area below the intersection of
the S and Z curves, which you can do with the magic of numerix and the
fill function

Actually, no. I don't need to fill the plots. That upper illustration is
the fuzzy set intersect; the logical "AND" of two sets. The lower
illustration shows "AND," "OR," and "X-OR." What I will need to be plotting,
however, is the result of aggregating fuzzy sets using any of several methods
(such as min-max). But, one step at a time as we re-write the code from the
ground up in python (and wxPython).

As these examples illustrate, matplotlib doesn't come with helper
functions for all the kinds of curves people want to plot, but
along with numerix and python, provides the basic tools to enable you
to build them yourself.

And I will be creating those Real Soon Now. And, I'll make them available
to anyone who wants them.

Hope this helps!

It certainly does; well beyond what I expected.

Now, my other current hangup is trying to identify what wxPython/wxWidget
widget to place in the UI for display of these plots. Working on this, too.

Well, my plotting library is now set. I know I can do all these things in
PSTricks -- as far as drawing for publication is concerned -- but I need to
use them in computations, too.

Again, thanks,

Rich

···

On Thu, 28 Jul 2005, John Hunter wrote:

--
Dr. Richard B. Shepard, President | Author of "Quantifying Environmental
Applied Ecosystem Services, Inc. (TM) | Impact Assessments Using Fuzzy Logic"
<http://www.appl-ecosys.com> Voice: 503-667-4517 Fax: 503-667-8863

Rich Shepard wrote:

Now, my other current hangup is trying to identify what wxPython/wxWidget
widget to place in the UI for display of these plots. Working on this, too.

Well, my plotting library is now set. I know I can do all these things in
PSTricks -- as far as drawing for publication is concerned -- but I need to
use them in computations, too.

You have a few different options available to you for embedding matplotlib in wxPython:

1. Embed one of the wxPython backend widgets (which subclass wx.Panel) directly and draw plots on it using matplotlib's object-oriented API. This approach is demonstrated by some of the examples that come with matplotlib (examples/embedding_in_wx*.py).

2. Embed the PlotPanel from Matt Newville's `MPlot' package and draw plots on it using its plot() and oplot() methods.

3. Embed the PlotPanel from my `wxmpl' module and draw plots on it using the matplotlib's object-oriented API.

Each of these approachs has different benefits and drawbacks, so I encourage you to evaluate each of them and select the one that best meets your needs.

Ken

Ken,

Thanks very much. I discovered matplotlib only yesterday so I have a lot of
studying to do. I suspected that the OO API would be better for my needs than
is the MatLab(TM)-style interface. Of course, I'm still brand-new to python
and wxPython (I keep thinking in terms of C and GTK+), but I'm working hard
at getting up to speed.

I appreciate the pointers,

Rich

···

On Thu, 28 Jul 2005, Ken McIvor wrote:

You have a few different options available to you for embedding matplotlib in wxPython:

Each of these approachs has different benefits and drawbacks, so I encourage you to evaluate each of them and select the one that best meets your needs.

--
Dr. Richard B. Shepard, President | Author of "Quantifying Environmental
Applied Ecosystem Services, Inc. (TM) | Impact Assessments Using Fuzzy Logic"
<http://www.appl-ecosys.com> Voice: 503-667-4517 Fax: 503-667-8863

John Hunter wrote:

I'm CC-ing my reply to the matplotlib-users list, so other people can
contribute to and benefit from the discussion.

WOW! now that's support!

John,

Are you keeping a collection of all these little examples around? In a Wiki, perhaps?

-Chris

···

--
Christopher Barker, Ph.D.
Oceanographer

NOAA/OR&R/HAZMAT (206) 526-6959 voice
7600 Sand Point Way NE (206) 526-6329 fax
Seattle, WA 98115 (206) 526-6317 main reception

Chris.Barker@...259...

Either a wiki or a tips-and-tricks page would be great. The advantage of
the wiki is that those of us who do different things to extend or enhance
matplotlib can post them there. The LyX wiki is a great example.

Rich

···

On Thu, 28 Jul 2005, Chris Barker wrote:

Are you keeping a collection of all these little examples around? In a
Wiki, perhaps?

--
Dr. Richard B. Shepard, President | Author of "Quantifying Environmental
Applied Ecosystem Services, Inc. (TM) | Impact Assessments Using Fuzzy Logic"
<http://www.appl-ecosys.com> Voice: 503-667-4517 Fax: 503-667-8863

Rich Shepard wrote:

Thanks very much. I discovered matplotlib only yesterday so I have a lot of studying to do. I suspected that the OO API would be better for my

> needs than is the MatLab(TM)-style interface.

The matplotlib FAQ links to several resources that I found useful when learning about the OO API.

My experience was that reading classdocs was the most helpful source of information. The matplotlib.axes.Axes class is where most of the plotting methods live, so it's probably a good place to start, once you've figured out how to create Figures:

The demos for my wxmpl module (demos/wxmpl-demos.py) may also help you out, as they are OO versions of several of the matplotlib examples. Look at all of the plot_XXX() functions at the beginning of the file. Since each of them takes a Figure as its only argument, they are backend-neutral.

Of course, I'm still brand-new to python and wxPython (I keep thinking in

> terms of C and GTK+), but I'm working hard at getting up to speed.

Ah, a glorious day! Our numbers grow! Ken