## Sunday, 13 June 2010

### Broken axis, once more

I have discussed this subject at least on two occasions, and in fact, most of the present post was already described in one of my very first posts. However, I thought that it might be worthwhile to dust it off, especially, that this is a scientifically relevant feature, which is still missing in gnuplot. But with a bit of work, we can rectify the problem. In short, there are situations, when we just have to break the axis, simply because there is a large gap between two relevant ranges of a plot. Take this example, for instance
```plot [0:40] 20.0*atan(x-20.0) + 32 + sin(x)
```
which would result in the following graph:

The problem is obvious: the function has some interesting modulation close to 0, and close to 60, but it is rather dull between these two extrema. The solution is to cut out the segment between 10, and 60, say.

What we will use is a very handy function in gnuplot 4.4, which lets the user set the position of the graph exactly. In a multiplot, we would usually set the position as
```set multiplot
set size 1, 0.3
set origin 0, 0.5
plot 'foo' using 1:2

set size 1, 0.2
set origin 0, 0
plot 'bar' using 3:4
```
which produces two graphs of size 1, 0.3, and 1, 0.2, respectively, and places them in such a way that their bottom left corner is at (0, 0.5), and (0, 0). But when we say "the bottom left corner", we actually mean the whole figure, tic marks, axis labels, everything. This means at least two things. One is that if we want to break the axis using a multiplot, the ranges will not necessarily be proportional on the figure, simply because the size referred to the size of the whole graph, and not only to the plotting area. Second, if the size of the tic labels is different in the two graphs, they will no longer be aligned properly. This would happen, e.g., if we were to plot over the ranges [0:9], and [1000:10009]: the first range requires labels of width 1, while the second labels of width 5, therefore, the second graph would be narrower, and its left vertical axis shifted to the right, at least, as far as the plotting area is concerned.

In gnuplot 4.4, however, one can set the positions of the plots, and not the whole graph. This is achieved by issuing a command similar to this
```set lmargin at screen 0.1
```
which aligns the left vertical axis of the graph with the point that is at screen position 0.1 along the horizontal direction. There are three more margins, rmargin, tmargin, and bmargin, setting the right hand side, the top, and the bottom of the graph. Specifying the plot's corners explicitly removes the above-mentioned problem with the alignments.

Having said this, our script could read as follows
```reset
unset key
bm = 0.15
lm = 0.12
rm = 0.95
gap = 0.03
size = 0.75
y1 = 0.0; y2 = 11.5; y3 = 58.5; y4 = 64.0

set multiplot
set xlabel 'Time [ns]'
set border 1+2+8
set xtics nomirror
set ytics nomirror
set lmargin at screen lm
set rmargin at screen rm
set bmargin at screen bm
set tmargin at screen bm + size * (abs(y2-y1) / (abs(y2-y1) + abs(y4-y3) ) )

set yrange [y1:y2]
plot [0:40] 20.0*atan(x-20.0) + 32 + sin(x)

unset xtics
unset xlabel
set border 2+4+8
set bmargin at screen bm + size * (abs(y2-y1) / (abs(y2-y1) + abs(y4-y3) ) ) + gap
set tmargin at screen bm + size + gap
set yrange [y3:y4]

set label 'Power [mW]' at screen 0.03, bm + 0.5 * (size + gap) offset 0,-strlen("Power [mW]")/4.0 rotate by 90

set arrow from screen lm - gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1)+abs(y4-y3) ) ) - gap / 4.0 to screen \
lm + gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1) + abs(y4-y3) ) ) + gap / 4.0 nohead

set arrow from screen lm - gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1)+abs(y4-y3) ) ) - gap / 4.0  + gap to screen \
lm + gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1) + abs(y4-y3) ) ) + gap / 4.0 + gap nohead

set arrow from screen rm - gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1)+abs(y4-y3) ) ) - gap / 4.0 to screen \
rm + gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1) + abs(y4-y3) ) ) + gap / 4.0 nohead

set arrow from screen rm - gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1)+abs(y4-y3) ) ) - gap / 4.0  + gap to screen \
rm + gap / 4.0, bm + size * (abs(y2-y1) / (abs(y2-y1) + abs(y4-y3) ) ) + gap / 4.0 + gap nohead

plot [0:40] 20.0*atan(x-20.0) + 32 + sin(x)

unset multiplot
```

The first couple of lines specify how big a figure we want to have: bm, lm, and rm are the bottom, left, and right margins, respectively. We also define the size of the gap, which we will have between the two plots. y1 through y4 are the definitions of our plot ranges. In other words, the interval between y2, and y3 will be cut out of our figure.

In the multiplot environment, we set the axes (for the bottom figure on the bottom, left, and right, while for the top figure on the top, left, and right), the axis labels (the vertical label we have to set by hand, for otherwise it would be centred on the vertical axis of the bottom or top figure, but not on the whole), and set the positions of the figures. Note that the definition used for tmargin and bmargin makes sure that the two plotted intervals are proportional. Before plotting the second curve, we also set four small headless arrows, which are meant to represent the break in the axes. It can be left out, if not desired, or they can be replaced by two dashed vertical lines.

This method can also be used, if one wants to plot a single curve or data set, but with logarithmic axis on one interval, and linear on the other.

## Saturday, 12 June 2010

### Ministry of Sillier Walks

In my last post, I have shown how we can define or read an array of number from a file. Having constructed the array, the question naturally arises: can we use it for something else, to do something that we could not do otherwise. The answer is yes, although in the present post, there will not be anything that I haven't discussed here or there. We have only got to put the pieces together, and with a little bit of work, we can produce three-dimensional column stacked histograms at ease. I will use the same datafile, but I repeat it here:

```"" France Germany Japan Nauru
Defense 9163 4857 2648 9437
Agriculture 3547 5378 1831 1948
Education 7722 7445 731 9822
Industry 4837 147 3449 6111
"Silly walks" 3441 7297 308 7386
```

Then, let us see, how we are going to make that histogram! For the sake of example, we will draw four cylinders, which will be striped according to the values that the columns in the data file take. We could do two things here. Provided that we have already read the values into an array, we could draw 4 times 5 cylinders of various size and colour. Alternatively, we could draw 4 cylinders, and colour them with stripes that we take from the palette. In this case, we have to find some way of defining the proper palette. Both methods have advantages and disadvantages. The advantage of the first one is that it is faster, because the cylinders are monocolour, which means that we need only two isosamples. On the other hand, we have to have "nested" for loops (which we would just simply write out). The advantage of the second method is that we can get away with one for loop, but we need many isosamples, because the cylinders contain many colour, though the transition between the colours is required to be abrupt. But since we do not know where the boundary between two colours is, we would need many isosamples. Consequently, it will be slow.

When defining our array, we will employ the trick from the last post, and we will do something very similar, when we determine our palette. In addition, we will also have to calculate the sum in the columns, for the cylinders' height should be proportional to that. If, on the other hand, we opt for re-scaling the cylinders to the same height, we, again, need the sum. With these preliminary remarks, the first version of our script could look like this

```reset

unset key
unset colorbox
unset xtics; unset xlabel
unset ytics; unset ylabel
unset ztics
file = 'marimekko.dat'
cylinder = 'cylinder.dat'

set ticslevel 0
set border 1+2+4+8
set parametric; set urange [0:2*pi]; set vrange [0:1]; set iso 2, 200
set table cylinder
splot cos(u), sin(u), v, cos(u)*v, sin(u)*v, 1
unset table
unset parametric

col = 4
row = 5
sm = 0.0
g(x,a) = (abs(x-a) < 0.1 ? 1 : 0)
h(x,a) = (x <= a ? 0.0 : 1.0)

ARRAY = "b(x, y) = 0"
SUM = "s(x,a) = 0"
PALETTE = "set palette defined (-1 0.7 0 0"

array(x, c) = (sm = h(\$0,0.5)*sm + x, ARRAY = ARRAY.sprintf(" + g(x,%d)*h(y,%.3f)", c, sm), \
SUM = SUM.sprintf(" + %f*g(x,%d)", x, c), x )
pal(x, c) = (PALETTE = PALETTE.sprintf(", %.1f %.3f %.3f %.3f", c-0.1, rand(0)*0.7, rand(0)*0.7, rand(0)*0.7), x
ff(x, c) = (array(x, c), x)

plot for [i=2:col+1] file every ::1 using 0:( ff(column(i), i) )
plot file every ::1 using 0:(pal(1, column(0)))

eval(ARRAY)
eval(PALETTE.")")
eval(SUM)

set xrange [4:5+3*col]
set yrange [-10:3*col-9]

splot for [i=2:col+1] cylinder using (\$1+3*i):2:(\$3*s(i,i)):(b(i,\$3*s(i,i))) with pm3d
```

The first couple of lines set up the graph, and they are trivial, as is the plot to the 'cylinder.dat'. The definition of g(x,a) should be familiar from the last post, and h(x,a) is nothing but the Heaviside function. We have to deal with a matrix, therefore, the string ARRAY begins with "b(x,y) = 0", which will, in due course, become the definition of a two-variable function. SUM, that will bring s(x,a) to life, also appears to define a two-variable function, but this is only apparent: s(x,a) is exactly as much of a two-variable function as is g(x,a). A mere convenience, nothing more. We also begin the definition of a palette, but we stop short of its completion: that will be done during the first plot.

In the array(x,c) function, we read in the numbers from the file, and at the same time, we also calculate the sums in the columns. This function is really similar to that from the last post. We also define a function for filling up the palette. For now, this function writes random colours into the palette at every integer.

Having defined the functions, we "plot" our data file, so as to read in the numbers. We plot the second column of the same file again, in order to prepare our palette. Note that this latter plot is really nothing but a strange way of creating a for loop: we do something as many times as there are elements in a column. In both plots, by applying the every keyword, we skip the first line, which is the column header.

At this point we are almost done: the only remaining thing is the evaluation of our new definitions, and the actual plot. The setting of the xrange and yrange is necessary only in order to ensure that the cylinders are cylinders, not elliptical.

Our first script results in the graph below:

This is OK for a start, but could we improve it a bit? With some work, we could. For one thing, we could add the sum at the top of each cylinder. This can easily be done by augmenting the last plot with the line
```for [i=2:col+1] file every ::0::0 using (3*i):(0):(s(i,i)+5e3):(sprintf("%d", s(i,i))) w labels
```

We can also give a title to each cylinder by reading out the values in the first row. This is done by adding
```for [i=2:col+1] file every ::0::0 using (3*i):(-2):(0):(stringcolumn(i)) w labels centre
```

Finally, we can easily add a legend to the figure. All we have to do is to read out the first column of our data file, and draw 5 cylinders with the appropriate colour. We can achieve this by invoking
```file using (0):(0):(2e4-\$0*5e3):1 w labels right, \
for [i=1:row] cylinder using (\$1+5):(\$2-5.0):(\$3*2e3+i*5e3):(i-1) with pm3d
```
in the last plot. With these modifications, the complete script would look like this
```reset

unset key
unset colorbox
unset xtics; unset xlabel
unset ytics; unset ylabel
unset ztics
file = 'marimekko.dat'
cylinder = 'cylinder.dat'

set ticslevel 0
set border 1+2+4+8
set parametric; set urange [0:2*pi]; set vrange [0:1]; set iso 2, 200
set table cylinder
splot cos(u), sin(u), v, cos(u)*v, sin(u)*v, 1
unset table
unset parametric

col = 4
row = 5
sm = 0.0
g(x,a) = (abs(x-a) < 0.1 ? 1 : 0)
h(x,a) = (x <= a ? 0.0 : 1.0)

ARRAY = "b(x, y) = 0"
SUM = "s(x,a) = 0"
PALETTE = "set palette defined (-1 0.7 0 0"

array(x, c) = (sm = h(\$0,0.5)*sm + x, ARRAY = ARRAY.sprintf(" + g(x,%d)*h(y,%.3f)", c, sm), \
SUM = SUM.sprintf(" + %f*g(x,%d)", x, c), x )
pal(x, c) = (PALETTE = PALETTE.sprintf(", %.1f %.3f %.3f %.3f", c-0.1, rand(0)*0.7, rand(0)*0.7, rand(0)*0.7), x
ff(x, c) = (array(x, c), x)

plot for [i=2:col+1] file every ::1 using 0:( ff(column(i), i) )
plot file every ::1 using 0:(pal(1, column(0)))

eval(ARRAY)
eval(PALETTE.")")
eval(SUM)

set xrange [4:5+3*col]
set yrange [-10:3*col-9]

splot for [i=2:col+1] cylinder using (\$1+3*i):2:(\$3*s(i,i)):(b(i,\$3*s(i,i))) with pm3d, \
for [i=2:col+1] file every ::0::0 using (3*i):(0):(s(i,i)+5e3):(sprintf("%d", s(i,i))) w labels, \
file using (0):(0):(2e4-\$0*5e3):1 w labels right, \
for [i=1:row] cylinder using (\$1+5):(\$2-5.0):(\$3*2e3+i*5e3):(i-1) with pm3d, \
for [i=2:col+1] file every ::0::0 using (3*i):(-2):(0):(stringcolumn(i)) w labels centre
```
and would the following figure:

Well, this is sort of OK, but what if we still do not like it? There are two things that we could easily implement, and would change the character of our graph completely. One is that we can give the cylinders a true 3D lookout, by adding phongs to them. The other one is that we could remove the legends, and add it to one of the cylinders.

So, let us see what we could do in the way of phonging. This is really simple: if we think about it, the phong is nothing but a white spot on our graph, where the saturation of the colours increases towards the centre of the spot. Therefore, all we have to do is to insert white into the palette, but to do it in a way that white saturates all little cylinders. We could, then, modify our palette function as

```pal(x, c) = (PALETTE = PALETTE.sprintf(", %.1f %.3f %.3f %.3f, %.1f 1 1 1", \
c-0.01, rand(0)*0.7, rand(0)*0.7, rand(0)*0.7, c+0.99), x)
```
and add a function that changes the saturation as

```colour(x,y) = 0.25*exp(-(x-0.7)**2/0.2-(y+0.7)**2/0.2)
```
If we look at the palette function, the random numbers will be between 0-0.7, and the function colour(x,y) will add 0.25 to those numbers at the centre of the spot, which, in this particular case, will be at 45 degrees with respect to the x axis. When plotting, we have to add this function to our cylinders.

As for the labels, we might want to place them at the centre of each coloured cylinder in the last cylinder, representing Nauru. That is, we use the first column, and add the labels from that at half of the height of the cylinders whose size is read from the fifth column. We could use this function
```lab(x) = (sm = sm + x, sm-0.5*x)
```
The value of sm is updated when a new value is read from the fifth column, and the return value of the function is just the cumulative sum minus half of the last value. Note that since we use sm, which was also utilised in array(x,c), we will have to re-set its value to zero before we use it.

With these modifications, the complete script reads as
```reset

unset key
unset colorbox
unset xtics; unset xlabel
unset ytics; unset ylabel
unset ztics
file = 'marimekko.dat'
cylinder = 'cylinder.dat'

set ticslevel 0
set border 1+2+4+8
set parametric; set urange [0:2*pi]; set vrange [0:1]; set iso 2, 200
set table cylinder
splot cos(u), sin(u), v, cos(u)*v, sin(u)*v, 1
unset table
unset parametric

col = 4
row = 5
sm = 0.0
g(x,a) = (abs(x-a) < 0.1 ? 1 : 0)
h(x,a) = (x <= a ? 0.0 : 1.0)

ARRAY = "b(x, y) = 0"
SUM = "s(x,a) = 0"
PALETTE = "set palette defined (-1 0.7 0 0, -0.5 1 1 1"

array(x, c) = (sm = h(\$0,0.5)*sm + x, ARRAY = ARRAY.sprintf(" + g(x,%d)*h(y,%.3f)", c, sm), \
SUM = SUM.sprintf(" + %f*g(x,%d)", x, c), x )
pal(x, c) = (PALETTE = PALETTE.sprintf(", %.1f %.3f %.3f %.3f, %.1f 1 1 1", \
c-0.01, rand(0)*0.7, rand(0)*0.7, rand(0)*0.7, c+0.99), x)
ff(x, c) = (array(x, c), x)

colour(x,y) = 0.25*exp(-(x-0.7)**2/0.2-(y+0.7)**2/0.2)
lab(x) = (sm = sm + x, sm-0.5*x)

plot for [i=2:col+1] file every ::1 using 0:( ff(column(i), i) )
plot file every ::1 using 0:(pal(1, column(0)))

eval(ARRAY)
eval(PALETTE.")")
eval(SUM)

set xrange [4:5+3*col]
set yrange [-10:3*col-9]

splot for [i=2:col+1] cylinder using (\$1+3*i):2:(\$3*s(i,i)):(b(i,\$3*s(i,i))+colour(\$1,\$2)) with pm3d, \
for [i=2:col+1] file every ::0::0 using (3*i):(0):(s(i,i)+5e3):(sprintf("%d", s(i,i))) w labels, \
sm = 0, file using (3*col+4):(1):(lab(\$5)):1 w labels left, \
for [i=2:col+1] file every ::0::0 using (3*i):(-2):(0):(stringcolumn(i)) w labels right rotate by 30
```

I think I cannot add more to this figure, we have explored and exhausted all possibilities here. Next time I will come back to an older topic from this blog, and show how that can be done in an elegant way, applying the new functionalities of gnuplot 4.4.
Cheers,
Zoltán