[Math] How to read a contour plot

graphing-functions

I am taking machine learning, and I have seen a few contour plots in the course.

It seems that I can't understand how to read this plot, I have tried looking it up in Wikipedia, but I don't even understand the first example (the figure on the right).

Another example is from the course:

enter image description here

The remark in the brackets say that it is easier to see that the function plotted is convex from the counter plot. but I don't see it at all.

What are those circles ? what is the meaning that the red x is in the inner circle (or even near its middle) ?

I would appreciate it if someone could please explain what information I should be able to read from the right drawing (the contour plot) and how ?

Best Answer

The circles you see are level sets: the function $J$ is equal where the colors are the same, i.e. around each circle. The red X is, presumably, the point where $J$ attains its minimum.

If you've ever seen a topographic map, it is also a form of contour plot, where the function being plotted is the height of the ground.

From the plot you should be able to see that the function $J$ is "bowl-shaped" and therefore convex. Moreover, from the spacing you can see that the function increases quickly starting from the X and moving up-right or down-left, and slowly moving up-left or down-right.

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[Math] How to plot a smooth curve function from given points

some options in R would be

t=seq(0,10,0.01)
y=sin(t)+rnorm(length(t))
plot(t,y,cex=0.1)

# puts a loess smoother through the points
lines(loess.smooth(t,y),col=2)

library(mgcv)
# puts a spline through the points - read Simon Woods work to learn more about this
g<-gam(y~s(t),data=data.frame(t=t,y=y))
lines(t,g$fitted.values,col=3)

and for your recently added data

t=1:400
y=rep(NA,length(t))
y[10]<-30
y[111]<-100
y[171]<-128
y[181]<-86
y[201]<-42
y[211]<-44
y[281]<-39
y[321]<-59
y[341]<-20
y[351]<-4
library(mgcv)

g<-gam(y~s(t),data=data.frame(t=t,y=y))
lm<-predict.gam(g,data.frame(t=t,y=y))

plot(t,y,cex=0.5,pch=16,ylim=c(min(lm),max(lm)))
lines(t,lm,col='red')

[Math] How to make a good “infinity plot”

I suspect there is no natural sigmoid function that will "preserve" in some sense your original function $f$.

In all generality, you want to "squeeze" the real line ($\mathbb{R}$) into an (open) interval of the form $(x_1,x_2)$. Let's try to squeeze both the $x$-axis and the $y$-axis so that we may fit one graph into a rectangle. Let this rectangle be, for simplicity, $(-a,a)\times(-b,b)$ where $a$ and $b$ are positive.

Let $D$ be the domain of $f$ and $R$ be its codomain (for the sake of simplicity, let's think of $R$ as $\mathbb{R}$)

To squeeze the $x$-axis we really need a function $\varphi:\ (-a,a)\longrightarrow D$, so that when we graph $f\circ\varphi$ we'll need to see what that thing does only on the interval $(-a,a)$.

To squeeze also the $y$-axis, we need another function $\psi:R\longrightarrow(-b,b)$. The end result is then \begin{equation} \tilde{f}:=\psi\circ f\circ\varphi \end{equation} Its graph will be contained in the rectangle $(-a,a)\times(-b,b)$.

The tricky part, as you pointed out, is finding the "right" sigmoid function(s). One can try various cocktails. If $D=R=\mathbb{R}$ for instance, one could take \begin{equation} \varphi(x):=k\tan\left(\frac{\pi}{2a}x\right)\qquad\psi(x):=\frac{2b}{\pi}\arctan(x/k) \end{equation} to do the job. Varying the parameter $k$ allows for trying to "match" the original function (for biggish $k$). It really depends on what characteristics of $f$ you want to study.

[EDIT: I personally like viewing functions "wrapped" on a sphere- but it's only an aesthetic approach. Here i posed a question about it.]

Best Answer

Related Solutions

[Math] How to plot a smooth curve function from given points

[Math] How to make a good “infinity plot”

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