Thursday, November 8, 2012

Linear Interpolation Made Convenient in Maxima



(I usually put the load statement in a separate cell in wxMaxima so that I am not re-executing the load every time I correct my main code and rerun the cell.  Minor point.)

Maxima can turn a set of points into a function for linear interpolation.  For that matter, it can make a cubic spline out of the same points.  I'm more interested in the linear interpolated function today—the other is for your reference.  There are three main points of interest in the above code:
  1. linearinterpol() takes points, sorts them by the first ordinate, and produces an expression.
  2. linearinterpol() produces expressions which use the characteristic function, rather than taking cases to make it transparent to other Maxima routines.
  3. The use of the double single quote operator ('') and the single quote operator (') helps me to define the function f(x) such that x is a formal variable and not embedded as an expression.
  4. linearinterpol() produces an expression which extrapolates beyond the specified points (from negative infinity to positive infinity) and it is left to the user's discretion to determine the limits to which this extrapolation is reasonable or applicable.
In a Nutshell

All you need is a list of 2D points.  There is more than one way to specify these, but I find it easiest to use a list of lists.  The main list elements are all lists of 2 representing x and y ordinates.  The function sorts these points according to the x-coordinate before proceeding.  You can specify the variable which will be used in the expression.  Provided that you precede that variable with a single quote operator, you can use a variable that is elsewhere assigned a value.  More on that below.

Transparent Output

The characteristic function, which is often assigned the Greek letter chi ("χ"), returns either 0 or 1.  It is a function of two variables, a set and a value.  If the value is an element of the set, it returns 1, otherwise it returns 0.  In the context of our linear interpolation expression, we have a linear expression for each interval in our set of points and each of these is multiplied by a characteristic function with a different set.  Since the sets used for the characteristic functions form a partition of the real numbers, we won't have more than one of the terms of our overall expression from linearinterpol() evaluate to a non-zero value at a time.  This expression can be integrated, differentiated, or combined with other expressions without having to take special cases.  Take the monkey off your back, let Maxima do it.

Cool, Useful, but Initially Abstruse Operators

The single quote operator (beside the key) is a way to tell Maxima, "Don't evaluate this thing."  If I write x and x is assigned value, then Maxima will do what is normally very helpful to me, namely, make the substitution for me.  If I don't want Maxima to do that, because I actually want the expression x, not the value that x stands for (because it has been assigned a value), I need to write 'x instead. In contrast, the double single quote operator ('') tells Maxima to replace the given expression with the value of the expression. This way, when I use the delayed assignment operator (:=) it does not delay the execution of the linearinterpol() function, only the evaluation of the expression as a function of x is delayed.  You may recall from my previous post that parameters in function declarations are formal.  Even if x is elsewhere assigned a value, f(x) := some_expression_including_x + x, will still work intuitively.

Here is some sample input followed by the output.  This is the best way to clarify in your mind the use of these operators:

x:2;
y:2*x;
z: 2*'x;
d(x) := z;
d(2);
d(3);
h(x) := ''(z);
h(2);
h(3);

(%o46) 2
(%o47) 4
(%o48) 2*x
(%o49) d(x):=z
(%o50) 2*x
(%o51) 2*x
(%o52) h(x):=2*x
(%o53) 4
(%o54) 6

Extrapolation

The expression that linearinterpol() produces does not protect you from yourself.  If will produce a result from negative to positive infinity without telling you that you have gone beyond the boundaries of your initial data set.  Actually, this is as it should be, in my opinion.  The program gives you the freedom to ask all the what-ifs your little heart desires, but it is up to you to discern what is really true to life and applicable to the particular problem you are trying to solve—whether the extrapolation is truly justified by the data is rightly left as a matter for user judgement.
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