Skip to main content

programming - Combinations of multiple matching patterns


I wonder if there is any nice way to combine DownValues (or any other suitable rule-/pattern-/function -based Mathematica construct) when multiple patterns match an expression. Let me explain what I mean with a somewhat silly example:


f[x_?(Mod[#,2]==0 &&Mod[#,3]==0&)]:= "FizzBuzz"
f[x_?(Mod[#,2]==0& )]:= "Fizz"
f[x_?(Mod[#,3]==0&)]:= "Buzz"

f[x_]:=x

Range@15 //Map@f
(* {1,Fizz,Buzz,Fizz,5,FizzBuzz,7,Fizz,Buzz,Fizz,11,FizzBuzz,13,Fizz,Buzz} *)

It would be really nice if one could do away with the first DownValue of f which checks if an expression is divisible by both 2 and 3 (since this is just a combination of the PatternTests used by the two DownValues defined beneath it). In this simple case one additional DownValue might not be an issue but if one adds more and more "rules" the number of additional combinations to check increases rapidly with the number of "rules". For instance:


g[x_?(Mod[#,2]==0 &&Mod[#,3]==0 && #<10&)]:= "FizzBuzzZapp"
g[x_?(Mod[#,2]==0 &&Mod[#,3]==0&)]:= "FizzBuzz"
g[x_?(Mod[#,2]==0&& #<10& )]:= "FizzZapp"
g[x_?(Mod[#,2]==0& )]:= "Fizz"

g[x_?(Mod[#,3]==0&& #<10& )]:= "BuzzZapp"
g[x_?(Mod[#,3]==0&)]:= "Buzz"
g[x_?(#<10&)]:= "Zapp"
g[x_]:=x

Range@15 //Map@g
(* {Zapp,FizzZapp,BuzzZapp,FizzZapp,Zapp,FizzBuzzZapp,Zapp,FizzZapp,
BuzzZapp,Fizz,11,FizzBuzz,13,Fizz,Buzz} *)

Is there an elegant idiom for this?



Edit:


I actually came up with this whole question when looking at this website about fizzbuzz in too much detail and thinking that the presented FP solution was not really comprehensible anymore. As this fizzbuzz task is all about rule-replacement one might assume that a pattern-matching/rule-replacement/functional approach should give the most natural, elegant and easy to understand representation but this seems not necessarily to be true.



Answer



This does not answers my own question fully (I am still interested to see if someone might come up with an truly elegant solution based on DownValues) but I found a rule-based solution that is imho. elegant non the less.


fizzbuzz[rls_]:= With[{res=ReplaceList[#, rls]}, If[res=={}, #, StringJoin@res]]& 

fizzbuzz[{_?(Mod[#,2]==0&) -> "Fizz",
          _?(Mod[#,3]==0&) -> "Buzz",
          _?(#<10&) -> "Zapp"}] /@ Range@15


Looks especially neat (or obfuscated, depending on your point of view) with the escfnesc glyph for Function


enter image description here


Update: Generalization of my solution and Example


Because some confusion arose in the comments on march's answer I though I should address those in my own answer and give another example to show that this approach can be easily extended to all kinds of rules. So here is a generalization of my function fizzbuzz. It takes three arguments:



  • a list of (possibly overlapping) replacement rules

  • a function to be applied to the expressions found via pattern matching (note that the function also has access to the actual variable not only the results from pattern matching, see example below)


  • an alternative function to be applied if no pattern matched





func[rls_, f_, alt_]:= With[{res=ReplaceList[#, rls]}, If[res=={}, alt@#, f[res,#]]]&

**Example**
regions=1/2*{{1, -Sqrt[3]/3}, {0, 2 Sqrt[3]/3}, {-1, -Sqrt[3]/3}} //Map@Disk;

f = func[{x_ /;RegionMember[regions[[1]],x]:> {1,0,0},
          x_ /;RegionMember[regions[[2]],x]:> {0,1,0},
          x_ /;RegionMember[regions[[3]],x]:> {0,0,1}},  
          {RGBColor@(Plus@@#1), Point[#2]}&, Point[#]& ] 


f/@ RandomPoint[Disk[{0,0}, 2], 5000] //Graphics

output


Comments

Post a Comment

Popular posts from this blog

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]
Post a Comment
Read more

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]
Post a Comment
Read more

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1....
Post a Comment
Read more