New in Mathematica 10: Associations as General-Purpose Associative Arrays

Associations are the most important data structure introduced in Mathematica in years. As the Guide says:

Along with lists, associations are fundamental constructs in the Wolfram Language. They associate keys with values, allowing highly efficient lookup and updating even with millions of elements. Associations provide generalizations of symbolically indexed lists, associative arrays, dictionaries, hashmaps, structs, and a variety of other powerful data structures. 

They of course will underlie databases and artificial intelligence data structures. They are intended to replace less structured expressions such as pairs in Lists or Rules. The code written for Association functions has been highly optimized for speed. I suspect that Associations are the core data structure of the Wolfram Alpha natural language processor, which is the front end for the curated data collections in Alpha, just as the Notebook is the front end for the kernel.

So when I needed a lookup table today I decided to try Associations. Here are some simple functions that define an axon's threshold in terms of the stimulating pulse seen at the axon, according to the Lapicque equation V_th = V_rh(1+τ_ch/pw).

(You don't need to understand these!)

absoluteRheobase@fiberDiameter_:=0.000589+0.01518 Exp[-fiberDiameter/2.3477]

chronaxie@fiberDiameter_:=102.764 +162.46744 Exp[-fiberDiameter/5.53435];

absoluteThreshold=.;absoluteThreshold[pulseWidth_,fiberDiameter_]:=Module[{},absoluteRheobase@fiberDiameter(1+chronaxie@fiberDiameter/pulseWidth)]

Now since fiber threshold is driven by their diameter and the pulse width of the stimulating pulse, we make sample tables of those. Esc + m + esc gives us the mu in microseconds and micrometers; no need to open a Palette.

pulseWidthTable50to500=Table[pw,{pw,100,500,100}] (* in μs *);

fiberDiameters1to15=Table[diameters,{diameters,5,10}] (* in μm *);

Now here is the function that creates the Associations. It took a bit of fiddling, but by making a Table of Rules in one step as should be done with good functional programming, it is then easy to Apply Association to the entire Table, simply replacing the Head, List, with Association, which converts the List of Rules into an Association. In the Table pulseWidth is the "i" iterator and fiberDiameter is the "j" iterator, and notice that we don't need to iterate numerically from 1 to Length@pulseWidthTable50to500, for example, we just directly iterate over the List itself, which is simpler.

absoluteThresholdTable=Association@@Table[Rule[{pulseWidth,fiberDiameter},absoluteThreshold[pulseWidth,fiberDiameter]],{pulseWidth,pulseWidthTable50to500},{fiberDiameter,fiberDiameters1to15}]

<|{100,5}->0.00642849,{100,6}->0.00455515,{100,7}->0.00337826,{100,8}->0.00263167,{100,9}->0.00215327,{100,10}->0.00184348,{200,5}->0.00441095,{200,6}->0.00316135,{200,7}->0.00236852,{200,8}->0.00186172,{200,9}->0.00153533,{200,10}->0.00132348,{300,5}->0.00373844,{300,6}->0.00269675,{300,7}->0.00203194,{300,8}->0.00160507,{300,9}->0.00132935,{300,10}->0.00115015,{400,5}->0.00340218,{400,6}->0.00246445,{400,7}->0.00186364,{400,8}->0.00147675,{400,9}->0.00122636,{400,10}->0.00106348,{500,5}->0.00320043,{500,6}->0.00232507,{500,7}->0.00176267,{500,8}->0.00139975,{500,9}->0.00116456,{500,10}->0.00101149|>

Here are several versions of lookup functions. The first one just uses the same syntax as we'd use to access elements of an associative array made with function@value, such as we'd construct with a memo function.

Clear@thresholdLookup; 
thresholdLookup[lookupTable_Association, pulseWidth_Integer, diameter_] := lookupTable@{pulseWidth, diameter}

absoluteThresholdTable@{100, 7}

0.00337826

Likely it's safer and more elegant to use the new Lookup function instead.

Clear@thresholdLookup2; 
thresholdLookup2[lookupTable_Association, pulseWidth_Integer, diameter_] := Lookup[lookupTable, {{pulseWidth, diameter}}]

thresholdLookup2[absoluteThresholdTable, 100, 7]

{0.00337826}

We see that using a List as the Key results in returning a List as the Value, like in solutions for equations returned by Solve, etc. So to access the Value we can use First@key to remove the List brackets, as we often do with solutions to equations.

Clear@thresholdLookup3; 
thresholdLookup3[lookupTable_Association, pulseWidth_Integer, diameter_] := Lookup[lookupTable, {{pulseWidth, diameter}}] // First

thresholdLookup3[absoluteThresholdTable, 100, 7]

0.00337826

Years ago I talked to Stephen Wolfram at an artificial life conference about artificial intelligence and his views were kind of naive. Things have changed. I expect that Mathematica and its Wolfram Language are headed inexorably toward large-scale AI applications,  approaching, and exceeding in some respects, human intelligence over the next two decades.

How to Lose List Brackets in Mathematica

First, Apply, Sequence, ReplaceAll

There come times in every Mathematica user's life when the all-pervasive List brackets get in the way. Sometimes it's easy to redesign the function to accept the arguments in a List. Here are other solutions.

First

For a single element enclosed by brackets, you can use First to extract the element. In this example, since all Mathematica arithmetic functions are Listable (they will automatically Map over a List) we specify that the argument must be an Integer so the function will fail.

enclosed1={2};
Clear@function1;function1@x_Integer:=x^3;
function1@enclosed1

function1[{2}]

function1@First@enclosed1

8

Apply

First, see if you can use Apply @@ to apply a function to the contents of your List. Apply will replace the List head with your desired function and Evaluate it.

In[107]:=  Range@15
Out[107]=  {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}

In[108]:=  Plus@@%
Out[108]=  120

Here's a single filename that is unnecessarily surrounded by List brackets. It's a String, so we just Apply ToString to it. It then stands alone – no List brackets – but its Head is String.

FileNames[]

{TableFiredFibers-Control-CathodeOnly-0p644.txt}

ToString@@%141

TableFiredFibers-Control-CathodeOnly-0p644.txt

Head@%
String

FullForm@%%
"TableFiredFibers-Control-CathodeOnly-0p644.txt"

Sequence

Mathematica was designed as a list processing language in which the list is the fundamental data structure. Allan Newell's IPL (Information Processing Language) was the original prototype, after which John McCarthy coupled the list as a universal data structure with the lambda calculus and logic functions into the major programming language, LISP.

We can Apply Sequence to a List to replace List as the container (the Head). Here are two examples, first to a single List, then to a List of Lists.

In[115]:= Sequence@@{0,4}

Out[115]= Sequence[0,4]

In[119]:= Sequence@@{{0,1,1},{0,2},{0,1,3},{0,4}}

Out[119]= Sequence[{0,1,1},{0,2},{0,1,3},{0,4}]

FileNameJoin is the safe way to assemble filepaths in Mathematica since it knows your operating system syntax. However FileNameJoin only accepts a single-level List as its argument. To feed it a nested List, use Sequence instead of List as the Head of the inner List.

In[350]:= twoPathElements={"webPages","index.html"};

In[351]:= FileNameJoin@{$HomeDirectory,twoPathElements}

Out[351]= FileNameJoin[{C:\Users\kwcarlso,{webPages,index.html}}]

Since it didn't see a List of Strings (filepath names) as the correct argument types, FileNameJoin returned the input. Using Sequence solves this.

In[352]:= twoPathElements=Sequence["webPages","index.html"];

In[353]:= FileNameJoin@{$HomeDirectory,twoPathElements}

Out[353]= C:\Users\kwcarlso\webPages\index.html

The context of one function I use to illustrate Sequence is worth mentioning (but skip this to get to more usages of Sequence). I have digitally sampled a curve from the literature (a paper by Holsheimer from 1999) in the graphing program Origin. In other words, you can import an image, in this case a graph for which Holsheimer doesn't give either the underlying data or an equation, and repeatedly click on it to harvest {x,y} values. We captured 163 sample points on the solid curve below, which is a vast improvement on the antiquated method of using a straightedge to pick up x and y values off the axes.

With that many sample points, finding the x value closest to the one desired is a reasonable alternative to fitting a curve and applying the fit equation to values you wish to interpolate or extrapolate.

So this is a function to find the nearest x value (a neuron axon diameter, fiberDiameter) to one for which we want to know the threshold as determined by Holsheimer (which is the y-value retrieved). The problem is Nearest returns values in List brackets, which don't match the un-bracketed x values in the List of data. Applying Sequence to the result of Nearest feeds the unbracketed value to Cases, which then performs the match.

Clear@diameterThreshold; diameterThreshold[fiberDiameter_, fiberThresholdSourceData_:holsheimerData1999] := Cases[fiberThresholdSourceData, {x_, y_} /; x == Sequence@@Nearest[fiberThresholdSourceData[[All, 1]], fiberDiameter] ]

So in sum this function says: Clear the function name (which I do habitually at the start of a function definition to prevent accidentally testing a previous version as I revise it). Then define it as the Cases in the source data file (fiberThresholdSourceData--default to the holsheimerData1999 file unless I specify another file) such that ( /; ) the x-value is Nearest to the fiberDiameter parameter I give it.

ReplaceAll

Here is a third way using rule patterns.

In[128]:= {{0,1,1},{0,2},{0,1,3},{0,4}}/. List[a__] ->a

Out[128]= Sequence[{0,1,1},{0,2},{0,1,3},{0,4}]