Page - 443 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 110 2.2. ParameterSpoiling In the theoryoferror-correctingcodes,oneconsiders spoilingoperationsonthecodeparameters. Appliedtoan [n,k,d]2-codeC, theseproduce, respectively,newcodeswith the followingdescription (seeSection1.1.1of [24]): • A code C1 = C i f in Fn+12 , for a map f : C → F2, whose code words are of the form (x1, . . . ,xi−1, f(x1, . . . ,xn),xi, . . . ,xn) forw = (x1, . . . ,xn) ∈ C. If f is a constant function, C1 isan [n+1,k,d]2-code. If allpairsw,w′ ∈CwithdH(w,w′)= dhave f(w) = f(w′), thenC1 is an [n+1,k,d+1]2-code. • AcodeC2=C i inFn−12 ,whosecodewordsaregivenbytheprojections (x1, . . . ,xi−1,xi+1, . . . ,xn) ofcodewords (x1, . . . ,xi−1,xi,xi+1, . . . ,xn) inC. This isan [n−1,k,d−1]2-code,exceptwhenall pairsw,w′ ∈CwithdH(w,w′)= dhavethesameletterxi, inwhichcase it isan [n−1,k,d]2-code. • AcodeC3 = C(a, i)⊂ C ⊂ Fn2, givenby the level setC(a, i) = {w = (xk)nk=1 ∈ C | xi = a}. TakingC(a, i) igivesan [n−1,k′,d′]2-codewithk−1≤ k′< k, andd′ ≥ d. Thesamespoilingoperationsholdforq-arycodesC⊂Fnq, foranyfixedq. Inoursetting,whereC is thecodeobtainedfromafamilyof languages,accordingtotheprocedure describedabove, thefirst spoilingoperationcanbeseenas theeffectof consideringonemoresyntactic parameter,which isdependenton theotherparameters, hencedescribinga function F :Fn2 →F2, whose restriction toC gives the function f : C → F2. In particular, the casewhere f is constant onC represents the situation inwhich thenewparameter addsnouseful comparison information for theselectedfamilyof languages. Thesecondspoilingoperationconsists in forgettingoneof the parameters, and the thirdcorresponds to formingsubfamiliesof thegiven familyof languages, by groupingtogether those languageswithasetvalueofoneof thesyntacticparameters. Thus,all these spoilingoperationshaveaclearmeaningfromthepointofviewof the linguisticPCM. 2.3. Examples Weconsider the same list of 63 parameters used in [3] (see Section 5.3.1 and TableA). This choiceofparameters followsthemodularizedglobalparameterizationmethodof [2], for theDeterminer Phrasemodule. They encompass parameters dealingwith person, number, and gender (1–6 on their list), parameters of definiteness (7–16 in their list), of countability (17–24), genitive structure (25–31), adjectivalandrelativemodification (32–14),positionandmovementof theheadnoun(42–50), demonstrativesandotherdeterminers(51–50and60–63),possessivepronouns(56–59); seeSection5.3.1 andSection5.3.2of [3] formoredetails. Our very simple examples here are justmeant to clarify our notation: they consist of some collectionsof languagesselectedfromthelistof28,mostlyIndo-European, languagesconsideredin[3]. Ineachgroupweconsiderweeliminate theparameters thatareentailed fromothers, andwefocuson ashorter list, amongtheremainingparameters, thatwill suffice to illustrateourviewpoint. Example1. ConsideracodeC formedoutofthelanguages 1= Italian, 2=Spanish,and 3=French, andletusconsideronly thefirst sixsyntacticparametersofTableAof [3], so thatC⊂Fn2 withn=6. Thecodewords for the three languagesare 1 1 1 1 0 1 1 2 1 1 1 1 1 1 3 1 1 1 0 1 0 Thishascodeparameters (R= log2(3)/6=0.2642,δ=1/6),whichsatisfyR<1−H2(δ),hence they liebelowtheGVcurve (seeEquation(8)below).Weuse thiscodeto illustrate the threespoiling operationsmentionedabove. 443