Correlation coefficient - mascotasohana

Definition

Correlationisanon-deterministicrelationship,andthecorrelationcoefficientistheamountoflinearcorrelationbetweenresearchvariables.Duetodifferentresearchobjects,thecorrelationcoefficientcanbedefinedinthefollowingways.

Simplecorrelationcoefficient:alsocalledcorrelationcoefficientorlinearcorrelationcoefficient,generallyrepresentedbytheletterr,usedtomeasurethelinearrelationshipbetweentwovariables.

Definingformula

Amongthem,Cov(X,Y)isthecovarianceofXandY,Var[X]isthevarianceofX,Var[Y]isthevarianceofY

Multiplecorrelationcoefficient:alsocalledmultiplecorrelationcoefficient.Multiplecorrelationreferstothecorrelationbetweenthedependentvariableandmultipleindependentvariables.Forexample,thereisacomplexcorrelationbetweentheseasonaldemandforacertaincommodityanditspricelevel,employeeincomelevelandotherphenomena.

Thecanonicalcorrelationcoefficient:firstlyconductprincipalcomponentanalysisontheoriginalsetofvariablestoobtainthecomprehensiveindexofthenewlinearrelationship,andthenusethelinearcorrelationcoefficientbetweenthecomprehensiveindexestostudythecorrelationbetweentheoriginalsetofvariablesrelation.

Nature

Here,,isonethatcancharacterizeand

Theamountofclosenessofthelinearrelationship.Ithastwoproperties:

(1)

(2)Thenecessaryandsufficientconditionofisthatthereisaconstanta,b,making

derivedfromproperties:

a.ThecorrelationcoefficientquantitativelydescribesthedegreeofcorrelationbetweenXandY,namely

Thebigger,thegreaterthedegreeofcorrelation;correspondstothelowestdegreeofcorrelation;

b.XandYarecompletelycorrelatedmeansthatthereisalinearrelationshipwiththeprobabilityof1,soisaquantitythatcancharacterizetheclosenessofthelinearrelationshipbetweenXandY.Whenislarge,itisusuallysaidthatXandYarerelatedbetter;whenissmall,itisusuallysaidthatXandYarelessrelated;whenXandYarenotrelated,ItisgenerallybelievedthatthereisnolinearrelationshipbetweenXandY,butitcannotberuledoutthattheremaybeotherrelationshipsbetweenXandY.

Irrelevantandindependent

IfXandYareirrelevant,,itisgenerallyconsideredthatthereisnolinearrelationshipbetweenXandY,butXcannotberuledoutTheremaybeotherrelationshipsbetweenXandY;if,XandYarenotrelated.

IfXandYareindependent,theremustbe,soXandYarenotrelated;ifXandYarenotrelated,thereissimplynolinearrelationship,andotherrelationshipsmayexist,Suchas,XandYarenotindependent.

Therefore,"irrelevant"isaweakerconceptthan"independent".

Examplesoflife

Softwarecompanieshavemanyagentsthroughoutthecountry.Inordertostudytherelationshipbetweenadvertisinginvestmentandsalesofitsfinancialsoftwareproducts,statisticiansrandomlyselected10agentstoconductresearch.Observe,collecttheannualadvertisingexpenditureandmonthlyaveragesalesdata,andcompilethemintorelatedtables,seeTable1:

Annualadvertisingexpenditure

12.5

15.3

23.2

26.4

33.5

34.4

39.4

45.2

55.4

60.9

monthAveragesales

Table1Advertisingexpensesandmonthlyaveragesalesrelatedtableunit:tenthousandYuan
21.2	23.9	32.9	34.1	42.5	43.2	49.0	52.8	59.4	63.5

RefertoTable1,andthecorrelationcoefficientcanbecalculatedasTable2:

Serialnumber	Advertisinginvestment(tenthousandyuan) x	Averagemonthlysales(tenthousandyuan)) y
1 2 3 4 5 6 7 8 9 10	12.5 15.3 23.2 26.4 33.5 34.4 39.4 45.2 55.4 60.9	21.2 23.9 32.9 34.1 42.5 43.2 49.0 52.8 59.4 63.5	156.25 234.09 538.24 696.96 1122.25 1183.36 1552.36 2043.04 3069.16 3708.81	449.44 571.21 1082.41 1162.81 1806.25 1866.24 2401.00 2787.84 3528.36 4032.25	265.00 365.67 763.28 900.24 1423.75 1486.08 1930.60 2386.56 3290.76 3867.15
Total	346.2	422.5	14304.52	19687.81	16679.09

Serialnumber

Advertisinginvestment(tenthousandyuan)

Averagemonthlysales(tenthousandyuan))

12.5

15.3

23.2

26.4

33.5

34.4

39.4

45.2

55.4

60.9

21.2

23.9

32.9

34.1

42.5

43.2

49.0

52.8

59.4

63.5

156.25

234.09

538.24

696.96

1122.25

1183.36

1552.36

2043.04

3069.16

3708.81

449.44

571.21

1082.41

1162.81

1806.25

1866.24

2401.00

2787.84

3528.36

4032.25

265.00

365.67

763.28

900.24

1423.75

1486.08

1930.60

2386.56

3290.76

3867.15

Total

346.2

422.5

14304.52

19687.81

16679.09

Thecorrelationcoefficientis0.9942,indicatingthatthereisahighdegreeoflinearpositivecorrelationbetweenadvertisingexpenditureandmonthlyaveragesales.

Application

ProbabilityTheory

[Example]Ifacoinistossedntimes,Xrepresentsthenumberofheadsinntrials,andYrepresentsnThenumberofnegativesinthistrial.Calculateρ_X_Y.

Solution:SinceX+Y=n,thenY=-X+n,inferringfromthenatureofthecorrelationcoefficient,ρ_X_Y=−1.

Enterpriselogistics

[Example]Anewproductislaunched.Beforelisting,thecompany’slogisticsdepartmentneedstoallocatethenewproductsto10warehousesacrossthecountry.Onemonthafterthenewproductsarelisted,itisnecessarytoevaluatewhethertheactualdistributionplanisbetterthantheotherdistributionplansconsideredbefore,orwhetherithasnotbeenused.Thedistributionplanisbetter.Throughthisevaluation,amoreaccurateproductdistributionplancanbeusedinthenextnewproductlaunchtoavoidbacklogandoutofstockduetodistribution.Table1isanumbertablebasedonactualdata.

Throughcalculations,itiseasytofindthatamongthethreeallocationplans,thecorrelationcoefficientofBisthelargest.Inthisway,itisestimatedthattheallocationplanofBisbetterthantheactualallocationplanA.InthenextnewproductInthelistingdistributionplan,thedistributionmethodBcanbeconsideredtocalculatetheactualdistributionplan.

Clusteranalysis

[Example]Ifthereareseveralsamples,andeachsamplehasnfeatures,thecorrelationcoefficientcanindicatethedegreeofsimilaritybetweenthetwosamples.Inthisway,distanceclusteringcanbeperformedontheclosenessandclosenessofthesamples.Forexample,9wheatvarieties(respectivelyuseA₁,A₂,...,A₉)The6traitsdataareshowninTable2,andthecorrelationcoefficientiscalculatedandtested.

Thecorrelationcoefficientamongthe6traitscanbecalculatedfromthecorrelationcoefficientcalculationformula.TheanalysisandtestresultsareshowninTable3.ItcanbeseenfromTable3thatthereisanegativecorrelationbetweentilleringinwinterandthenumberofgrainsperspike(ρ=−0.8982),thatis,themoretillersinwinter,thelessthenumberofgrainsperspike,andtherelationshipbetweenothertraitsisnotSignificantly.

Disadvantages

Itshouldbepointedoutthatthecorrelationcoefficienthasanobviousdisadvantage,thatis,itscloseto1isrelatedtothenumberofdatasetsn,whichiseasytogivepeopleanillusion.Thisisbecausewhennissmall,thecorrelationcoefficientfluctuatesgreatly,andtheabsolutevalueofthecorrelationcoefficientiseasilycloseto1forsomesamples;whennislarge,theabsolutevalueofthecorrelationcoefficientislikelytobesmall.Especiallywhenn=2,theabsolutevalueofthecorrelationcoefficientisalways1.Therefore,whenthesamplesizenissmall,itisnotappropriateforustojudgethatthereisacloselinearrelationshipbetweenthevariablesxandybasedonthelargecorrelationcoefficient.