transformations. transformations to linearity many non-linear curves can be put into a linear form...
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Transformations
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Transformations to Linearity Many non-linear curves can be put into a linear form by appropriate transformations of the either the dependent variable Y or some (or all) of the independent variables X1, X2, ... , Xp . This leads to the wide utility of the Linear model. We have seen that through the use of dummy variables, categorical independent variables can be incorporated into a Linear Model. We will now see that through the technique of variable transformation that many examples of non-linear behaviour can also be converted to linear behaviour.
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Intrinsically Linear (Linearizable) Curves 1 Hyperbolas y = x/(ax-b)Linear form: 1/y = a -b (1/x) or Y = b0 + b1 XTransformations: Y = 1/y, X=1/x, b0 = a, b1 = -b
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2. Exponentialy = a ebx = aBxLinear form: ln y = lna + b x = lna + lnB x or Y = b0 + b1 XTransformations: Y = ln y, X = x, b0 = lna, b1 = b = lnB
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3. Power Functionsy = a xbLinear from: ln y = lna + blnx or Y = b0 + b1 X
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Logarithmic Functionsy = a + b lnxLinear from: y = a + b lnx or Y = b0 + b1 XTransformations: Y = y, X = ln x, b0 = a, b1 = b
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Other special functionsy = a e b/x Linear from: ln y = lna + b 1/x or Y = b0 + b1 XTransformations: Y = ln y, X = 1/x, b0 = lna, b1 = b
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Sheet: Sheet2 (2)
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-
Polynomial Modelsy = b0 + b1x + b2x2 + b3x3 Linear form Y = b0 + b1 X1 + b2 X2 + b3 X3Variables Y = y, X1 = x , X2 = x2, X3 = x3
-
Exponential Models with a polynomial exponent Linear form lny = b0 + b1 X1 + b2 X2 + b3 X3+ b4 X4Y = lny, X1 = x , X2 = x2, X3 = x3, X4 = x4
-
Trigonometric Polynomial Modelsy = b0 + g1cos(2pf1x) + d1sin(2pf1x) + +gkcos(2pfkx) + dksin(2pfkx)Linear form Y = b0 + g1 C1 + d1 S1 + + gk Ck + dk Sk Variables Y = y, C1 = cos(2pf1x) , S2 = sin(2pf1x) , Ck = cos(2pfkx) , Sk = sin(2pfkx)
Chart1
02040117
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2-02-04-0117
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3-02-04-0117
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3.064.41749463221.85955297184.10728263562.91587450971.80965410490.425779291625.5356381457
3.085.78104408921.75261336015.0659675532.14330717991.99605345690.062790519526.8017761586
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3.128.21456527111.45793725485.98816037060.25116207811.5410264856-0.637423989726.8154274705
3.149.24615891331.27484797955.8937235044-0.74952525830.9635073482-0.8763066825.752405807
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3.50-2-04-0-111
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4-02-04-0117
Sheet1
Sheet2
Sheet3
-
Response Surface modelsDependent variable Y and two independent variables x1 and x2. (These ideas are easily extended to more the two independent variables)The Model (A cubic response surface model)or Y = b0 + b1 X1 + b2 X2 + b3 X3 + b4 X4 + b5 X5 + b6 X6 + b7 X7 + b8 X8 + b9 X9+ ewhere
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The Box-Cox Family of Transformations
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The Transformation Staircase
-
The Bulging Rule
-
Non-Linear ModelsNonlinearizable models
-
Non-Linear Growth models many models cannot be transformed into a linear model The Mechanistic Growth Model Equation:or (ignoring e) rate of increase in Y =
Sheet: Mechanistic Growth
Sheet: Sheet1
Sheet: Sheet2
Sheet: Sheet2 (2)
Sheet: Sheet3
Sheet: Sheet4
Sheet: Sheet5
Sheet: Sheet6
Sheet: Sheet7
Sheet: Sheet8
Sheet: Sheet9
Sheet: Sheet10
Sheet: Sheet11
Sheet: Sheet12
Sheet: Sheet13
Sheet: Sheet14
Sheet: Sheet15
Sheet: Sheet16
alpha
1.0
k
1.0
beta
0.1
0.819032516392808
0.6380650327856161
0.45709754917842427
0.2761300655712323
0.09516258196404048
0.2
0.8362538493844036
0.6725076987688072
0.508761548153211
0.34501539753761445
0.18126924692201818
0.3
0.8518363558636564
0.7036727117273128
0.5555090675909693
0.40734542345462565
0.2591817793182821
0.4
0.8659359907928721
0.7318719815857442
0.5978079723786165
0.46374396317148847
0.3296799539643607
0.5
0.8786938680574733
0.7573877361149466
0.63608160417242
0.5147754722298932
0.3934693402873666
0.6
0.8902376727811947
0.7804753455623894
0.6707130183435841
0.5609506911247788
0.4511883639059736
0.7
0.900682939241718
0.8013658784834362
0.7020488177251543
0.6027317569668724
0.5034146962085905
0.8
0.9101342071765557
0.8202684143531114
0.7304026215296671
0.6405368287062227
0.5506710358827784
0.9
0.9186860680518801
0.8373721361037604
0.7560582041556405
0.6747442722075208
0.5934303402594009
1.0
0.9264241117657115
0.8528482235314231
0.7792723352971346
0.7056964470628462
0.6321205588285577
1.1
0.9334257832603841
0.8668515665207681
0.8002773497811523
0.7337031330415363
0.6671289163019205
1.2
0.9397611576175595
0.8795223152351191
0.8192834728526788
0.7590446304702383
0.6988057880877978
1.3
0.9454936413931975
0.8909872827863949
0.8364809241795924
0.7819745655727899
0.7274682069659875
1.4
0.9506806072116787
0.9013612144233574
0.8520418216350361
0.8027224288467147
0.7534030360583935
1.5
0.955373967970314
0.9107479359406281
0.8661219039109421
0.8214958718812562
0.7768698398515702
1.6
0.959620696401069
0.9192413928021379
0.8788620892032067
0.8384827856042757
0.7981034820053446
1.7
0.9634632951894531
0.9269265903789061
0.8903898855683592
0.8538531807578122
0.8173164759472653
1.8
0.9669402223556827
0.9338804447113653
0.9008206670670481
0.8677608894227308
0.8347011117784134
1.9
0.9700862761554729
0.940172552310946
0.9102588284664189
0.880345104621892
0.8504313807773649
2.0
0.9729329433526774
0.9458658867053549
0.9187988300580324
0.8917317734107099
0.8646647167633873
2.1
0.9755087143494037
0.9510174286988072
0.9265261430482109
0.9020348573976145
0.8775435717470181
2.2
0.9778393683275333
0.9556787366550664
0.9335181049825997
0.9113574733101328
0.8891968416376661
2.3
0.9799482312554393
0.9598964625108785
0.9398446937663177
0.919792925021757
0.8997411562771962
2.4
0.9818564093421175
0.963712818684235
0.9455692280263525
0.92742563736847
0.9092820467105875
2.5
0.9835830002752203
0.9671660005504404
0.9507490008256607
0.934332001100881
0.9179150013761012
2.6
0.9851452843571332
0.9702905687142664
0.9554358530713997
0.9405811374285329
0.9257264217856661
2.7
0.98655889745205
0.9731177949041001
0.9596766923561502
0.9462355898082002
0.9327944872602503
2.8
0.9878379874749564
0.9756759749499128
0.9635139624248692
0.9513519498998256
0.9391899373747821
2.9
0.9889953559887186
0.9779907119774371
0.9669860679661557
0.9559814239548742
0.9449767799435927
3.0
0.9900425863264272
0.9800851726528544
0.9701277589792816
0.9601703453057089
0.950212931632136
3.1
0.9909901595212884
0.9819803190425769
0.9729704785638653
0.9639606380851538
0.9549507976064422
3.2
0.9918475592043268
0.9836951184086535
0.9755426776129803
0.9673902368173071
0.9592377960216338
3.3
0.992623366519752
0.9852467330395039
0.977870099559256
0.970493466079008
0.96311683259876
3.4
0.9933253460079348
0.9866506920158695
0.9799760380238044
0.9733013840317392
0.9666267300396739
3.5
0.9939605233155363
0.9879210466310726
0.9818815699466089
0.9758420932621452
0.9698026165776815
3.6
0.9945352555105414
0.989070511021083
0.9836057665316245
0.9781410220421659
0.9726762775527075
3.7
0.9950552947059321
0.9901105894118643
0.9851658841177964
0.9802211788237285
0.9752764735296606
3.8
0.9955258456287669
0.9910516912575338
0.9865775368863007
0.9821033825150676
0.9776292281438344
3.9
0.9959516177108392
0.9919032354216782
0.9878548531325174
0.9838064708433565
0.9797580885541957
4.0
0.9963368722222532
0.9926737444445063
0.9890106166667595
0.9853474888890127
0.9816843611112658
4.1
0.9966854649196477
0.9933709298392955
0.9900563947589432
0.986741859678591
0.9834273245982388
4.2
0.9970008846359044
0.9940017692718089
0.9910026539077134
0.9880035385436179
0.9850044231795223
4.3
0.9972862881975598
0.9945725763951196
0.9918588645926795
0.9891451527902393
0.9864314409877991
4.4
0.9975445320193863
0.9950890640387726
0.9926335960581589
0.9901781280775452
0.9877226600969315
4.5
0.9977782006923516
0.995556401384703
0.9933346020770546
0.9911128027694062
0.9888910034617577
4.6
0.9979896328510733
0.9959792657021466
0.9939688985532198
0.9919585314042931
0.9899481642553665
4.7
0.9981809445796609
0.9963618891593217
0.9945428337389826
0.9927237783186433
0.9909047228983042
4.8
0.998354050590196
0.996708101180392
0.995062151770588
0.993416202360784
0.99177025295098
4.9
0.9985106833858152
0.9970213667716302
0.9955320501574454
0.9940427335432606
0.9925534169290756
5.0
0.998652410600183
0.9973048212003658
0.9959572318005487
0.9946096424007316
0.9932620530009145
a
1.0
b
0.1
3.162277660168379
10.0
31.62277660168379
0.2
2.23606797749979
5.0
11.180339887498945
0.3
1.8257418583505538
3.3333333333333335
6.085806194501847
0.4
1.5811388300841895
2.5
3.9528470752104736
0.5
1.414213562373095
2.0
2.82842712474619
0.6
1.2909944487358056
1.6666666666666667
2.151657414559676
0.7
1.1952286093343936
1.4285714285714286
1.7074694419062768
0.8
1.118033988749895
1.25
1.3975424859373686
0.9
1.0540925533894598
1.1111111111111112
1.1712139482105108
1.0
1.0
1.0
1.0
1.1
0.9534625892455922
0.9090909090909091
0.8667841720414474
1.2
0.9128709291752769
0.8333333333333334
0.7607257743127307
1.3
0.8770580193070292
0.7692307692307692
0.6746600148515609
1.4
0.8451542547285166
0.7142857142857143
0.6036816105203691
1.5
0.8164965809277261
0.6666666666666666
0.5443310539518174
1.6
0.7905694150420948
0.625
0.4941058844013092
1.7
0.7669649888473704
0.5882352941176471
0.4511558757925709
1.8
0.7453559924999299
0.5555555555555556
0.41408666249996107
1.9
0.7254762501100117
0.5263157894736842
0.38182960532105886
2.0
0.7071067811865475
0.5
0.3535533905932738
2.1
0.6900655593423541
0.47619047619047616
0.328602647305883
2.2
0.674199862463242
0.45454545454545453
0.3064544829378372
2.3
0.6593804733957871
0.4347826086956522
0.2866871623459944
2.4
0.6454972243679028
0.4166666666666667
0.26895717681995956
2.5
0.6324555320336759
0.4
0.25298221281347033
2.6
0.6201736729460422
0.3846153846153846
0.2385283357484778
2.7
0.6085806194501845
0.37037037037037035
0.22540022942599425
2.8
0.5976143046671968
0.35714285714285715
0.21343368023828466
2.9
0.5872202195147035
0.3448275862068966
0.20248973086713917
3.0
0.5773502691896258
0.3333333333333333
0.19245008972987526
a
1.0
b
-1.5
0.1
4.453877639491068
0.2
3.4141568686511503
0.3
2.8059592064889043
0.4
2.374436097811232
0.5
2.039720770839918
0.6
1.7662384356489862
0.7
1.5350124159080987
0.8
1.3347153269713146
0.9
1.1580407734867395
1.0
1.0
1.1
0.8570347302935126
1.2
0.7265176648090681
1.3
0.6064536032987634
1.4
0.4952916450681807
1.5
0.3918023378377534
1.6
0.2949945561313966
1.7
0.20405762340674438
1.8
0.11832000264682141
1.9
0.037219170741408014
2.0
-0.039720770839917874
2.1
-0.11290601709406589
2.2
-0.18268604054640547
2.3
-0.24936368440265588
2.4
-0.3132031060308498
2.5
-0.37443609781123266
2.6
-0.4332671675411546
2.7
-0.4898776595154253
2.8
-0.5444291257717371
2.9
-0.5970661054886424
3.0
-0.6479184330021646
3.1
-0.697103167236651
3.2
-0.7447262147085212
3.3
-0.7908837027086517
3.4
-0.8356631474331735
3.5
-0.8791444527430521
3.6
-0.9214007681930965
3.7
-0.9624992294752683
3.8
-1.00250160009851
3.9
-1.041464829703401
4.0
-1.0794415416798357
4.1
-1.116480460565393
4.2
-1.152626787933984
4.3
-1.187922534049275
4.4
-1.2224068113863233
4.5
-1.256116095164411
4.6
-1.2890844552425738
4.7
-1.3213437630740197
4.8
-1.3529238768707676
4.9
-1.3838528076748715
5.0
-1.4141568686511503
a
1.0
b
0.5
0.1
148.4131591025766
0.2
12.182493960703473
0.3
5.29449005047003
0.4
3.4903429574618414
0.5
2.718281828459045
0.6
2.300975890892825
0.7
2.042727070266142
0.8
1.8682459574322223
0.9
1.7429089986334578
1.0
1.6487212707001282
1.1
1.5754571033903182
1.2
1.5168967963882134
1.3
1.4690491938490164
1.4
1.4292400324179777
1.5
1.3956124250860895
1.6
1.3668379411737963
1.7
1.3419417698828804
1.8
1.3201927884341202
1.9
1.3010321288603353
2.0
1.2840254166877414
2.1
1.2688300280258125
2.2
1.2551721409393686
2.3
1.2428303319121348
2.4
1.2316236423470497
2.5
1.2214027581601699
2.6
1.2120433960254957
2.7
1.2034412789899398
2.8
1.1955082736719047
2.9
1.188169387804344
3.0
1.1813604128656459
3.1
1.1750260558957077
3.2
1.1691184461695043
3.3
1.1635959319175064
3.4
1.1584221035023807
3.5
1.1535649948951077
3.6
1.148996426641145
3.7
1.1446914619352877
3.8
1.1406279537431718
3.9
1.136786165687872
4.0
1.1331484530668263
4.1
1.1296989931682577
4.2
1.126423556228212
4.3
1.12330931006362
4.4
1.1203446527472556
4.5
1.1175190687418637
4.6
1.1148230047465537
4.7
1.1122477621768179
4.8
1.1097854037367088
4.9
1.1074286719757578
5.0
1.1051709180756477
-
The Logistic Growth Model
10
8
6
4
2
0
0.0
0.5
1.0
Logistic Growth Model
x
y
k=1/4
k=1/2
k=1
k=2
k=4
-
The Gompertz Growth Model:or (ignoring e) rate of increase in Y =
10
8
6
4
2
0
0.0
0.2
0.4
0.6
0.8
1.0
Gompertz Growth Model
x
y
k = 1
-
Example: daily auto accidents in Saskatchewan to 1984 to 1992Data collected:DateNumber of AccidentsFactors we want to consider:TrendYearly Cyclical EffectDay of the week effectHoliday effects
-
TrendThis will be modeled by a Linear function :Y = b0 +b1 X(more generally a polynomial)Y = b0 +b1 X +b2 X2 + b3 X3 + .
Yearly Cyclical TrendThis will be modeled by a Trig Polynomial Sin and Cos functions with differing frequencies(periods) :Y = d1 sin(2pf1X) + g1 cos(2pf2X) + d1 sin(2pf2X) + g2 cos(2pf2X) +
-
Day of the week effect:This will be modeled using dummyvariables :a1 D1 + a2 D2 + a3 D3 + a4 D4 + a5 D5 + a6 D6
Di = (1 if day of week = i, 0 otherwise)Holiday EffectsAlso will be modeled using dummyvariables :
-
Independent variablesX = day,D1,D2,D3,D4,D5,D6,S1,S2,S3,S4,S5, S6,C1,C2,C3,C4,C5,C6,NYE,HW,V1,V2,cd,T1,T2.
Si=sin(0.017202423838959*i*day). Ci=cos(0.017202423838959*i*day).
Dependent variableY = daily accident frequency
-
Independent variables ANALYSIS OF VARIANCE SUM OF SQUARES DF MEAN SQUARE F RATIO REGRESSION 976292.38 18 54238.46 114.60 RESIDUAL 1547102.1 3269 473.2646 VARIABLES IN EQUATION FOR PACC . VARIABLES NOT IN EQUATION STD. ERROR STD REG F . PARTIAL F VARIABLE COEFFICIENT OF COEFF COEFF TOLERANCE TO REMOVE LEVEL. VARIABLE CORR. TOLERANCE TO ENTER LEVEL (Y-INTERCEPT 60.48909 ) . day 1 0.11107E-02 0.4017E-03 0.038 0.99005 7.64 1 . IACC 7 0.49837 0.78647 1079.91 0 D1 9 4.99945 1.4272 0.063 0.57785 12.27 1 . Dths 8 0.04788 0.93491 7.51 0 D2 10 9.86107 1.4200 0.124 0.58367 48.22 1 . S3 17 -0.02761 0.99511 2.49 1 D3 11 9.43565 1.4195 0.119 0.58311 44.19 1 . S5 19 -0.01625 0.99348 0.86 1 D4 12 13.84377 1.4195 0.175 0.58304 95.11 1 . S6 20 -0.00489 0.99539 0.08 1 D5 13 28.69194 1.4185 0.363 0.58284 409.11 1 . C6 26 -0.02856 0.98788 2.67 1 D6 14 21.63193 1.4202 0.273 0.58352 232.00 1 . V1 29 -0.01331 0.96168 0.58 1 S1 15 -7.89293 0.5413 -0.201 0.98285 212.65 1 . V2 30 -0.02555 0.96088 2.13 1 S2 16 -3.41996 0.5385 -0.087 0.99306 40.34 1 . cd 31 0.00555 0.97172 0.10 1 S4 18 -3.56763 0.5386 -0.091 0.99276 43.88 1 . T1 32 0.00000 0.00000 0.00 1 C1 21 15.40978 0.5384 0.393 0.99279 819.12 1 . C2 22 7.53336 0.5397 0.192 0.98816 194.85 1 . C3 23 -3.67034 0.5399 -0.094 0.98722 46.21 1 . C4 24 -1.40299 0.5392 -0.036 0.98999 6.77 1 . C5 25 -1.36866 0.5393 -0.035 0.98955 6.44 1 . NYE 27 32.46759 7.3664 0.061 0.97171 19.43 1 . HW 28 35.95494 7.3516 0.068 0.97565 23.92 1 . T2 33 -18.38942 7.4039 -0.035 0.96191 6.17 1 . ***** F LEVELS( 4.000, 3.900) OR TOLERANCE INSUFFICIENT FOR FURTHER STEPPING
-
Day of the week effects
D14.99945D29.86107D39.43565D413.84377D528.69194D621.63193
-
Chart1
65.48854
70.35016
69.92474
74.33286
89.18103
82.12102
60.48909
Day of Week Effect
Graph
S1S2S4C1C2C3C4C5
-7.89293-3.41996-3.5676315.409787.53336-3.67034-1.40299-1.36866
112.3349672349
212.0219155888
311.7286149407
411.454849376
511.2002722837
610.9644105571
710.7466694701
810.5463381894
910.3625958832
1010.1945183799
1110.0410853298
129.9011878194
139.7736363868
149.6571693832
159.5504616268
169.4521332904
179.3607589679
189.2748768624
199.1929980368
209.1136156733
219.0352142845
228.9562788215
238.8753036284
248.7908011894
258.701310622
268.6054058705
278.5017035536
288.3888704299
298.2656304404
308.1307712971
317.9831505856
327.8217013574
337.6454371868
347.4534566765
357.2449473954
367.0191892381
376.7755572008
386.5135235692
396.23265952
405.9326361422
415.6132248844
425.2742974431
434.9158251049
444.5378775628
454.1406212285
463.7243170637
473.2893179592
482.8360656899
492.3650874765
501.8769921889
511.3724662234
520.8522690905
530.3172287501
54-0.2317632701
55-0.793756935
56-1.3677489034
57-1.9526878347
58-2.5474797631
59-3.1509935047
60-3.7620660623
61-4.3795079952
62-5.002108722
63-5.6286417259
64-6.2578696364
65-6.8885491588
66-7.51943583
67-8.1492885783
68-8.7768740682
69-9.4009708144
70-10.0203730515
71-10.6338943467
72-11.2403709501
73-11.8386648728
74-12.4276666942
75-13.006298093
76-13.5735141084
77-14.1283051315
78-14.669698637
79-15.1967606599
80-15.708597032
81-16.2043543863
82-16.6832209458
83-17.1444271116
84-17.5872458657
85-18.0109930071
86-18.4150272389
87-18.798750126
88-19.1616059413
89-19.5030814209
90-19.8227054473
91-20.1200486787
92-20.3947231441
93-20.6463818209
94-20.8747182118
95-21.0794659372
96-21.2603983574
97-21.4173282368
98-21.5501074621
99-21.6586268252
100-21.742815876
101-21.8026428549
102-21.838114705
103-21.8492771705
104-21.8362149771
105-21.7990520968
106-21.7379520889
107-21.6531185157
108-21.5447954214
109-21.4132678672
110-21.2588625106
111-21.0819482149
112-20.8829366766
113-20.6622830532
114-20.4204865753
115-20.1580911252
116-19.8756857632
117-19.573905182
118-19.2534300697
119-18.9149873623
120-18.559350364
121-18.1873387179
122-17.799818206
123-17.3977003615
124-16.9819418753
125-16.5535437796
126-16.1135503954
127-15.6630480279
128-15.2031633999
129-14.7350618127
130-14.2599450259
131-13.7790488511
132-13.2936404561
133-12.8050153782
134-12.3144942483
135-11.8234192303
136-11.3331501814
137-10.8450605443
138-10.3605329822
139-9.8809547711
140-9.4077129679
141-8.9421893727
142-8.4857553088
143-8.0397662433
144-7.6055562768
145-7.1844325299
146-6.7776694572
147-6.3865031216
148-6.0121254619
149-5.6556785892
150-5.3182491478
151-5.0008627776
152-4.7044787145
153-4.4299845671
154-4.1781913056
155-3.9498285017
156-3.7455398535
157-3.5658790326
158-3.4113058865
159-3.2821830288
160-3.1787728486
161-3.1012349673
162-3.0496241691
163-3.0238888298
164-3.0238698641
165-3.0493002104
166-3.0998048674
167-3.1749014939
168-3.2740015812
169-3.3964122011
170-3.5413383312
171-3.707885754
172-3.8950645233
173-4.1017929874
174-4.3269023535
175-4.5691417766
176-4.8271839493
177-5.0996311669
178-5.3850218404
179-5.6818374214
180-5.9885097063
181-6.3034284784
182-6.6249494479
183-6.9514024442
184-7.2810998148
185-7.6123449827
186-7.9434411099
187-8.2726998186
188-8.5984499144
189-8.9190460608
190-9.2328773487
191-9.5383757106
192-9.8340241224
193-10.1183645437
194-10.3900055435
195-10.6476295622
196-10.8899997605
197-11.1159664119
198-11.3244727932
199-11.5145605335
200-11.6853743848
201-11.8361663792
202-11.9662993437
203-12.0752497458
204-12.1626098478
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