Transformations. Transformations to Linearity Many non-linear curves can be put into a linear form by appropriate transformations of the either – the

Transformations

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.

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

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

3. Power Functionsy = a xbLinear from: ln y = lna + blnx or Y = b0 + b1 X

Sheet: Sheet1

Sheet: Sheet2

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Sheet: Sheet15

Sheet: Sheet16

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0.4472135954999579

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0.08944271909999162

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0.5477225575051661

0.3

0.16431676725154978

0.4

0.6324555320336759

0.4

0.2529822128134704

0.5

0.7071067811865476

0.5

0.3535533905932738

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0.7745966692414834

0.6

0.46475800154489

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0.8366600265340756

0.7

0.5856620185738528

0.8

0.8944271909999159

0.8

0.7155417527999327

0.9

0.9486832980505138

0.9

0.8538149682454624

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1.0

1.0

1.0

1.1

1.0488088481701516

1.1

1.153689732987167

1.2

1.0954451150103321

1.2

1.3145341380123987

1.3

1.140175425099138

1.3

1.4822280526288794

1.4

1.1832159566199232

1.4

1.6565023392678924

1.5

1.224744871391589

1.5

1.8371173070873836

1.6

1.2649110640673518

1.6

2.023857702507763

1.7

1.3038404810405297

1.7

2.2165288177689004

1.8

1.3416407864998738

1.8

2.414953415699773

1.9

1.378404875209022

1.9

2.6189692628971417

2.0

1.4142135623730951

2.0

2.82842712474619

2.1

1.449137674618944

2.1

3.0431891166997818

2.2

1.4832396974191326

2.2

3.2631273343220926

2.3

1.51657508881031

2.3

3.488122704263713

2.4

1.5491933384829668

2.4

3.7180640123591195

2.5

1.5811388300841898

2.5

3.9528470752104745

2.6

1.61245154965971

2.6

4.192374029115246

2.7

1.6431676725154984

2.7

4.436552715791846

2.8

1.6733200530681511

2.8

4.6852961485908216

2.9

1.70293863659264

2.9

4.938522046118655

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1.7320508075688772

3.0

5.196152422706632

Sheet: Sheet1

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Sheet: Sheet9

Sheet: Sheet10

Sheet: Sheet11

Sheet: Sheet12

Sheet: Sheet13

Sheet: Sheet14

Sheet: Sheet15

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3.162277660168379

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31.62277660168379

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2.23606797749979

5.0

11.180339887498945

0.3

1.8257418583505538

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1.5811388300841895

2.5

3.9528470752104736

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1.414213562373095

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2.82842712474619

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1.2909944487358056

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1.1952286093343936

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1.118033988749895

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1.3975424859373686

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1.0540925533894598

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1.0

1.0

1.0

1.1

0.9534625892455922

0.9090909090909091

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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

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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

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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

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0.25298221281347033

2.6

0.6201736729460422

0.3846153846153846

0.2385283357484778

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0.6085806194501845

0.37037037037037035

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0.5976143046671968

0.35714285714285715

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2.9

0.5872202195147035

0.3448275862068966

0.20248973086713917

3.0

0.5773502691896258

0.3333333333333333

0.19245008972987526

Logarithmic Functionsy = a + b lnxLinear from: y = a + b lnx or Y = b0 + b1 XTransformations: Y = y, X = ln x, b0 = a, b1 = b

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

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3.9528470752104736

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1.414213562373095

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2.82842712474619

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1.2909944487358056

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1.1952286093343936

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1.3975424859373686

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1.0540925533894598

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0.8333333333333334

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0.6746600148515609

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0.8164965809277261

0.6666666666666666

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0.7905694150420948

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0.7669649888473704

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0.7453559924999299

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0.7254762501100117

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0.7071067811865475

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0.3535533905932738

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0.6454972243679028

0.4166666666666667

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0.6324555320336759

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0.6201736729460422

0.3846153846153846

0.2385283357484778

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0.6085806194501845

0.37037037037037035

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0.5976143046671968

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0.5872202195147035

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0.20248973086713917

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0.5773502691896258

0.3333333333333333

0.19245008972987526

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1.0

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0.1

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0.6

0.23376156435101392

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0.46498758409190133

0.8

0.6652846730286854

0.9

0.8419592265132606

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1.0

1.1

1.1429652697064874

1.2

1.273482335190932

1.3

1.3935463967012365

1.4

1.5047083549318194

1.5

1.6081976621622465

1.6

1.7050054438686035

1.7

1.7959423765932556

1.8

1.8816799973531786

1.9

1.962780829258592

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2.039720770839918

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2.112906017094066

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2.1826860405464057

2.3

2.249363684402656

2.4

2.3132031060308496

2.5

2.3744360978112327

2.6

2.4332671675411546

2.7

2.4898776595154253

2.8

2.544429125771737

2.9

2.5970661054886426

3.0

2.6479184330021646

3.1

2.697103167236651

3.2

2.744726214708521

3.3

2.7908837027086517

3.4

2.8356631474331735

3.5

2.879144452743052

3.6

2.9214007681930965

3.7

2.9624992294752683

3.8

3.00250160009851

3.9

3.041464829703401

4.0

3.0794415416798357

4.1

3.116480460565393

4.2

3.152626787933984

4.3

3.187922534049275

4.4

3.2224068113863233

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3.256116095164411

4.6

3.2890844552425738

4.7

3.3213437630740197

4.8

3.3529238768707676

4.9

3.3838528076748715

5.0

3.4141568686511503

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1.0

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0.1

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11.180339887498945

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1.414213562373095

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1.2909944487358056

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1.1952286093343936

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1.118033988749895

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1.3975424859373686

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1.0540925533894598

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1.0

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0.6746600148515609

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0.7142857142857143

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0.8164965809277261

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0.7905694150420948

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0.4941058844013092

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0.7669649888473704

0.5882352941176471

0.4511558757925709

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0.7453559924999299

0.5555555555555556

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0.7254762501100117

0.5263157894736842

0.38182960532105886

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0.7071067811865475

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0.3535533905932738

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0.6900655593423541

0.47619047619047616

0.328602647305883

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0.674199862463242

0.45454545454545453

0.3064544829378372

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0.6593804733957871

0.4347826086956522

0.2866871623459944

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0.6454972243679028

0.4166666666666667

0.26895717681995956

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0.6324555320336759

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0.25298221281347033

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0.6201736729460422

0.3846153846153846

0.2385283357484778

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0.6085806194501845

0.37037037037037035

0.22540022942599425

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0.5976143046671968

0.35714285714285715

0.21343368023828466

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0.5872202195147035

0.3448275862068966

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0.5773502691896258

0.3333333333333333

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1.0

b

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0.1

4.453877639491068

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2.039720770839918

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1.7662384356489862

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1.5350124159080987

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1.3347153269713146

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1.1580407734867395

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1.0

1.1

0.8570347302935126

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0.7265176648090681

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0.6064536032987634

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0.4952916450681807

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0.3918023378377534

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0.2949945561313966

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0.20405762340674438

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0.11832000264682141

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0.037219170741408014

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-0.039720770839917874

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-0.11290601709406589

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-0.5970661054886424

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-0.6479184330021646

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-0.697103167236651

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-0.8791444527430521

3.6

-0.9214007681930965

3.7

-0.9624992294752683

3.8

-1.00250160009851

3.9

-1.041464829703401

4.0

-1.0794415416798357

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-1.116480460565393

4.2

-1.152626787933984

4.3

-1.187922534049275

4.4

-1.2224068113863233

4.5

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4.6

-1.2890844552425738

4.7

-1.3213437630740197

4.8

-1.3529238768707676

4.9

-1.3838528076748715

5.0

-1.4141568686511503

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

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet2 (2)

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Sheet: Sheet4

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Sheet: Sheet7

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Sheet: Sheet11

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Sheet: Sheet15

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0.1

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10.0

31.62277660168379

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11.180339887498945

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1.8257418583505538

3.3333333333333335

6.085806194501847

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1.5811388300841895

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3.9528470752104736

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1.414213562373095

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2.82842712474619

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1.2909944487358056

1.6666666666666667

2.151657414559676

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1.1952286093343936

1.4285714285714286

1.7074694419062768

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1.118033988749895

1.25

1.3975424859373686

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1.0540925533894598

1.1111111111111112

1.1712139482105108

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1.0

1.0

1.1

0.9534625892455922

0.9090909090909091

0.8667841720414474

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0.9128709291752769

0.8333333333333334

0.7607257743127307

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0.8770580193070292

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0.6746600148515609

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0.6036816105203691

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0.8164965809277261

0.6666666666666666

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0.7905694150420948

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0.7669649888473704

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0.7453559924999299

0.5555555555555556

0.41408666249996107

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0.7254762501100117

0.5263157894736842

0.38182960532105886

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0.7071067811865475

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0.3535533905932738

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0.6900655593423541

0.47619047619047616

0.328602647305883

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0.674199862463242

0.45454545454545453

0.3064544829378372

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0.6593804733957871

0.4347826086956522

0.2866871623459944

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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

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Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet2 (2)

Sheet: Sheet3

Sheet: Sheet4

Sheet: Sheet5

Sheet: Sheet6

Sheet: Sheet7

Sheet: Sheet8

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Sheet: Sheet10

Sheet: Sheet11

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Sheet: Sheet15

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5.0

2.7693490391599074

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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1.2211.78744700870.37476262922.2087473161-3.7191059436-1.688655851-0.53582679518.4273683645

1.2411.97632074110.12558103910.7519994014-3.9684588053-1.9645745015-0.187381314616.7334865603

1.2611.9763207411-0.1255810391-0.7519994014-3.9684588053-1.96457450150.187381314615.3530883086

1.2811.7874470087-0.3747626292-2.2087473161-3.7191059436-1.6886558510.53582679514.3320020639

1.311.4126781955-0.6180339887-3.5267115138-3.2360679775-1.17557050460.809016994413.6653112053

1.3210.8579246296-0.8515585831-4.6230794567-2.549695959-0.49737977430.968583161113.3047940176

1.3410.131935106-1.07165359-5.4289623148-1.70311716630.25066646710.992114701313.1709832035

1.369.2461589133-1.2748479795-5.8937235044-0.74952525830.96350734820.8763066813.1678761993

1.388.2145652711-1.4579372548-5.98816037060.25116207811.54102648560.637423989713.1980801991

1.47.0534230275-1.6180339887-5.70633909781.23606797751.90211303260.309016994413.1762479455

1.425.7810440892-1.7526133601-5.0659675532.14330717991.9960534569-0.062790519513.0390332934

1.444.4174946322-1.8595529718-4.10728263562.91587450971.8096541049-0.425779291612.7504083479

1.462.984278646-1.9371663223-2.89052204463.50522672021.3690942119-0.728968627412.3019425837

1.481.5039988028-1.9842294026-1.4921393233.87433264450.7362491054-0.929776485911.7084353411

1.50-2-040-111

1.52-1.5039988028-1.98422940261.4921393233.8743326445-0.7362491054-0.929776485910.2122181708

1.54-2.984278646-1.93716632232.89052204463.5052267202-1.3690942119-0.72896862749.3762409573

1.56-4.4174946322-1.85955297184.10728263562.9158745097-1.8096541049-0.42577929168.5106761448

1.58-5.7810440892-1.75261336015.0659675532.1433071799-1.9960534569-0.06279051957.6167733072

1.6-7.0534230275-1.61803398875.70633909781.2360679775-1.90211303260.30901699446.6778540208

1.62-8.2145652711-1.45793725485.98816037060.2511620781-1.54102648560.63742398975.6632174269

1.64-9.2461589133-1.27484797955.8937235044-0.7495252583-0.96350734820.876306684.5359906851

1.66-10.131935106-1.071653595.4289623148-1.7031171663-0.25066646710.99211470133.2637046867

1.68-10.8579246296-0.85155858314.6230794567-2.5496959590.49737977430.96858316111.8298632204

1.7-11.4126781955-0.61803398873.5267115138-3.23606797751.17557050460.80901699440.2445188509

1.72-11.7874470087-0.37476262922.2087473161-3.71910594361.6886558510.535826795-1.4480856194

1.74-11.9763207411-0.12558103910.7519994014-3.96845880531.96457450150.1873813146-3.166405368

1.76-11.97632074110.1255810391-0.7519994014-3.96845880531.9645745015-0.1873813146-4.7940047219

1.78-11.78744700870.3747626292-2.2087473161-3.71910594361.688655851-0.535826795-6.1877085832

1.8-11.41267819550.6180339887-3.5267115138-3.23606797751.1755705046-0.8090169944-7.1908701878

1.82-10.85792462960.8515585831-4.6230794567-2.5496959590.4973797743-0.9685831611-7.6503448489

1.84-10.1319351061.07165359-5.4289623148-1.7031171663-0.2506664671-0.9921147013-7.4351421656

1.86-9.24615891331.2748479795-5.8937235044-0.7495252583-0.9635073482-0.87630668-6.4543737248

1.88-8.21456527111.4579372548-5.98816037060.2511620781-1.5410264856-0.6374239897-4.6720767841

1.9-7.05342302751.6180339887-5.70633909781.2360679775-1.9021130326-0.3090169944-2.116790186

1.92-5.78104408921.7526133601-5.0659675532.1433071799-1.99605345690.06279051951.1156459604

1.94-4.41749463221.8595529718-4.10728263562.9158745097-1.80965410490.42577929164.8667754003

1.96-2.9842786461.9371663223-2.89052204463.5052267202-1.36909421190.72896862748.9274667674

1.98-1.50399880281.9842294026-1.4921393233.8743326445-0.73624910540.929776485913.0559513019

2-02-04-0117

2.021.50399880281.98422940261.4921393233.87433264450.73624910540.929776485920.5207257642

2.042.9842786461.93716632232.89052204463.50522672021.36909421190.728968627423.4152565723

2.064.41749463221.85955297184.10728263562.91587450971.80965410490.425779291625.5356381457

2.085.78104408921.75261336015.0659675532.14330717991.99605345690.062790519526.8017761586

2.17.05342302751.61803398875.70633909781.23606797751.9021130326-0.309016994427.2069601297

2.128.21456527111.45793725485.98816037060.25116207811.5410264856-0.637423989726.8154274705

2.149.24615891331.27484797955.8937235044-0.74952525830.9635073482-0.8763066825.752405807

2.1610.1319351061.071653595.4289623148-1.70311716630.2506664671-0.992114701324.1879856103

2.1810.85792462960.85155858314.6230794567-2.549695959-0.4973797743-0.968583161122.3169037749

2.211.41267819550.61803398873.5267115138-3.2360679775-1.1755705046-0.809016994420.3367682216

2.2211.78744700870.37476262922.2087473161-3.7191059436-1.688655851-0.53582679518.4273683645

2.2411.97632074110.12558103910.7519994014-3.9684588053-1.9645745015-0.187381314616.7334865603

2.2611.9763207411-0.1255810391-0.7519994014-3.9684588053-1.96457450150.187381314615.3530883086

2.2811.7874470087-0.3747626292-2.2087473161-3.7191059436-1.6886558510.53582679514.3320020639

2.311.4126781955-0.6180339887-3.5267115138-3.2360679775-1.17557050460.809016994413.6653112053

2.3210.8579246296-0.8515585831-4.6230794567-2.549695959-0.49737977430.968583161113.3047940176

2.3410.131935106-1.07165359-5.4289623148-1.70311716630.25066646710.992114701313.1709832035

2.369.2461589133-1.2748479795-5.8937235044-0.74952525830.96350734820.8763066813.1678761993

2.388.2145652711-1.4579372548-5.98816037060.25116207811.54102648560.637423989713.1980801991

2.47.0534230275-1.6180339887-5.70633909781.23606797751.90211303260.309016994413.1762479455

2.425.7810440892-1.7526133601-5.0659675532.14330717991.9960534569-0.062790519513.0390332934

2.444.4174946322-1.8595529718-4.10728263562.91587450971.8096541049-0.425779291612.7504083479

2.462.984278646-1.9371663223-2.89052204463.50522672021.3690942119-0.728968627412.3019425837

2.481.5039988028-1.9842294026-1.4921393233.87433264450.7362491054-0.929776485911.7084353411

2.50-2-040-111

2.52-1.5039988028-1.98422940261.4921393233.8743326445-0.7362491054-0.929776485910.2122181708

2.54-2.984278646-1.93716632232.89052204463.5052267202-1.3690942119-0.72896862749.3762409573

2.56-4.4174946322-1.85955297184.10728263562.9158745097-1.8096541049-0.42577929168.5106761448

2.58-5.7810440892-1.75261336015.0659675532.1433071799-1.9960534569-0.06279051957.6167733072

2.6-7.0534230275-1.61803398875.70633909781.2360679775-1.90211303260.30901699446.6778540208

2.62-8.2145652711-1.45793725485.98816037060.2511620781-1.54102648560.63742398975.6632174269

2.64-9.2461589133-1.27484797955.8937235044-0.7495252583-0.96350734820.876306684.5359906851

2.66-10.131935106-1.071653595.4289623148-1.7031171663-0.25066646710.99211470133.2637046867

2.68-10.8579246296-0.85155858314.6230794567-2.5496959590.49737977430.96858316111.8298632204

2.7-11.4126781955-0.61803398873.5267115138-3.23606797751.17557050460.80901699440.2445188509

2.72-11.7874470087-0.37476262922.2087473161-3.71910594361.6886558510.535826795-1.4480856194

2.74-11.9763207411-0.12558103910.7519994014-3.96845880531.96457450150.1873813146-3.166405368

2.76-11.97632074110.1255810391-0.7519994014-3.96845880531.9645745015-0.1873813146-4.7940047219

2.78-11.78744700870.3747626292-2.2087473161-3.71910594361.688655851-0.535826795-6.1877085832

2.8-11.41267819550.6180339887-3.5267115138-3.23606797751.1755705046-0.8090169944-7.1908701878

2.82-10.85792462960.8515585831-4.6230794567-2.5496959590.4973797743-0.9685831611-7.6503448489

2.84-10.1319351061.07165359-5.4289623148-1.7031171663-0.2506664671-0.9921147013-7.4351421656

2.86-9.24615891331.2748479795-5.8937235044-0.7495252583-0.9635073482-0.87630668-6.4543737248

2.88-8.21456527111.4579372548-5.98816037060.2511620781-1.5410264856-0.6374239897-4.6720767841

2.9-7.05342302751.6180339887-5.70633909781.2360679775-1.9021130326-0.3090169944-2.116790186

2.92-5.78104408921.7526133601-5.0659675532.1433071799-1.99605345690.06279051951.1156459604

2.94-4.41749463221.8595529718-4.10728263562.9158745097-1.80965410490.42577929164.8667754003

2.96-2.9842786461.9371663223-2.89052204463.5052267202-1.36909421190.72896862748.9274667674

2.98-1.50399880281.9842294026-1.4921393233.8743326445-0.73624910540.929776485913.0559513019

3-02-04-0117

3.021.50399880281.98422940261.4921393233.87433264450.73624910540.929776485920.5207257642

3.042.9842786461.93716632232.89052204463.50522672021.36909421190.728968627423.4152565723

3.064.41749463221.85955297184.10728263562.91587450971.80965410490.425779291625.5356381457

3.085.78104408921.75261336015.0659675532.14330717991.99605345690.062790519526.8017761586

3.17.05342302751.61803398875.70633909781.23606797751.9021130326-0.309016994427.2069601297

3.128.21456527111.45793725485.98816037060.25116207811.5410264856-0.637423989726.8154274705

3.149.24615891331.27484797955.8937235044-0.74952525830.9635073482-0.8763066825.752405807

3.1610.1319351061.071653595.4289623148-1.70311716630.2506664671-0.992114701324.1879856103

3.1810.85792462960.85155858314.6230794567-2.549695959-0.4973797743-0.968583161122.3169037749

3.211.41267819550.61803398873.5267115138-3.2360679775-1.1755705046-0.809016994420.3367682216

3.2211.78744700870.37476262922.2087473161-3.7191059436-1.688655851-0.53582679518.4273683645

3.2411.97632074110.12558103910.7519994014-3.9684588053-1.9645745015-0.187381314616.7334865603

3.2611.9763207411-0.1255810391-0.7519994014-3.9684588053-1.96457450150.187381314615.3530883086

3.2811.7874470087-0.3747626292-2.2087473161-3.7191059436-1.6886558510.53582679514.3320020639

3.311.4126781955-0.6180339887-3.5267115138-3.2360679775-1.17557050460.809016994413.6653112053

3.3210.8579246296-0.8515585831-4.6230794567-2.549695959-0.49737977430.968583161113.3047940176

3.3410.131935106-1.07165359-5.4289623148-1.70311716630.25066646710.992114701313.1709832035

3.369.2461589133-1.2748479795-5.8937235044-0.74952525830.96350734820.8763066813.1678761993

3.388.2145652711-1.4579372548-5.98816037060.25116207811.54102648560.637423989713.1980801991

3.47.0534230275-1.6180339887-5.70633909781.23606797751.90211303260.309016994413.1762479455

3.425.7810440892-1.7526133601-5.0659675532.14330717991.9960534569-0.062790519513.0390332934

3.444.4174946322-1.8595529718-4.10728263562.91587450971.8096541049-0.425779291612.7504083479

3.462.984278646-1.9371663223-2.89052204463.50522672021.3690942119-0.728968627412.3019425837

3.481.5039988028-1.9842294026-1.4921393233.87433264450.7362491054-0.929776485911.7084353411

3.50-2-04-0-111

3.52-1.5039988028-1.98422940261.4921393233.8743326445-0.7362491054-0.929776485910.2122181708

3.54-2.984278646-1.93716632232.89052204463.5052267202-1.3690942119-0.72896862749.3762409573

3.56-4.4174946322-1.85955297184.10728263562.9158745097-1.8096541049-0.42577929168.5106761448

3.58-5.7810440892-1.75261336015.0659675532.1433071799-1.9960534569-0.06279051957.6167733072

3.6-7.0534230275-1.61803398875.70633909781.2360679775-1.90211303260.30901699446.6778540208

3.62-8.2145652711-1.45793725485.98816037060.2511620781-1.54102648560.63742398975.6632174269

3.64-9.2461589133-1.27484797955.8937235044-0.7495252583-0.96350734820.876306684.5359906851

3.66-10.131935106-1.071653595.4289623148-1.7031171663-0.25066646710.99211470133.2637046867

3.68-10.8579246296-0.85155858314.6230794567-2.5496959590.49737977430.96858316111.8298632204

3.7-11.4126781955-0.61803398873.5267115138-3.23606797751.17557050460.80901699440.2445188509

3.72-11.7874470087-0.37476262922.2087473161-3.71910594361.6886558510.535826795-1.4480856194

3.74-11.9763207411-0.12558103910.7519994014-3.96845880531.96457450150.1873813146-3.166405368

3.76-11.97632074110.1255810391-0.7519994014-3.96845880531.9645745015-0.1873813146-4.7940047219

3.78-11.78744700870.3747626292-2.2087473161-3.71910594361.688655851-0.535826795-6.1877085832

3.8-11.41267819550.6180339887-3.5267115138-3.23606797751.1755705046-0.8090169944-7.1908701878

3.82-10.85792462960.8515585831-4.6230794567-2.5496959590.4973797743-0.9685831611-7.6503448489

3.84-10.1319351061.07165359-5.4289623148-1.7031171663-0.2506664671-0.9921147013-7.4351421656

3.86-9.24615891331.2748479795-5.8937235044-0.7495252583-0.9635073482-0.87630668-6.4543737248

3.88-8.21456527111.4579372548-5.98816037060.2511620781-1.5410264856-0.6374239897-4.6720767841

3.9-7.05342302751.6180339887-5.70633909781.2360679775-1.9021130326-0.3090169944-2.116790186

3.92-5.78104408921.7526133601-5.0659675532.1433071799-1.99605345690.06279051951.1156459604

3.94-4.41749463221.8595529718-4.10728263562.9158745097-1.80965410490.42577929164.8667754003

3.96-2.9842786461.9371663223-2.89052204463.5052267202-1.36909421190.72896862748.9274667674

3.98-1.50399880281.9842294026-1.4921393233.8743326445-0.73624910540.929776485913.0559513019

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

The Box-Cox Family of Transformations

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

205-12.2280891522

206-12.2715151251

207-12.2928331882

208-12.2921059773

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Transformations. Transformations to Linearity Many non-linear curves can be put into a linear form by appropriate transformations of the either – the