Neural Networks Demystified

by Louise Francis

Francis Analytics and Actuarial Data Mining, Inc.louise_francis@msn.com

Objectives of PaperIntroduce actuaries to neural networksShow that neural networks are a lot like some conventional statisticsIndicate where use of neural networks might be helpfulShow how to interpret neural network models

Data MiningNeural networks are one of a number of data mining techniquesMethods primarily developed in artificial intelligence and statistical disciplines to find patterns in dataTypically applied to large databases with complex relationships

Some Other Data Mining MethodsDecision treesClusteringRegression splinesAssociation rules

Some Data Mining AdvantagesNonlinear relationshipsInteractionsMulticollinearity

Data Mining: Neural NetworksOne of more established approachesSomewhat glamorousAI description: they function like neurons in the brain

Neural Networks: DisadvantagesThey are a black boxUser gets a prediction from them, but the form of the fitted function is not revealedDont know which variables are the most important in the prediction

Kinds of Neural NetworksSupervised learningMultilayer perceptronAlso known as backpropagation neural networkPaper explains this kind of NNUnsupervised learningKohonen neural networks

The MLP Neural Network

The Activation FunctionThe sigmoid logistic function

The Logistic Function

The Logistic Function

The Logistic Function

OtherData is usually normalizedUsually both independent and dependent variables transformed to lie in range between 0 and 1

Logistic Function

Fitting the curveTypically use a procedure which is like gradient descent

Fitting a nonlinear function

Graph of nonlinear function

Fitted Weights

Hidden Layer

EMBED SPLUSGraphSheetFileType

_1023804577.bin

Selected Fitted Values for function

Hidden and Output Layer

EMBED SPLUSGraphSheetFileType

_1023873834.bin

Fit of Curve with 2 Nodes

EMBED SPLUSGraphSheetFileType

_1023874210.bin

Fit of Curve with 3 Nodes

_1039702918.bin

Universal Function ApproximatorThe multilayer perceptron neural network with one hidden layer is a universal function approximator Theoretically, with a sufficient number of nodes in the hidden layer, any nonlinear function can be approximated

Correlated VariablesVariables used in model building are often correlated.It is difficult to isolate the effect of the individual variables because of the correlation between the variables.

Example of correlated variables

Chart1

0.0110599078

0.0110497238

0.0110395584

0.0119485294

0.0110192837

0.0119375574

0.0110091743

0.0165289256

0.0137362637

0.0137362637

0.0128087832

0.0146520147

0.0100273473

0.0081967213

0.008189263

0.0063578565

0.0072661217

0.0063520871

0.0072595281

0.0027100271

0.0036133695

0.0045167118

0.0045167118

0.0027075812

0.0045126354

0.0081300813

0.0081227437

0.0072202166

0.007213706

0.007213706

0.0063063063

0.0072072072

0.0063006301

0.0053956835

0.0035971223

0.0054005401

0.0035938904

0

-0.0008952551

0.0008960573

0.0008952551

0.0008952551

0.0044762757

0.0035778175

0.0035778175

0.0053667263

0.0080645161

0.0080572963

0.0098478066

0.0116487455

0.0134408602

0.0125335721

0.0143112701

0.0152057245

0.0115864528

0.0115864528

0.013368984

0.012455516

0.016

0.0159857904

0.0115248227

0.0106288751

0.0097259063

0.0097259063

0.0070546737

0.0052863436

0.0061674009

0.0052863436

0.0008795075

0

-0.0052493438

-0.0061188811

-0.0070113935

-0.0061349693

-0.0070052539

-0.0087565674

-0.0087565674

-0.0096406661

-0.0105078809

-0.0105170903

-0.007029877

-0.0079086116

-0.0070360598

-0.0061565523

-0.0052956752

-0.0061728395

-0.0061728395

-0.0044169611

-0.0044169611

-0.0026548673

-0.003539823

-0.0035429584

-0.0044247788

-0.0035429584

-0.0035429584

-0.0053097345

-0.0053238687

-0.0044365572

-0.0053238687

-0.0062111801

-0.0062111801

-0.0062111801

-0.0053285968

-0.0044444444

-0.0044444444

-0.0044444444

-0.0053333333

-0.0044483986

-0.0035682426

-0.004456328

-0.002676182

-0.0035714286

-0.0044642857

CarPartsRate

Car Body Rates

Car Parts Rate

CarPartsRate vs CarBody Inflation Rates

BLS2

RowNamesSeriesYearMoCPIMedCareHourlyEarnMed.ServiceCarBodiesOther.ServicesCarPartsCPIRateMedCareRateHourlyEarnRateMedServRateCarBodyRateOther.ServicesRateCarPartsRateFactor1Factor11CarBodyRateCarPartsRate

13CUUR0000SA01991.00M01134.60171.0010.18171.10125.10166.50109.700.060.100.040.100.060.080.01-1.992.400.05658783780.0110599078

14CUUR0000SA01991.00M02134.80172.5010.19172.80125.90167.40109.800.050.100.030.100.060.080.01-1.922.410.06244725740.0110497238SUMMARY OUTPUT

15CUUR0000SA01991.00M03135.00173.7010.22173.80125.50167.90109.900.050.090.030.100.060.080.01-1.872.280.0635593220.0110395584

16CUUR0000SA01991.00M04135.20174.4010.26174.50124.40168.80110.100.050.090.030.090.060.080.01-1.842.190.05602716470.0119485294Regression Statistics

17CUUR0000SA01991.00M05135.60175.2010.30175.10123.20169.10110.100.050.090.030.090.050.080.01-1.812.000.05029838020.0110192837Multiple R0.6167101872

18CUUR0000SA01991.00M06136.00176.2010.32176.10122.90170.00110.200.050.090.030.090.030.080.01-1.771.970.03277310920.0119375574R Square0.380331455

19CUUR0000SA01991.00M07136.20177.5010.35177.50122.80170.80110.200.040.090.030.090.030.070.01-1.721.770.03020134230.0110091743Adjusted R Square0.3747488555

20CUUR0000SA01991.00M08136.60178.9010.37178.90122.60172.20110.700.040.080.030.080.040.070.02-1.661.660.03722504230.0165289256Standard Error0.0172655433

21CUUR0000SA01991.00M09137.20179.7010.39179.70120.30175.80110.700.030.080.030.080.030.080.01-1.631.640.02645051190.0137362637Observations113

22CUUR0000SA01991.00M10137.40180.7010.39180.80131.00176.20110.700.030.080.030.080.050.080.01-1.581.520.0480.0137362637

23CUUR0000SA01991.00M11137.80181.8010.42181.90130.30176.90110.700.030.080.030.080.040.080.01-1.541.400.04407051280.0128087832ANOVA

24CUUR0000SA01991.00M12137.90182.6010.45182.80129.50177.60110.800.030.080.030.080.040.080.01-1.511.440.03766025640.0146520147dfSSMSFSignificance F

25CUUR0000SA01992.00M01138.10184.3010.45184.60129.20178.60110.800.030.080.030.080.030.070.01-1.441.400.0327737810.0100273473Regression10.02030889390.020308893968.12802076870

26CUUR0000SA01992.00M02138.60186.2010.47186.40129.00179.40110.700.030.080.030.080.020.070.01-1.361.390.02462271640.0081967213Residual1110.03308898740.000298099

27CUUR0000SA01992.00M03139.30187.3010.50187.40129.30179.80110.800.030.080.030.080.030.070.01-1.311.360.03027888450.008189263Total1120.0533978812

28CUUR0000SA01992.00M04139.50188.1010.52188.10129.20180.30110.800.030.080.030.080.040.070.01-1.281.350.0385852090.0063578565

29CUUR0000SA01992.00M05139.70188.7010.54188.90128.90181.30110.900.030.080.020.080.050.070.01-1.251.390.04626623380.0072661217CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

30CUUR0000SA01992.00M06140.20189.4010.57189.70128.30181.50110.900.030.070.020.080.040.070.01-1.231.310.04393816110.0063520871Intercept0.01045437840.00173436466.02778601620.00000002220.00701762040.01389113650.00701762040.0138911365

31CUUR0000SA01992.00M07140.50190.7010.59191.10128.40182.30111.000.030.070.020.080.050.070.01-1.171.280.04560260590.0072595281CarPartsRate1.77367688350.21488773088.253970000501.34786263462.19949113241.34786263462.1994911324

32CUUR0000SA01992.00M08140.90191.5010.62192.20127.50183.90111.000.030.070.020.070.040.070.00-1.141.170.03996737360.0027100271

33CUUR0000SA01992.00M09141.30192.3010.62192.90123.50187.00111.100.030.070.020.070.030.060.00-1.111.130.02660016630.0036133695

34CUUR0000SA01992.00M10141.80193.3010.65194.20132.10187.90111.200.030.070.030.070.010.070.00-1.061.160.00839694660.0045167118

35CUUR0000SA01992.00M11142.00194.3010.67195.20132.10188.00111.200.030.070.020.070.010.060.00-1.021.110.01381427480.0045167118

36CUUR0000SA01992.00M12141.90194.7010.69195.60131.80189.10111.100.030.070.020.070.020.060.00-1.010.950.01776061780.0027075812

37CUUR0000SA01993.00M01142.60196.4010.72197.50132.10191.00111.300.030.070.030.070.020.070.00-0.940.950.02244582040.0045126354

38CUUR0000SA01993.00M02143.10198.0010.73199.10132.90191.50111.600.030.060.020.070.030.070.01-0.870.860.03023255810.0081300813

39CUUR0000SA01993.00M03143.60198.6010.78199.70133.00192.00111.700.030.060.030.070.030.070.01-0.850.730.02861562260.0081227437

40CUUR0000SA01993.00M04144.00199.4010.78200.70132.90192.40111.600.030.060.020.070.030.070.01-0.810.800.02863777090.0072202166

41CUUR0000SA01993.00M05144.20200.5010.80202.00132.60193.20111.700.030.060.020.070.030.070.01-0.770.920.0287044220.007213706

42CUUR0000SA01993.00M06144.40201.1010.82202.60132.80193.10111.700.030.060.020.070.040.060.01-0.740.850.03507404520.007213706

43CUUR0000SA01993.00M07144.40202.2010.84203.80132.90193.70111.700.030.060.020.070.040.060.01-0.700.780.0350467290.0063063063

44CUUR0000SA01993.00M08144.80202.9010.86204.50132.60193.40111.800.030.060.020.060.040.050.01-0.670.650.040.0072072072

45CUUR0000SA01993.00M09145.10203.3010.89205.00128.00193.10111.800.030.060.030.060.040.030.01-0.650.590.0364372470.0063006301

46CUUR0000SA01993.00M10145.70204.4010.91206.20135.70193.40111.800.030.060.020.060.030.030.01-0.610.540.02725208180.0053956835

47CUUR0000SA01993.00M11145.80204.9010.93206.80136.20193.80111.600.030.050.020.060.030.030.00-0.590.430.03103709310.0035971223

48CUUR0000SA01993.00M12145.80205.2010.96207.10136.30194.20111.700.030.050.030.060.030.030.01-0.570.390.03414264040.0054005401

49CUUR0000SA01994.00M01146.20206.4010.99208.40137.30195.10111.700.030.050.030.060.040.020.00-0.520.210.03936411810.0035938904

50CUUR0000SA01994.00M02146.70207.7011.02209.80137.50195.20111.600.030.050.030.050.030.020.00-0.470.140.03461249060

51CUUR0000SA01994.00M03147.20208.3011.03210.40137.40195.50111.600.030.050.020.050.030.02-0.00-0.440.130.0330827068-0.0008952551

52CUUR0000SA01994.00M04147.40209.2011.05211.40137.40196.40111.700.020.050.030.050.030.020.00-0.410.120.03386004510.0008960573

53CUUR0000SA01994.00M05147.50209.7011.07212.00138.30197.10111.800.020.050.020.050.040.020.00-0.39-0.070.04298642530.0008952551

54CUUR0000SA01994.00M06148.00210.4011.09212.60138.10197.60111.800.020.050.020.050.040.020.00-0.36-0.080.03990963860.0008952551

55CUUR0000SA01994.00M07148.40211.5011.12213.80138.20198.00112.200.030.050.030.050.040.020.00-0.31-0.090.03987960870.0044762757

56CUUR0000SA01994.00M08149.00212.2011.14214.70138.20199.40112.200.030.050.030.050.040.030.00-0.28-0.050.04223227750.0035778175

57CUUR0000SA01994.00M09149.40212.8011.17215.40133.90201.40112.200.030.050.030.050.050.040.00-0.26-0.010.046093750.0035778175

58CUUR0000SA01994.00M10149.50214.0011.21216.80140.00201.90112.400.030.050.030.050.030.040.01-0.210.020.03168754610.0053667263

59CUUR0000SA01994.00M11149.70214.7011.22217.50139.30202.30112.500.030.050.030.050.020.040.01-0.180.040.02276064610.0080645161

60CUUR0000SA01994.00M12149.70215.3011.25218.20139.80202.40112.600.030.050.030.050.030.040.01-0.160.130.025678650.0080572963

61CUUR0000SA01995.00M01150.30216.6011.27219.80140.10203.00112.800.030.050.030.050.020.040.01-0.100.190.02039329930.0098478066

62CUUR0000SA01995.00M02150.90217.9011.30221.30139.90204.10112.900.030.050.030.050.020.050.01-0.050.190.01745454550.0116487455

63CUUR0000SA01995.00M03151.40218.4011.33221.80139.30204.00113.100.030.050.030.050.010.040.01-0.030.160.01382823870.0134408602

64CUUR0000SA01995.00M04151.90218.9011.35222.40139.10204.30113.100.030.050.030.050.010.040.01-0.010.060.01237263460.0125335721

65CUUR0000SA01995.00M05152.20219.3011.37223.00138.70204.90113.400.030.050.030.050.000.040.010.010.050.00289226320.0143112701

66CUUR0000SA01995.00M06152.50219.8011.41223.50137.80205.30113.500.030.040.030.05-0.000.040.020.030.02-0.00217233890.0152057245

67CUUR0000SA01995.00M07152.50220.8011.45224.60137.70205.70113.500.030.040.030.05-0.000.040.010.07-0.02-0.0036179450.0115864528

68CUUR0000SA01995.00M08152.90221.6011.47225.60137.20207.70113.500.030.040.030.05-0.010.040.010.10-0.01-0.007235890.0115864528

69CUUR0000SA01995.00M09153.20222.1011.50226.10132.90210.20113.700.030.040.030.05-0.010.040.010.13-0.06-0.00746825990.013368984

70CUUR0000SA01995.00M10153.70222.9011.54226.90141.70210.70113.800.030.040.030.050.010.040.010.16-0.220.01214285710.012455516

71CUUR0000SA01995.00M11153.60223.5011.56227.40142.40211.20114.300.030.040.030.050.020.040.020.18-0.270.02225412780.016

72CUUR0000SA01995.00M12153.50223.8011.59227.80142.40211.10114.400.030.040.030.040.020.040.020.20-0.350.01859799710.0159857904

73CUUR0000SA01996.00M01154.40225.2011.63229.30141.20212.00114.100.030.040.030.040.010.040.010.25-0.380.00785153460.0115248227

74CUUR0000SA01996.00M02154.90226.2011.64230.30141.10212.60114.100.030.040.030.040.010.040.010.29-0.510.00857755540.0106288751

75CUUR0000SA01996.00M03155.70226.6011.67230.70141.10213.00114.200.030.040.030.040.010.040.010.31-0.540.01292175160.0097259063

76CUUR0000SA01996.00M04156.30227.0011.72231.10140.50213.30114.200.030.040.030.040.010.040.010.33-0.590.01006470170.0097259063

77CUUR0000SA01996.00M05156.60227.4011.75231.60140.60214.10114.200.030.040.030.040.010.040.010.34-0.620.01369863010.0070546737

78CUUR0000SA01996.00M06156.70227.8011.81231.90140.70214.00114.100.030.040.040.040.020.040.010.36-0.670.02104499270.0052863436

79CUUR0000SA01996.00M07157.00228.7011.83232.90139.50214.60114.200.030.040.030.040.010.040.010.40-0.700.01307189540.0061674009

80CUUR0000SA01996.00M08157.30229.2011.87233.40139.20216.30114.100.030.030.030.030.010.040.010.42-0.820.01457725950.0052863436

81CUUR0000SA01996.00M09157.80229.4011.91233.60135.60218.30113.800.030.030.040.030.020.040.000.43-0.890.02031602710.0008795075

82CUUR0000SA01996.00M10158.30230.1011.93234.20142.20218.80113.800.030.030.030.030.000.040.000.46-0.940.00352858150

83CUUR0000SA01996.00M11158.60230.5011.99234.90141.90219.20113.700.030.030.040.03-0.000.04-0.010.47-0.90-0.003511236-0.0052493438

84CUUR0000SA01996.00M12158.60230.6012.02235.00141.60218.70113.700.030.030.040.03-0.010.04-0.010.48-0.96-0.0056179775-0.0061188811

85CUUR0000SA01997.00M01159.10231.8012.06236.30141.70220.00113.300.030.030.040.030.000.04-0.010.53-1.020.0035410765-0.0070113935

86CUUR0000SA01997.00M02159.60232.7012.10237.10141.50220.70113.400.030.030.040.030.000.04-0.010.56-1.070.0028348689-0.0061349693

87CUUR0000SA01997.00M03160.00233.4012.14237.70141.20221.40113.400.030.030.040.030.000.04-0.010.59-1.030.0007087172-0.0070052539

88CUUR0000SA01997.00M04160.20233.8012.16238.10140.30222.70113.200.020.030.040.03-0.000.04-0.010.61-1.03-0.0014234875-0.0087565674

89CUUR0000SA01997.00M05160.10234.2012.21238.50138.40223.10113.200.020.030.040.03-0.020.04-0.010.62-1.05-0.0156472262-0.0087565674

90CUUR0000SA01997.00M06160.30234.4012.23238.70138.00223.10113.000.020.030.040.03-0.020.04-0.010.63-1.08-0.0191897655-0.0096406661

91CUUR0000SA01997.00M07160.50234.8012.27239.20135.90223.50113.000.020.030.040.03-0.030.04-0.010.65-1.19-0.0258064516-0.0105078809

92CUUR0000SA01997.00M08160.80235.2012.33239.80135.90225.70112.900.020.030.040.03-0.020.04-0.010.67-1.17-0.0237068966-0.0105170903

93CUUR0000SA01997.00M09161.20235.4012.37240.00133.70228.10113.000.020.030.040.03-0.010.04-0.010.67-1.17-0.0140117994-0.007029877

94CUUR0000SA01997.00M10161.60235.8012.43240.50140.60229.40112.900.020.020.040.03-0.010.05-0.010.69-1.20-0.0112517581-0.0079086116

95CUUR0000SA01997.00M11161.50236.4012.48241.20139.40229.90112.900.020.030.040.03-0.020.05-0.010.72-1.20-0.0176180409-0.0070360598

96CUUR0000SA01997.00M12161.30237.1012.52241.80137.80230.10113.000.020.030.040.03-0.030.05-0.010.74-1.10-0.0268361582-0.0061565523

97CUUR0000SA01998.00M01161.60238.1012.54242.90137.80231.30112.700.020.030.040.03-0.030.05-0.010.79-1.15-0.0275229358-0.0052956752

98CUUR0000SA01998.00M02161.90239.3012.60244.20138.00233.10112.700.010.030.040.03-0.020.06-0.010.84-1.05-0.0247349823-0.0061728395

99CUUR0000SA01998.00M03162.20239.8012.64244.80137.80232.40112.700.010.030.040.03-0.020.05-0.010.86-1.05-0.0240793201-0.0061728395

100CUUR0000SA01998.00M04162.50240.7012.69245.40137.20234.70112.700.010.030.040.03-0.020.05-0.000.89-1.01-0.0220955096-0.0044169611

101CUUR0000SA01998.00M05162.80241.4012.73245.90135.40236.70112.700.020.030.040.03-0.020.06-0.000.92-0.99-0.0216763006-0.0044169611

102CUUR0000SA01998.00M06163.00242.0012.76246.50134.30236.40112.700.020.030.040.03-0.030.06-0.000.95-0.91-0.0268115942-0.0026548673

103CUUR0000SA01998.00M07163.20242.7012.79247.40134.80237.80112.600.020.030.040.03-0.010.06-0.000.98-0.83-0.0080941869-0.003539823

104CUUR0000SA01998.00M08163.40243.5012.85248.20133.90238.00112.500.020.040.040.04-0.010.05-0.001.01-0.79-0.0147167035-0.0035429584

105CUUR0000SA01998.00M09163.60243.9012.87248.40132.70240.40112.500.010.040.040.03-0.010.05-0.001.03-0.79-0.0074794316-0.0044247788

106CUUR0000SA01998.00M10164.00244.3012.91249.00140.60241.30112.500.010.040.040.040.000.05-0.001.04-0.780-0.0035429584

107CUUR0000SA01998.00M11164.00244.7012.95249.30140.40240.50112.500.020.040.040.030.010.05-0.001.06-0.870.0071736011-0.0035429584

108CUUR0000SA01998.00M12163.90245.2012.99249.60139.20250.30112.400.020.030.040.030.010.09-0.011.08-0.930.0101596517-0.0053097345

109CUUR0000SA01999.00M01164.30246.6013.04251.30138.20255.40112.100.020.040.040.030.000.10-0.011.14-0.820.0029027576-0.0053238687

110CUUR0000SA01999.00M02164.50247.7013.06252.60138.70255.00112.200.020.040.040.030.010.09-0.001.18-0.820.0050724638-0.0044365572

111CUUR0000SA01999.00M03165.00248.3013.10253.10137.70253.30112.100.020.040.040.03-0.000.09-0.011.21-0.85-0.0007256894-0.0053238687

112CUUR0000SA01999.00M04166.20249.1013.14253.50138.00256.10112.000.020.030.040.030.010.09-0.011.24-0.890.0058309038-0.0062111801

113CUUR0000SA01999.00M05166.20249.5013.19254.00137.20255.80112.000.020.030.040.030.010.08-0.011.26-0.900.0132939439-0.0062111801

114CUUR0000SA01999.00M06166.20250.2013.23254.60136.00255.90112.000.020.030.040.030.010.08-0.011.29-0.900.0126582278-0.0062111801

115CUUR0000SA01999.00M07166.70251.1013.27255.50134.70258.30112.000.020.030.040.03-0.000.09-0.011.32-0.91-0.0007418398-0.0053285968

116CUUR0000SA01999.00M08167.10251.9013.30256.20134.40257.60112.000.020.030.040.030.000.08-0.001.36-0.930.0037341299-0.0044444444

117CUUR0000SA01999.00M09167.90252.3013.35256.60133.40262.60112.000.030.030.040.030.010.09-0.001.37-0.890.0052750565-0.0044444444

118CUUR0000SA01999.00M10168.20252.8013.38257.10142.10263.20112.000.030.030.040.030.010.09-0.001.39-0.920.0106685633-0.0044444444

119CUUR0000SA01999.00M11168.30253.3013.41257.70140.80263.00111.900.030.040.040.030.000.09-0.011.41-0.860.0028490028-0.0053333333

120CUUR0000SA01999.00M12168.30254.2013.44258.50140.50263.00111.900.030.040.030.040.010.05-0.001.45-0.760.0093390805-0.0044483986

121CUUR0000SA02000.00M01168.70255.5013.49260.10139.90264.70111.700.030.040.030.040.010.04-0.001.50-0.790.012301013-0.0035682426

122CUUR0000SA02000.00M02169.70257.0013.54262.00139.40266.70111.700.030.040.040.040.010.05-0.001.57-0.680.0050468637-0.004456328

123CUUR0000SA02000.00M03171.10258.1013.58263.20139.00268.00111.800.040.040.040.040.010.06-0.001.61-0.550.0094408134-0.002676182

124CUUR0000SA02000.00M04171.20258.8013.64263.90139.20271.90111.600.030.040.040.040.010.06-0.001.64-0.490.0086956522-0.0035714286

125CUUR0000SA02000.00M05171.30259.4013.66264.40138.80270.20111.500.030.040.040.040.010.06-0.001.67-0.500.0116618076-0.0044642857

BLS2

CarPartsRate

Car Body Rates

Car Parts Rate

CarPartsRate vs CarBody Inflation Rates

A Solution: Principal Components & Factor Analysis

One Factor Model

F1

X1

X2

X3

U1

U2

U3

Factor Analysis: An Example

Factor Analysis Diagram

Social InflationFactor

Litigation Rates

Size of Jury Awards

Index of State Litigation Environment

U1

U2

U3

Factor Analysis

Factor Analysis Result used for Prediction

X1

X2

X2

F1

Y

Input Variables

Factor

Dependent Variable

Factor Analysis

Three Layer Neural Network With One Hidden Node

InputLayer

HiddenLayer

OutputLayer

Correlated Variables: An ExampleWorkers Compensation LineProduce an economic inflation indexWage InflationMedical InflationBenefit Level IndexIn simplified example no other variable drives severity results

Factor Analysis Example

X1 = b1 Factor1X2 = b2 Factor1X3 = b3 Factor1

Index =.395 (Wage Inflation)+.498(Medical Inflation)+.113(Benefit Level Inflation)

Sheet1

Table 8

VariableLoadingWeights

Wage Inflation Index0.9850.395

Medical Inflation Index0.9880.498

Benefit Level Inflation Index0.9470.113

Factor Analysis Example

Interpreting Neural NetworkLook at weights to hidden layerCompute sensitivities: a measure of how much the predicted values error increases when the variables are excluded from the model one at a time

Interpretation of Neural Network

Interactions: Another Modeling ProblemImpact of two variables is more or less than the sum of their independent impacts.

Interactions: Simulated Data

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Interactions: Neural Network

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Interactions: Regression

EMBED SPLUSGraphSheetFileType

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Example With Messy Data

Table 15

Variable

Variable Type

Number of Categories

Missing Data

Age of Driver

Continuous

No

Territory

Categorical

45

No

Age of Car

Continuous

Yes

Car Type

Categorical

4

No

Credit Rating

Continuous

Yes

Auto BI Inflation Factor

Continuous

No

Auto PD and Phys Dam Inflation Factor

Continuous

No

Law Change

Categorical

2

No

Bogus

Continuous

No

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Example With Messy Data

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Visualizing Neural Network Result

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Visualizing Neural Network Result

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Visualization of Law Change Effect

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Visualization of Inflation

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How Good Was the Fit?

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How Good Was the Fit?

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