Statistical Analysis: The Vibrating Beam Example . Surajit Ray Minjung Kyung Jiezhun (Sherry) Gu Ray SAMSI, June 2 2005 - slide #1
Statistical Analysis Our Goal Statistical Analysis Statistical Analysis ■ Estimate σ 2 (the variance of the measurement error) Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error ■ Estimate the standard errors for the estimates of the (Change) Sensitivity Equations(Change) parameters Sensitivity Equations (Change) Sensitivity Equations (Change) ◆ Two Parameter setup C and K Checking the Model Assumptions ◆ Two Parameter setup C , K , y ( 0 ) , v ( 0 ) . ■ Graphically examine whether the least squares assumptions hold. Ray SAMSI, June 2 2005 - slide #2
Statistical Analysis Our Model Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear ■ The mass-spring-dashpot model Regression Estimation of the S. E.: Our Model Estimation of the Standard Error d 2 y ( t ) (Change) + Cdy ( t ) Sensitivity Equations(Change) + Ky ( t ) = 0 Sensitivity Equations (Change) dt 2 dt Sensitivity Equations (Change) Checking the Model Assumptions ■ Let a ( t ) = d 2 y ( t ) and v ( t ) = dy ( t ) dt , then dt 2 a ( t ) = − Cv ( t ) − Ky ( t ) (1) Ray SAMSI, June 2 2005 - slide #3
Estimation of the S. E.: Linear Regression Recall that for the simple linear model Statistical Analysis Statistical Analysis Y i = β 0 + β 1 X i + ε i , we estimated the covariance matrix Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model of � β 0 and � β 1 using Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Cov ( � β 0 , � β 1 ) = ( X ′ X ) − 1 � σ 2 Sensitivity Equations (Change) Checking the Model Assumptions where ∂ Y 1 ∂ Y 1 1 X 1 ∂β 0 ∂β 1 ∂ Y 2 ∂ Y 2 1 X 2 ∂β 0 ∂β 1 X = = . . . . . . . . . . . . . ∂ Y n ∂ Y n 1 X n ∂β 0 ∂β 1 Ray SAMSI, June 2 2005 - slide #4
Estimation of the S. E.: Our Model Statistical Analysis σ 2 Cov ( � C , � K ) = ( X ′ X ) − 1 � Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model where Estimation of the Standard Error ∂ y ( t 1 ) ∂ y ( t 1 ) (Change) Sensitivity Equations(Change) ∂ C ∂ K Sensitivity Equations (Change) ∂ y ( t 2 ) ∂ y ( t 2 ) Sensitivity Equations (Change) Checking the Model Assumptions ∂ C ∂ K X = . . . . . . . ∂ y ( t n ) ∂ y ( t n ) ∂ C ∂ K The standard errors of � C and � K are the square roots of the diagonal elements of Cov ( � C , � K ) . Ray SAMSI, June 2 2005 - slide #5
Estimation of the Standard Error (Change) Statistical Analysis ■ To compute the standard errors of � C and � K , first, we Statistical Analysis Estimation of the S. E.: Linear need to compute ∂ y ( t ) ∂ C and ∂ y ( t ) Regression ∂ K (to get the columns of Estimation of the S. E.: Our Model Estimation of the Standard Error X matrix). (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) ■ Using the chain rule for differentiation, we get the Sensitivity Equations (Change) Checking the Model Assumptions relation ∂ a ( t ) = − v ( t ) − C ∂ v ( t ) ∂ C − K ∂ y ( t ) ∂ C ∂ C and ∂ a ( t ) = − C ∂ v ( t ) ∂ K − y ( t ) − K ∂ y ( t ) ∂ K ∂ K ■ We need to compute ∂ y ( t ) ∂ C , ∂ y ( t ) ∂ K , ∂ v ( t ) ∂ K and ∂ v ( t ) ∂ K . Ray SAMSI, June 2 2005 - slide #6
Sensitivity Equations(Change) How do we compute ∂ y ( t ) ∂ C , ∂ y ( t ) ∂ K , ∂ v ( t ) ∂ K and ∂ v ( t ) Statistical Analysis ∂ K , if we Statistical Analysis Estimation of the S. E.: Linear Regression don’t have an analytical expression for y ( t ) or v ( t ) ? Estimation of the S. E.: Our Model Estimation of the Standard Error - Solve a new system of differential equations, called the (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) sensitivity equations . Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #7
Sensitivity Equations (Change) ■ To make the following derivation clearer, we will omit Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear from our notation the dependence of y and v on t . Regression Estimation of the S. E.: Our Model Estimation of the Standard Error dy (Change) Sensitivity Equations(Change) dt = v (2) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions and dv dt = − Cv − Ky . (3) ■ Differentiating Equation (3) with respect to C and K and interchanging the order of derivatives on the left hand side gives � ∂ y = ∂ v � d (4) ∂ C ∂ C . dt � ∂ y = ∂ v � d (5) ∂ K ∂ K . dt Ray SAMSI, June 2 2005 - slide #8
Sensitivity Equations (Change) ■ Differentiating Equation (3) with respect to C and K Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear gives Regression Estimation of the S. E.: Our Model Estimation of the Standard Error � ∂ v � ∂ v � ∂ y � � � (Change) d Sensitivity Equations(Change) = − v − C − K (6) ∂ C ∂ C ∂ C Sensitivity Equations (Change) . dt Sensitivity Equations (Change) Checking the Model Assumptions � ∂ v � ∂ v � ∂ y � � � d = − C − y − K (7) ∂ K ∂ K ∂ K .. dt ■ Equations (5)-(8) are the four sensitivity equations . ■ The sensitivity equations, along with the original two equations for y and v can be solved by the Matlab function, ode . Ray SAMSI, June 2 2005 - slide #9
Checking the Model Assumptions ■ If the model is appropriate for the data at hand, the Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear observed residuals e i should reflect the properties Regression assumed for the ε i . Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) ■ Residuals can be used to detect departures from the Sensitivity Equations (Change) Sensitivity Equations (Change) model Checking the Model Assumptions ◆ A residual plot against the fitted values can be used to determine if the error terms have a constant variance. ◆ A plot of the residuals with time can be used to check for non-independence over time. When the error terms are independent, we expect them to fluctuate in a random pattern around 0. ◆ Plot of quantiles of residuals against the quantiles of a normal: QQPlot to check normality of errors. Ray SAMSI, June 2 2005 - slide #10
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