. Surajit Ray Minjung Kyung Jiezhun (Sherry) Gu Ray SAMSI, June - - PowerPoint PPT Presentation

Ray SAMSI, June 2 2005 - slide #1

Statistical Analysis: The Vibrating Beam Example

.

Surajit Ray Minjung Kyung Jiezhun (Sherry) Gu

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #2

Statistical Analysis

Our Goal

■ Estimate σ2(the variance of the measurement error) ■ Estimate the standard errors for the estimates of the

parameters

◆ Two Parameter setup C and K ◆ Two Parameter setup C, K, y(0), v(0). ■ Graphically examine whether the least squares

assumptions hold.

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #3

Statistical Analysis

Our Model

■ The mass-spring-dashpot model

d2y(t) dt2 +Cdy(t) dt +Ky(t) = 0

■ Let a(t) = d2y(t)

dt2

and v(t) = dy(t)

dt , then

a(t) = −Cv(t)−Ky(t) (1)

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #4

Estimation of the S. E.: Linear Regression

Recall that for the simple linear model

Yi = β0 +β1Xi +εi, we estimated the covariance matrix

• f

β0 and β1 using Cov( β0, β1) = (X′X)−1 σ2

where

X =       1 X1 1 X2

. . . . . .

1 Xn       =      

∂Y1 ∂β0 ∂Y1 ∂β1 ∂Y2 ∂β0 ∂Y2 ∂β1

. . . . . .

∂Yn ∂β0 ∂Yn ∂β1

      .

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #5

Estimation of the S. E.: Our Model

Cov( C, K) = (X′X)−1 σ2

where

X =      

∂y(t1) ∂C ∂y(t1) ∂K ∂y(t2) ∂C ∂y(t2) ∂K

. . . . . .

∂y(tn) ∂C ∂y(tn) ∂K

      .

The standard errors of

C and K are the square roots of

the diagonal elements of Cov(

C, K) .

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #6

Estimation of the Standard Error (Change)

■ To compute the standard errors of

C and K, first, we

need to compute ∂y(t)

∂C and ∂y(t) ∂K (to get the columns of

X matrix).

■ Using the chain rule for differentiation, we get the

relation

∂a(t) ∂C = −v(t)−C∂v(t) ∂C −K ∂y(t) ∂C

and

∂a(t) ∂K = −C∂v(t) ∂K −y(t)−K ∂y(t) ∂K

■ We need to compute ∂y(t)

∂C , ∂y(t) ∂K , ∂v(t) ∂K and ∂v(t) ∂K .

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #7

Sensitivity Equations(Change)

How do we compute ∂y(t)

∂C , ∂y(t) ∂K , ∂v(t) ∂K and ∂v(t) ∂K , if we

don’t have an analytical expression for y(t) or v(t)?

• Solve a new system of differential equations, called the

sensitivity equations.

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #8

Sensitivity Equations (Change)

■ To make the following derivation clearer, we will omit

from our notation the dependence of y and v on t.

dy dt = v (2)

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

d dt ∂y ∂C

• = ∂v

∂C. (4) d dt ∂y ∂K

• = ∂v

∂K . (5)

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #9

Sensitivity Equations (Change)

■ Differentiating Equation (3) with respect to C and K

gives

d dt ∂v ∂C

• = −v−C

∂v ∂C

• −K

∂y ∂C

• .

(6) d dt ∂v ∂K

• = −C

∂v ∂K

• −y−K

∂y ∂K

• ..

(7)

■ 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.

Statistical Analysis Statistical Analysis Estimation of the S. E.: Linear Regression Estimation of the S. E.: Our Model Estimation of the Standard Error (Change) Sensitivity Equations(Change) Sensitivity Equations (Change) Sensitivity Equations (Change) Checking the Model Assumptions Ray SAMSI, June 2 2005 - slide #10

Checking the Model Assumptions

■ If the model is appropriate for the data at hand, the

• bserved residuals ei should reflect the properties

assumed for the εi.

■ Residuals can be used to detect departures from the

model

◆ 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

• f a normal: QQPlot to check normality of errors.