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Environmentally Conscious Design & Manufacturing

Environmentally Conscious Design & Manufacturing. Class 26: Data Dependent Systems. Prof. S. M. Pandit. Autoregressive Model - AR(1). AR(1) model:. Estimation of parameters:. Checks of Adequacy. To check the dependence of a t on a t-1. Prediction or Forecasting.

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Environmentally Conscious Design & Manufacturing

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  1. Environmentally Conscious Design & Manufacturing Class26: Data Dependent Systems Prof. S. M. Pandit

  2. Autoregressive Model - AR(1) AR(1) model: Estimation of parameters:

  3. Checks of Adequacy To check the dependence of at on at-1

  4. Prediction or Forecasting One step ahead prediction at time t-1: One step ahead prediction error: Conditional distribution: The probability limits (e.g. 95 %):

  5. Conditional Distribution of Xt

  6. Autoregressive Model - AR(2) AR(2) model: or

  7. Estimate of Parameters Use the collected data: Then,

  8. Estimate of Parameters (cont.) The conditional least square estimates of The conditional least square estimates of

  9. Extension of AR (2) In practice, it is enough to use AR model in many applications. The AR model can also be extended to multiple inputs single output. For one input- one output, model takes the form In general form:

  10. Estimates of Parameters Use the collected data: The conditional least square estimates of

  11. Estimates of Parameters (cont.) Use the collected data:

  12. Estimates of Parameters (cont.) The conditional least square estimates of

  13. Example A model fitted to the data of system A for the purpose of developing a maintenance strategy is given below. where X2t is the measured pH and X1t is the caustic added. a). Express the model in transfer function form

  14. b). Show a block diagram of the system including the above transfer functions? c). Determine the ratio of steady state output to steady state input. d). Given the following observations, determine the 95% probability limit on the 5th observation.

  15. Backshift Operator Backshift operator is defined by In general,

  16. Transfer Function Using B operator It follows

  17. Block Diagram a2t X1t X2t

  18. Steady State When a system reaches a STEADY state, EX2t=EX2t-1=…=EX2t-n=say X2s and EX1t=EX1t-1=…=EX1t-n=say X1s i.e. B=1, and Eat=0 since at has zero mean.Therefore, taking the mean or expectation on both sides the model becomes i.e., same as obtained before substituting B=1 in the transfer function.

  19. Prediction The prediction of X2t at t=5 is given by

  20. Probability Interval 95% probability interval on the observation That is

  21. How to decide n? Increase n until does not reduce significantly.

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