Recursive Least squares modeling

Procedures on how to make inferences on the parameters and the response variables are introduced in Chapter 11. The design of experiments has a direct impact on the quality of the estimated parameters and is presented in Chapter 12. The emphasis is on sequential experimental design for parameter estimation and for model discrimination. Recursive least squares estimation, used for on-line data analysis, is briefly covered in Chapter 13. 

An alternative SPM framework for autocorrelated data is developed by monitoring variations in time series model parameters that are updated at each new measurement instant. Parameter change detection with recursive weighted least squares was used to detect changes in the parameters and the order of a time series model that describes stock prices in financial markets [263]. Here, the recursive least squares is extended with adaptive forgetting. 

Structure of model of identification system ARX Identification method recursive least squares Adaptation algorithm parametric decreasing gain Te=3s Delay D=0... 

This chapter describes two new methods for obtaining frequency response and step response models from processes operating under relay feedback control. Both methods are based on the frequency sampling filter model structure and a recursive least squares estimator. 

The second type of model is called recurrent model or N-step-ahead prediction model the recursive least squares algorithm can be used to identify this type of model. Only after the pH has been predicted at a particular time can the next predicted value be calculated. In addi-... 

The most simple model identification procedure for recurrent model identification is to assume fixed membership functions and use recursive least squares to identify the consequence part of the model. The model structure in this case is the model as shown in Eqn. (29.25b). 

F2912.m using recursive least squares for pH recurrent model development... 

Before we introduce the Kalman filter, we reformulate the least-squares algorithm discussed in Chapter 8 in a recursive way. By way of illustration, we consider a simple straight line model which is estimated by recursive regression. Firstly, the measurement model has to be specified, which describes the relationship between the independent variable x, e.g., the concentrations of a series of standard solutions, and the dependent variable, y, the measured response. If we assume a straight line model, any response is described by ... 

In this chapter different aspects of data processing and reconciliation in a dynamic environment were briefly discussed. Application of the least square formulation in a recursive way was shown to lead to the classical Kalman filter formulation. A simpler situation, assuming quasi-steady-state behavior of the process, allows application of these ideas to practical problems, without the need of a complete dynamic model of the process. 

If Yi equals 0, the model becomes partially recursive. The first equation becomes a regression which can be estimated by ordinary least squares. However, the second equation continues to fail the order condition. To see the problem, consider that even with the restriction, any linear combination of the two equations has the same variables as the original second eqation. 

Keywords soft-sensor, Just-In-Time modeling, recursive partial least squares regression, principal component analysis, estimation... 

The common procedure used to calculate the SLD profile from the reflectivity curve is to assume a model profile, calculate the theoretical reflectivity curve using the optical matrix or recursion method, and compare calculated and experimental curves. A least-squares iterative procedure is then used to vary the parameters of the SLD profile until a good fit between the calculated curve and the experimental data is achieved. Although the inversion of the reflectivity data is not unique and... 

Chapter 11 presents the use of sequential least squares techniques for the recursive estimation of uncertain model parameters. There is a statistical advantage in taking this approach to model parameter identification over that of incorporating model parameter estimation directly into Kalman filtering. 

Fig. 1. Pattern recognition methods. ANN, artificial neural networks BP ANN, back-propagation ANN CA, cluster analysis CART, classification and regression trees (recursive partitioning) CCA, canonical correlation analysis CVA, canonical variate analysis kNN, -nearest neighbor methods LDA, linear discriminant analysis PCA, principal component analysis PLS DA, partial least squares regression discriminant analysis SIMCA, soft independent modeling of class analogy SOM, self-organizing maps.

The objective of state estimation is to estimate the system states from a limited number of response measurements (e.g., acceleration measurements) and a system model. The state vector can be determined using a deterministic approach (e.g., least squares estimation of the state vector in the frequency domain) or using a combined deterministic-stochastic approach, which mostly results in recursive time-domain algorithms which can be applied for online state estimation. The best-known recursive state estimation algorithm for linear systems is the Kalman filter algorithm (Kalman 1960), which is outlined next. 

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