This monograph, "Orthogonal Least Squares Methods and Their Application to Nonlinear System Identification," by S. Chen, S.A. Billings, and W. Luo, published in 1988, focuses on the development of identification algorithms that combine structure determination and parameter estimation for multivariable discrete-time nonlinear stochastic systems. The NARMAX (Nonlinear AutoRegressive Moving Average with eXogenous inputs) model is introduced as a basis for these algorithms, which can provide information about the structure of the model and the final parameter estimates in a simple and efficient manner. The authors review methods for solving least squares problems, including normal equation methods, orthogonal decomposition methods, and singular value decomposition methods. They discuss the advantages and disadvantages of each method, emphasizing the importance of avoiding ill-conditioning issues that can arise with normal equation methods. The paper then presents orthogonal algorithms for structure selection and parameter estimation, which are derived by augmenting well-known techniques of orthogonal decomposition of the regression matrix. These algorithms are shown to be effective for both polynomial NARX and NARMAX models. The algorithms are designed to select a subset of the full model set in a forward-regression manner, maximizing the reduction in the sum of squares of residuals at each step. The monograph also includes practical applications and simulation studies demonstrating the effectiveness of the proposed algorithms in fitting parsimonious models to real systems. The authors conclude by discussing the advantages and disadvantages of using different orthogonal decomposition techniques and comparing the resulting structure selection algorithms.This monograph, "Orthogonal Least Squares Methods and Their Application to Nonlinear System Identification," by S. Chen, S.A. Billings, and W. Luo, published in 1988, focuses on the development of identification algorithms that combine structure determination and parameter estimation for multivariable discrete-time nonlinear stochastic systems. The NARMAX (Nonlinear AutoRegressive Moving Average with eXogenous inputs) model is introduced as a basis for these algorithms, which can provide information about the structure of the model and the final parameter estimates in a simple and efficient manner. The authors review methods for solving least squares problems, including normal equation methods, orthogonal decomposition methods, and singular value decomposition methods. They discuss the advantages and disadvantages of each method, emphasizing the importance of avoiding ill-conditioning issues that can arise with normal equation methods. The paper then presents orthogonal algorithms for structure selection and parameter estimation, which are derived by augmenting well-known techniques of orthogonal decomposition of the regression matrix. These algorithms are shown to be effective for both polynomial NARX and NARMAX models. The algorithms are designed to select a subset of the full model set in a forward-regression manner, maximizing the reduction in the sum of squares of residuals at each step. The monograph also includes practical applications and simulation studies demonstrating the effectiveness of the proposed algorithms in fitting parsimonious models to real systems. The authors conclude by discussing the advantages and disadvantages of using different orthogonal decomposition techniques and comparing the resulting structure selection algorithms.