Least squares problems with inequality constraints as quadratic constraints

Mead, Jodi; Renaut, Rosemary A.

doi:10.1016/j.laa.2009.04.017

Cited by 29 publications

(28 citation statements)

References 16 publications

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“…The linear regression model calculates the coefficients X s k of each metric k for a particular segment s accompanied with an interception coefficient C. Coefficients are found using the least squares problem method by minimising the sum of the squared error (min Ax − b 2 2 ) where in our case A represents a matrix of our metrics observations and b is their corresponding time metrics (Mead and Renaut 2010;Lawson and Hanson 1974). We experimented with six different approaches using this model (The name in the brackets specifies the id given to each of the approaches):…”

Section: Performance Metrics Weightsmentioning

confidence: 99%

Adaptive user modelling in car racing games using behavioural and physiological data

Georgiou

Demiris

2017

User Model User-Adap Inter

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Personalised content adaptation has great potential to increase user engagement in video games. Procedural generation of user-tailored content increases the self-motivation of players as they immerse themselves in the virtual world. An adaptive user model is needed to capture the skills of the player and enable automatic game content altering algorithms to fit the individual user. We propose an adaptive user modelling approach using a combination of unobtrusive physiological data to identify strengths and weaknesses in user performance in car racing games. Our system creates user-tailored tracks to improve driving habits and user experience, and to keep engagement at high levels. The user modelling approach adopts concepts from the Trace Theory framework; it uses machine learning to extract features from the user's physiological data and game-related actions, and cluster them into low level primitives. These primitives are transformed and evaluated into higher level abstractions such as experience, exploration and attention. These abstractions are subsequently used to provide track alteration decisions for the player. Collection of data and feedback from 52 users allowed us to associate key model variables and outcomes to user responses, and to verify that the model provides statistically significant decisions personalised to the individual player. Tailored game content variations between users in our experiments, as well as the correlations with user satisfaction demonstrate that our algorithm is able to automatically incorporate user feedback in subsequent procedural content generation.

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Section: Performance Metrics Weightsmentioning

confidence: 99%

Adaptive user modelling in car racing games using behavioural and physiological data

Georgiou

Demiris

2017

User Model User-Adap Inter

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“…This is an extension of the scalar χ 2 method [21,22,23,31] which can be viewed as a regularization method. The new method amounts to solving multiple χ 2 tests to give an equal number of equations as the number of unknowns in the diagonal weighting matrix for data or parameter misfits.…”

Section: Discussionmentioning

confidence: 99%

“…This statistical interpretation of the weights in (3) and multipliers in (8) gives us a method for calculating them, and we term it the χ 2 method for parameter estimation and uncertainty quantification [21,22,23,31].…”

Section: Regularization and Constrained Optimizationmentioning

confidence: 99%

“…Weights for the estimator are found by ensuring its minimum follows a χ 2 distribution. Previous work with this approach has shown that the it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [23]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization parameter.…”

Section: Introductionmentioning

confidence: 99%

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Discontinuous parameter estimates with least squares estimators

Mead

2013

Applied Mathematics and Computation

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We discuss weighted least squares estimates of ill-conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices. Least squares estimators are the maximum likelihood estimate for normally distributed data and parameters, but here we do not assume particular probability distributions. Weights for the estimator are found by ensuring its minimum follows a χ 2 distribution. Previous work with this approach has shown that the it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [23]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization parameter. Diagonal weighting matrices are advantageous because they give piecewise smooth least squares estimates and hence are a mechanism through which least squares can be used to estimate discontinuous parameters. This is explained by viewing least squares estimation as a constrained optimization problem. Results with diagonal weighting matrices are given for a benchmark discontinuous inverse problem from [15]. In addition, the method is used to estimate soil moisture from data collected in the Dry Creek Watershed near Boise, Idaho. Parameter estimates are found that combine two different types of measurements, and weighting matrices are found that incorporate uncertainty due to spatial variation so that the parameters can be used over larger scales than those that were measured.

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“…For a problem solution the generalized method of least squares is applied to an estimation of parameters. In [14] the problem o f an estimation of parameters of plant with applicat ion of a method of least squares and square restrictions is studied. For construction of restrictions physical knowledge and a regularization method are used.…”

Section: Introductionmentioning

confidence: 99%

Identification of Parametrical Restrictions in Staic Systems in Conditions of Uncertainty

Karabutov¹

2013

IJISA

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Abstract-The approach to an estimation of area of parametrical restrictions (APR) for static linear system on parameters in the conditions of uncertainty is offered. For decision-making indicators of do mination o f an exit of model over an exit of system and the special indicator setting admissible level of errors of domination are used. The case of the representation of area of restrictions in the form of boundaries fro m belo w and fro m above on a mod ification of parameters of system is considered. The iteration algorith m of identification of restrictions and decision-making is offered. The adaptive algorithm of an estimation of boundaries of area of parametrical restrictions is synthesized. Procedure of estimat ion APR on the basis of the analysis of a field of secants of system is described. Method development on a case of representation APR in the form of restriction on norm of a modification of parameters of system is given. Various forms of vectorial norms and algorith ms of construction of area of parametrical restrictions corresponding to them are considered.

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Least squares problems with inequality constraints as quadratic constraints

Cited by 29 publications

References 16 publications

Adaptive user modelling in car racing games using behavioural and physiological data

Adaptive user modelling in car racing games using behavioural and physiological data

Discontinuous parameter estimates with least squares estimators

Identification of Parametrical Restrictions in Staic Systems in Conditions of Uncertainty

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