THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Statistics for Spatial Data.

http://ee.sharif.edu/~miap/Files/A%20Survey%20of%20Medical%20Image%20Registration.pdf

A survey of medical image registration.

The successful application of a conceptual rainfall-runoff (CRR) model depends on how well it is calibrated. Despite the popularity of CRR models, reports in the literature indicate that it is typically difficult, if not impossible, to obtain unique optimal values for their parameters using automatic calibration methods. Unless the best set of parameters associated with a given calibration data set can be found, it is difficult to determine how sensitive the parameter estimates (and hence the model forecasts) are to factors such as input and output data error, model error, quantity and quality of data, objective function used, and so on. Results are presented that establish clearly the nature of the multiple optima problem for the research CRR model SIXPAR. These results suggest that the CRR model optimization problem is more difficult than had been previously thought and that currently used local search procedures have a very low probability of successfully finding the optimal parameter sets. Next, the performance of three existing global search procedures are evaluated on the model SIXPAR. Finally, a powerful new global optimization procedure is presented, entitled the shuffled complex evolution (SCE-UA) method, which was able to consistently locate the global optimum of the SIXPAR model, and appears to be capable of efficiently and effectively solving the CRR model optimization problem.

Effective and efficient global optimization for conceptual rainfall‐runoff models

We present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set, which can be either images (two-dimensional) or volumes (three-dimensional). It uses an explicit spline representation of the images in conjunction with spline processing, and is based on a coarse-to-fine iterative strategy (pyramid approach). The minimization is performed according to a new variation (ML*) of the Marquardt-Levenberg algorithm for nonlinear least-square optimization. The geometric deformation model is a global three-dimensional (3-D) affine transformation that can be optionally restricted to rigid-body motion (rotation and translation), combined with isometric scaling. It also includes an optional adjustment of image contrast differences. We obtain excellent results for the registration of intramodality positron emission tomography (PET) and functional magnetic resonance imaging (fMRI) data. We conclude that the multiresolution refinement strategy is more robust than a comparable single-stage method, being less likely to be trapped into a false local optimum. In addition, our improved version of the Marquardt-Levenberg algorithm is faster.

/pdf/a-pyramid-approach-to-subpixel-registration-based-on-41xhw0es6x.pdf

A pyramid approach to subpixel registration based on intensity

A standard objective in computer experiments is to approximate the behaviour of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.

Bayesian quadrature and energy minimization for space-filling design

Covariate-adaptive treatment allocation is considered in the situation when a compromise must be made between information (about the dependency of the probability of success of each treatment upon influential covariates) and cost (in terms of number of subjects receiving the poorest treatment). Information is measured through a design criterion for parameter estimation, the cost is additive and is related to the success probabilities. Within the framework of approximate design theory, the determination of optimal allocations forms a compound design problem. We show that when the covariates are i.i.d. with a probability measure µ, its solution possesses strong similarities with the construction of optimal design measures bounded by µ. We characterize optimal designs through an Equivalence Theorem and construct a covariate-adaptive sequential allocation strategy that converges to the optimum. Our new optimal designs can be used as benchmarks for other, more usual, allocation methods. A response-adaptive implementation is possible for practical applications with unknown model parameters. Several illustrative examples are provided.

/pdf/information-regret-compromise-in-covariate-adaptive-5f4t63w7sb.pdf

Information-regret compromise in covariate-adaptive treatment allocation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy
 k
 . This may correspond to the case of a regression model, where one observesy
 k
 =f(θ,x
 k
 )+e
 k
 , with e
 k
 some random error, or to the Bernoulli case wherey
 k
 ∈{0, 1}, with Pr[y
 k
 =1|θ,x
 k
 |=f(θ,x
 k
 ). Special attention is given to sequences given by
 
$$x_{k + 1} = \arg \max _x f(\hat \theta ^k ,x) + \alpha _k d_k (x)$$

 , with
 
$$\hat \theta ^k $$

 an estimated value of θ obtained from (x1, y1),...,(x
 k
 ,y
 k
 ) andd
 k
 (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑
 =1
 w
 i
 f(θ, x
 i
 ) with {w
 i
 } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.

/pdf/sequential-experimental-design-and-response-optimisation-3v1pze1xt9.pdf

Sequential experimental design and response optimisation

We study the consistency and asymptotic normality of the LS estimator of a function h(θ) of the parameters θ in a nonlinear regression model with observations $y_i=\eta(x_i,\theta) +\varepsilon_i$, $i=1,2\ldots$ and independent errors e i . Optimum experimental design for the estimation of h(θ) frequently yields singular information matrices, which corresponds to the situation considered here. The difficulties caused by such singular designs are illustrated by a simple example: depending on the true value of the model parameters and on the type of convergence of the sequence of design points $x_1,x_2\ldots$ to the limiting singular design measure ξ, the convergence of the estimator of h(θ) may be slower than $1/\sqrt{n}$, and, when convergence is at a rate of $1/\sqrt{n}$ and the estimator is asymptotically normal, its asymptotic variance may differ from that obtained for the limiting design ξ (which we call irregular asymptotic normality of the estimator). For that reason we focuss our attention on two types of design sequences: those that converge strongly to a discrete measure and those that correspond to sampling randomly from ξ. We then give assumptions on the limiting expectation surface of the model and on the estimated function h which, for the designs considered, are sufficient to ensure the regular asymptotic normality of the LS estimator of h(θ).

/pdf/asymptotic-normality-of-nonlinear-least-squares-under-23uiiz53hg.pdf

Luc Pronzato

Papers

Bayesian quadrature and energy minimization for space-filling design

Information-regret compromise in covariate-adaptive treatment allocation

Sequential experimental design and response optimisation

Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs

A Dirac-function method for densities of nonlinear statistics and for marginal densities in nonlinear regression