optDesign: A MATLAB toolbox to compute optimal regression designs

Chi-Kuang Yeh, Julie Zhou

May 08, 2024

Home | Description | Status | Example | TODO | Reference

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Description

A MATLAB toolbox to compute optimal regression designs.

Consider a general regression model, $$y_i=\eta(\mathbf{x}_i, \mathbf{\theta})+ \epsilon_i, \quad i=1, \ldots, n,$$ where $y_i$ is the $i$-th observation of a response variable $y$ at design point $\mathbf{x}_i \in S \subset \mathbb{R}^p$, $S$ is a design space, $\mathbf{\theta} \in \mathbb{R}^q$ is the unknown regression parameter vector, response function $\eta(\mathbf{x}_i, \mathbf{\theta})$ can be a linear or nonlinear function of $\mathbf{\theta}$, and the errors $\epsilon_i$ are assumed to be uncorrelated with mean zero and finite variance $\sigma^2$.

Let $\hat{\mathbf{\theta}}$ be an estimator of $\mathbf{\theta}$, such as the least squares estimator. Various optimal designs are defined by minimizing $\phi\left( \mathbb{c}ov(\hat{\mathbf{\theta}}) \right)$ over the design points $\mathbf{x}_1, \ldots, \mathbf{x}_n$, where function $\phi(\cdot)$ can be determinant, trace, or other scalar functions. The resulting designs are called optimal exact designs (OEDs), which depend on the response function $\eta(\cdot,\cdot)$, the design space $S$, the estimator $\hat{\mathbf{\theta}}$, the scalar function $\phi(\cdot)$, and the number of points $n$.

Status

Currently, this toolbox is under development. I will update it whenever I have leisure time.

Installation

1. Download

From a command line:

git clone https://github.com/chikuang/optDesign

2. Install in MATLAB

From a MATLAB command window:

addpath( genpath('AbsolutePathToToolbox') )

where AbsolutePathToToolbox is the name of the absolute path where you cloned this toolbox. Adding this command to your MATLAB startup.m file will make sure these tools are available every time you use MATLAB.

Example

Example 1. Optimal approximate design under ordinary least-squares estimator

The information matrix of a design $\xi(\mathbf{x})\in \Xi_S$ for model on top is given by

$$\boldsymbol{\mathrm{I}}(\xi, \mathbf{\theta}^*) =\sum_{i=1}^k w_i \boldsymbol{\mathrm{f}}({\bf v}_i, \mathbf{\theta}^*)\boldsymbol{\mathrm{f}}^\top({\bf v}_i, \mathbf{\theta}^*),$$

where vector $\boldsymbol{\mathrm{f}}(\mathbf{x}, \mathbf{\theta}) =\frac{\partial \eta(\mathbf{x}, \mathbf{\theta})} {\partial \mathbf{\theta}}$, and $\mathbf{\theta}^*$</code> is the true value of $<code>\mathbf{\theta}$. The covariance matrix of $\hat{\mathbf{\theta}}$, $\mathbb{c}ov(\hat{\mathbf{\theta}})$, is proportional to $\boldsymbol{\mathrm{I}}^{-1}(\xi, \mathbf{\theta}^*)$. An OAD is defined as the minimizer of $\phi\left\{ \boldsymbol{\mathrm{I}}^{-1}(\xi, \mathbf{\theta}^*)\right\}$ over all possible designs $\xi \in \Xi_S$ for a given function $\phi(\cdot)$.

[optval, opt_design] = opt_OLS("D", 1001, [0, 4], [1,1]', @mm)

Results are as follows:

Optimal value (cvx_optval): +2.74887
opt_val =
    2.7489
opt_design =
    0.6680    4.0000
    0.5000    0.5000

Example 2. Optimal approximate design under second least-squares estimator

Second order least-squares estimator is defined as

$$(\boldsymbol{\hat{\theta}}^\top,\hat{\sigma}^2)^\top:=\underset{\boldsymbol{\theta},\sigma^2}{\mathrm{argmin}}\sum_{i=1}^n \begin{pmatrix} y_i-\eta(\boldsymbol{x}_i;\boldsymbol{\theta})\\\ y_i^2-\eta^2(\boldsymbol{x}_i;\boldsymbol{\theta})-\sigma^2 \end{pmatrix}^\top W(\boldsymbol{x_i}) \begin{pmatrix} y_i-\eta(\boldsymbol{x_i};\boldsymbol{\theta})\\\ y_i^2-\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})-\sigma^2 \end{pmatrix}.$$

Note that $W(\boldsymbol{x_i})$ is a $2\times 2$ non-negative semi-definite matrix which may or may not depend on $\boldsymbol{x_i}$ \Wang and Leblanc (2008). It is clear that SLSE is a natural extension of the OLSE which is defined based on the first-order difference function (i.e. $y_i-\mathbb{E}[y_i]=y_i-\eta(\boldsymbol{x_i};\boldsymbol{\theta})$). On the other hand, SLSE is defined using not only the first-order difference function, but also second-order difference function (i.e. $y_i^2-\mathbb{E}[y_i^2]=y_i^2-(\eta^2(\boldsymbol{x_i};\boldsymbol{\theta})+\sigma^2))$. One might think about the downsides of the SLSE after talking about the advantages of SLSE over OLSE. SLSE does have its disadvantages indeed. It is not a linear estimator and there is no closed-form solution. It requires more computational resources compared to the OLSE due to the nonlinearity. However, numerical results can be easily computed for SLSE nowadays. As a result, SLSE is a powerful alternative estimator to be considered in research studies and real-life applications.

[opt_val, opt_design, error] = opt_SLSE("D", 101, 0, [0.05, 0.5]', [0,180]',@peleg)

Results are as follows:

opt_val =
   -2.7447
opt_design =
    1.8000  180.0000
    0.5000    0.5000
error =
   1.1299e-10

TODO

• Second-order least squares estimator with A-, D- and c-criteria
• Optimal approximate design for various regression models, including the following:
  1. Michaelis-Menton
  2. Gompertz
  3. EMax
  4. Compartment
  5. Peleg
  6. Arrhenius equation
  7. Second-order regression
• Various optimalities including A-, D-, c-, K- and E-.
• Implementation of the equivalence theorem under ordinary least-squares estimator
• Optimal exact design -- CVXSADes paper

Reference

1. Berger, M.P.F. and Wong, W.K. (2009). An Introduction to Optimal Designs for Social and Biomedical Research.
2. Wang, L. and Leblac, A. (2008). Second-order nonlinear least squares estimation. Annals of the Institute of Statistical Mathematics, 60, 883-900.
3. Wong, W.K. and Zhou, J. (2019). CVX‐based algorithms for constructing various optimal regression designs. Canadian Journal of Statistics, 47, 374-391.
4. Yeh, C.-K. (2018). Optimal Regression Design under Second-order Least Squares Estimator: Theory, Algorithm and Applications. Masters Thesis, University of Victoria.
5. Yeh, C.-K. and Zhou, J. (2024+). CVXSADes: a stochastic algorithm for constructing optimal exact regression designs with single or multiple objectives, arXiv preprint.
6. Yeh, C.-K. and Zhou, J. (2021). Properties of optimal regression designs under the second-order least squares estimator. Statistical Papers, 62, 75-92.

Contact or suggestions

Please contact me (Chi-Kuang Yeh) at [email protected].