Examples

We will revise some examples we have seen in previous lectures as examples.

X-ray

Now let us consider the X-ray data

Data: \((y_i, x_i), \quad i=1,\dots,n; \quad n=15.\)

\(y_i = \log(N(x_i)/N(0))\),

\(x_i\) = exposure time.

Possible model: \(y_i = \beta x_i + \epsilon_i\) for some \(\beta\).

We can write this model in vector-matrix form.

\[\begin{aligned} \mbox{response } \mathbf{Y} &=\left( \begin{array}{c} y_{1} \\ \vdots \\ y_{15} \\ \end{array} \right), \\ \quad \mbox{parameter } \boldsymbol{\beta} &=\left( \begin{array}{c} \beta \end{array} \right), \\ \quad \mbox{design matrix } \mathbf{X} &= \left( \begin{array}{c} x_{1} \\ \vdots\\ x_{15} \end{array} \right), \\ \quad \mbox{errors } \boldsymbol{\epsilon} &=\left( \begin{array}{c} \epsilon_{1} \\ \vdots \\ \epsilon_{15} \\ \end{array} \right).\\ \end{aligned}\]

Potatoes

Now let us now write the model for the potato data in vector-matrix notation.

Data: \((y_i, x_i), \quad i=1,\dots,n; \quad n=14\)

\(y_i\) = glucose concentration,

\(x_i\) = storage time.

Possible model: \(y_i = \alpha + \beta x_i+ \gamma x_i^2+ \epsilon_i, \quad i=1,\dots,n.\)

The vector-matrix representation of the above model is as follows: \[\begin{aligned} \mathbf{Y} &=\left( \begin{array}{c} y_{1} \\ \vdots \\ y_{14} \\ \end{array} \right), \\ \quad \boldsymbol{\beta} &=\left( \begin{array}{c} \alpha \\ \beta \\ \gamma \end{array} \right), \\ \quad \mathbf{X} &= \left( \begin{array}{ccc} 1 & x_{1} & x^2_{1}\\ \vdots & \vdots & \vdots\\ 1 & x_{14} & x^2_{14} \end{array} \right), \\ \quad \boldsymbol{\epsilon} &=\left( \begin{array}{c} \epsilon_{1} \\ \vdots \\ \epsilon_{14} \\ \end{array} \right).\\ \end{aligned}\]