Left curly brace in tabular environment

Question

I have following code:

    \begin{IEEEeqnarray}{c} 
    y = \beta_0 + \sum\limits_{i=1}^{m}{\beta_i x_i} + \sum\limits_{i = 1}^{m} \sum\limits_{j \geq 1}^{m}{\beta_{ij} x_i x_j} + \epsilon
    \label{doe_eq}
\end{IEEEeqnarray}


\begin{tabular}{lll}
    where, & $m$ &= total number of input parameters,\\
     & $x_{i}$ &= $i$th system input,\\
     & $y$ &= system output,\\
     & $\beta_{0}$ &= model constant,\\
     & $\beta_{i}$ &= linear constant,\\
     & $\beta_{ij}$ &= \left \{ \begin{tabular}{l}
                                    interaction constant when $i$\neq$j$,\\
                                    quadratic constant when $i$=$j$,\\
                                 \end{tabular} \\                              
     & $\epsilon$ &= model bias  
\end{tabular}

With this I get following errors:

    line 553: Missing $ inserted. & $\beta_{ij}$ &= \left
line 554: Missing $ inserted. interaction constant when $i$\neq
line 555: Missing $ inserted. q
line 557: Missing \right. inserted. &
line 557: Missing $ inserted. &
: Unsupported document class (or package) detected,(caption) usage of the caption package is not recommended.
line 501: Underfull \hbox (badness 10000) in paragraph

I am not sure what is causing this exactly. Any idea?

Thank you.

Answer 1

As I said in my comment, \left and \neq can be used only in math mode, and \left always requires a matching \right. Furthermore I wouldn't write $i$=$j$, i.e. I wouldn't exit math mode for the equal sign. I'd type your tabular as

\documentclass{article}
\usepackage{amsmath}
\begin{document}
where
\begin{tabular}[t]{ll}
$m$ &= total number of input parameters,\\
$x_{i}$ &= $i$th system input,\\
$y$ &= system output,\\
$\beta_{0}$ &= model constant,\\
$\beta_{i}$ &= linear constant,\\
$\beta_{ij}$ &= $\begin{cases}
                 \text{interaction constant when } i\neq j,\\
                 \text{quadratic constant when } i = j,
                 \end{cases}$\\
$\epsilon$ &= model bias  
\end{tabular}
\end{document}

I have removed one column, putting the word "where" outside the tabular, and I have used the alignment option [t] to match the first line of the tabular with the word "where" (but that's a matter of taste...). Furthermore, I used the cases environment from amsmath for the conditional definition. The spacing around the = still doesn't look good to me, but I think the issue was to explain the errors...

EDIT: to improve the spacing you could use an align environment

\begin{align*}
m & =  \text{total number of input parameters,}\\
x_i & =  i\text{th system input,}\\
y & = \text{system output,}\\
\beta_0&=\text{model constant,}\\
\beta_{ij}&= 
\begin{cases}
  \text{interaction constant when }&i\neq j,\\
  \text{quadratic constant when }&i=j
\end{cases},\\
\epsilon&=\text{model bias}
\end{align*}

or if you want the symbols to be left-aligned, an alignat environment (but that's really a matter of taste)

\begin{alignat*}{2}
&m &&= \text{total number of input parameters,}\\
&x_{i} &&= i\text{-th system input,}\\
&y &&= \text{system output,}\\
&\beta_{0} &&= \text{model constant,}\\
&\beta_{i} &&= \text{linear constant,}\\
&\beta_{ij} &&= \begin{cases}
                 \text{interaction constant when } i\neq j,\\
                 \text{quadratic constant when } i = j,
                 \end{cases}\\
&\epsilon &&= \text{model bias} 
\end{alignat*}

Answer 2

Use align*:

where
\begin{align*}
    m & = & \text{total number of input parameters},\\
    x_i & = & i\text{th system input},\\
    y & =& \text{system output},\\
    \beta_0&=&\text{model constant},\\
    \beta_{ij}&=& 
    \begin{cases}
      \text{interaction constant when }&i\neq j,\\
      \text{quadratic constant when }&i=j
    \end{cases},\\
    \epsilon&=&\text{model bias}
\end{align*}