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\title{The Reservoir Model: \\\bigskip Differential Equation Model of Psychological Capacity}

%Modeling How Inputs and Outputs to Constructs Add Up}

% Differential Equation Model of Psychological Capacity

%\author{}
\twoauthors{Pascal R. Deboeck}{C. S.  Bergeman}
%\affiliation{\mbox{ }}
\twoaffiliations{Department of Psychology \\ University of Kansas}{Department of Psychology \\ University of Notre Dame}

\rightheader{Reservoir Model}
 \shorttitle{Reservoir Model}

\abstract{ }

\acknowledgements{ }

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\singlespace
\section{Supplementary Material A}

The following tables present bias and RMSE information for Figures~7,~9, and~10. Bias has been calculated as
\begin{eqnarray}
\nonumber Bias = \frac{1}{N}\sum_{i=1}^N (\hat{\beta_{i}}-\beta_{True})
\end{eqnarray}
and RMSE has been calculated as
\begin{eqnarray}
\nonumber  RMSE=\sqrt{ \frac{1}{N}\sum_{i=1}^N (\hat{\beta_{i}}-\beta_{True})^2},
\end{eqnarray}
where $\beta$ is being used to indicate any parameter of interest (i.e., dissipation $\beta$, mean input $\mu$, or input variance $\sigma^2$), and $N$ is the total number of estimates 

\begin{table}[ht]
\small
\begin{center}
\caption{Estimates of Bias and RMSE for the Dissipation Parameter $\beta$}
\begin{tabular}{|c|cccc|cccc|}
   \multicolumn{9}{c}{}\\
 \multicolumn{1}{c}{}&\multicolumn{8}{c}{Input Frequency}\\
 \multicolumn{1}{c}{}& 12.5\% & 25\% & 50\% &  \multicolumn{1}{c}{100\%} &  12.5\% & 25\% & 50\% &  \multicolumn{1}{c}{100\%}  \\\cline{2-9}
\multicolumn{1}{|c|}{$\beta$ True }&\multicolumn{4}{c|}{Bias}&\multicolumn{4}{c|}{RMSE}\\\hline
   \multicolumn{9}{c}{}\\
  \multicolumn{9}{l}{Time Series Length = 25}\\\hline
-0.10 & -0.03 & -0.02 & -0.01 & 0.01 & 0.10 & 0.09 & 0.09 & 0.07 \\ 
-0.20 & -0.00 & 0.01 & 0.02 & 0.06 & 0.13 & 0.11 & 0.12 & 0.12 \\ 
-0.30 & 0.00 & 0.03 & 0.06 & 0.12 & 0.14 & 0.14 & 0.15 & 0.17 \\ 
-0.40 & -0.02 & 0.03 & 0.10 & 0.16 & 0.22 & 0.18 & 0.18 & 0.22 \\ 
-0.50 & -0.05 & 0.04 & 0.13 & 0.22 & 0.28 & 0.24 & 0.22 & 0.28 \\ 
-0.60 & -0.09 & 0.02 & 0.16 & 0.28 & 0.36 & 0.28 & 0.28 & 0.34 \\ 
-0.70 & -0.14 & -0.00 & 0.17 & 0.34 & 0.44 & 0.32 & 0.31 & 0.41 \\ 
-0.80 & -0.19 & -0.03 & 0.20 & 0.39 & 0.48 & 0.37 & 0.35 & 0.47 \\ 
-0.90 & -0.22 & -0.07 & 0.20 & 0.45 & 0.49 & 0.42 & 0.39 & 0.53 \\ \hline
  \multicolumn{9}{c}{}\\
  \multicolumn{9}{l}{Time Series Length = 50}\\\hline
 -0.10 & 0.01 & 0.02 & 0.02 & 0.04 & 0.05 & 0.06 & 0.06 & 0.06 \\ 
 -0.20 & 0.02 & 0.04 & 0.07 & 0.10 & 0.08 & 0.08 & 0.10 & 0.12 \\ 
 -0.30 & 0.03 & 0.06 & 0.11 & 0.16 & 0.09 & 0.11 & 0.14 & 0.19 \\ 
 -0.40 & 0.01 & 0.07 & 0.14 & 0.22 & 0.14 & 0.13 & 0.18 & 0.25 \\ 
 -0.50 & 0.01 & 0.07 & 0.17 & 0.29 & 0.17 & 0.15 & 0.22 & 0.32 \\ 
 -0.60 & -0.02 & 0.06 & 0.19 & 0.35 & 0.21 & 0.18 & 0.25 & 0.39 \\ 
 -0.70 & -0.05 & 0.06 & 0.21 & 0.42 & 0.27 & 0.20 & 0.29 & 0.46 \\ 
 -0.80 & -0.08 & 0.05 & 0.22 & 0.49 & 0.30 & 0.23 & 0.31 & 0.53 \\ 
 -0.90 & -0.08 & 0.06 & 0.26 & 0.55 & 0.30 & 0.25 & 0.35 & 0.59 \\ \hline
  \multicolumn{9}{c}{}\\
  \multicolumn{9}{l}{Time Series Length = 100}\\\hline
 -0.10 & 0.02 & 0.03 & 0.04 & 0.05 & 0.04 & 0.05 & 0.06 & 0.07 \\ 
 -0.20 & 0.03 & 0.05 & 0.08 & 0.12 & 0.06 & 0.07 & 0.10 & 0.13 \\ 
 -0.30 & 0.03 & 0.06 & 0.11 & 0.19 & 0.08 & 0.09 & 0.14 & 0.20 \\ 
 -0.40 & 0.02 & 0.06 & 0.14 & 0.25 & 0.10 & 0.11 & 0.17 & 0.27 \\ 
 -0.50 & 0.01 & 0.07 & 0.16 & 0.31 & 0.12 & 0.12 & 0.19 & 0.33 \\ 
 -0.60 & 0.00 & 0.07 & 0.19 & 0.39 & 0.13 & 0.14 & 0.23 & 0.40 \\ 
 -0.70 & 0.00 & 0.07 & 0.21 & 0.45 & 0.15 & 0.15 & 0.25 & 0.47 \\ 
 -0.80 & -0.01 & 0.08 & 0.23 & 0.51 & 0.16 & 0.17 & 0.28 & 0.54 \\ 
 -0.90 & -0.01 & 0.08 & 0.25 & 0.58 & 0.17 & 0.19 & 0.31 & 0.60 \\ 
   \hline
    \multicolumn{9}{c}{}
\end{tabular}
\end{center}
\begin{flushleft}
\footnotesize 
\end{flushleft}
\end{table}

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% latex table generated in R 2.13.0 by xtable 1.6-0 package
% Thu Feb 16 07:29:38 2012
\begin{table}[ht]
\small
\begin{center}
\caption{Estimates of Bias and RMSE for the Mean of the Input Distribution $\mu$}
\begin{tabular}{|c|c|cccc|cccc|}
   \multicolumn{10}{c}{}\\
 \multicolumn{2}{c}{}&\multicolumn{8}{c}{True $\beta$}\\
 \multicolumn{2}{c}{}& -0.2 & -0.4 & -0.6 &  \multicolumn{1}{c}{-0.8} & -0.2 & -0.4 & -0.6 &  \multicolumn{1}{c}{-0.8}  \\\cline{3-10}
Input Freq&\multicolumn{1}{c|}{$\mu$ True }&\multicolumn{4}{c|}{Bias}&\multicolumn{4}{c|}{RMSE}\\\hline
   \multicolumn{10}{c}{}\\
  \multicolumn{10}{l}{Time Series Length = 25}\\\hline
12.5\% \& 25\% & 0.12 & -0.00 & 0.01 & 0.03 & 0.04 & 0.12 & 0.12 & 0.16 & 0.15 \\ 
12.5\% \& 25\% & 0.17 & 0.00 & 0.02 & 0.04 & 0.08 & 0.15 & 0.19 & 0.18 & 0.29 \\ 
12.5\% \& 25\% & 0.25 & 0.01 & 0.00 & 0.02 & 0.06 & 0.25 & 0.21 & 0.23 & 0.26 \\ 
12.5\% \& 25\% & 0.33 & -0.03 & -0.02 & -0.01 & 0.03 & 0.25 & 0.20 & 0.23 & 0.25 \\
12.5\% \& 25\% & 0.50 & -0.03 & -0.04 & -0.01 & 0.07 & 0.36 & 0.35 & 0.35 & 0.41 \\ \hline
50\% \& 100\%   & 0.50 & -0.06 & -0.11 & -0.13 & -0.11 & 0.33 & 0.27 & 0.28 & 0.25 \\ 
50\% \& 100\%   & 0.67 & -0.12 & -0.15 & -0.17 & -0.16 & 0.42 & 0.36 & 0.41 & 0.36 \\ 
50\% \& 100\%   & 1.00 & -0.20 & -0.33 & -0.34 & -0.37 & 0.64 & 0.55 & 0.56 & 0.54 \\ 
50\% \& 100\%   & 1.33 & -0.38 & -0.56 & -0.64 & -0.63 & 0.85 & 0.78 & 0.78 & 0.79 \\  
50\% \& 100\%   &  2.00 & -0.56 & -0.77 & -0.92 & -0.97 & 1.20 & 1.17 & 1.18 & 1.22 \\  \hline
    \multicolumn{10}{c}{}\\
  \multicolumn{10}{l}{Time Series Length = 50}\\\hline
12.5\% \& 25\% &  0.12 & -0.02 & 0.00 & 0.02 & 0.03 & 0.08 & 0.08 & 0.09 & 0.10 \\ 
12.5\% \& 25\% &  0.17 & -0.01 & 0.01 & 0.02 & 0.04 & 0.11 & 0.12 & 0.12 & 0.14 \\ 
12.5\% \& 25\% &  0.25 & -0.04 & -0.01 & 0.02 & 0.04 & 0.14 & 0.14 & 0.15 & 0.17 \\ 
12.5\% \& 25\% &  0.33 & -0.06 & -0.06 & -0.01 & 0.00 & 0.17 & 0.15 & 0.14 & 0.14 \\ 
12.5\% \& 25\% &  0.50 & -0.10 & -0.08 & -0.02 & 0.01 & 0.27 & 0.22 & 0.24 & 0.23 \\  \hline
50\% \& 100\%  &  0.50 & -0.17 & -0.17 & -0.15 & -0.13 & 0.27 & 0.24 & 0.21 & 0.22 \\ 
50\% \& 100\%  &  0.67 & -0.25 & -0.22 & -0.21 & -0.18 & 0.36 & 0.31 & 0.30 & 0.28 \\ 
50\% \& 100\%  &  1.00 & -0.43 & -0.46 & -0.45 & -0.45 & 0.58 & 0.56 & 0.56 & 0.56 \\ 
50\% \& 100\%  &  1.33 & -0.69 & -0.73 & -0.79 & -0.80 & 0.82 & 0.82 & 0.85 & 0.87 \\ 
50\% \& 100\%  &  2.00 & -1.03 & -1.11 & -1.18 & -1.23 & 1.29 & 1.24 & 1.30 & 1.33 \\  \hline
    \multicolumn{10}{c}{}\\
  \multicolumn{10}{l}{Time Series Length = 100}\\\hline
12.5\% \& 25\% &  0.12 & -0.02 & 0.00 & 0.01 & 0.03 & 0.05 & 0.06 & 0.06 & 0.07 \\ 
12.5\% \& 25\% &  0.17 & -0.02 & 0.00 & 0.02 & 0.03 & 0.08 & 0.07 & 0.08 & 0.09 \\ 
12.5\% \& 25\% &  0.25 & -0.04 & -0.01 & 0.01 & 0.02 & 0.11 & 0.10 & 0.10 & 0.11 \\ 
12.5\% \& 25\% &  0.33 & -0.08 & -0.04 & -0.01 & 0.00 & 0.14 & 0.12 & 0.11 & 0.11 \\  
12.5\% \& 25\% &  0.50 & -0.11 & -0.07 & -0.03 & -0.00 & 0.22 & 0.16 & 0.17 & 0.16 \\  \hline
50\% \& 100\% &  0.50 & -0.21 & -0.17 & -0.16 & -0.13 & 0.26 & 0.21 & 0.20 & 0.18 \\ 
50\% \& 100\% &  0.67 & -0.27 & -0.23 & -0.20 & -0.17 & 0.34 & 0.29 & 0.26 & 0.24 \\ 
50\% \& 100\% &  1.00 & -0.52 & -0.50 & -0.46 & -0.45 & 0.60 & 0.56 & 0.54 & 0.54 \\ 
50\% \& 100\% &  1.33 & -0.84 & -0.83 & -0.87 & -0.86 & 0.90 & 0.88 & 0.91 & 0.90 \\ 
50\% \& 100\% &  2.00 & -1.21 & -1.27 & -1.28 & -1.27 & 1.34 & 1.34 & 1.34 & 1.34 \\ 
   \hline
    \multicolumn{10}{c}{}
\end{tabular}
\end{center}
\begin{flushleft}
\footnotesize 
\end{flushleft}
\end{table}

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% latex table generated in R 2.13.0 by xtable 1.6-0 package
% Thu Feb 16 07:29:38 2012
\begin{table}[ht]
\small
\begin{center}
\caption{Estimates of Bias and RMSE for the Variance of the Input Distribution $\sigma^2$}
\begin{tabular}{|c|c|cccc|cccc|}
   \multicolumn{10}{c}{}\\
 \multicolumn{2}{c}{}&\multicolumn{8}{c}{True $\beta$}\\
 \multicolumn{2}{c}{}& -0.2 & -0.4 & -0.6 &  \multicolumn{1}{c}{-0.8} & -0.2 & -0.4 & -0.6 &  \multicolumn{1}{c}{-0.8}  \\\cline{3-10}
Input Freq&\multicolumn{1}{c|}{$\mu$ True }&\multicolumn{4}{c|}{Bias}&\multicolumn{4}{c|}{RMSE}\\\hline
   \multicolumn{10}{c}{}\\
  \multicolumn{10}{l}{Time Series Length = 25}\\\hline
12.5\% \& 25\% &0.23 & -0.01 & 0.00 & 0.03 & 0.02 & 0.31 & 0.40 & 0.50 & 0.40 \\ 
12.5\% \& 25\% &0.41 & -0.01 & 0.04 & -0.01 & 0.12 & 0.55 & 0.93 & 0.54 & 1.01 \\ 
12.5\% \& 25\% &0.44 & -0.02 & -0.02 & -0.07 & 0.02 & 0.44 & 0.46 & 0.38 & 0.55 \\ 
12.5\% \& 25\% &0.78 & -0.07 & 0.04 & -0.04 & -0.03 & 0.81 & 0.97 & 1.00 & 0.94 \\ 
12.5\% \& 25\% &0.93 & 0.15 & 0.01 & 0.12 & 0.05 & 1.65 & 1.62 & 1.78 & 1.67 \\  
12.5\% \& 25\% &1.75 & -0.02 & -0.07 & -0.12 & 0.14 & 2.16 & 2.22 & 2.10 & 2.41 \\  \hline
50\% \& 100\% & 0.75 & -0.05 & -0.11 & -0.13 & -0.06 & 0.57 & 0.54 & 0.62 & 0.66 \\ 
50\% \& 100\% & 1.00 & -0.03 & -0.13 & -0.26 & -0.22 & 0.79 & 0.66 & 0.62 & 0.75 \\ 
50\% \& 100\% & 1.33 & -0.18 & -0.16 & -0.20 & -0.22 & 0.89 & 1.13 & 1.31 & 1.27 \\ 
50\% \& 100\% & 1.77 & -0.12 & -0.26 & -0.48 & -0.43 & 1.26 & 1.34 & 1.15 & 1.37 \\ 
50\% \& 100\% & 3.00 & -0.35 & -0.50 & -0.28 & -0.52 & 2.20 & 2.23 & 3.02 & 2.50 \\ 
50\% \& 100\% & 4.01 & -0.22 & -0.65 & -0.95 & -1.03 & 2.96 & 2.88 & 2.67 & 2.99 \\ \hline
    \multicolumn{10}{c}{}\\
  \multicolumn{10}{l}{Time Series Length = 50}\\\hline
12.5\% \& 25\%  &0.23 & -0.02 & -0.01 & -0.00 & 0.02 & 0.23 & 0.22 & 0.24 & 0.30 \\  
12.5\% \& 25\%  &0.41 & -0.01 & 0.01 & -0.00 & 0.02 & 0.41 & 0.49 & 0.48 & 0.45 \\  
12.5\% \& 25\%  &0.44 & -0.04 & -0.02 & -0.02 & 0.00 & 0.29 & 0.32 & 0.36 & 0.34 \\  
12.5\% \& 25\%  &0.78 & -0.07 & -0.12 & -0.06 & -0.06 & 0.53 & 0.53 & 0.52 & 0.54 \\ 
12.5\% \& 25\%  &0.93 & -0.05 & -0.05 & 0.07 & 0.01 & 0.89 & 0.89 & 1.23 & 1.02 \\  
12.5\% \& 25\%  &1.75 & -0.21 & -0.16 & -0.04 & -0.11 & 1.22 & 1.35 & 1.40 & 1.26 \\ \hline
50\% \& 100\%   &0.75 & -0.11 & -0.13 & -0.12 & -0.07 & 0.44 & 0.40 & 0.45 & 0.53 \\ 
50\% \& 100\%   &1.00 & -0.20 & -0.28 & -0.34 & -0.38 & 0.57 & 0.50 & 0.53 & 0.59 \\ 
50\% \& 100\%   &1.33 & -0.23 & -0.23 & -0.21 & -0.25 & 0.73 & 0.72 & 0.72 & 0.76 \\ 
50\% \& 100\%   &1.77 & -0.39 & -0.47 & -0.58 & -0.63 & 0.88 & 0.86 & 0.95 & 1.02 \\ 
50\% \& 100\%   &3.00 & -0.49 & -0.54 & -0.64 & -0.60 & 1.66 & 1.60 & 1.62 & 1.69 \\ 
50\% \& 100\%   &4.01 & -0.83 & -1.07 & -1.27 & -1.49 & 2.05 & 1.97 & 2.20 & 2.36 \\ \hline
    \multicolumn{10}{c}{}\\
  \multicolumn{10}{l}{Time Series Length = 100}\\\hline 
12.5\% \& 25\%   & 0.23 & -0.02 & -0.00 & -0.00 & 0.01 & 0.18 & 0.16 & 0.17 & 0.19 \\ 
12.5\% \& 25\%   & 0.41 & -0.01 & -0.02 & -0.01 & -0.01 & 0.29 & 0.28 & 0.30 & 0.31 \\ 
12.5\% \& 25\%   & 0.44 & -0.05 & -0.02 & -0.03 & -0.04 & 0.23 & 0.23 & 0.23 & 0.24 \\ 
12.5\% \& 25\%   & 0.78 & -0.08 & -0.04 & -0.03 & -0.04 & 0.38 & 0.42 & 0.39 & 0.43 \\ 
12.5\% \& 25\%   & 0.93 & -0.06 & -0.03 & -0.01 & -0.02 & 0.63 & 0.61 & 0.65 & 0.72 \\ 
12.5\% \& 25\%   & 1.75 & -0.21 & -0.18 & -0.10 & -0.05 & 0.83 & 0.82 & 0.99 & 0.98 \\ \hline
50\% \& 100\%    & 0.75 & -0.15 & -0.13 & -0.14 & -0.11 & 0.33 & 0.31 & 0.32 & 0.31 \\ 
50\% \& 100\%    & 1.00 & -0.28 & -0.33 & -0.36 & -0.38 & 0.46 & 0.46 & 0.48 & 0.50 \\ 
50\% \& 100\%    & 1.33 & -0.24 & -0.24 & -0.24 & -0.18 & 0.60 & 0.52 & 0.52 & 0.59 \\ 
50\% \& 100\%    & 1.77 & -0.49 & -0.56 & -0.67 & -0.65 & 0.80 & 0.81 & 0.85 & 0.87 \\ 
50\% \& 100\%    & 3.00 & -0.59 & -0.64 & -0.57 & -0.40 & 1.30 & 1.23 & 1.23 & 1.21 \\ 
50\% \& 100\%    & 4.01 & -1.22 & -1.33 & -1.46 & -1.53 & 1.87 & 1.87 & 1.97 & 2.13 \\\hline
   \hline
    \multicolumn{10}{c}{}
\end{tabular}
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