results.tex

\begin{table}
\caption{Results: Back-Propagation with Stochastic Gradient Descent}
\resizebox{\textwidth}{!}{
\label{tab:bp_results}
	\begin{tabular}{llllll}
	\toprule
	Stock              & Hidden Nodes & Training Time (sec) & Mean Squared Error & Hits/Misses = hit\% (type 1) & Hits/Misses = hit\% (type 2) \\\midrule
	Apple Inc          & 40           & 17.7                & 4.47               & $283/214=56.9\%$       & $57/82=41.0\%$         \\
	Walmart            & 40           & 23.6                & 0.339              & $349/148=70.2\%$       & $123/69=64.1\%$       \\
	Bank of America    & 40           & 61.8                & 0.06               & $312/185=62.8\%$       & $91/78=53.8\%$         \\
	Ford Motor Company & 40           & 35.3                & 0.036              & $324/173=65.2\%$       & $71/93=43.3\%$         \\
	Coca-Cola          & 40           & 34.2                & 0.07               & $327/170=65.8\%$       & $82/83=49.7\%$        \\\midrule
	Average            & 40           & 35.5                & 0.995               & $64.2\%$              & $50.4\%$               \\
	\bottomrule
	\end{tabular}}
\end{table}

\begin{table}
\caption{Results: Extreme Learning Machine}
\resizebox{\textwidth}{!}{%
\label{tab:elm_results}
	\begin{tabular}{llllll}
	\toprule
	Stock              & Hidden Nodes & Training Time (sec) & Mean Squared Error & Hits/Misses= hit\% (type 1) & Hits/Misses= hit\% (type 2) \\\midrule
	Apple Inc          & 497          & 1.44                & 1.454              & 372/125=74.8\%              & 124/69=64.2\%               \\
	Walmart            & 890          & 1.63                & 0.267              & 370/127=74.4\%              & 146/51=74.1\%               \\
	Bank of America    & 440          & 1.5                 & 0.039              & 356/141=71.6\%              & 116/77=60.1\%               \\
	Ford Motor Company & 276          & 1.38                & 0.027              & 366/131=73.6\%              & 128/58=68.8\%               \\
	Coca-Cola          & 590          & 1.91                & 0.063              & 363/134=73.0\%              & 134/45=74.9\%               \\\midrule
	Average            & 539          & 1.57                & 0.37               & 73.5\%                      & 68.4\%                      \\
	\bottomrule
	\end{tabular}}
\end{table}

The error measurements to quantify the results include the number of nodes, training time, mean-squared error, and a couple of hit or miss ratios.
The number of nodes corresponds to the nodes within the single hidden layer.
The training time is the amount of time the script took between beginning and ending the training phase.
Mean-squared error is the average of squared deviation from the real value of the stock.
For $N$ testing samples, real values $y$, and forecasted values $\hat{y}$, mean-squared error is expressed as

\begin{equation}
	MSE=\dfrac{1}{N}\sum_{i=1}^{N} (\hat{y_i} - y_i)^2\label{eq:equation11}
\end{equation}

While mean-squared error is useful for comparisons between different models, mean-squared error is not relevant for measuring classification successes.
Two other hit or miss ratios were adopted for the purpose of better defining a successful prediction.
For the first type, a hit signifies that the stock and prediction have the same sign of change between any two adjacent days.
For the second type, a hit signifies the model predicting an inflection point correctly; given any adjacent three days, if the model predicted the sign of change correctly between days 1 and 2, the type 2 hit or miss ratio measures whether a model correctly forecasts the opposite sign change between days 2 and 3.
Both hit or miss ratios were implemented because classification accuracy is necessary for profiting within a market.

The results of training, validating, and testing each stock for Back-Propagation and Extreme Learning Machine models are provided within tables\ \ref{tab:bp_results} and\ \ref{tab:elm_results}, respectively.
Each stock was trained, validated, and tested on only its own historical stock information.

% Discussion