## 4 Performance Analysis
### The Effect of Interval Prediction by 2D Kernel Density Estimation  
The 2D kernel density estimation proposed in this paper can be used to predict the location of the predicted results according to the confidence level. Interval prediction measures the variability of the accumulated data and models the probability distribution to see how much difference there can be from the predicted results. The following experiment confirms the effect of the interval prediction by estimating the 2D kernel density based on the variability. First, we check the estimated quantity of demand and the quantity of demand for randomly generated data. The increase in noise means that the volatility increases accordingly. As can be seen in Figure 6, as the noise increases, the prediction interval becomes wider. That is, the increase in the variability because of noise adversely affects the prediction result by means of the LSTM, which makes it difficult to obtain accurate predicted values. Since the interval prediction has an advantage, in that it can secure the reliability of the prediction result, it can be suitably used to obtain the prediction result by means of the data of the SMEs with high volatility. In Figure 7, it can be seen whether the actual value of is included in a prediction interval. Generally, the higher the reliability, the wider the range, and the lower the reliability, the narrower the range. Therefore, the fact that the actual value is located in the lower reliability means that the prediction result is similar to the actual value. By means of \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Noise** & **0.5** & & & & & \\ \hline  & \#1 & \#2 & \#3 & \#4 & \#5 & Average \\ \hline RMSE & 81.26246 & 117.6317 & 51.70315 & 82.43249 & 88.79546 & 84.36505 \\ \hline MAPE & 3.498915 & 4.735661 & 2.440641 & 5.03232 & 5.401816 & 4.22187 \\ \hline
**Noise** & **1.0** & & & & & \\ \hline  & \#1 & \#2 & \#3 & \#4 & \#5 & Average \\ \hline RMSE & 139.8895 & 199.44 & 231.1574 & 173.0788 & 173.7208 & 183.4573 \\ \hline MAPE & 5.077092 & 7.807049 & 11.14622 & 9.198444 & 8.848312 & 8.415424 \\ \hline
**Noise** & **1.5** & & & & & \\ \hline  & \#1 & \#2 & \#3 & \#4 & \#5 & Average \\ \hline RMSE & 320.0853 & 269.0784 & 262.721 & 269.6649 & 193.4566 & 263.0013 \\ \hline MAPE & 11.6281 & 9.711254 & 11.57804 & 12.08087 & 10.46633 & 11.09292 \\ \hline
**Noise** & **2.0** & & & & & \\ \hline  & \#1 & \#2 & \#3 & \#4 & \#5 & Average \\ \hline RMSE & 317.9036 & 300.7283 & 335.1237 & 272.3147 & 228.4517 & 290.9044 \\ \hline MAPE & 12.67522 & 10.22078 & 13.586 & 14.58546 & 10.63073 & 12.33964 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of LSTM prediction for random signal. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Warehousing Out** & & & & \\ \hline  & \#1 & \#2 & \#3 & \#4 & Average \\ \hline RMSE & 653.9884 & 727.7047 & 1051.383 & 1094.891 & 881.992 \\ \hline MAPE & 90.57324 & 32.43212 & 37.1611 & 37.0847 & 49.285 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of LSTM prediction for warehousing out data. this experiment, we confirmed that only two values out of the 200 actual values (Dataset: 4; length of sequence: 10; number of sequence: 5) were out of the range. Figure 6: The result of interval prediction for random period data. Figure 7: Random value: What prediction range the actual value belongs to. As can be seen from the previous prediction performance experiments, the warehousing out data is more variable than the randomly generated periodic signals. Accordingly, the prediction interval by means of the interval prediction also has a very wide range. The experimental results show that all of the 40 real values exist in the predictor, the rest, except for 2, are located in the 80% confidence interval, and the result is very stable.