# A Scientific Document Retrieval and Reordering Method by Incorporating HFS and LSD
## 4 Experimental Process and Result Analysis
### System Experiment
#### 4.2.2 Retrieval Results of Scientific Documents by Incorporating HFS and LSD  
Similar to many search engines and retrieval tasks that yield a sequence of results, this research, focusing on the NTCIR dataset and CSD dataset, generates a list of HTML file names. The principle followed here is that the more relevant documents should appear towards the front of the list, meaning they are sorted in descending order of  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**No.** & **Query Expression** & **Query Text** \\ \hline
1 & \(\sin^{2}\theta+\cos^{2}\theta=1\) & The basic relationship between sines and cosines is called the Pythagorean theorem \\ \hline
2 & \(f(x)=ae^{-(x-b)^{2}/2}\alpha^{2}\) & The mathematical representation of the Gaussian function \\ \hline
3 & \(\mathrm{x}=\frac{-bx+\sqrt{2}-4 cx}{24}\) & Two solutions of any quadratic polynomial can be expressed as follows \\ \hline
4 & \(E=\frac{1}{2}mv^{2}\) & The relationship between the change in kinetic energy of an object and the work performed by the resultant external force \\ \hline
5 & \(\lim\limits_{x\to 0}f(x)=0\) & \(f(\mathrm{x})\) can be made as close as desired to 0 by making x close enough but not equal to a \\ \hline
6 & \(F_{g}=G\frac{M_{\mathrm{m}}}{\pi^{2}}\) & Any two particles have a mutual attractive force in the direction of the line connecting their centers \\ \hline
7 & \(y=nx^{n-1}\) & Derivatives of power functions \\ \hline
8 & \(f(x)=\frac{\infty}{n-0}\frac{f^{(n)}(x_{0})}{\pi^{2}}(x-x_{0})^{n}\) & The Taylor formula uses the information of a function at a certain point to describe its nearby values \\ \hline
9 & \(\sqrt{ab}\leq\frac{c+b}{2}\) & The arithmetic mean of two non-negative real numbers is greater than or equal to their geometric mean \\ \hline
10 & \(ax^{2}+bx=c\) & first explicit solution of the quadratic equation \\ \hline \end{tabular}
\end{table}
Table 2: Statistical expressions and their query statements.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Dataset** & **MAP\_5** & **MAP\_10** & **MAP\_20** \\ \hline English dataset & 0.912 & 0.833 & 0.784 \\ Chinese dataset & 0.907 & 0.827 & 0.776 \\ \hline \end{tabular}
\end{table}
Table 3: The MAP values of the expression under different datasets.  
relevance. Henceforth, the evaluation of the sorted retrieval results is conducted using DCG (Discounted Cumulative Gain). The calculation formula for DCG is as follows:  
\[DCG=rel_{1}+\sum_{i=2}^{P}\frac{rel_{i}}{\log_{2}i}(P\geq 2), \tag{8}\]  
where \(i\) represents the ranking number of the retrieval result; \(rel_{i}\) denotes the relevance of the \(i\)-th search result, categorized as completely relevant (scored as 4), relatively relevant (scored as 3), partially relevant (scored as 1), or completely irrelevant (scored as 0); and \(P\) refers to the total number of retrieval results.  
To facilitate the comparison of retrieval result scores across different levels, nDCG (normalized Discounted Cumulative Gain) is employed to standardize the DCG values. The calculation method is as follows:  
\[nDCG=\frac{DCG}{IDCG}, \tag{9}\]  
where \(IDCG\) is the \(DCG\) value achieved when the returned list is in its ideal state.  
In this experiment, the mathematical expressions and query texts from Table 2 are employed for the statistical analysis of scientific document retrieval. Figure 4 displays the nDCG@10 scores obtained for various mathematical expressions.