## 4 Results
### LDPS Algorithm
#### 4.5.3 Search for Global Linear Structures  
A quantification \(L\) of the property \(\mathfrak{L}\) is made. Hence, it is possible to involve a GDPS scheme based on \(L\) for the weak solution of Task \(1\) in this case, which leads to the GDPS\((P,\alpha,\gamma)\) algorithm, where \(\alpha\) is the expression level of property \(L\) and \(\gamma\) is its representativity degree. Research shows that in the generic case its result \(Z\) on space \(Y=Q^{1}(L,\alpha)\) needs additional filtering, which is done by the MDPS algorithm with a solidity level \(\varepsilon\). Its result \(\mathrm{MDPS}(Z,\varepsilon)\) is considered final in the search for linear structures within the space. The LDPS algorithm is the described superposition of GDPS and MDPS:  
\[Q\to Q^{1}(L,\alpha)=Y\to\mathrm{GDPS}(Y,\gamma)=Z\to\mathrm{MDPS}(Z, \varepsilon)=\mathrm{LDPS}(Q).\]  
Its parameters will be (parameters \(L\)) + (parameters \(M\))+(\(\alpha,\gamma,\varepsilon\)), and the result is the global linear structures in \(Q\) relative to them. The LDPS algorithm has four stages:  
* the first of them with the selected parameters of local linearity \(r\) and \(h\) constructs its quantification \(L_{Q}(q|r,h)\) at each point \(q\) of space \(Q\), and the best corridor \(K_{Q}(q|r,h)\), where the estimate \(L_{Q}(q|r,h)\) is reached;
* the second stage includes constructing the basis for application of the GDPS scheme, namely coverage \(U_{\alpha}(L)=\{K_{Q}(q|r,h):L_{Q}(q|r,h)\geq\alpha\}\) for a given level of local linearity \(\alpha\);
* the third stage is GDPS scheme working on \(U_{\alpha}(L)\) data. Its result will represent linear structures in \(Q\). On space \[Y=\mathrm{Supp}\;U_{\alpha}(L)=\{q\in Q:L_{Q}(q|r,h)\geq\alpha\}\] a measure \(G=G(U_{\alpha}(L))\) is constructed (9) \[G_{B}(y)=\frac{|D_{B}(y,r)\cap K_{Q}(q|r,h)|}{|K_{Q}(q|r,h)|}.\] The result will represent the raw linear structures on \(Q\);
* the fourth stage is their filtering by solidity using the MDPS algorithm with a level of \(\varepsilon\). In conclusion, we will address the operation of the LDPS algorithm on two arrays, while the operation in the first instance will be explained in detail, and in the second instance only the result is shown. **Example 10**. : _The initial array \(Q\) is shown in Figure 11(a)_  
* _At the first stage with the chosen parameters at each blue point_ \(q\) _a linear corridor_ \(K_{Q}(q|r,h)\) _is constructed with parameters_ \(r=1.09\) _and_ \(h=0.44\)_, then its separability_ \(L_{Q}(q|r,h)\) _is calculated. Figure 11(b),c show corridors in green with centers at black points. Their separability equals to_ \(0.64\) _and_ \(0.5\)_, respectively. In the second instance, it proves to be insufficient to overcome the second stage._
* _Second stage. The separability level_ \(\alpha\) _is assumed to be_ \(0.6\)_. Figure 12(a) shows the points_ \(y\) _in red, that passed this selection and formed the basis_ \(Y\)_--the first half for application of the GDPS scheme. The second half GDPS scheme: corridors_ \(K_{Q}(q|r,h)\) _are shown for the already familiar point on the left (Figure 12(b)) and at some point on the right (Figure 12(c)). It can be seen from the figures that the relative density of red dots in the left corridor is higher than in the right one. This circumstance will help the left point overcome the third stage and move into the lower linear structure, while the right point will not stand the test with GDPS operation and will not be included in the final result. Figure 12: The first stage of the LDPS algorithm. (**a**) original array; (**b**) a corridor with a separability equal to 0.64; and (**c**) a corridor with a separation of 0.5. The position of the fragments of the original array shown in figures (**b,c**) is indicated by the leaders in figure (**a**). Figure 13: The second stage of the algorithm. (**a**) The original array with the selected base (red dots); (**b**) points from in the corridor from Figure 12b; and (**c**) points from in the corridor from Figure 12c.