#
Efficacy of M_{split} Estimation in Displacement Analysis

^{*}

## Abstract

**:**

_{split}estimation, which is based on a fundamental assumption that each observation is related to several competitive functional models. The optimal assignment of every observation to the respective functional model is automatic during the estimation process. Considering deformation analysis, each observation is assigned to several functional models, each of which is related to one measurement epoch. This paper focuses on the efficacy of the method in detecting point displacements. The research is based on example observation sets and the application of Monte Carlo simulations. The results were compared with the classical deformation analysis, which shows that the M

_{split}estimation seems to be an interesting alternative for conventional methods. The most promising are results obtained for disordered observation sets where the M

_{split}estimation reveals its natural advantage over the conventional approach.

## 1. Introduction and Motivation

_{split}estimation surely belongs in the latter group. The M

_{split}estimation was proposed by Wiśniewski [19,20] and has been applied to some practical problems in which each observation could be assigned to several different functional models. For example, it was used in remote sensing (terrestrial laser scanning or ALS data) for data modeling [21] and in some geodetic problems, for example, in deformation analysis [22,23,24,25], and robust estimation (e.g., [26]). Automatic assignment of each observation to the best fitted model is one of the most important features of M

_{split}estimation. It is also very useful in deformation analysis, when the observation set might include observations from all measurement epochs (the set is an unrecognized mixture of such observations). Note that there is usually no problem with separating observations from different epochs and hence with separate analyses. However, there are some cases when the application of the M

_{split}estimation is advisable. For example, when a point is displaced during an observation session, thus, one should consider two pseudo-epochs, and the M

_{split}estimation allows us to estimate the parameters of the functional models for such pseudo-epochs. Such models can also be applied when an observation set is disturbed by outliers [23,24]. Note that the method under investigation can be applied in all observation sets that are an unrecognized (and/or unordered) mixture of observation aggregations. Such data can result from different sources or instrumentations. In fact, the source of data does not matter here. It can be, for example, geodetic instruments: total stations, GNSS receivers, etc., or in remote sensing such as terrestrial or airborne laser scanners.

_{split}estimation are discussed in the papers cited above; that study focused on the efficacy of the method in estimating parameters of the competitive functional models, hence also in estimating point displacements. The analyses were based on simulations of the crude Monte Carlo method and the application of elementary functional models or models of a leveling network. The results were compared with the results of the LS-method.

## 2. Theoretical Foundations

_{split}estimation, we assumed that each observation belonged to either of two sets ${\mathsf{\Phi}}_{1}$ or ${\mathsf{\Phi}}_{2}$; however, there is one observation set $\mathsf{\Phi}={\mathsf{\Phi}}_{1}\cup {\mathsf{\Phi}}_{2}$ and one observation vector $y={[{y}_{1},\cdots ,{y}_{n}]}^{T}$, $n={n}_{1}+{n}_{2}$. There are two competitive functional models

**y**.

_{split}estimation is an assumption that every observation ${y}_{i}$ can be assigned to either of two density function $f({y}_{i};{X}_{(1)})$ or $f({y}_{i};{X}_{(2)})$. If ${y}_{i}$ occurs, it brings the f-information ${I}_{f}({y}_{i};{X}_{(1)})=-\mathrm{ln}f({y}_{i};{X}_{(1)})$ or the f-information ${I}_{f}({y}_{i};{X}_{(2)})=-\mathrm{ln}f({y}_{i};{X}_{(2)})$, which are competitive to each other. M

_{split}estimates of the parameters ${X}_{(1)}$ and ${X}_{(2)}$, namely ${\widehat{X}}_{(1)}$ and ${\widehat{X}}_{(2)}$, minimize the following global information that is brought by all elements of the vector $y$ [19]

_{split}estimation is a special development of the ML-method. Huber [27,28] generalized the ML-method to M-estimation by introducing ${\phi}_{M}(X)={\displaystyle {\sum}_{i=1}^{n}\rho ({y}_{i};X)}$, where $\rho ({y}_{i};X)$ is an arbitrary function for which estimators obtain the desired properties (for example, they are robust against outliers). A similar generalization was also proposed for the M

_{split}estimation [19,20]. The objective function of Equation (4) is replaced by the following function

_{split}estimation is also a development of classical M-estimation, if only ${X}_{(1)}={X}_{(2)}=X$, and hence ${\phi}_{\rho}({X}_{(1)},{X}_{(2)})={\phi}_{M}(X)$.

_{split}estimation that differ from one another in the objective function or assumed parameters [19,22,29]. So far, the most popular is the squared M

_{split}estimation for which ${\rho}_{(1)}({y}_{i};{X}_{(1)})={p}_{i}{v}_{i(1)}^{2}$ and ${\rho}_{(2)}({y}_{i};{X}_{(2)})={p}_{i}{v}_{i(2)}^{2}$. Hence, one can write the following optimization problem of such a method [19,30] as

_{split}estimation is a development of the LS method. Considering such a relationship and the range of the practical applications of the M

_{split}estimation, we will only discuss the squared M

_{split}estimation. To compute M

_{split}estimates, one can use the sufficient conditions for the minimum of the objective function. Considering the optimization problem (7), one can write the following equations

_{split}estimators

_{split}estimates is shown. Here, we consider a basic example that precedes the more detail analysis presented in the next section. Let us assume the functional model ${y}_{i}=X+{v}_{i}$, $i=1,\dots ,7$, and the observation set $\mathsf{\Phi}$ as a following vector $y=\begin{array}{ccccccc}[1.2& 0.9& 1.8& 1.3& 2.2& 1.1& 1.9{]}^{T}\end{array}$ (Figure 1a). Then, ${\widehat{X}}_{LS}=1.49$. For the sake of comparison, let the robust M-estimate be computed. By applying the Huber method [27,28], where the weight function is $w(v)=\mathrm{min}\{1,k/|v\left|\right\}$ and $k=3$, one can obtain ${\widehat{X}}_{M}=1.27$ (Figure 1b). Both estimates in question are not satisfactory, and do not reflect the nature of the observation set. The robust estimate ${\widehat{X}}_{M}$ lies closer to the “bigger” aggregation of observations. Next, the question of how to treat the observations that are furthest from that estimate arises. In the classical approach, such observations are regarded as outliers (for example, affected by gross errors), and we are no longer interested in such observations. Different conclusions follow the M

_{split}estimation where ${\widehat{X}}_{(1)}=1.10$ and ${\widehat{X}}_{(2)}=2.00$ (Figure 1c).

_{split}estimates show that set $\mathsf{\Phi}$ consists of two subsets ${\mathsf{\Phi}}_{1}$ and ${\mathsf{\Phi}}_{2}$ (Figure 1d), whose elements can be regarded as realizations of two different random variables that differ from each other in location parameters ${X}_{1}$ and ${X}_{2}$, respectively. Similar assumptions can also be found in other estimation problems, for example, cluster analysis (e.g., [32,33]); or in a mixed model estimation applied in geosciences (e.g., [34,35]). Such approaches can be regarded as alternatives; however, we should have some understanding that they differ significantly in their general ideas.

_{split}estimation. This property can be applied in the analysis of network deformation where there are two functional models:$\hspace{0.17em}{y}_{1}={A}_{1}{X}_{1}+{v}_{1}$ and $\hspace{0.17em}{y}_{2}={A}_{2}{X}_{2}+{v}_{2}$ for two measurement epochs, respectively. Thus, one can create one common observation vector $y={[{y}_{1}^{T},{y}_{2}^{T}]}^{T}$, the common weight matrix $P=\mathrm{Diag}\left({P}_{(1)},{P}_{(2)}\right)$, and the coefficient matrix $A={\left[{A}_{1}^{T},{A}_{2}^{T}\right]}^{T}$. It is noteworthy that the order of the observation within vector $y$ can be arbitrary. The actual order of the observations must coincide with the order of the rows within matrix $A$ and order of the weights in weight matrix $P$. Here, the shift $\mathsf{\Delta}{X}_{(1,2)}$ can be estimated by $\mathsf{\Delta}{\widehat{X}}_{(1,2)}={\widehat{X}}_{(2)}-{\widehat{X}}_{(1)}$. It is worth noting that $\mathsf{\Delta}{X}_{(1,2)}$ can also be estimated directly by applying the Shift-M

_{split}estimation proposed by Duchnowski and Wiśniewski [22].

## 3. Empirical Analyses

#### 3.1. Elementary Tests

_{split}estimation, we assumed the model of Equation (3), taking $y={[{y}_{1}^{T},{y}_{2}^{T}]}^{T}\in {R}^{n}$, $n={n}_{1}+{n}_{2}$, and $A={[{1}_{{n}_{1}}^{T},{1}_{{n}_{2}}^{T}]}^{T}={1}_{n}$. We also applied the iterative procedure of Equation (13) by taking LS-estimates as the starting point (note that the starting point can usually be arbitrary).

_{split}estimation is more satisfactory for bigger shifts than for smaller ones. Thus, let us examine how efficient the M

_{split}estimation is for different shifts.

_{split}estimate is closer to the theoretical value than the LS estimate. Now, we can define the following function of an elementary success of M

_{split}estimation

#### 3.2. Vertical Displacement Analysis

_{split}estimates in a more practical example, namely the analysis of vertical displacements within the leveling network, which is presented in Figure 5. Such a network has already been under investigation in previous papers [24,25].

_{split}estimation, we should apply the functional model of Equation (3) for which $y={[{y}_{1}^{T},{y}_{2}^{T}]}^{T}\in {R}^{64}$, $A={[{A}_{\otimes}^{T},{A}_{\otimes}^{T}]}^{T}\in {R}^{64,5}$, and ${X}_{(1)},{X}_{(2)}\in {R}^{5}$ are the competitive versions of the parameter vector, hence ${v}_{(1)},{v}_{(2)}\in {R}^{64}$ are the respective competitive versions of the measurement errors.

_{split}estimation, we can use two measures, namely the local measure of the distance between the LS and M

_{split}estimates

_{split}estimations and $\mathsf{\Delta}{H}_{5(1,2)}=-50$, $\mathsf{\Delta}{H}_{5(1,2)}=-100$, or $\mathsf{\Delta}{H}_{5(1,2)}=-200$ mm are presented in Table 1, which also presents the local and global SRs.

_{split}estimates that were obtained for all of the MC simulations. Generally, this confirmed the correctness of both estimation methods; however, differences between these two estimation methods were also apparent. The main difference was the dispersion, which was larger in the case of the M

_{split}estimation, especially for the stable points, which suggests that the accuracy of the M

_{split}estimation was worse than LS estimation. It is also worth noting that the SRs of the M

_{split}estimation achieved bigger values in this variant. In the case of point ${P}_{5}$, the results of the M

_{split}estimation were better than the results of the classical approach in almost one third of the simulations.

_{split}estimation, yielded satisfactory solutions. However, such a conclusion was valid for the ordered observation sets, namely when each observation was properly assigned to its measurement epoch. If such a condition is not met, then the observation from another epoch will usually be regarded as an outlier. Since LS estimation as well as M

_{split}estimation are not robust against outliers, they both break down (please note that M

_{split}estimation is generally not robust unless we introduce an additional virtual model for outliers). Note that in the context addressed here, the outliers result from the assignment of an observation to the wrong measurement epoch, but not from gross errors. The natural feature of M

_{split}estimation is the automatic assignment of each observation to the proper epoch. Thus, we can suppose that this estimation method will not break down if such outliers occur. To illustrate this feature of M

_{split}estimation, we simulated that point ${P}_{5}$ was displaced and that $\mathsf{\Delta}{H}_{5(1,2)}=-50$ mm. Now, let us consider the following variants of the observation sets: variant A, where both observation sets were correct (all observations were assigned to their epochs properly); variant B, where the observation ${h}_{16}$ at the second epoch was equal to ${h}_{16}$ at the first one, namely ${h}_{16}^{2}={h}_{16}^{1}$; and variant C, where ${h}_{16}^{2}={h}_{16}^{1}$, but also ${h}_{15}^{2}={h}_{15}^{1}$. Thus, in variants B and C, we simulated that some observations that were assigned to the second measurement epoch should be related to the first one. The results obtained for all variants are presented in Table 3. In the case of variant A, the results were very close to the respective results presented in Table 1. If the observation sets are not ordered correctly, then the local SRs at the second epoch are close to 1, which means that almost always, the height of point ${P}_{5}$ at the second measurement epoch is better assessed by the M

_{split}estimation than by LS estimation. Additionally, the global SRs were very high at the second epoch, hence one can say that the heights of all network points were better estimated by the application of M

_{split}estimation.

## 4. Conclusions

_{split}estimation can be successfully applied in deformation analysis. The results were generally similar to the results of the more conventional LS estimation; however, the latter method usually yielded slightly better outcomes. The elementary tests showed that the efficacy of the M

_{split}estimation grew with an increasing shift between the observation sets. In the case of geodetic networks, where a parameter vector usually consists of several point coordinates, the shift of one or two such coordinates between measurement epochs does not influence the efficacy of the M

_{split}estimation in a significant way. The real advantage of the M

_{split}estimation was revealed for the disordered observation sets, for example, when the observations from at least two measurement epochs were mixed for some reason. Note that the LS estimates break down in such cases, in contrast with the M

_{split}estimation, for which the ordering of all observations within the combined observation set can be arbitrary and does not influence the final results of the method as well as its iterative process. Such a feature results directly from the theoretical foundations of the method, which are based on the concept of the split potential. In short, each observation “chooses” the functional model that fits it best. In this context, M

_{split}estimates are robust against some kind of “outliers”, namely observations that come from other observation sets. Referring to the presented example, there were four height differences regarding the height of network point ${P}_{5}$. If one of them does not fit the other, then the method tries to fit such an “outlying” observation into another epoch. If it works, then the whole estimation process succeeds. However, if such an observation is in fact affected by a gross error, then it does not fit any epoch, and the estimation must break down. The introduction of a virtual epoch, which is not related to any real measurements, is one solution to this problem. One can say that such an epoch can collect all “loners” that do not fit any real measurement epochs. Generally speaking, one can say that the M

_{split}estimation is not robust against outliers, which results from the occurrence of gross errors. However, if one assumes an additional competitive functional model (dedicated to outliers), then the M

_{split}estimation can estimate the location parameters for “good” observation aggregations as well as outlier(s). Increasing the number of competitive functional models protects the estimation of location parameters of good observations from the bad influence of outliers. Note that in this context, outliers are no longer “outlying”, and become regular observations of the third (or more generally next) aggregation. This concept, which is out of the scope of this paper, was discussed in [23,24,30].

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Least squares, robust M-estimate and both M

_{split}estimates within a sample observation set (

**a**) observation set; (

**b**) classical estimates; (

**c**) M

_{split}estimates; (

**d**) observation subsets.

**Figure 2.**LS estimates, M

_{split}estimates, and respective residuals (elementary functional models for $\mathsf{\Delta}{X}_{(1,2)}=5$).

**Figure 3.**Location of the Monte Carlo estimates for $\mathsf{\Delta}{X}_{(1,2)}=5\hspace{0.17em}$ or $\mathsf{\Delta}{X}_{(1,2)}=20$ (for $N=5000$).

**Figure 4.**The success rates of M

_{split}estimates ${\widehat{X}}_{(1)}$ and ${\widehat{X}}_{(2)}$ for the growing value of $\mathsf{\Delta}{X}_{(1,2)}$.

**Figure 6.**The LS and M

_{split}estimates of the MC simulations ($\mathsf{\Delta}{H}_{5(1,2)}=-50$ mm).

**Figure 7.**The LS and M

_{split}estimates of the MC simulations ($\mathsf{\Delta}{H}_{4(1,2)}=-50$ and $\mathsf{\Delta}{H}_{5(1,2)}=-100$ mm).

$\mathbf{\Delta}{\mathit{H}}_{5(1,2)}=-50$ | $\mathbf{\Delta}{\mathit{H}}_{5(1,2)}=-100$ | $\mathbf{\Delta}{\mathit{H}}_{5(1,2)}=-200$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ |

0.2 | −3.1 | 0.4 | 2.9 | −0.5 | −1.1 | −0.6 | 0.6 | 0.9 | −1.8 | −0.7 | −0.4 |

1.4 | −1.2 | −1.0 | 0.9 | −0.4 | −1.3 | 0.7 | 2.9 | 0.5 | 0.7 | 0.5 | −0.8 |

2.1 | −0.6 | −0.6 | −0.6 | 0.1 | −0.6 | 0.5 | 1.3 | −0.3 | −2.8 | −0.1 | 1.1 |

-0.8 | −3.6 | −0.6 | 0.6 | 1.1 | −0.9 | −1.0 | 1.2 | 0.0 | −0.4 | −1.5 | 1.5 |

0.8 | −1.9 | −50.4 | −49.1 | 0.3 | −1.2 | −99.8 | −98.7 | 0.8 | −0.8 | −200.1 | −199.7 |

${\gamma}_{(1)}=0.018$ ${\gamma}_{(1),5}=0.172$ | ${\gamma}_{(2)}=0.019$ ${\gamma}_{(2),5}=0.182$ | ${\gamma}_{(1)}=0.020$ ${\gamma}_{(1),5}=0.177$ | ${\gamma}_{(2)}=0.017$ ${\gamma}_{(2),5}=0.187$ | ${\gamma}_{(1)}=0.025$ ${\gamma}_{(1),5}=0.171$ | ${\gamma}_{(2)}=0.024$ ${\gamma}_{(2),5}=0.196$ |

$\mathbf{\Delta}{\mathit{H}}_{5(1,2)}=-50;\mathbf{\Delta}{\mathit{H}}_{4(1,2)}=-50$ | $\mathbf{\Delta}{\mathit{H}}_{5(1,2)}=-100;\mathbf{\Delta}{\mathit{H}}_{4(1,2)}=-50$ | $\mathbf{\Delta}{\mathit{H}}_{5(1,2)}=-200;\mathbf{\Delta}{\mathit{H}}_{4(1,2)}=-50$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ |

−0.2 | −0.5 | −0.3 | 0.6 | −0.5 | 0.1 | −0.4 | 0.3 | 0.0 | −1.3 | −0.3 | −1.1 |

−0.4 | −2.0 | −0.1 | −0.1 | 0.2 | 0.5 | −0.2 | −0.4 | 0.0 | −0.2 | −0.1 | 0.9 |

−0.4 | −0.4 | −0.9 | 0.2 | 0.1 | −0.3 | 0.2 | −0.3 | −0.2 | −0.2 | −0.3 | −0.4 |

−0.1 | −0.3 | −50.5 | −50.1 | 0.4 | 0.5 | −49.9 | −49.6 | −0.1 | −0.4 | −50.0 | −50.1 |

−0.5 | −1.4 | −50.1 | −50.2 | −0.6 | −0.4 | −100.1 | −99.8 | −0.5 | −0.8 | −200.3 | −200.2 |

${\gamma}_{(1)}=0.070$ ${\gamma}_{(1),5}=0.272$ | ${\gamma}_{(2)}=0.070$ ${\gamma}_{(2),5}=0.268$ | ${\gamma}_{(1)}=0.080$ ${\gamma}_{(1),5}=0.281$ | ${\gamma}_{(2)}=0.080$ ${\gamma}_{(2),5}=0.288$ | ${\gamma}_{(1)}=0.103$ ${\gamma}_{(1),5}=0.314$ | ${\gamma}_{(2)}=0.105$ ${\gamma}_{(2),5}=0.312$ |

Variant A: Correct Order | $\mathbf{Variant}\mathbf{B}:{\mathit{h}}_{16}^{2}={\mathit{h}}_{16}^{1}$ | $\mathbf{Variant}\mathbf{C}:{\mathit{h}}_{15}^{2}={\mathit{h}}_{15}^{1}$$,{\mathit{h}}_{16}^{2}={\mathit{h}}_{16}^{1}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},1}$ | ${\widehat{\mathbf{X}}}_{(1)}$ | ${\widehat{\mathbf{X}}}_{\mathit{L}\mathit{S},2}$ | ${\widehat{\mathbf{X}}}_{(2)}$ |

0.0 | 2.2 | 0.3 | −1.1 | 0.4 | −1.5 | −6.8 | 0.3 | −0.8 | 0.4 | −4.5 | −5.2 |

0.4 | −0.1 | 1.1 | 0.4 | −0.5 | −1.5 | 2.1 | 1.8 | −0.2 | −0.8 | −5.3 | −7.7 |

0.6 | 0.8 | 0.3 | −1.5 | −0.6 | −3.6 | 3.4 | 1.6 | −0.1 | −1.0 | 4.9 | 7.4 |

−0.7 | −0.9 | 0.0 | 1.0 | 0.3 | −1.4 | 2.0 | 2.4 | −1.3 | −0.6 | 5.2 | 7.1 |

−0.2 | 0.5 | −49.8 | −50.3 | 0.4 | −1.5 | −36.2 | −46.5 | −2.0 | −1.0 | 25.3 | −42.6 |

${\gamma}_{(1)}=0.018$ ${\gamma}_{(1),5}=0.172$ | ${\gamma}_{(2)}=0.019$ ${\gamma}_{(2),5}=0.210$ | ${\gamma}_{(1)}=0.127$ ${\gamma}_{(1),5}=0.321$ | ${\gamma}_{(2)}=0.875$ ${\gamma}_{(2),5}=0.986$ | ${\gamma}_{(1)}=0.263$ ${\gamma}_{(1),5}=0.474$ | ${\gamma}_{(2)}=0.887$ ${\gamma}_{(2),5}=0.998$ |

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**MDPI and ACS Style**

Wiśniewski, Z.; Duchnowski, R.; Dumalski, A.
Efficacy of M_{split} Estimation in Displacement Analysis. *Sensors* **2019**, *19*, 5047.
https://doi.org/10.3390/s19225047

**AMA Style**

Wiśniewski Z, Duchnowski R, Dumalski A.
Efficacy of M_{split} Estimation in Displacement Analysis. *Sensors*. 2019; 19(22):5047.
https://doi.org/10.3390/s19225047

**Chicago/Turabian Style**

Wiśniewski, Zbigniew, Robert Duchnowski, and Andrzej Dumalski.
2019. "Efficacy of M_{split} Estimation in Displacement Analysis" *Sensors* 19, no. 22: 5047.
https://doi.org/10.3390/s19225047