# Orthogonal Msplit Estimation for Consequence Disaster Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Msplit Estimation Description

_{α}{y

_{i}} = a

_{i}X

_{α}

_{β}{y

_{i}} = a

_{i}X

_{β}**V**=

**AX**+

**L**is split into two competitive ones which concern the same vector of observation

**L**:

**A**= $\left[{a}_{1}^{T},\dots ,{a}_{n}^{T}\right]$

**is a common coefficient matrix,**

^{T}**V**and

_{α}**V**are competitive vectors of random variables, and

_{β}**X**and

_{α}**X**are competitive parameter vectors. Estimating competitive vectors (

_{β}**X**and

_{α}**X**) of parameters using the same observation vector

_{β}**L**requires appropriate objective function formulation. The Msplit method replaces function

**ρ**(

**v**) with functions

**ρ**(

_{α}**v**) and

_{α}**ρ**(

_{β}**v**) according to Equation (3) and in compliance with the cross-weighting

_{β}**V**and

_{α}**V**. Before the Msplit estimation, the weights are assumed to follow the standard weighting procedure. So, the first step is the least-squares method. In the next step, the weights of observations are modified according to the following equation:

_{β}#### 2.2. Orthogonal Regression

#### 2.3. Nelder–Mead Simplex Method

#### 2.4. Processing with Orthogonal Msplit Estimation

_{i—}the distance of a single point from the plane, calculated as:

_{i}denotes the weight of the i-th point and the distance from the plane to point (x,y,z) can be expressed as:

_{βi}) of the observation are modified according to Equation (3). The parameters and free terms are updated. In the next step, the second (beta) solution is calculated using the Nelder–Mead method using p

_{βi}. The new weights for the α solution are computed with Equation (4). Then, the procedure goes back to the α solution, and the process is iteratively repeated until the increase in estimated parameters reaches a satisfying level.

#### 2.5. Description of Research Objects

**z1 = 7x + 2y − 9.5**

**z2 = x + 2y +3**

## 3. Results

#### 3.1. Simulated Objects Results

#### 3.2. Real Object Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Correct (

**a**) and incorrect (

**b**) data fitting based on the same functional model depending on the data distribution.

**Figure 6.**The concrete slabs and a gap formed between them after a storm: (

**a**) the fragment of the retaining wall; (

**b**) the fragment of the wall is used to fit the planes.

**Figure 7.**The point cloud representing the fragment of the retaining wall: (

**a**) front view, (

**b**) cross-section.

Characteristics | Plane Parameter Values | |
---|---|---|

α Solution | β Solution | |

$\mathrm{A}$ | −2.0063 | −2.8107 |

$\mathrm{B}$ | 5.9106 | 4.9235 |

C | 8.1475 | 6.7818 |

D | −1.2617 | 1.1191 |

Number of points | 151 | 134 |

RMS [m] | 0.0287 m | 0.0246 m |

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**MDPI and ACS Style**

Janicka, J.; Rapinski, J.; Błaszczak-Bąk, W.
Orthogonal Msplit Estimation for Consequence Disaster Analysis. *Remote Sens.* **2023**, *15*, 421.
https://doi.org/10.3390/rs15020421

**AMA Style**

Janicka J, Rapinski J, Błaszczak-Bąk W.
Orthogonal Msplit Estimation for Consequence Disaster Analysis. *Remote Sensing*. 2023; 15(2):421.
https://doi.org/10.3390/rs15020421

**Chicago/Turabian Style**

Janicka, Joanna, Jacek Rapinski, and Wioleta Błaszczak-Bąk.
2023. "Orthogonal Msplit Estimation for Consequence Disaster Analysis" *Remote Sensing* 15, no. 2: 421.
https://doi.org/10.3390/rs15020421