# Estimation of Usable Area of Flat-Roof Residential Buildings Using Topographic Data with Machine Learning Methods

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data on Single-Family Houses in Koszalin

#### 2.2. Data from Design Offices

#### 2.3. Formula Based on Architectural Assumptions

- the building is located 30 cm above the ground,
- the structural ceiling is 30 cm thick,
- the internal staircase occupies 4.5 m${}^{2}$ per story,
- external construction walls w/o lining are 40 cm thick, internal construction walls—24 cm thick, partition walls—12 cm thick,
- a chimney occupies 1 m${}^{2}$ per story,
- the covered area of a one-spot garage is 20 m${}^{2}$, two-spot 30 m${}^{2}$,
- the story height is minimum 2.5 m,
- the lining thickness is 2.5 cm,
- the boiler room area is 5 m${}^{2}$,
- the balcony area is 5 m${}^{2}$,
- the length of partition walls is equal to half of the building perimeter.

#### 2.4. ML Methods

#### 2.5. Uncertainty of Data and Estimation Results

## 3. Results

#### 3.1. Formula Based on Architectural Assumptions for Model Houses from the Design Offices

#### 3.2. The Design Offices’ Buildings: Without Garages and Extensions

#### 3.3. The Design Offices’ Buildings: Full Dataset

#### 3.4. Koszalin Buildings

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BDOT10k | Database of Topographic Objects (pol. Baza Danych Obiektów Topologicznych) |

LiDAR | Light Detection and Ranging |

LoD | Level of Detail |

ML | Machine Learning |

NN | Neural Network |

PS | Polish Standard (pol. Polska Norma) |

PVR | Price and Value Register (pol. Rejestr Cen i Wartości) |

SGD | Stochastic Gradient Descent |

## References

- Dawid, L. Characteristics of the Residential Real Estate Market and Their Valuations in 2010–2015 on the Example of Mielno Commune. Appl. Sci. Rev.
**2017**, 14, 56–71. (In Polish) [Google Scholar] - Dawid, L. Analysis of Completeness of Data from the Price and Value Register on the Example of Kołobrzeg and Koszalin Districts in Years 2010–2017. Stud. Res. FEM SU
**2018**, 1, 91–102. (In Polish) [Google Scholar] - Dawid, L. Analysis of Data Completeness in the Register of Real Estate Prices and Values Used for Real Estate Evaluation on the Example of Koszalin District in the Years 2010–2016. Folia Econ. Stetin.
**2018**, 18, 17–26. [Google Scholar] [CrossRef] - The Ordinance of 10 June 2016 on the Promulgation of the Consolidated Text of the Ordinance of the Minister of Regional Development and Construction, Warsaw. Journal of Laws of 2016, Item 1034. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU20160001034 (accessed on 5 July 2019). (In Polish)
- Bydłosz, J.; Cichociński, P.; Piotr, P. Possibilities of the Register of Real Estates Prices and Values Restrictions Overcoming Applying GIS Tools. Stud. Inform.
**2010**, 31, 229–244. (In Polish) [Google Scholar] - Budzyński, T. Calculating the Area of Newly-Built Apartments and Buildings According to Uniform Rules. Geod. R.
**2012**, 84, 31. (In Polish) [Google Scholar] - Hopfer, A. The Real Estate Price and Value Register in the Light of the Project of the Regulation of the Council of Ministers on Integrated Information System on Real Estate—Impact of the Mentioned Regulation on Quality and Reliability of PVR. Real Estate Apprais.
**2012**, 74, 4–11. (In Polish) [Google Scholar] - Williamson, I. Land Administration for Sustainable Development; ESRI Press Academic: Beijing, China, 2010. [Google Scholar]
- Enemark, S. Building Modern Land Markets in Developed Economies. J. Spat. Sci.
**2005**, 50, 51–68. [Google Scholar] [CrossRef] - Enemark, S. From Cadastre to Land Governance: The Role of Land Professionals and FIG. 2010. Available online: https://www.semanticscholar.org/paper/From-Cadastre-to-Land-Governance%3A-The-role-of-land-Enemark/227a1e96079aeccd6ee59be4f84fbd7e2ec5372c (accessed on 1 July 2019).
- Felcenloben, D. Real Estate Cadastre; Gall: Katowice, Poland, 2009; pp. 29–42. (In Polish) [Google Scholar]
- Hycner, R. Basics of the Cadastre; AGH University of Science and Technology Press: Lesser Poland, Poland, 2004; pp. 241–282. (In Polish) [Google Scholar]
- Henssen, J. Basic Principles of the Main Cadastral Systems in the World. In Proceedings of the One Day Seminar held during the Annual Meeting of Commission 7, Cadastre and Rural Land Management, of the International Federation of Surveyors (FIG), Delft, The Netherlands, 16 May 1995. [Google Scholar]
- Bennett, R. On the Nature and Utility of Natural Boundaries for Land and Marine Administration. Land Use Policy
**2010**, 27, 772–779. [Google Scholar] [CrossRef] - Bennett, R. Cadastral Futures: Building a New Vision for the Nature and Role of Cadastres. FIG Congr.
**2010**, 1–11. Available online: https://www.fig.net/resources/monthly_articles/2011/june_2011/june_2011_bennett_rajabifard_et_al.pdf (accessed on 1 July 2019). - Larsson, G. Land Registration and Cadastral Systems; Longman Scientific and Technical: Harlow, UK, 1991; pp. 21–65. [Google Scholar]
- Kaufmann, J. Cadastre 2014: A Vision for a Future Cadastral System; International Federation of Surveyors: Copenhagen, Denmark, 1998; pp. 1–38. [Google Scholar]
- Stoter, J. Towards a 3D Cadastre: Where Do Cadastral Needs and Technical Possibilities Meet? Comput. Environ. Urban Syst.
**2003**, 27, 395–410. [Google Scholar] [CrossRef] - Konowalczuk, J. The Corporate Real Estate Market in Public Statistics in Poland. Real Estate Manag. Valuat.
**2014**, 22, 41–51. [Google Scholar] [CrossRef][Green Version] - Foryś, I.; Kokot, S. Problems with Real Estate Market Analysis. In Microeconomy in Theory and Practice; Res. Bull. Univ. Szczec.: Szczecin, Poland, 2001; pp. 175–182. (In Polish) [Google Scholar]
- Kokot, S. Data Quality of Transaction Prices in Real Estate Market. Acta Sci. Adm. Locorum
**2015**, 14, 43–49. (In Polish) [Google Scholar] - Benduch, P.; Hanus, P. The Concept of Estimating Usable Floor Area of Buildings Based on Cadastral Data. Rep. Geod. Geoinform.
**2018**, 105, 29–41. [Google Scholar] [CrossRef] - Benduch, P. Legal and Standard Principles of Buildings and Their Parts Usable Floor Area Quantity Surveying. Infrastruct. Ecol. Rural Areas
**2018**, 1, 225–238. (In Polish) [Google Scholar] - Act of January 12, 1991 on Local Taxes and Charges. Journal of Laws of 1991, Item 31. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU19910090031 (accessed on 25 June 2019). (In Polish)
- Act of July 28, 1983 on Tax on Inheritance and Donations. Journal of Laws of 1983, Item 207. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU19830450207 (accessed on 25 June 2019). (In Polish)
- Act of June 21, 2001 on Tenants Rights, Municipal Housing Stock and the Civil Code Amendment. Journal of Laws of 2001, Item 733. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU20010710733 (accessed on 25 June 2019). (In Polish)
- Regulation of the Minister of Justice of February 15, 2016 on the Establishment and Maintenance of Land and Mortgage Registers in an IT System. Journal of Laws of 2016, Item 312. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU20160000312 (accessed on 25 June 2019). (In Polish)
- Normalization Act of April 3, 1993. Journal of Laws of 1993, Item 251. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU19930550251 (accessed on 26 June 2019). (In Polish)
- Normalization Act of September 12, 2002. Journal of Laws of 2002, Item 1386. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU20021691386 (accessed on 26 June 2019). (In Polish)
- Polish Commitee of Standardization. PN-70/B-02365 Surface Area of Buildings—Classification, Definitions, and Methods of Measurement 1970. Available online: http://rzeczoznawca-zachodniopomorskie.pl/pliki/PN_70_B_02365.pdf (accessed on 26 June 2019). (In Polish).
- Zbroś, D. The Rules for Calculating the Usable Area by Two Current Polish Standards. Saf. Eng. Anthropog. Objects
**2016**, 3, 19–22. (In Polish) [Google Scholar] - Polish Commitee of Standardization. PN-ISO 9836:1997 Performance Standards in Building—Definition and Calculation of Area and Space Indicators 1997. Available online: http://rzeczoznawca-zachodniopomorskie.pl/pliki/PN_ISO_9836_1997.pdf (accessed on 26 June 2019). (In Polish).
- International Organization for Standardization. ISO-9836:1992 Performance Standards in Building—Definition and Calculation of Area and Space Indicators 1992. Available online: https://www.sis.se/api/document/preview/608742/ (accessed on 22 June 2019).
- Regulation of the Minister of Transport, Construction and Maritime Economy of April 25, 2012 on Detailed Scope and Form of a Construction Project. Journal of Laws of 2012, Item 462. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU20120000462 (accessed on 25 June 2019). (In Polish)
- Polish Commitee of Standardization. PN-ISO 9836:2015-12 Performance Standards in Building—Definition and Calculation of Area and Space Indicators 2015. Available online: http://sklep.pkn.pl/pn-iso-9836-2015-12p.html (accessed on 5 July 2019). (In Polish).
- Garcia-Gutierrez, J.; Martínez-Álvarez, F.; Troncoso, A.; Riquelme, J.C. A Comparative Study of Machine Learning Regression Methods on LiDAR Data: A Case Study; Herrero, Á., Baruque, B., Klett, F., Abraham, A., Snášel, V., de Carvalho, A.C., Bringas, P.G., Zelinka, I., Quintián, H., Corchado, E., Eds.; International Joint Conference SOCO’13-CISIS’13-ICEUTE’13; Springer International Publishing: Cham, Switzerland, 2014; pp. 249–258. [Google Scholar]
- Nahhas, F.H.; Shafri, H.Z.M.; Sameen, M.I.; Pradhan, B.; Mansor, S. Deep Learning Approach for Building Detection Using LiDAR–Orthophoto Fusion. J. Sens.
**2018**, 2018. [Google Scholar] [CrossRef] - Verschoof-van der Vaart, W.; Lambers, K. Learning to Look at LiDAR: The Use of R-CNN in the Automated Detection of Archaeological Objects in LiDAR Data from the Netherlands. J. Comput. Appl. Archaeol.
**2019**, 2, 31–40. [Google Scholar] [CrossRef][Green Version] - Marrs, J.; Ni-Meister, W. Machine Learning Techniques for Tree Species Classification Using Co-Registered LiDAR and Hyperspectral Data. Remote Sens.
**2019**, 11, 819. [Google Scholar] [CrossRef] - Lipińscy, M.L. Design Office. Houses Projects. Available online: https://lipinscy.pl/ (accessed on 10 July 2019).
- Mendel, B. ARCHON+ Project Office. Available online: https://www.archon.pl/(accessed on 10 July 2019).
- Head Office of Land Surveying and Cartography. Geoportal of National Spatial Data Infrastructure. Available online: https://www.geoportal.gov.pl/ (accessed on 20 July 2019).
- QGIS Development Team. QGIS Geographic Information System. Open Source Geospatial Foundation Project. Available online: http://qgis.osgeo.org. (accessed on 21 July 2019).
- Chrobak, T.; Łabaj, A.; Bolibok, A. (Eds.) Textbook for Participants of Trainings on Possibilities, Forms, and Methods of Using the Database of Topographic Objects; Head Office of Land Surveying and Cartography: Cracow, Poland, 2015. (In Polish) [Google Scholar]
- Rubinowicz, P. Generation of CityGML LoD1 City Models Using BDOT10k and LiDAR Data. Space Form
**2017**, 31, 61–74. [Google Scholar] - Wężyk, P. (Ed.) Textbook for Participants of Trainings on Using LiDAR Products; Head Office of Land Surveying and Cartography: Cracow, Poland, 2015. (In Polish) [Google Scholar]
- Schabowicz, K.; Gorzelańczyk, T. General Architecture. Basics of Designing and Calculating Buildings Construction; Lower Silesian Educational Publishing House: Wrocław, Poland, 2017; pp. 50–199. (In Polish) [Google Scholar]
- Michalak, H.; Pyrak, S. Single-Family Houses. Construction and Calculation; ARKADY: Warsaw, Poland, 2006; pp. 164–307. (In Polish) [Google Scholar]
- Piotrowski, R.; Dominiak, P. Construction of a Passive House, Step by Step; Construction Guide: Warsaw, Poland, 2008; pp. 135–198. (In Polish) [Google Scholar]
- Korzeniewski, W. Single-Family Buildings. Usage Requirements and Technical Conditions; Central Office on Construction Information: Warsaw, Poland, 1998; pp. 152–191. (In Polish) [Google Scholar]
- Buczkowski, W. (Ed.) General Architecture, Buildings Construction; ARKADY: Warsaw, Poland, 2009; Volume 4, pp. 7–98. (In Polish)
- Gołuch, A. Architectural-Construction Design; KANON: Gdańsk, Poland, 1998; pp. 241–252. (In Polish) [Google Scholar]
- Draper, N.R.; Smith, H. Applied Regression Analysis, 3rd ed.; Wiley Series in Probability and Statistics; Wiley: Hoboken, NJ, USA, 1998; p. 33. [Google Scholar]
- Aggarwal, C.C. Neural Networks and Deep Learning. A Textbook; Springer: Berlin, Germany, 2018; pp. 4–20. [Google Scholar]
- Singh, A.; Thakur, N.; Sharma, A. A review of supervised machine learning algorithms. In Proceedings of the 3rd INDIACom, New Delhi, India, 16–18 March 2016; pp. 1310–1315. [Google Scholar]
- Kotsiantis, S.B.; Zaharakis, I.D.; Pintelas, P.E. Machine Learning: A Review of Classification and Combining Techniques. Artif. Intell. Rev.
**2006**, 26, 159. [Google Scholar] [CrossRef] - Hutter, F.; Hoos, H.; Leyton-Brown, K. An Efficient Approach for Assessing Hyperparameter Importance. In Proceedings of the 31st International Conference on Machine Learning, Beijing, China, 21–26 June 2014; Volume 32, pp. 754–762. [Google Scholar]
- Huber, P.J. Robust Estimation of a Location Parameter. Ann. Stat.
**1964**, 53. [Google Scholar] [CrossRef] - Kennedy, P. A Guide to Econometrics; The MIT Press: Cambridge, MA, USA, 2003; pp. 205–206. [Google Scholar]
- Ng, A.; Katanforoosh, K.; Mourri, Y.B. Addressing Data Mismatch. Structuring Machine Learning Projects. 2019. Available online: https://www.coursera.org/lecture/machine-learning-projects/addressing-data-mismatch-biLiy (accessed on 15 August 2019).
- Hołub, A. PN-ISO 9836:2015-12—Further Confusion and Uncertainty. 2016. Available online: https://resources.geodetic.co/norma-pn-iso-98362015-12-kolejne-zamieszanie-i-niepewnosc/ (accessed on 15 August 2019). (In Polish).
- Polish Association of Development Companies. Are Differences in a Flat’s Usable Area Acceptable? Available online: http://pzfd.pl/pzfd-dla-kupujacego/pytania/ (accessed on 1 September 2019). (In Polish).
- The Ordinance of the Council of Ministers of September 21, 2004 on Real Estate Valuation and Preparation of Valuation Survey. Journal Laws of 2004, Item 2109. Available online: http://prawo.sejm.gov.pl/isap.nsf/DocDetails.xsp?id=WDU20042072109 (accessed on 20 June 2019). (In Polish)

**Figure 1.**The results of the formula estimation with distinction between buildings with and without garages. The coefficients of determination are as following: without garages ${R}^{2}$ = 98.28%, with garages ${R}^{2}$ = 94.77%, total ${R}^{2}$ = 95.37%

**Figure 2.**Predictions of (

**a**) the linear regression ($\lambda =0.02$) and (

**b**) neural network with eight hidden units ($\lambda =0.015$) with distinction between one- and two-story buildings. Input features: the covered area, ${A}_{C}$ and the number of stories, ${S}_{N}$.

**Figure 3.**Predictions of the NN (4-64-8-1), $\lambda =0.015$ for 29 Koszalin flat-roof single-family buildings with distinction between buildings with and without balconies. Input features: the covered area, ${A}_{C}$, number of stories, ${S}_{N}$, number of garage spots, ${G}_{N}$, and percentage of extension area in ${A}_{C}$, ${E}_{\%}$.

PN-B-02365:1970 | PN-ISO 9836:1997 | |
---|---|---|

state of walls during measurement | unplastered * | plastered |

measuring height | 1 m above floor level | at floor level |

niches | included if over 0.1 m${}^{2}$ | not included |

wall protrusions | deducted if over 0.1 m${}^{2}$ | not deducted |

if $H>2.2$ m—included in 100% | if $H>=1.9$ m—included in 100% | |

room height (H) | if $1.4<=H<=2.2$ m—included in 50% | if $H<1.9$ m—not included |

if $H<1.4$ m—not included | ||

calculation precision | 0.1 m${}^{2}$ | 0.01 m${}^{2}$ |

exterior areas, unclosed from all sides, | ||

available from a given room | not included | included, indicated separately |

e.g., balconies, terraces, loggias | ||

internal staircase | not included | included |

**Table 2.**Recorded features of 29 single-family houses in Koszalin with the source and range of values in the dataset.

No. | Feature | Symbol | Range of Values | Source |
---|---|---|---|---|

0 | Usable area | ${A}_{U}$ | 90–213.8 m${}^{2}$ | PVR |

1 | Covered area | ${A}_{C}$ | 73.87–150 m${}^{2}$ | BDOT10k or PVR |

2 | Estimated number of garage spots | $e{G}_{N}$ | 0–1 | Google Street View |

3 | Height | H | 6.41–9.08 m | LiDAR |

4 | Number of stories | ${S}_{N}$ | 0–3 | Google Street View |

5 | Perimeter | P | 34.79–56.77 m | BDOT10k |

6 | Estimated percentage of extension area in ${A}_{C}$ | ${E}_{\%}$ | 0–42.91% | BDOT10k + Google Street View |

7 | Number of balconies | ${B}_{N}$ | 0-3 | Google Street View |

**Table 3.**Recorded features of 96 single-family houses from the design offices with the range of values present in the dataset.

No. | Feature | Symbol | Range of Values |
---|---|---|---|

0 | Usable area | ${A}_{U}$ | 43.19–357.32 m${}^{2}$ |

1 | Covered area | ${A}_{C}$ | 54.93–334.46 m${}^{2}$ |

2 | Garage area | ${A}_{G}$ | 0–44.19 m${}^{2}$ |

3 | Number of garage spots | ${G}_{N}$ | 0–2 |

4 | Number of stories | ${S}_{N}$ | 1–2 |

5 | Height | H | 3.7–7.8 m |

6 | Perimeter | P | 29.9–112.26 m |

7 | Percentage of extension area in ${A}_{C}$ | ${E}_{\%}$ | 0–80.02% |

8 | Width | W | 5.64–19.24 m |

9 | Number of chimneys | ${C}_{N}$ | 0–4 |

10 | Presence of a boiler room | B | 0–1 |

11 | Number of balconies | ${B}_{N}$ | 0 |

**Table 4.**Mathematical formula based on architectural assumptions to estimate usable area of flat-roof houses for three standards.

Usable Area Formula | ||
---|---|---|

${A}_{U}={S}_{N}*({A}_{C}-{A}_{{\mathrm{walls}}_{\mathrm{ext}}}-{A}_{{\mathrm{walls}}_{\mathrm{int}}}-{A}_{\mathrm{chim}})-{A}_{G}-{A}_{E}-{A}_{\mathrm{balc}}-{A}_{\mathrm{boiler}}$ | ||

where | ||

${S}_{N}=\mathrm{floor}(\frac{(H-0.3\phantom{\rule{0.166667em}{0ex}}\mathrm{m})}{2.8\phantom{\rule{0.166667em}{0ex}}\mathrm{m}})$ | ${A}_{{\mathrm{walls}}_{\mathrm{ext}}}=P*\left(0.4\phantom{\rule{0.222222em}{0ex}}\mathrm{or}\phantom{\rule{0.222222em}{0ex}}0.45\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | ${A}_{G}=20\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{or}\phantom{\rule{0.222222em}{0ex}}30\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ |

${A}_{\mathrm{chim}}={C}_{N}*1\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$, ${A}_{\mathrm{balc}}={B}_{N}*5\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ | ${A}_{{\mathrm{walls}}_{\mathrm{int}}}=W*\left(0.24\phantom{\rule{0.222222em}{0ex}}\mathrm{or}\phantom{\rule{0.222222em}{0ex}}0.29\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | ${A}_{\mathrm{boiler}}=B*5\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ |

PN-B-02365:1970 | PN-ISO 9836:1997 | PN-ISO 9836:1997 (+2012) |

${A}_{U}+=-{A}_{\mathrm{stairs}}-{S}_{N}*{A}_{{\mathrm{walls}}_{\mathrm{part}}}$ | ${A}_{U}+=-{S}_{N}*{A}_{{\mathrm{walls}}_{\mathrm{part}}}$ | |

where | ||

${A}_{\mathrm{stairs}}={S}_{N}*4.5\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ | ${A}_{{\mathrm{walls}}_{\mathrm{part}}}=\frac{P}{2}*\left(0.12\phantom{\rule{0.222222em}{0ex}}\mathrm{or}\phantom{\rule{0.222222em}{0ex}}0.17\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

**Table 5.**Statistical measures of predictions of the linear regression model and the best found neural network models for 16 buildings without garages and extensions from the test and bridge sets, trained on 24 examples, validated on 8, with different input features.

Measure [%] | Input: ${\mathit{A}}_{\mathit{C}}$ and ${\mathit{N}}_{\mathit{S}}$ | Input: ${\mathit{A}}_{\mathit{C}}$, ${\mathit{N}}_{\mathit{S}}$, P | Input: ${\mathit{A}}_{\mathit{C}}$, H, P | Input: ${\mathit{A}}_{\mathit{C}}$, ${\mathit{N}}_{\mathit{S}}$, P, W | ||||
---|---|---|---|---|---|---|---|---|

LinReg | NN (2-8-1) | LinReg | NN (3-8-1) | LinReg | NN (3-32-1) | LinReg | NN (4-8-1) | |

$\mathbf{\lambda}=\mathbf{0.02}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | $\mathbf{\lambda}=\mathbf{0.05}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | $\mathbf{\lambda}=\mathbf{0}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | $\mathbf{\lambda}=\mathbf{0.05}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | |

mean error | 5.4 | 3.34 | 5.52 | 3.15 | 7.36 | 5.24 | 5.34 | 3.25 |

errors’ median | 3.57 | 2.74 | 2.95 | 2.89 | 4.66 | 3.9 | 2.64 | 2.88 |

max error | 20.46 | 7.34 | 19.76 | 6.96 | 24.3 | 13.47 | 20.28 | 8.83 |

min error | 0.3 | 0.03 | 0.24 | 0.16 | 0.11 | 0.09 | 0.22 | 0.19 |

${R}^{2}$ | 89.99 | 97.09 | 89.36 | 91.21 | 80.23 | 92.75 | 89.49 | 96.31 |

**Table 6.**Statistical measures of predictions of the best found neural network models for 15 buildings from the test set, trained on 66 examples, validated on 15, with different input features.

Measure [%] | Input Features, Architecture, and L2 Regularization Strengths | ||||
---|---|---|---|---|---|

${\mathit{A}}_{\mathit{C}}$, ${\mathit{N}}_{\mathit{S}}$ | ${\mathit{A}}_{\mathit{C}}$, ${\mathit{N}}_{\mathit{S}}$, ${\mathit{G}}_{\mathit{N}}$ | ${\mathit{A}}_{\mathit{C}}$, ${\mathit{N}}_{\mathit{S}}$, ${\mathit{G}}_{\mathit{N}}$, ${\mathit{E}}_{\%}$ | ${\mathit{A}}_{\mathit{C}}$, H, ${\mathit{G}}_{\mathit{N}}$, ${\mathit{E}}_{\%}$ | ${\mathit{A}}_{\mathit{C}}$, ${\mathit{N}}_{\mathit{S}}$, ${\mathit{A}}_{\mathit{G}}$, ${\mathit{E}}_{\%}$ | |

NN (2-8-1) | NN (3-32-1) | NN (4-64-8-1) | NN (4-64-8-1) | NN (4-64-8-1) | |

$\mathbf{\lambda}=\mathbf{0}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | $\mathbf{\lambda}=\mathbf{0}$ | $\mathbf{\lambda}=\mathbf{0.015}$ | |

mean error | 9.71 | 5.95 | 3.37 | 8.41 | 2.3 |

errors’ median | 6.29 | 4.14 | 2.29 | 7.4 | 0.95 |

max error | 30.62 | 18.09 | 10.71 | 21.28 | 7.02 |

min error | 1.77 | 0.32 | 0.23 | 1.15 | 0.07 |

${R}^{2}$ | 82.41 | 94.34 | 97.74 | 87.55 | 98.99 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dawid, L.; Tomza, M.; Dawid, A. Estimation of Usable Area of Flat-Roof Residential Buildings Using Topographic Data with Machine Learning Methods. *Remote Sens.* **2019**, *11*, 2382.
https://doi.org/10.3390/rs11202382

**AMA Style**

Dawid L, Tomza M, Dawid A. Estimation of Usable Area of Flat-Roof Residential Buildings Using Topographic Data with Machine Learning Methods. *Remote Sensing*. 2019; 11(20):2382.
https://doi.org/10.3390/rs11202382

**Chicago/Turabian Style**

Dawid, Leszek, Michał Tomza, and Anna Dawid. 2019. "Estimation of Usable Area of Flat-Roof Residential Buildings Using Topographic Data with Machine Learning Methods" *Remote Sensing* 11, no. 20: 2382.
https://doi.org/10.3390/rs11202382