The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction
To model dynamic systems in various situations results in an ordinary differential equation of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle=&qu...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/20/4233 |
| _version_ | 1797573086392877056 |
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| author | Óscar Cornejo Sebastián Muñoz-Herrera Felipe Baesler Rodrigo Rebolledo |
| author_facet | Óscar Cornejo Sebastián Muñoz-Herrera Felipe Baesler Rodrigo Rebolledo |
| author_sort | Óscar Cornejo |
| collection | DOAJ |
| description | To model dynamic systems in various situations results in an ordinary differential equation of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>t</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>θ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>g</i> denotes a function and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> stands for a parameter or vector of unknown parameters that require estimation from observations. In order to consider environmental fluctuations and numerous uncontrollable factors, such as those found in forestry, a stochastic noise process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ϵ</mi><mi>t</mi></msub></semantics></math></inline-formula> may be added to the aforementioned equation. Thus, a stochastic differential equation is obtained: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><msub><mi>Y</mi><mi>t</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>f</mi><mrow><mo>(</mo><msub><mi>Y</mi><mi>t</mi></msub><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>t</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>ϵ</mi><mi>t</mi></msub></mrow></semantics></math></inline-formula>. This paper introduces a method and procedure for parameter estimation in a stochastic differential equation utilising the Richards model, facilitating growth prediction in a forest’s tree population. The fundamental concept of the approach involves assuming that a deterministic differential equation controls the development of a forest stand, and that randomness comes into play at the moment of observation. The technique is utilised in conjunction with the logistic model to examine the progression of an agricultural epidemic induced by a virus. As an alternative estimation method, we present the Random Time Transformation (RTT) method. Thus, this paper’s primary contribution is the application of the RTT method to estimate the Richards model, which has not been conducted previously. The literature often uses the logistic or Gompertz models due to difficulties in estimating the parameter form of the Richards model. Lastly, we assess the effectiveness of the RTT Method applied to the Chapman–Richards model using both simulated and real-life data. |
| first_indexed | 2024-03-10T21:05:44Z |
| format | Article |
| id | doaj.art-5f17f5c01a32496ab2f0eab58fef444c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T21:05:44Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5f17f5c01a32496ab2f0eab58fef444c2023-11-19T17:13:03ZengMDPI AGMathematics2227-73902023-10-011120423310.3390/math11204233The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth PredictionÓscar Cornejo0Sebastián Muñoz-Herrera1Felipe Baesler2Rodrigo Rebolledo3Departament of Industrial Engineering, Faculty of Engineering, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepcion 4090541, ChileDepartament of Industrial Engineering, Faculty of Engineering, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepcion 4090541, ChileDepartament of Industrial Engineering, Faculty of Engineering, Universidad del Bio Bio, Avenida Ignacio Collao 1202, Concepcion 4051381, ChileDepartament of Industrial Engineering, Faculty of Engineering, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepcion 4090541, ChileTo model dynamic systems in various situations results in an ordinary differential equation of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>t</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>θ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>g</i> denotes a function and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> stands for a parameter or vector of unknown parameters that require estimation from observations. In order to consider environmental fluctuations and numerous uncontrollable factors, such as those found in forestry, a stochastic noise process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ϵ</mi><mi>t</mi></msub></semantics></math></inline-formula> may be added to the aforementioned equation. Thus, a stochastic differential equation is obtained: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><msub><mi>Y</mi><mi>t</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>f</mi><mrow><mo>(</mo><msub><mi>Y</mi><mi>t</mi></msub><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>t</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>ϵ</mi><mi>t</mi></msub></mrow></semantics></math></inline-formula>. This paper introduces a method and procedure for parameter estimation in a stochastic differential equation utilising the Richards model, facilitating growth prediction in a forest’s tree population. The fundamental concept of the approach involves assuming that a deterministic differential equation controls the development of a forest stand, and that randomness comes into play at the moment of observation. The technique is utilised in conjunction with the logistic model to examine the progression of an agricultural epidemic induced by a virus. As an alternative estimation method, we present the Random Time Transformation (RTT) method. Thus, this paper’s primary contribution is the application of the RTT method to estimate the Richards model, which has not been conducted previously. The literature often uses the logistic or Gompertz models due to difficulties in estimating the parameter form of the Richards model. Lastly, we assess the effectiveness of the RTT Method applied to the Chapman–Richards model using both simulated and real-life data.https://www.mdpi.com/2227-7390/11/20/4233stochastic differential equationsRichards modelRandom Time Transformation Techniquemaximum likelihood estimategrowth models |
| spellingShingle | Óscar Cornejo Sebastián Muñoz-Herrera Felipe Baesler Rodrigo Rebolledo The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction Mathematics stochastic differential equations Richards model Random Time Transformation Technique maximum likelihood estimate growth models |
| title | The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction |
| title_full | The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction |
| title_fullStr | The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction |
| title_full_unstemmed | The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction |
| title_short | The Application of the Random Time Transformation Method to Estimate Richards Model for Tree Growth Prediction |
| title_sort | application of the random time transformation method to estimate richards model for tree growth prediction |
| topic | stochastic differential equations Richards model Random Time Transformation Technique maximum likelihood estimate growth models |
| url | https://www.mdpi.com/2227-7390/11/20/4233 |
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