Hyperbolastic functions

The hyperbolastic rate equation of type I, denoted H1, is given by

$$\backslash frac\{dP(x)\}\{dx\}=\; \backslash frac\{P(x)\}\{M\}\; \backslash left(M-P\; \backslash left(x\backslash right)\backslash right)\backslash left(\backslash delta+\backslash frac\{\backslash theta\}\{\backslash sqrt\{1+x^2\}\}\backslash right),$$

where $x$ is any real number and

$P\backslash left(x\; \backslash right)$ is the population size at $x$. The parameter $M$ represents carrying capacity, and parameters $\backslash delta$ and $\backslash theta$ jointly represent growth rate. The parameter $\backslash theta$ gives the distance from a symmetric sigmoidal curve. Solving the hyperbolastic rate equation of type I for $P\; \backslash left(x\; \backslash right)$ gives

$$P(x)=\; \backslash frac\{M\}\{1+\; \backslash alpha\; e^\{-\backslash delta\; x-\; \backslash theta\; \backslash operatorname\{arsinh\}(x)\}\},$$

where $\backslash operatorname\{arsinh\}$ is the inverse hyperbolic sine function. If one desires to use the initial condition $P\backslash left(x\_0\backslash right)=P\_0$, then $\backslash alpha$ can be expressed as

:$\backslash alpha=\backslash frac\{M-P\_0\}\{P\_0\}\; e^\{\backslash delta\; x\_0+\; \backslash theta\; \backslash operatorname\{arsinh\}(x\_0)\}$.

If $x\_0=0$, then $\backslash alpha$ reduces to

:$\backslash alpha=\; \backslash frac\{M-P\_0\}\{P\_0\}$.

In the event that a vertical shift is needed to give a better model fit, one can add the shift parameter $\backslash zeta$, which would result in the following formula

:$P(x)=\; \backslash frac\{M\}\{1+\; \backslash alpha\; e^\{-\backslash delta\; x-\; \backslash theta\; \backslash operatorname\{arsinh\}(x)\}\}\; +\; \backslash zeta$.

The hyperbolastic function of type I generalizes the logistic function. If the parameters $\backslash theta\; =\; 0$, then it would become a logistic function. This function $P(x)$ is a hyperbolastic function of type I. The standard hyperbolastic function of type I is

:$P(x)=\; \backslash frac\{1\}\{1+\; e^\{-x-\backslash theta\; \backslash operatorname\{arsinh\}(x)\}\}$.

The hyperbolastic rate equation of type II, denoted by H2, is defined as

:$\backslash frac\{dP(x)\}\{dx\}=\; \backslash frac\{\backslash alpha\; \backslash delta\; \backslash gamma\; P^2\; (x)\; x^\{\backslash gamma\; -\; 1\}\}\{M\}\; \backslash tanh\; \backslash left(\backslash frac\{M-P(x)\}\{\backslash alpha\; P(x)\}\backslash right),$

where $\backslash tanh$ is the hyperbolic tangent function, $M$ is the carrying capacity, and both $\backslash delta$ and $\backslash gamma>0$ jointly determine the growth rate. In addition, the parameter $\backslash gamma$ represents acceleration in the time course. Solving the hyperbolastic rate function of type II for $P\backslash left(x\backslash right)$ gives

:$P(x)=\backslash frac\{M\}\{1+\backslash alpha\backslash operatorname\{arsinh\}\; \backslash left(e^\{-\backslash delta\; x^\backslash gamma\}\backslash right)\}$

.

If one desires to use initial condition $$

P(x_0)=P_0, then $\backslash alpha$ can be expressed as

:$\backslash alpha=\backslash frac\{M-P\_0\}\{P\_0\; \backslash operatorname\{arsinh\}\; \backslash left(e^\{-\backslash delta\; x\_0^\backslash gamma\}\backslash right)\}$

.

If $x\_0=0$, then $\backslash alpha$ reduces to

:$\backslash alpha=\backslash frac\{M-P\_0\}\{P\_0\; \backslash operatorname\{arsinh\}\; (1)\}$.

Similarly, in the event that a vertical shift is needed to give a better fit, one can use the following formula

:$P(x)=\backslash frac\{M\}\{1+\backslash alpha\backslash operatorname\{arsinh\}\; \backslash left(e^\{-\backslash delta\; x^\backslash gamma\}\backslash right)\}+\backslash zeta$

.

The standard hyperbolastic function of type II is defined as

:$P(x)=\backslash frac\{1\}\{1+\; \backslash operatorname\{arsinh\}\; \backslash left(e^\{-x\}\backslash right)\}$

.

The hyperbolastic rate equation of type III is denoted by H3 and has the form

:$\backslash frac\{dP(t)\}\{dt\}=\; \backslash left(M-P\; \backslash left(t\; \backslash right)\backslash right)\backslash left(\backslash delta\; \backslash gamma\; t^\{\backslash gamma\; -\; 1\}+\; \backslash frac\{\backslash theta\}\{\backslash sqrt\{1+\; \backslash theta^2\; t^2\}\}\backslash right)$,

where $t$ > 0. The parameter $M$ represents the carrying capacity, and the parameters $\backslash delta,$ $\backslash gamma,$ and $\backslash theta$ jointly determine the growth rate. The parameter $\backslash gamma,$ represents acceleration of the time scale, while the size of $\backslash theta$ represents distance from a symmetric sigmoidal curve. The solution to the differential equation of type III is

:$P(t)=\; M-\; \backslash alpha\; e^\{-\backslash delta\; t^\backslash gamma-\; \backslash operatorname\{arsinh\}(\backslash theta\; t)\}$,

with the initial condition $$

P\left(t_0\right)=P_0 we can express $\backslash alpha$ as

:$\backslash alpha=\backslash left(M-P\_0\; \backslash right)\; e^\{\backslash delta\; t\_0^\backslash gamma+\; \backslash operatorname\{arsinh\}(\backslash theta\; t\_0)\}$.

The hyperbolastic distribution of type III is a three-parameter family of continuous probability distributions with scale parameters $\backslash delta$ > 0, and $\backslash theta$ ≥ 0 and parameter $\backslash gamma$ as the shape parameter. When the parameter $\backslash theta$ = 0, the hyperbolastic distribution of type III is reduced to the weibull distribution.Kamar SH, Msallam BS. Comparative Study between Generalized Maximum Entropy and Bayes Methods to Estimate the Four Parameter Weibull Growth Model. Journal of Probability and Statistics. 2020 Jan 14;2020:1–7. The hyperbolastic cumulative distribution function of type III is given by

:$F(x;\; \backslash delta,\; \backslash gamma,\; \backslash theta)=$

\begin{cases}

1- e^{-\delta x^\gamma - \operatorname{arsinh}(\theta x)} & x\geq0 ,\\

0 & x < 0

\end{cases}

,

and its corresponding probability density function is

:$$

f(x; \delta, \gamma, \theta) =

\begin{cases}

e^{- \delta x^\gamma - \operatorname{arsinh}(\theta x)}\left(\delta \gamma x^{\gamma-1}+ \frac{\theta}{\sqrt{1+\theta^2 x^2}}\right) & x\geq0 ,\\

0 & x<0

\end{cases}.

The hazard function $h$ (or failure rate) is given by

:$h\backslash left(x;\; \backslash delta,\; \backslash gamma,\; \backslash theta\; \backslash right)\; =\; \backslash delta\backslash gamma\; x^\{\backslash gamma-1\}\; +\; \backslash frac\{\backslash theta\}\{\backslash sqrt\{1+\; x^2\backslash theta^2\}\}.$

The survival function $S$ is given by

:$S(x;\; \backslash delta,\; \backslash gamma,\; \backslash theta)=\; e^\{-\; \backslash delta\; x^\backslash gamma-\; \backslash operatorname\{arsinh\}(\backslash theta\; x)\}.$

The standard hyperbolastic cumulative distribution function of type III is defined as

:$F\backslash left(x\backslash right)=1-e^\{-x-\; \backslash operatorname\{arsinh\}(x)\}$,

and its corresponding probability density function is

:$$

f(x) = e^{- x - \operatorname{arsinh}(x)}\left(1+ \frac{1}{\sqrt{1+ x^2}}\right) .

If one desires to calculate the point $x$ where the population reaches a percentage of its carrying capacity $M$, then

one can solve the equation

:$P(x)\; =\; k\; M$

for $x$, where . For instance, the half point can be found by setting $k=\; \backslash frac\{1\}\{2\}$.

File:3D Hyperbolastic Graph of Phytoplankton Biomass.png

According to stem cell researchers at McGowan Institute for Regenerative Medicine at the University of Pittsburgh, "a newer model [called the hyperbolastic type III or] H3 is a differential equation that also describes the cell growth. This model allows for much more variation and has been proven to better predict growth."{{cite web |vauthors=Roehrs T, Bogdan P, Gharaibeh B, et al. |date=n.d. |title=Proliferative heterogeneity in stem cell populations |url=https://www.engineering.pitt.edu/Departments/Bioengineering/_Documents/Undergraduate-research/Intramural-Internships/(2131)---Fall2012/Abstract_Roehrs |publisher=Live Cell Imaging Laboratory, McGowan Institute for Regenerative Medicine}}

The hyperbolastic growth models H1, H2, and H3 have been applied to analyze the growth of solid Ehrlich carcinoma using a variety of treatments.{{Cite journal |doi = 10.1186/1471-2407-10-509|title = Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulphoxide|year = 2010|last1 = Eby|first1 = Wayne M.|last2 = Tabatabai|first2 = Mohammad A.|last3 = Bursac|first3 = Zoran|journal = BMC Cancer|volume = 10|pages = 509|pmid = 20863400|pmc = 2955040 | doi-access=free }}

In animal science,{{cite book |editor-last=France |editor-first=James |editor-last2=Kebreab |editor-first2=Ermias |title=Mathematical Modelling in Animal Nutrition |url=https://www.researchgate.net/publication/40802178 |location=Wallingford |publisher=CABI |year=2008 |isbn=9781845933548}} the hyperbolastic functions have been used for modeling broiler chicken growth.{{Cite journal |doi = 10.3382/ps.2007-00086|title = Hyperbolastic Models as a New Powerful Tool to Describe Broiler Growth Kinetics|year = 2007|last1 = Ahmadi|first1 = H.|last2 = Mottaghitalab|first2 = M.|journal = Poultry Science|volume = 86|issue = 11|pages = 2461–2465|pmid = 17954598|doi-access = free}}{{Cite journal |doi = 10.31548/ujvs2021.04.002|title = Approximation of Growth Indicators and Analysis of Individual Growth Curves by Linear Dimensions of Tubular Bones in Chickens of Meat Production Direction During Postnatal Period of Ontogenesis|year = 2021|last1 = Tkachuk|first1 = S. A.|last2 = Pasnichenko|first2 = O. S.| last3 = Savchok | first3 = L. B. |journal = Ukrainian Journal of Veterinary Sciences | volume = 12|issue = 4| s2cid=245487460 |doi-access = free}} The hyperbolastic model of type III was used to determine the size of the recovering wound.{{Cite journal | doi=10.1155/2014/965036| title=Novel Real-Time Facial Wound Recovery Synthesis Using Subsurface Scattering| year=2014| last1=Choi| first1=Taeyoung| last2=Chin| first2=Seongah| journal=The Scientific World Journal| volume=2014| pages=1–8| pmid=25197721| pmc=4146479| doi-access=free}}

In the area of wound healing, the hyperbolastic models accurately representing the time course of healing. Such functions have been used to investigate variations in the healing velocity among different kinds of wounds and at different stages in the healing process taking into consideration the areas of trace elements, growth factors, diabetic wounds, and nutrition.{{Cite journal |doi = 10.1016/j.mcm.2010.10.013|title = Hyperbolastic modeling of wound healing|year = 2011|last1 = Tabatabai|first1 = M.A.|last2 = Eby|first2 = W.M.|last3 = Singh|first3 = K.P.|journal = Mathematical and Computer Modelling|volume = 53|issue = 5–6|pages = 755–768|doi-access = free}}{{Cite journal | doi=10.1038/s41598-019-52523-9| title=Physicochemically Tuned Myofibroblasts for Wound Healing Strategy| year=2019| last1=Ko| first1=Ung Hyun| last2=Choi| first2=Jongjin| last3=Choung| first3=Jinseung| last4=Moon| first4=Sunghwan| last5=Shin| first5=Jennifer H.| journal=Scientific Reports| volume=9| issue=1| pages=16070| pmid=31690789| pmc=6831678| bibcode=2019NatSR...916070K}}

Another application of hyperbolastic functions is in the area of the stochastic diffusion process,{{Cite journal |doi = 10.3390/math9161835|title = Hyperbolastic Models from a Stochastic Differential Equation Point of View|year = 2021|last1 = Barrera|first1 = Antonio|last2 = Román-Román|first2 = Patricia|last3 = Torres-Ruiz |first3 = Francisco|journal = Mathematics|volume = 9|issue = 16|pages = 1835|doi-access = free}} whose mean function is a hyperbolastic curve. The main characteristics of the process are studied and the maximum likelihood estimation for the parameters of the process is considered.{{Cite book |doi = 10.1007/978-3-030-45093-9_25|chapter = Diffusion Processes for Weibull-Based Models|title = Computer Aided Systems Theory – EUROCAST 2019|series = Lecture Notes in Computer Science|year = 2020|last1 = Barrera|first1 = Antonio|last2 = Román-Román|first2 = Patricia|last3 = Torres-Ruiz|first3 = Francisco|volume = 12013|pages = 204–210|isbn = 978-3-030-45092-2| s2cid=215792096 }}

To this end, the firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stage wise procedure. Some examples based on simulated sample paths and real data illustrate this development. A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion.{{Cite journal |doi = 10.1016/j.biosystems.2017.11.001|title = A hyperbolastic type-I diffusion process: Parameter estimation by means of the firefly algorithm|year = 2018|last1 = Barrera|first1 = Antonio|last2 = Román-Román|first2 = Patricia|last3 = Torres-Ruiz|first3 = Francisco|journal = Biosystems|volume = 163|pages = 11–22|pmid = 29129822|arxiv = 2402.03416| bibcode=2018BiSys.163...11B }}{{Cite journal | doi=10.3934/mbe.2020043| title=Hyperbolastic type-III diffusion process: Obtaining from the generalized Weibull diffusion process| year=2020| last1=Barrera| first1=Antonio| last2=Román-Roán| first2=Patricia| last3=Torres-Ruiz| first3=Francisco| journal=Mathematical Biosciences and Engineering| volume=17| issue=1| pages=814–833| pmid=31731379| doi-access=free| hdl=10481/58209| hdl-access=free}}{{Cite journal | doi=10.3390/math8020155| title=Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior| year=2020| last1=Barrera| first1=Antonio| last2=Román-Román| first2=Patricia| last3=Torres-Ruiz| first3=Francisco| journal=Mathematics| volume=8| issue=2| pages=155| doi-access=free| hdl=10481/61054| hdl-access=free}}{{Cite book |doi = 10.3390/books978-3-03921-729-8|title = Stochastic Processes with Applications|year = 2019|isbn = 978-3-03921-729-8|url = https://digitalcommons.uri.edu/cgi/viewcontent.cgi?article=1006&context=nonequilibrium_statistical_physics| doi-access=free }}{{Cite journal | doi=10.3390/math9090959| title=T-Growth Stochastic Model: Simulation and Inference via Metaheuristic Algorithms| year=2021| last1=Barrera| first1=Antonio| last2=Román-Román| first2=Patricia| last3=Torres-Ruiz| first3=Francisco| journal=Mathematics| volume=9| issue=9| pages=959| doi-access=free| hdl=10481/68288| hdl-access=free}} The hyperbolastic function of type III was used to model the proliferation of both adult mesenchymal and embryonic stem cells;{{Cite journal | doi=10.1007/s11517-010-0686-y| title=Mathematical modeling of stem cell proliferation| year=2011| last1=Tabatabai| first1=Mohammad A.| last2=Bursac| first2=Zoran| last3=Eby| first3=Wayne M.| last4=Singh| first4=Karan P.| journal=Medical & Biological Engineering & Computing| volume=49| issue=3| pages=253–262| pmid=20953843| s2cid=33828764}}{{Cite book |doi = 10.1007/978-94-017-8032-2_18|chapter = Methods in Mathematical Modeling for Stem Cells|title = Stem Cells and Cancer Stem Cells, Volume 12 |year = 2014|last1 = Eby|first1 = Wayne M.|last2 = Tabatabai|first2 = Mohammad A.|volume = 12|pages = 201–217|isbn = 978-94-017-8031-5}}{{Cite journal | doi=10.1007/s42452-020-2070-3|pmc=7391994| title=The recent advances in the mathematical modelling of human pluripotent stem cells| year=2020| last1=Wadkin| first1=L. E.| last2=Orozco-Fuentes| first2=S.| last3=Neganova| first3=I.| last4=Lako| first4=M.| last5=Shukurov| first5=A.| last6=Parker| first6=N. G.| journal=SN Applied Sciences| volume=2| issue=2|page=276 |pmid=32803125 | doi-access=free}}{{Cite book | doi=10.1007/978-94-017-8032-2| title=Stem Cells and Cancer Stem Cells, Volume 12| year=2014| volume=12| isbn=978-94-017-8031-5| s2cid=34446642}} and, the hyperbolastic mixed model of type II has been used in modeling cervical cancer data.{{Cite journal |doi = 10.1371/journal.pone.0107242|title = Disparities in Cervical Cancer Mortality Rates as Determined by the Longitudinal Hyperbolastic Mixed-Effects Type II Model|year = 2014|last1 = Tabatabai|first1 = Mohammad A.|last2 = Kengwoung-Keumo|first2 = Jean-Jacques|last3 = Eby|first3 = Wayne M.|last4 = Bae|first4 = Sejong|last5 = Guemmegne|first5 = Juliette T.|last6 = Manne|first6 = Upender|last7 = Fouad|first7 = Mona|last8 = Partridge|first8 = Edward E.|last9 = Singh|first9 = Karan P.|journal = PLOS ONE|volume = 9|issue = 9|pages = e107242|pmid = 25226583|pmc = 4167327|bibcode = 2014PLoSO...9j7242T| doi-access=free }} Hyperbolastic curves can be an important tool in analyzing cellular growth, the fitting of biological curves, the growth of phytoplankton, and instantaneous maturity rate.{{Cite journal |doi = 10.1186/1471-2105-14-283|title = BGFit: Management and automated fitting of biological growth curves|year = 2013|last1 = Veríssimo|first1 = André|last2 = Paixão|first2 = Laura|last3 = Neves|first3 = Ana|last4 = Vinga|first4 = Susana|journal = BMC Bioinformatics|volume = 14|pages = 283|pmid = 24067087|pmc = 3848918 | doi-access=free }}{{Cite journal |doi = 10.3934/mbe.2013.10.913|title = A flexible multivariable model for Phytoplankton growth|journal = Mathematical Biosciences and Engineering|year = 2013|volume = 10|issue = 3|pages = 913–923|pmid = 23906155|last1 = Tabatabai|first1 = M. A.|last2 = Eby|first2 = W. M.|last3 = Bae|first3 = S.|last4 = Singh|first4 = K. P.|doi-access = free}}{{Cite journal |doi = 10.1177/00080683211037203|title = A New Growth Rate Measure in Identifying Extended Gompertz Growth Curve and Development of Goodness-of-fit Test|journal = Calcutta Statistical Association Bulletin|year = 2021|volume = 73|issue = 2|pages = 127–145|last1 = Yeasmin|first1 = Farhana|last2 = Daw|first2 = Ranadeep|last3 = Chakraborty|first3 = Bratati|doi-access = free}}{{cite thesis |last= Arif|first= Samiur|date= 2014|title= Modeling Stem Cell Population Dynamics|publisher= Old Dominion University|url= https://digitalcommons.odu.edu/cgi/viewcontent.cgi?article=1048&context=computerscience_etds|doi = 10.25777/thnx-6q07}}

In forest ecology and management, the hyperbolastic models have been applied to model the relationship between DBH and height.{{Cite journal | doi=10.1016/j.foreco.2017.04.015| title=A new nonlinear model applied to the height-DBH relationship in Gmelina arborea| year=2017| last1=Eby| first1=Wayne M.| last2=Oyamakin| first2=Samuel O.| last3=Chukwu| first3=Angela U.| journal=Forest Ecology and Management| volume=397| pages=139–149| bibcode=2017ForEM.397..139E}}

The multivariable hyperbolastic model type III has been used to analyze the growth dynamics of phytoplankton taking into consideration the concentration of nutrients.{{Cite journal | doi=10.3934/mbe.2013.10.913| title=A flexible multivariable model for Phytoplankton growth| journal=Mathematical Biosciences and Engineering| year=2013| volume=10| issue=3| pages=913–923| pmid=23906155| last1=Tabatabai| first1=M. A.| last2=Eby| first2=W. M.| last3=Bae| first3=S.| last4=Singh| first4=K. P.| doi-access=free}}

File:Cumulative Distribution Function of Hyperbolastic Type I, Logistic, and Hyperbolastic Type II.png

File:PDF of H1 Logistic and H2.png

Hyperbolastic regressions are statistical models that utilize standard hyperbolastic functions to model a dichotomous or multinomial outcome variable. The purpose of hyperbolastic regression is to predict an outcome using a set of explanatory (independent) variables. These types of regressions are routinely used in many areas including medical, public health, dental, biomedical, as well as social, behavioral, and engineering sciences. For instance, binary regression analysis has been used to predict endoscopic lesions in iron deficiency anemia.{{Cite journal |doi = 10.1186/1471-230X-8-52|title = Predictors of gastrointestinal lesions on endoscopy in iron deficiency anemia without gastrointestinal symptoms|year = 2008|last1 = Majid|first1 = Shahid|last2 = Salih|first2 = Mohammad|last3 = Wasaya|first3 = Rozina|last4 = Jafri|first4 = Wasim|journal = BMC Gastroenterology|volume = 8|pages = 52|pmid = 18992171|pmc = 2613391 | doi-access=free }} In addition, binary regression was applied to differentiate between malignant and benign adnexal mass prior to surgery.{{Cite journal |doi = 10.1200/JCO.2005.01.7632|title = Logistic Regression Model to Distinguish Between the Benign and Malignant Adnexal Mass Before Surgery: A Multicenter Study by the International Ovarian Tumor Analysis Group|year = 2005|last1 = Timmerman|first1 = Dirk|last2 = Testa|first2 = Antonia C.|last3 = Bourne|first3 = Tom|last4 = Ferrazzi|first4 = Enrico|last5 = Ameye|first5 = Lieveke|last6 = Konstantinovic|first6 = Maja L.|last7 = Van Calster|first7 = Ben|last8 = Collins|first8 = William P.|last9 = Vergote|first9 = Ignace|last10 = Van Huffel|first10 = Sabine|last11 = Valentin|first11 = Lil|journal = Journal of Clinical Oncology|volume = 23|issue = 34|pages = 8794–8801|pmid = 16314639|doi-access = free}}

Let $Y$ be a binary outcome variable which can assume one of two mutually exclusive values, success or failure. If we code success as $Y=1$ and failure as $Y=0$, then for parameter $\backslash theta\; \backslash geq\; -1$, the hyperbolastic success probability of type I with a sample of size $n$ as a function of parameter $\backslash theta$ and parameter vector $\backslash boldsymbol\{\backslash beta\}\; =\; (\backslash beta\_0,\; \backslash beta\_1,\backslash ldots,\; \backslash beta\_p)$ given a $p$-dimensional vector of explanatory variables is defined as $\backslash mathbf\{x\}\_i=(x\_\{i1\},\backslash \; x\_\{i2\},\backslash ldots\; ,\backslash \; x\_\{ip\})^T$, where $i\; =\; 1,2,\backslash ldots,n$, is given by

:$\backslash pi(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\; =\; P(y\_i=1|\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\backslash frac\{1\}\{1+e^\{-(\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\})-\backslash theta\; \backslash operatorname\{arsinh\}(\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\})\; \}\; \}$.

The odds of success is the ratio of the probability of success to the probability of failure. For binary hyperbolastic regression of type I, the odds of success is denoted by $Odds\_\{H1\}$ and expressed by the equation

:$Odds\_\{H1\}=e^\{\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\}+\backslash theta\; \backslash operatorname\{arsinh\}(\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\})\; \}$.

The logarithm of $Odds\_\{H1\}$ is called the logit of binary hyperbolastic regression of type I. The logit transformation is denoted by $L\_\{H1\}$ and can be written as

:$L\_\{H1\}=\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\}\; +\backslash theta\; \backslash operatorname\{arsinh\}[\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\}]$.

The Shannon information for the random variable $Y$ is defined as

:$I(y)=-\{log\}\_bP(y)$

where the base of logarithm $b\; >\; 0$ and $b\; \backslash neq\; 1$. For binary outcome, $b$ is equal to $2$.

For the binary hyperbolastic regression of type I, the information $I(y)$ is given by

:$I(y)=$

\begin{cases}

-log_b\frac{1}{1+e^{-Z-\theta \operatorname{arsinh}(Z)}} & y = 1 ,\\

-log_b\frac{e^{-Z-\theta \operatorname{arsinh}(Z)}}{1+e^{-Z-\theta \operatorname{arsinh}(Z)}} & y = 0

\end{cases}

,

where $Z=\; \backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\backslash beta\_sx\_s$, and $x\_s$ is the $s^\{th\}$ input data.

For a random sample of binary outcomes of size $n$, the average empirical information for hyperbolastic H1 can be estimated by

:$\backslash overline\{I(y)\}=$

\begin{cases}

-\frac{1}{n}\sum_{i=1}^{n}{log_b\frac{1}{1+e^{-Z_i-\theta \operatorname{arsinh}(Z_i)}}} & y = 1 ,\\

-\frac{1}{n}\sum_{i=1}^{n}{log_b\frac{e^{-Z_i-\theta \operatorname{arsinh}(Z_i)}}{1+e^{-Z_i-\theta \operatorname{arsinh}(Z_i)}}} & y = 0

\end{cases}

,

where $Z\_i=\; \backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\backslash beta\_sx\_\{is\}$, and $x\_\{is\}$ is the $s^\{th\}$ input data for the $i^\{th\}$ observation.

====Information Entropy for hyperbolastic H1====

Information entropy measures the loss of information in a transmitted message or signal. In machine learning applications, it is the number of bits necessary to transmit a randomly selected event from a probability distribution. For a discrete random variable $Y$, the information entropy $H$ is defined as

:$H=-\backslash sum\_\{y\backslash in\; Y\}\{P(y)\backslash \; \{log\}\_bP(y)\}$

where $P(y)$ is the probability mass function for the random variable $Y$.

The information entropy is the mathematical expectation of $I(y)$ with respect to probability mass function $P(y)$. The Information entropy has many applications in machine learning and artificial intelligence such as classification modeling and decision trees. For the hyperbolastic H1, the entropy $H$ is equal to

:$$

\begin{align}

H & = -\sum_{y \in \{0,1\}}{P(Y=y;\mathbf{x},\boldsymbol{\beta})log_b(P(Y=y;\mathbf{x},\boldsymbol{\beta}))} \\

& = -[\pi(\mathbf{x};\boldsymbol{\beta})\ log_b(\pi(\mathbf{x};\boldsymbol{\beta})+(1-\pi(\mathbf{x};\boldsymbol{\beta}))log_b(1-\pi(\mathbf{x};\boldsymbol{\beta}))] \\

& = {log}_b(1+e^{-Z-\theta \operatorname{arsinh}(Z)})-\frac{e^{-Z-\theta \operatorname{arsinh}(Z)}{log}_b(e^{-Z-\theta \operatorname{arsinh}(Z)})}{1+e^{-Z-\theta \operatorname{arsinh}(Z)}}

\end{align}

The estimated average entropy for hyperbolastic H1 is denoted by $\backslash bar\{H\}$ and is given by

:$$

\bar{H}=\frac{1}{n}\sum_{i=1}^{n}{[log_b(1+e^{{-Z}_i-\theta \operatorname{arsinh}(Z_i)})-}\frac{e^{{-Z}_i-\theta \operatorname{arsinh}(Z_i)}\ {log}_b(e^{{-Z}_i-\theta \operatorname{arsinh}((Z_i)})}{1+e^{{-Z}_i-\theta \operatorname{arsinh}(Z_i)}}]

====Binary Cross-entropy for hyperbolastic H1====

The binary cross-entropy compares the observed $y\; \backslash in\; \backslash \{0,1\backslash \}$ with the predicted probabilities. The average binary cross-entropy for hyperbolastic H1 is denoted by $\backslash overline\{C\}$ and is equal to

:$$

\begin{align}

\overline{C} & =-\frac{1}{n}\sum_{i=1}^{n}{{[y}_i log_b(\pi(x_i;\boldsymbol{\beta}))+}{(1-y}_i)log_b(1-\pi(x_i;\boldsymbol{\beta}))] \\

&=\frac{1}{n}\sum_{i=1}^{n}{[log_b(1+e^{{-Z}_i-\theta \operatorname{arsinh}(Z_i)})-}{(1-y}_i)log_b(e^{{-Z}_i-\theta \operatorname{arsinh}(Z_i)})]

\end{align}

The hyperbolastic regression of type II is an alternative method for the analysis of binary data with robust properties. For the binary outcome variable $Y$, the hyperbolastic success probability of type II is a function of a $p$-dimensional vector of explanatory variables $\backslash mathbf\{x\}\_i$ given by

:$\backslash pi(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\; =\; P(y\_i=1|\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\; \backslash frac\{1\}\{1\; +\; \backslash operatorname\{arsinh\}[e^\{\; -\; (\backslash beta\_0\; +\; \backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\})\; \}]\}$ ,

For the binary hyperbolastic regression of type II, the odds of success is denoted by $Odds\_\{H2\}$ and is defined as

:$Odds\_\{H2\}\; =\; \backslash frac\{1\}\{\backslash operatorname\{arsinh\}[e^\{-(\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\})\; \}\; ]\; \}.$

The logit transformation $L\_\{H2\}$ is given by

:$L\_\{H2\}=\; -\; \backslash log\{(\; \backslash operatorname\{arsinh\}[\; e^\{-(\backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\{\backslash beta\_s\; x\_\{is\}\})\}])\}$

For the binary hyperbolastic regression H2, the Shannon information $I(y)$ is given by

:$I(y)\; =$

\begin{cases}

-log_b \frac{1}{1+arsinh(e^{-Z})} & y = 1 \\

-log_b \frac{arsinh(e^{-Z})}{1+arsinh(e^{-Z})} & y = 0

\end{cases}

where $Z=\; \backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\backslash beta\_sx\_s$, and $x\_s$ is the $s^\{th\}$ input data.

For a random sample of binary outcomes of size $n$, the average empirical information for hyperbolastic H2 is estimated by

:$\backslash overline\{I(y)\}=$

\begin{cases}

-\frac{1}{n}\sum_{i=1}^{n}log_b \frac{1}{1+arsinh(e^{-Z_i})} & y = 1 \\

-\frac{1}{n}\sum_{i=1}^{n}log_b \frac{arsinh(e^{-Z_i})}{1+arsinh(e^{-Z_i})} & y=0

\end{cases}

where $Z\_i=\; \backslash beta\_0+\backslash sum\_\{s=1\}^\{p\}\; \backslash beta\_sx\_\{is\}$, and $x\_\{is\}$ is the $s^\{th\}$ input data for the $i^\{th\}$ observation.

====Information Entropy for hyperbolastic H2====

For the hyperbolastic H2, the information entropy $H$ is equal to

:$$

\begin{align}

H& = -\sum_{y\in \{0,1\}}{P(Y=y;\mathbf{x} , \boldsymbol{\beta}) log_b(P(Y=y;\mathbf{x} ,\boldsymbol{\beta}))} \\

& =-[\pi(\mathbf{x};\boldsymbol{\beta})\ log_b(\pi(\mathbf{x};\boldsymbol{\beta}))+(1-\pi(\mathbf{x};\boldsymbol{\beta}))log_b(1-\pi(\mathbf{x};\boldsymbol{\beta}))] \\

& =log_b(1+arsinh(e^{-Z}))-\frac{arsinh(e^{-Z}) log_b (arsinh(e^{-Z}))}{1+arsinh(e^{-Z})}

\end{align}

and the estimated average entropy $\backslash bar\{H\}$ for hyperbolastic H2 is

:$$

\bar{H}=\frac{1}{n}\sum_{i=1}^{n}{[log_b(1+{arsinh(e}^{{-Z}_i}))-}\frac{{arsinh(e}^{{-Z}_i})\ {log}_b{(arsinh(e}^{{-Z}_i}))}{1+{arsinh(e}^{{-Z}_i})}]

====Binary Cross-entropy for hyperbolastic H2====

The average binary cross-entropy $\backslash overline\{C\}$ for hyperbolastic H2 is

:$$

\begin{align}

\overline{C} & =-\frac{1}{n}\sum_{i=1}^{n}{{[y}_ilog_b(\pi(x_i;\beta))+}{(1-y}_i)log_b(1-\pi(x_i;\beta))] \\

& =\frac{1}{n}\sum_{i=1}^{n}{[log_b(1+{arsinh(e}^{{-Z}_i}))-}{(1-y}_i)log_b({arsinh(e}^{{-Z}_i}))]

\end{align}

The estimate of the parameter vector $\backslash boldsymbol\{\backslash beta\}$ can be obtained by maximizing the log-likelihood function

:$\backslash hat\{\backslash beta\}\; =\; \backslash underset\{\backslash boldsymbol\{\backslash beta\}\}\backslash operatorname\{argmax\}\{\backslash sum\_\{i\; =\; 1\}^\{n\}[y\_iln(\backslash pi(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\}))+(1-y\_i)ln(1-\backslash pi(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\}))]\}$

where $\backslash pi(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})$ is defined according to one of the two types of hyberbolastic functions used.

The generalization of the binary hyperbolastic regression to multinomial hyperbolastic regression has a response variable $y\_i$ for individual $i$ with $k$ categories (i.e. $y\_i\; \backslash in\; \backslash \{1,2,\backslash ldots,k\backslash \}$). When $k=2$, this model reduces to a binary hyperbolastic regression.

For each $i=1,2,\backslash ldots,n$, we form $k$ indicator variables $y\_\{ij\}$ where

:$y\_\{ij\}=$

\begin{cases}

1 & \text{if } y_i = j,\\

0 & \text{if } y_i \neq j

\end{cases}

,

meaning that $y\_\{ij\}=1$ whenever the $i^\{th\}$ response is in category $j$ and $0$ otherwise.

Define parameter vector $\backslash boldsymbol\{\backslash beta\}\_j=(\backslash beta\_\{j0\},\backslash beta\_\{j1\},\backslash ldots,\backslash beta\_\{jp\})$ in a $p+1$-dimensional Euclidean space and $\backslash boldsymbol\{\backslash beta\}=(\backslash boldsymbol\{\backslash beta\}\_1,\backslash ldots,\backslash boldsymbol\{\backslash beta\}\_\{k-1\})^T$.

Using category 1 as a reference and $\backslash pi\_1(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})$ as its corresponding probability function, the multinomial hyperbolastic regression of type I probabilities are defined as

:$\backslash pi\_1(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=P(y\_i=1|\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\backslash frac\{1\}\{1+\backslash sum\_\{s=2\}^\{k\}e^\{-\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})-\backslash theta\; \backslash operatorname\{arsinh\}[\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})]\}\}$

and for $j\; =\; 2,\backslash ldots,k$,

:$\backslash pi\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=P(y\_i=j|\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\backslash frac\{e^\{-\backslash eta\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})-\backslash theta\; \backslash operatorname\{arsinh\}[\backslash eta\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})]\}\}\{1+\backslash sum\_\{s=2\}^\{k\}e^\{-\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})-\backslash theta\; \backslash operatorname\{arsinh\}[\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})]\}\}$

Similarly, for the multinomial hyperbolastic regression of type II we have

:$\backslash pi\_1(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=P(y\_i=1|\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\backslash frac\{1\}\{1+\backslash sum\_\{s=2\}^\{k\}arsinh[e^\{-\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\}]\}$

and for $j\; =\; 2,\backslash ldots,k$,

:$\backslash pi\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=P(y\_i=j|\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\backslash frac\{arsinh[e^\{-\backslash eta\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\}]\}\{1+\backslash sum\_\{s=2\}^\{k\}arsinh[e^\{-\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\}]\}$

where $\backslash eta\_s(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})=\backslash beta\_\{s0\}+\backslash sum\_\{l=1\}^\{p\}\backslash beta\_\{sl\}x\_\{il\}$ with $s\; =\; 2,\; \backslash dots,\; k$ and $i\; =\; 1,\backslash dots,n$.

The choice of $\backslash pi\_i(\backslash mathbf\{x\_i\};\backslash boldsymbol\{\backslash beta\})$ is dependent on the choice of hyperbolastic H1 or H2.

For the multiclass $(j=1,\; 2,\; \backslash dots,\; k)$, the Shannon information $I\_j$ is

:$I\_j=-log\_b(\backslash pi\_j(\backslash mathbf\{x\};\backslash boldsymbol\{\backslash beta\}))$.

For a random sample of size $n$, the empirical multiclass information can be estimated by

:$\backslash overline\{I\_j\}=-\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^\{n\}\{log\_b(\backslash pi\_j(\backslash mathbf\{x\_i\};\backslash boldsymbol\{\backslash beta\}))\}$.

====Multiclass Entropy in Information Theory====

For a discrete random variable $Y$, the multiclass information entropy is defined as

:$H=-\backslash sum\_\{y\; \backslash in\; Y\}\{P(y)\backslash \; \{log\}\_bP(y)\}$

where $P(y)$ is the probability mass function for the multiclass random variable $Y$.

For the hyperbolastic H1 or H2, the multiclass entropy $H$ is equal to

:$H=-\backslash sum\_\{j=1\}^\{k\}\{[\backslash pi\_j(\backslash mathbf\{x\};\backslash boldsymbol\{\backslash beta\})\; log\_b(\backslash pi\_j(\backslash mathbf\{x\};\backslash boldsymbol\{\backslash beta\}))]\}$

The estimated average multiclass entropy $\backslash overline\{H\}$ is equal to

:$\backslash overline\{H\}=-\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^\{n\}\{\backslash sum\_\{j=1\}^\{k\}\{[\backslash pi\_j(\backslash mathbf\{x\_i\};\backslash boldsymbol\{\backslash beta\})\; log\_b(\backslash pi\_j(\backslash mathbf\{x\_i\};\backslash boldsymbol\{\backslash beta\}))]\}\}$

====Multiclass Cross-entropy for hyperbolastic H1 or H2====

Multiclass cross-entropy compares the observed multiclass output with the predicted probabilities. For a random sample of multiclass outcomes of size $n$, the average multiclass cross-entropy $\backslash overline\{C\}$ for hyperbolastic H1 or H2 can be estimated by

:$\backslash overline\{C\}=-\backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^\{n\}\{\backslash sum\_\{j=1\}^\{k\}\{[y\_\{ij\}\; log\_b(\backslash pi\_j(\backslash mathbf\{x\_i\};\backslash boldsymbol\{\backslash beta\}))]\}\}$

The log-odds of membership in category $j$ versus the reference category 1, denoted by $\backslash omicron\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})$, is equal to

:$\backslash omicron\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\; =\; ln[\backslash frac\{\backslash pi\_j(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\}\{\backslash pi\_1(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})\}]$

where $j=2,\backslash ldots,k$ and $i=1,\backslash ldots,n$. The estimated parameter matrix $\backslash hat\backslash boldsymbol\{\backslash beta\}$ of multinomial hyperbolastic regression is obtained by maximizing the log-likelihood function. The maximum likelihood estimates of the parameter matrix $\backslash boldsymbol\backslash beta$ is

:$\backslash boldsymbol\{\backslash hat\{\backslash beta\}\}\; =\; \backslash underset\{\backslash boldsymbol\{\backslash beta\}\}\backslash operatorname\{argmax\}\{\backslash sum\_\{i=1\}^n(y\_\{i1\}ln[\backslash pi\_1(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})]+y\_\{i2\}ln[\backslash pi\_2(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})]+\backslash ldots+y\_\{ik\}ln[\backslash pi\_k(\backslash mathbf\{x\}\_i;\backslash boldsymbol\{\backslash beta\})])\}$

{{reflist}}

Category:Medical models

Category:Population models

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