Differential Equations Arising from Simple Nonlinear Regression Models Volume 1 - Issue 5

Michele Nichelatti*

• Service of Biostatistics, Fondazione Malattie del Sangue, Milan, Italy

Received: October 10, 2019;   Published: October 22, 2019

*Corresponding author: Michele Nichelatti, Service of Biostatistics, Fondazione Malattie del Sangue, Milan, Italy

DOI: 10.32474/CTBB.2018.01.000124

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Abstract

It is presented a method useful to form ordinary differential equations from nonlinear regression models of the type y=a_nxⁿ+a_n-1x^n-1+...+a₁x allowing to directly retrieve the differential model from raw data after fitting, to be compared with the differential model expected for the biological system which is studied.

Introduction

In some cases, to obtain an estimate of the behavior of dependent variable with respect to an independent one can be just the first task, and we would be much more interested in retrieving from the regression the differential equation governing the system [1]. For example, when dealing with a linear regression model given by the simple equation

y = a₁x + a₀ (1)

where is the independent variable, and is the dependent one, being two arbitrary constants, the differential equation governing the system may be easily obtained deriving with respect to both sides of equation (1)

∂_xy = a₁ (2)

so that, substitution of the value in equation (1) gives the differential equation

∂_xy − y + a₀ = 0 (3)

having equation (1) as general solution. The differential equation (3) is a linear first order ordinary differential equation, also known as Clairaut’s equation.

Differential Equations from Nonlinear Regression Models

The approach to nonlinear regression model is quite more complicated. Assuming a regression of the type

y = a₂x² + a₁x (4)

with two arbitrary constants a₁, a₂, such that the first and second derivatives respectively are

∂_x y = 2a₂x + a₁ (5)

∂²xx y = 2a₂.(6)

In this case, and in all next cases, we tacitly assume that is continuous and derivable at least times over its domain and that none of the coefficients are zero, so that we can eliminate the first constant since we have

(7)

and therefore equation (5) reads

∂_x y = x∂^{2xx y + a₁ (8)}

with

a₁ = −x∂²xx y + ∂_x y(9)

thus, inserting a₁, a₂ in equation (4), we obtain

(10)

hence

(11)

equivalent to

x²∂²_xxy −2x∂_x y + 2y = 0 (12)

which is the differential equation having equation (4) as general solution.

Now, using the third-degree regression result

y = a₃ x³ + a₂ x² + a₁ x (13)

and calculating the derivatives

we get from equation (16)

(17)

so that, inserting this value into equation (15) leads to

∂²_xx y = x∂³_xxx y + 2a₂ (18)

giving

(19)

which, inserted into equation (14) gives

(20)

and

(21)

At this point, we can use the values of a₁, a₂ and a₃ respectively found in equations (21), (19) and (17), ad insert them into equation (13), to obtain

thus, the differential equation is

(23)

Now, let us take into accounts the fourth-degree regression model:

y = a₄x⁴ + a₃ x + a₂ x + a₁ x (24)

with the usual procedure, we calculate

Thus, after a little algebra, avoiding intermediate passages:

hence, we obtain the differential equation having equation (24) as general solution

(33)

A Possible Generalized Method

In finding the differential equation ruling the nonlinear regression model we observed an evident pattern, which can be invoked for whichever degree of the regression model we are dealing with.

Indeed, for the nonlinear regression model of unspecified degree m:

(34)

the differential equation is

(35)

but, since m!/m!=1 and since m!/1!= m!/0!= m! to reduce the mathematical formalism, in the differential equation (35), we can rewrite it as

This model works for equations (33), (23) and (12), where was respectively equal to 4, 3, and 2. We can now try to see what will occur if m = 5 : in this case, equation (36) becomes

which is a Euler-Cauchy linear homogeneous differential equation, which can be solved at least with two methods. Using the simpler one, which is to assume that a solution will be of the form y = xζ (in which ζ > n is an arbitrary constant), and imposing this substitution in equation (37) also assuming , we will find that, in general,

and the individual solutions y₁ = a₁x, y₂= a₂x², y₃ = a₃x³, y₄ = a₄x⁴ , and y₅ = a₅x⁵ , so that the general solution of the differential equation (37) is

(40)

exactly corresponding to what expected. If regression model contains a constant term, so that

(41)

hence, if we assume a₀ = k , then it is immediate to see that the differential equation formed from the model given by equation (41) is

(42)

For example, if the regression model is

y = a₅x⁵ + a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀(43)

the differential equation is

(44)

Conclusion

When dealing with the analysis of a biological system, usually one expects to apply a given dynamical model [2] to the data. In this paper, the data are used after fitting to retrieve the ordinary differential equation having the fitted equation as general solution. However, in this paper, the case is limited to a simple nonlinear univariable model, but the differential equation obtained “from scratch” can be used to be compared to the differential equation one expects to rule the system.

References

1. Taubes CH (2001) Modeling differential equations in biology. Prentice Hall, Upper Saddle River, USA.
2. Gros C (2015) Complex and adaptive dynamical systems,(4^thEdn),Cham CH, Springer International Publishing, Germany.