Estimating the Coefﬁcients of a System of Ordinary Differential Equations Based on Inaccurate Observations

: In this paper, we solve the problem of estimating the parameters of a system of ordinary differential equations from observations on a short interval of argument values. By analogy with linear regression analysis, a sufﬁciently large number of observations are selected on this segment and the values of the functions on the right side of the system and the values of the derivatives are estimated. According to the obtained estimates, unknown parameters are determined, using the differential equations system. The consistency of the estimates obtained in this way is proved with an increase in the number of observations over a short period of argument values. Here, an algorithm for estimating parameters acts as a system. The error of the obtained estimate is an indicator of its quality. A sequence of inaccurate measurements is a random process. The method of linear regression analysis applied to an almost linear regression function is used as an optimization procedure.


Introduction
The problem of estimating the parameters of a system of nonlinear ordinary differential equations, based on inaccurate deterministic observations, using known optimization algorithms, is solved in the papers [1][2][3].An alternative approach for estimating the parameters of a deterministic recurrent sequence, observed with random additive and multiplicative errors, based on the relationships between the trajectory averages and their approximation from inaccurate observations, is proposed in [4,5].
The advantage of the first approach is the possibility of using known optimization algorithms, and the disadvantage of it is the lack of analytical estimates of the convergence rate to the estimated parameters.The advantage of the second approach is the availability of theoretical estimates of the convergence rate to the estimated parameters, and the disadvantage of it is the need to establish limit cycles or limit distributions for recurrent sequences.
Despite all the differences in these approaches, the common fact is that by increasing in the length of the observation segment, the accuracy of estimates increases and, under certain conditions, may tend to zero.At the same time, the problem of estimating parameters over a small observation interval is interesting, which is closely related to discrete optimization methods of experiment planning (see, for example, [6,7]).
In this paper, this problem is solved for a system of non-linear ordinary differential equations.At the same time, the estimation of the parameters of this system, based on inaccurate observations, is solved under the assumption that a large number of observations may be carried out over a relatively short segment.To estimate the parameters, the method of linear regression analysis is used in relation to a regression function that slightly deviates from the original function in a small neighbourhood of some time moment [8][9][10][11][12][13].
This method is based on minimizing the standard deviation of a sequence of observations from a linear regression function.In this case, such a relationship is selected between the number of observations and the interval between neighbouring observations so that the resulting error in determining the parameters tends to zero when the number of observations tends to infinity.
The final stage of the parameter estimation algorithm is the substitution of estimates of the values of functions and the values of their derivatives into the original system of equations at the selected point.Further, by analogy with the method of moments, unknown parameters of the system of equations are estimated and the consistency of the estimates obtained is proved.This paper also uses the implicit function theorem, which allows us to establish that the obtained parameter estimates are consistent depending on the number of observations.Based on the results obtained, computational experiments were carried out.
Thus, elements of system analysis have been introduced into the solution of the task.Here, an algorithm for estimating parameters acts as a system.The error of the obtained estimate is an indicator of its quality.A sequence of inaccurate measurements is a random process.Furthermore, the process and the method of linear regression analysis applied to an almost linear regression function is used as an optimization procedure.It is evaluated using the theorem on the existence and uniqueness of the solution of a system of ordinary differential equations and with the help of the implicit function theorem.Additionally, known error estimates in the linear regression analysis method are used.

Preliminaries
Consider a system of ordinary differential equations with fixed values of parameters where x 1 = x 1 (t), . . ., x m = x m (t) are unknown functions.In well-known monographs on the theory of ordinary differential equations (see, for example, [14,15]), the theorem of the existence and uniqueness of the solution of this system in a small neighbourhood of a certain point is formulated and proved in Theorem 1.
., m} together with their partial derivatives ∂F i ∂x i , i = 1, ..., m.Then there is a segment t 0 − r ≤ t ≤ t 0 + r, on which the system of Equation ( 1) has a unique solution satisfying the initial conditions x i (t 0 ) = x 0 i , i = 1, . . ., m.

Remark 1.
From the Weierstrass theorem for continuous functions on a compact, it follows that the functions x i (t), i = 1, ..., m, on the segment [t 0 − r, t 0 + r] (continuity follows from differentiability) and function F i • ∂F i ∂x i , i = 1, ..., m, on a set Q (due to the continuity of the multipliers) reach their highest final values C i . Denote where F i are described in Theorem 1, and consider the system of equations In monographs on mathematical analysis (see, for example, [16,17]), conditions are formulated, under which the system (2) may be resolved with respect to variables β 1 , . . ., β m (see for example Theorem 2).

Ordinary Differential Equation
Consider the differential equation for a fixed value of the parameter with the initial condition x(0) = x 0 , assuming that the function F(x, β) is continuously differentiable in the neighbourhood of a point M 0 = (x 0 , β 0 ) and ∂F ∂β M 0 = 0. Let the inaccurate observations y(t) = x(t) + ε(t) are known for the state of x(t) at the moments t = kh, k = 0, ±1, . . ., ±n, hn ≤ r.Denote and suppose that ε k , k = 0, ±1, . . ., ±n, is a set of independent and identically distributed random variables with zero mean and variance σ 2 .The problem of estimating the parameter β 0 of the differential Equation ( 4) from these observations is posed.The solution of this problem is carried out in two stages.First, they are constructed using a modification of the least squares estimation method x 0 , F 0 and their convergence to the estimated parameters x 0 , F 0 is investigated.Then, by analogy with the method of moments, an estimate of β 0 is constructed and its convergence to the estimated parameter β 0 is investigated.
Evaluation of values x 0 , F 0 .Let us introduce the notations, outlining the method for defining x 0 , F 0 (5) Theorem 3. If σ 2 < ∞ and h = n −α , then, for α > 1, the estimate of x 0 is an asymptotically unbiased and consistent estimate of the parameter x 0 .The estimate F 0 is an asymptotically unbiased estimate of the parameter F 0 .At 1 < α < 3/2; the estimate F 0 is a consistent estimate of F 0 .
Proof of Theorem 3.
Estimates of x0 , F0 are obtained by the least squares method for coefficients x 0 , F 0 of linear regression [9] and satisfy the following relations Here, Ex is mathematical expectation of arbitrary random variable x and Varx = E(x − Ex) 2 is its variance.In turn, the following equalities are almost certainly fulfilled Moreover, the differences y k − ỹk = x k − x 0 − F 0 kh, k = 0, ±1, . . ., ±n are deterministic quantities.
The Remark 1 implies the existence of a number C satisfying the inequality Then, from the Taylor formula with a residual term in the Lagrange form, From the Formulas ( 7) and ( 8) for n → ∞, the relations follow The Formulas ( 6), ( 9) and ( 10) lead to the relations Here a n b n means that lim sup and the Relations ( 11) and (12) we have that x 0 , F 0 are asymptotic unbiased estimates of x 0 , F 0 .
From the Bieneme-Chebyshev inequality, the Relations ( 9) and ( 11) and the conditions h = n −α , α > 1, we get for any δ > 0 Thus, for h = n −α , α > 1, estimate x 0 is a consistent estimate of x 0 .At the same time, from the Relations (10), ( 12) and ( 13) for h = n −α , 1 < α < 3/2, we get for any δ > 0 Therefore, if the condition h = n −α , 1 < α < 3/2, is true, the estimate F 0 is a consistent estimate of F 0 .Remark 3. It is worth noting that Theorem 3 is true for any distribution of random variables ε k with finite variance σ 2 .Indeed it is necessary to prove limit relation However, the most reasonable way to solve this question is to consider such distributions of random variables ε k as normal for σ 2 < ∞/ or stable for σ 2 = ∞, because H n has normal/stable distribution also.

Evaluation of parameter β 0 . Consider the equation
Theorem 4. In conditions of Theorem 3, Equation ( 16) has a unique solution β 0 , which is a consistent estimate of the parameter β 0 .
The proofs of the Theorems 5 and 6 almost verbatim repeat the proofs of the Theorems 3 and 4.
Remark 4. Theorems 3-6 are devoted to ordinary differential equations of the first order and their systems.However, it is possible to spread them to ordinary differential equations and their systems of arbitrary order.For this purpose it is possible to use for examples results of [12,13].

Computational Experiment
Example 1.The computational experiment was conducted first for the Cauchy problem The solution of this equation has the form x = e b 0 t .We assumed that by observing the process described by this equation, ±kh, k = 0, 1, ..., n, h = n −5/4 , n = 10, 000, inaccurate observations were obtained at time points y ±k = e ±b 0 hk + ε ±k , k = 0, 1, ..., n.
Here, independent random variables ε ±k , k = 0, 1, ..., n, are distributed uniformly on the segment [−1/2, 1/2] left and on the segment [−1/4, 1/4] right.According to the Formula (5), the parameters x 0 , F 0 = F(x 0 , β 0 ) in our notation x 0 , F 0 , were evaluated first, then the formula for evaluating the parameter β 0 was found from the equation F 0 = x 0 β 0 .Table 1 shows the results of a computational experiment conducted 1000 times, namely, the interval distribution (5 intervals) of relative frequencies β 0 .Consequently, a decrease in variance σ 2 improves the quality of the obtained estimates sufficiently clearly.Now, consider the case in which independent random variables ε ±k , k = 0, 1, ..., n, are distributed normally with mean 0 and variance σ 2 .Table 2 shows the results of a computational experiment conducted 1000 times, namely, the interval distribution (five intervals) of relative frequencies β 0 .Consequently, the quality of obtained results for disturbances distributed normally behaves like in a case of uniform distribution.