Least squares estimation of high-order uncertain differential equations

Abstract

Parameter estimation of high-order uncertain differential equations is an inevitable problem in practice. In this paper, the equivalent equations of high-order uncertain differential equations are obtained by transformation, and the parameters of the first-order uncertain differential equation including Liu process are estimated. Based on the least squares estimation method, this paper proposes a means to minimize the residual sum of squares to obtain an estimate of the parameters in the drift term, and make the noise sum of squares equal to the residual sum of squares to obtain an estimate of the parameters in the diffusion term. In addition, some numerical examples are given to illustrate the proposed method. Finally, applications of the high-order uncertain spring vibration equations verify the viability of our method.

Keywords

Uncertainty theory high-order uncertain differential equation parameter estimation least squares estimation

1 Introduction

The stochastic differential equation was proposed by Ito [6] in 1944, and it is non-deterministic equation involving stochastic process. The stochastic process is a sequence of random variables indexed by the time. The stochastic differential equations can be used for modeling stochastic dynamic phenomena, and they are widely applied in biology, natural science, finance and many other fields. The coefficients of stochastic differential equation usually contain unknown parameters that need to be estimated. Some scholars have proposed the most commonly used parameter estimation methods. For example, the maximal likelihood estimation was proposed by Breton [2] in 1976, the least squares estimation was proposed by Kasonga [10] in 1988, the method of moments was proposed by Chan et al. [3] in 1992. For more parameter estimation problems can consult the book written by Bishwal [1] in 2008.

In order to deal with the likelihood of an event more reasonably, based on the normality, duality, subadditivity and product axioms, the uncertainty theory was proposed by Liu [13] in 2007 and perfected by Liu [15] in 2009. The uncertain differential equation was proposed by Liu [14] in 2008, and it is non-deterministic equation involving Liu process. The Liu process is a sequence of uncertain variables indexed by the time, and its existence was proved by Liu [16] in 2010. The solution of uncertain differential equations is also a key research aspect. Analytic methods of solving uncertain differential equations were founded by Liu [17] in 2012 and Yao [26] in 2013. The numerical method to solve uncertain differential equation was designed by Yao and Chen [23] in 2013. In recent years, many scholars have made research on uncertain differential equations. For example, multi-dimensional uncertain differential equation was proposed by Yao [25] in 2014. More types of uncertain differential equations was founded by Yao [24] in 2016. In addition, uncertain heat conduction equation was proposed by Yang and Yao [21] in 2017, uncertain wave equation was proposed by Gao [5] in 2017, uncertain spring vibration equation was proposed by Jia et al. [8] in 2018. Except for general first-order uncertain differential equations, high-order uncertain differential equations also play a vital role in practice. High-order uncertain differential equations involving the high-order derivatives of uncertain processes which was proposed by Yao [24] in 2016. The numerical solution method for solving high-order uncertain differential equation using the α-paths was designed by Yao [24] in 2016. Next, the Runge-Kutta method for solving high-order uncertain differential equations was designed by Ji and Zhou [7] in 2018. Moreover, stability of high-order uncertain differential equations was studied by Sheng [19] in 2017.

How to estimate the unknown parameters in the drift terms and diffusion terms of uncertain differential equations is a key issue. The least-square method for simple uncertain differential equations was proposed by Li [11] in 2019. The method of moments for estimating uncertain differential equations was first proposed by Yao and Liu [22] in 2020. The least squares estimation method for estimating uncertain differential equations was proposed by Sheng et al. [18] in 2020. The α-path method for estimating uncertain differential equation and application to financial market was presented by Yang et al. [20] in 2020. Parameter estimation of uncertain SIR model was proposed by Chen et al. [4] in 2020. Using three methods to estimate the parameters in uncertain differential equations was proposed by Li et al. [12] in 2020. Jia and Chen [9] applied the parameter estimation method to the uncertain SEIAR model in 2020. However, in practice, the coefficients of high-order uncertain differential equations with unknown parameters hard be estimated. One question is how to estimate the parameters in a high-order uncertain differential equation, and another question is how to test a high-order uncertain differential equation is appropriate to model a dynamic system involving the high-order derivatives of uncertain processes. This paper is only concerned with parameter estimation of high-order uncertain differential equation using the method of the least squares estimation.

The rest of this paper is organized as follows. In Section 2, preliminaries of uncertainty theory are reviewed. In Section 3, the equivalent equation of high-order uncertain differential equation will be obtained, and then the unknown parameters in the coefficients of the high-order uncertain differential equation are estimated based on least square estimation method. In Section 4, some numerical examples to illustrate the method will be given. After that, in Section 5, applications of high-order uncertain spring vibration equations will be given. Finally, in Section 6 some conclusions will be made.

2 Preliminary

In this section, some concepts about uncertainty theory are reviewed.

Definition 1. (Liu [13, 15]) Let Ł be a σ-algebra on a nonempty set Γ. A set function ℳ is called an uncertain measure if it satisfies the following axioms:

Axiom 1. (Normality Axiom) M {Γ} =1 for the universal set Γ.

Axiom 2. (Duality Axiom) M {Λ} + M {Λ^c} =1 for any event Λ .

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$ Axiom 4. (Product Axiom) Let $(Γ_{k}, L_{k}, M_{k})$ be uncertainty spaces for k = 1, 2, ⋯ . Then the product uncertain measure ℳ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}},$ where Λ_k are arbitrarily chosen events from $L_{k}$ for k = 1, 2, ⋯ , respectively.

Definition 2. (Liu [15]) An uncertain variable ξ is a measurable function from the uncertainty space $(Γ, L, M)$ to the set of real numbers. The uncertain variables ξ₁, ξ₂, ⋯ , ξ_n are said to be independent if $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = min_{1 \leq i \leq n} M {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, ⋯ , B_n of real numbers.

Definition 3. (Liu [13]) Let ξ be an uncertain variable. Then its uncertainty distribution is defined by $Φ (x) = M {ξ \leq x}$ for any real number x.

An uncertain variable ξ is called normal if it has an uncertainty distribution $Φ (x) = {(1 + exp (\frac{π (μ - x)}{\sqrt{3} σ}))}^{- 1}, x \in R$ denoted by N (μ, σ). If μ = 0 and σ = 1, then ξ is called a standard normal uncertain variable. Then inverse uncertainty distribution of a standard normal uncertain variable is $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}, α \in (0, 1) .$

Definition 4. (Liu [15]) An uncertain process C_t is called a Liu process if

(i) C₀ = 0 and almost all sample paths are Lipschitz continuous,

(ii) C_t has stationary and independent increments,

(iii) the increment C_s+t - C_s has a normal uncertainty distribution $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Definition 5. (Yao [24]) Suppose that C_t is a Liu process, and f (t, x₁, x₂, ⋯ , x_n) and g (t, x₁, x₂, ⋯ , x_n) are two measurable functions. Then $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ (1) is called an n-order uncertain differential equation. An uncertain process that satisfies (refe1) identically at each time t is called a solution of the high-order uncertain differential equation.

Definition 6. (Jia et al. [8]) A high-order uncertain spring vibration equation ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t) + σ (t) \frac{d C_{t}}{d t}, \\ t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] \\ X_{t} |_{t = 0} = 0, \frac{d X_{t}}{d t} |_{t = 0} = 0, \end{matrix}$ where $δ = \frac{c}{2 m}$ , c is a damping constant, m is the mass of the vibration object, $ω^{2} = \frac{K}{m}$ , K is a Hookes constant, X_t is a displacement, f (t) is an external force, σ (t) is a diffusion term of external force, l^e, l^l, l^s are the lengths of equilibrium position, longest, and shortest of the spring, respectively.

3 Parameter estimation

In this section, we present a theorem for the least squares estimation method of n-order uncertain differential equation.

Theorem 1. Suppose the n-order uncertain differential equation $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}; μ) \\ + & g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}; σ) \frac{d C_{t}}{d t} \end{matrix}$ (2) where μ and σ are unknown parameters to be estimated. Let x_1t₁, ⋯ , x_{1t_N}, ⋯ , x_{nt
₁}, ⋯ , x_{nt
_N} be sample observations data of the n-order uncertain differential equation (refe2) at the times t₁, ⋯ , t_N with t₁ < ⋯ < t_N, respectively. Then the least squares estimation μ ^* of equation (refe2) optimal solution of the following minimization problem $\begin{matrix} min_{μ} \sum_{j = 1}^{N - 1} (x_{{nt}_{j + 1}} - x_{{nt}_{j}} - f (t_{j}, x_{{it}_{j}}; μ) \\ {(t_{j + 1} - t_{j}))}^{2}, \end{matrix}$ and the least squares estimation σ ^* of equation (refe2) solution of the following equation $\begin{matrix} E [\sum_{j = 1}^{N - 1} {(g (t_{j}, x_{{it}_{j}}; σ) (C_{t_{j + 1}} - C_{t_{j}}))}^{2}] \\ = & \sum_{j = 1}^{N - 1} {(x_{{nt}_{j + 1}} - x_{{nt}_{j}} - f (t_{j}, x_{{it}_{j}}; μ^{*}) (t_{j + 1} - t_{j}))}^{2} . \end{matrix}$

Proof. For the n-order uncertain differential equation (2), we write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} \\ ⋮ \\ X_{nt} = \frac{d^{n - 1} X_{t}}{d t^{n - 1}} . \end{matrix}$ Then we have ${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ d X_{2 t} = X_{3 t} d t \\ ⋮ \\ \begin{matrix} d X_{nt} = & f (t, X_{1 t}, X_{2 t}, \dots, X_{nt}; μ) d t \\ + & g (t, X_{1 t}, X_{2 t}, \dots, X_{nt}; σ) d C_{t} . \end{matrix} \end{matrix}$ (3) The parameter estimation of equation (2) is transformed into the parameters estimation of the following equivalent equation based on observations, $\begin{matrix} d X_{nt} = & f (t, X_{1 t}, X_{2 t}, \dots, X_{nt}; μ) d t \\ + & g (t, X_{1 t}, X_{2 t}, \dots, X_{nt}; σ) d C_{t} \end{matrix}$ (4) where μ and σ are unknown parameters to be estimated. Note that the equation (4) has the following discrete form

$\begin{matrix} X_{{nt}_{j + 1}} = & X_{{nt}_{j}} + f (t_{j}, X_{1 t_{j}}, X_{2 t_{j}}, \dots, X_{{nt}_{j}}; μ) (t_{j + 1} - t_{j}) \\ + & g (t_{j}, X_{1 t_{j}}, X_{2 t_{j}}, \dots, X_{{nt}_{j}}; σ) (C_{t_{j + 1}} - C_{t_{j}}) \end{matrix}$

which can be rewritten as

$\begin{matrix} g (t_{j}, X_{1 t_{j}}, X_{2 t_{j}}, \dots, X_{{nt}_{j}}; σ) (C_{t_{j + 1}} - C_{t_{j}}) \\ = & X_{{nt}_{j + 1}} - X_{{nt}_{j}} - f (t_{j}, X_{1 t_{j}}, X_{2 t_{j}}, \dots, X_{{nt}_{j}}; μ) (t_{j + 1} - t_{j}) . \end{matrix}$

Assume that there are observations x_{it
_j} (i = 1, 2, ⋯ , n) (j = 1, 2, ⋯ , N) are obtained by the system (3) of equations at the times t₁, ⋯ , t_N with t₁ < ⋯ < t_N, respectively, $\begin{matrix} x_{{it}_{j}} = (\begin{matrix} x_{{1 t}_{1}} x_{{1 t}_{2}} \dots x_{{1 t}_{N}} \\ x_{{2 t}_{1}} x_{{2 t}_{2}} \dots x_{{2 t}_{N}} \\ ⋮ \\ x_{{nt}_{1}} x_{{nt}_{2}} \dots x_{{nt}_{N}} \end{matrix}) \end{matrix} .$ The estimate of μ solves the following optimization problem $min_{μ} \sum_{j = 1}^{N - 1} {(x_{{nt}_{j + 1}} - x_{{nt}_{j}} - f (t_{j}, x_{{it}_{j}}; μ) (t_{j + 1} - t_{j}))}^{2} .$ (5) Let μ ^* denote the estimate of μ obtained from optimization problem (5). Then the estimate of σ solves the following equation

$\begin{matrix} E [\sum_{j = 1}^{N - 1} {(g (t_{j}, x_{{it}_{j}}; σ) (C_{t_{j + 1}} - C_{t_{j}}))}^{2}] \\ = & \sum_{j = 1}^{N - 1} {(x_{{nt}_{j + 1}} - x_{{nt}_{j}} - f (t_{j}, x_{{it}_{j}}; μ^{*}) (t_{j + 1} - t_{j}))}^{2} . \end{matrix}$ Since C_t is an independent increment process, according to Definition 2, the increment C_t - C_s is a normal uncertain variable with an expected value 0 and variance (t - s) ², we have

$\begin{matrix} E [\sum_{j = 1}^{N - 1} {(g (t_{j}, x_{{it}_{j}}; σ) (C_{t_{j + 1}} - C_{t_{j}}))}^{2}] \\ = & \sum_{j = 1}^{N - 1} E [{(g (t_{j}, x_{{it}_{j}}; σ) (C_{t_{j + 1}} - C_{t_{j}}))}^{2}] \\ = & \sum_{j = 1}^{N - 1} g (t_{j}, x_{{it}_{j}}; σ)^{2} (t_{j + 1} - t_{j})^{2} . \end{matrix}$ Hence, the estimate of σ is a solution of the following equation

$\begin{matrix} \sum_{j = 1}^{N - 1} g (t_{j}, x_{{it}_{j}}, σ)^{2} (t_{j + 1} - t_{j})^{2} \\ = & \sum_{j = 1}^{N - 1} {(x_{{nt}_{j + 1}} - x_{{nt}_{j}} - f (t_{j}, x_{{it}_{j}}; μ^{*}) (t_{j + 1} - t_{j}))}^{2} . \end{matrix}$ (6) The theorem is proved.

Remark 1. In order to improve the accuracy of parameter estimation, Theorem refTh1 is based on thought of the classic least squares estimation and the criterion of minimizing the residual sum of squares. Through the above theorem, we can obtain the estimated value of the unknown parameter in the coefficient of n-order uncertain differential equation.

Example 1. Consider the second-order uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = μ_{1} X_{t} + μ_{2} \frac{d X_{t}}{d t} + σ \frac{d C_{t}}{d t}$ with three parameters μ₁, μ₂ and σ > 0 to be estimated. We write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} . \end{matrix}$ Then we have ${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ d X_{2 t} = μ_{1} X_{1 t} d t + μ_{2} X_{2 t} d t + σ d C_{t} . \end{matrix}$ Assume that x_1t₁, ⋯ , x_{1t_N}, x_2t₁, ⋯ , x_{2t_N} are observations of the second-order uncertain differential equation at the times t₁, ⋯ , t_N with t₁ < ⋯ < t_N. According to the Equation (refe5), the estimates μ₁ and μ₂ solve the following optimization problem

$\begin{matrix} min_{μ_{1}, μ_{2}} \sum_{j = 1}^{N - 1} & (x_{2 t_{j + 1}} - x_{2 t_{j}} \\ {- μ_{1} x_{1 t_{j}} (t_{j + 1} - t_{j}) - μ_{2} x_{2 t_{j}} (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$

we get that the estimate of parameters μ₁ and μ₂ are $\begin{matrix} [\begin{matrix} μ_{1}^{*} \\ μ_{2}^{*} \end{matrix}] \\ = & {[\begin{matrix} \sum_{j = 1}^{N - 1} {(x_{1 t_{j}} h)}^{2} & \sum_{j = 1}^{N - 1} x_{1 t_{j}} x_{2 t_{j}} h^{2} \\ \sum_{j = 1}^{N - 1} x_{1 t_{j}} x_{2 t_{j}} h^{2} & \sum_{j = 1}^{N - 1} {(x_{2 t_{j}} h)}^{2} \end{matrix}]}^{- 1} \\ \times & [\begin{matrix} \sum_{j = 1}^{N - 1} (x_{2 t_{j + 1}} - x_{2 t_{j}}) x_{1 t_{j}} h \\ \sum_{j = 1}^{N - 1} (x_{2 t_{j + 1}} - x_{2 t_{j}}) x_{2 t_{j}} h \end{matrix}] \end{matrix}$ where h = t_j+1 - t_j.

Then according to the equation (refequal), estimate of σ is the solution of equation

$\begin{matrix} \sum_{j = 1}^{N - 1} σ^{2} (t_{j + 1} - t_{j})^{2} \\ = & \sum_{j = 1}^{N - 1} (x_{2 t_{j + 1}} - x_{2 t_{j}} - μ_{1}^{*} x_{1 t_{j}} (t_{j + 1} - t_{j}) - μ_{2}^{*} x_{2 t_{j}} (t_{j + 1} - t_{j}))^{2}, \end{matrix}$

which implies

$\begin{matrix} σ^{*} = \\ {(\frac{\sum_{j = 1}^{N - 1} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ_{1}^{*} x_{1 t_{j}} (t_{j + 1} - t_{j}) - μ_{2}^{*} x_{2 t_{j}} (t_{j + 1} - t_{j})}^{2}}{\sum_{j = 1}^{N - 1} (t_{j + 1} - t_{j})^{2}})}^{\frac{1}{2}} . \end{matrix}$

Example 2. Consider the third-order uncertain differential equation $\begin{matrix} \frac{d^{3} X_{t}}{d t^{3}} = & μ_{1} sin (\frac{d^{2} X_{t}}{d t^{2}}) + μ_{2} cos (\frac{d X_{t}}{d t}) \\ + & μ_{3} X_{t}^{2} + σ t \frac{d C_{t}}{d t} \end{matrix}$ with four parameters μ₁, μ₂, μ₃ and σ > 0 to be estimated. We write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} \\ X_{3 t} = \frac{d^{2} X_{t}}{d t^{2}} . \end{matrix}$ Then we have

${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ d X_{2 t} = X_{3 t} d t \\ d X_{3 t} = μ_{1} sin (X_{3 t}) d t + μ_{2} cos (X_{2 t}) d t + μ_{3} X_{1 t}^{2} d t + σ t d C_{t} . \end{matrix}$

Assume that x_1t₁, ⋯ , x_{1t_N}, ⋯ , x_3t₁, ⋯ , x_{3t_N} are observations of the third-order uncertain differential equation at the times t₁, ⋯ , t_N with t₁ < ⋯ < t_N. According to the equation (refe5), the estimates μ₁, μ₂ and μ₃ solve the following optimization problem

$\begin{matrix} min_{μ_{1}, μ_{2}, μ_{3}} \sum_{j = 1}^{N - 1} & (x_{3 t_{j + 1}} - x_{3 t_{j}} - μ_{1} sin (x_{3 t_{j}}) (t_{j + 1} - t_{j}) \\ - μ_{2} cos (x_{2 t_{j}}) (t_{j + 1} - t_{j}) {- μ_{3} x_{1 t_{j}}^{2} (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$

we get that the estimates of parameters μ₁, μ₂ and μ₃ are $\begin{matrix} [\begin{matrix} μ_{1}^{*} \\ μ_{2}^{*} \\ μ_{3}^{*} \end{matrix}] & = [\begin{matrix} \sum_{j = 1}^{N - 1} (sin (x_{3 t_{j}}) (t_{j + 1} - t_{j}))^{2} \\ \sum_{j = 1}^{N - 1} sin (x_{3 t_{j}}) cos (x_{2 t_{j}}) (t_{j + 1} - t_{j})^{2} \\ \sum_{j = 1}^{N - 1} sin (x_{3 t_{j}}) x_{1 t_{j}}^{2} (t_{j + 1} - t_{j})^{2} \end{matrix} \\ \begin{matrix} \sum_{j = 1}^{N - 1} sin (x_{3 t_{j}}) cos (x_{2 t_{j}}) (t_{j + 1} - t_{j})^{2} \\ \sum_{j = 1}^{N - 1} (cos (x_{2 t_{j}}) (t_{j + 1} - t_{j}))^{2} \\ \sum_{j = 1}^{N - 1} cos (x_{2 t_{j}}) x_{1 t_{j}}^{2} (t_{j + 1} - t_{j})^{2} \end{matrix} \\ {\begin{matrix} \sum_{j = 1}^{N - 1} sin (x_{3 t_{j}}) x_{1 t_{j}}^{2} (t_{j + 1} - t_{j})^{2} \\ \sum_{j = 1}^{N - 1} cos (x_{2 t_{j}}) x_{1 t_{j}}^{2} (t_{j + 1} - t_{j})^{2} \\ \sum_{j = 1}^{N - 1} (x_{1 t_{j}}^{2} (t_{j + 1} - t_{j}))^{2} \end{matrix}]}^{- 1} \\ \times & [\begin{matrix} \sum_{j = 1}^{N - 1} (x_{3 t_{j + 1}} - x_{3 t_{j}}) sin (x_{3 t_{j}}) (t_{j + 1} - t_{j}) \\ \sum_{j = 1}^{N - 1} (x_{3 t_{j + 1}} - x_{3 t_{j}}) cos (x_{2 t_{j}}) (t_{j + 1} - t_{j}) \\ \sum_{j = 1}^{N - 1} (x_{3 t_{j + 1}} - x_{3 t_{j}}) x_{1 t_{j}}^{2} (t_{j + 1} - t_{j}) \end{matrix}] . \end{matrix}$ Then according to the equation (refequal), the estimate of σ is the solution of equation

$\begin{matrix} \sum_{j = 1}^{N - 1} σ^{2} t_{j}^{2} (t_{j + 1} - t_{j})^{2} \\ = & \sum_{j = 1}^{N - 1} (x_{3 t_{j + 1}} - x_{3 t_{j}} - μ_{1}^{*} sin (x_{3 t_{j}}) (t_{j + 1} - t_{j}) \\ {- μ_{2}^{*} cos (x_{2 t_{j}}) (t_{j + 1} - t_{j}) - μ_{3}^{*} x_{1 t_{j}}^{2} (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$

which implies

$\begin{matrix} σ^{*} = \\ {(\frac{\sum_{j = 1}^{N - 1} {(x_{3 t_{j + 1}} - x_{3 t_{j}} - μ_{1}^{*} sin (x_{3 t_{j}}) h - μ_{2}^{*} cos (x_{2 t_{j}}) h - μ_{3}^{*} x_{1 t_{j}}^{2} h)}^{2}}{\sum_{j = 1}^{N - 1} t_{j}^{2} h^{2}})}^{\frac{1}{2}} \end{matrix}$

where h = t_j+1 - t_j.

4 Numerical examples

In this section, we present some numerical examples for parameter estimation in high-order uncertain differential equations to illustrate the method.

Example 3. Consider the second-order uncertain differential equation ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} = μ exp (- \frac{d X_{t}}{d t}) + σ exp (- \frac{d X_{t}}{d t}) \cdot \frac{d C_{t}}{d t} \\ X_{t} |_{t = 0} = 0, \frac{d X_{t}}{d t} |_{t = 0} = 0, \end{matrix}$ with two parameters μ and σ > 0 to be estimated. Assume that we have 16 groups of observed data as shown in Table 1.

Table 1
Observed Data in Example 4

i 1 2 3 4 5 6

t _i 0 0.42 0.53 0.66 0.85 1.16

x _{2t_i} 0 0.82 1.23 1.53 1.68 1.91

i 7 8 9 10 11 12

t _i 1.37 1.58 1.99 2.20 2.51 2.68

x _{2t_i} 2.19 2.36 2.46 2.26 1.91 1.88

i 13 14 15 16

t _i 2.97 3.12 3.65 3.99

x _{2t_i} 1.73 1.56 1.44 1.31

i	1	2	3	4	5	6
t _i	0	0.42	0.53	0.66	0.85	1.16
x _{2t_i}	0	0.82	1.23	1.53	1.68	1.91
i	7	8	9	10	11	12
t _i	1.37	1.58	1.99	2.20	2.51	2.68
x _{2t_i}	2.19	2.36	2.46	2.26	1.91	1.88
i	13	14	15	16
t _i	2.97	3.12	3.65	3.99
x _{2t_i}	1.73	1.56	1.44	1.31

We write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} . \end{matrix}$ Then we have ${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ d X_{2 t} = μ exp (- X_{2 t}) d t + σ exp (- X_{2 t}) d C_{t} . \end{matrix}$ According to the equation (refe5), the estimate μ solves the following optimization problem $min_{μ} \sum_{j = 1}^{15} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ exp (- x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2},$ we get that the estimate of parameter μ is $μ^{*} = \frac{\sum_{j = 1}^{15} (x_{2 t_{j + 1}} - x_{2 t_{j}}) exp (- x_{2 t_{j}}) (t_{j + 1} - t_{j})}{\sum_{j = 1}^{15} {(exp (- x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2}},$ which gives $μ^{*} = 1.7046 .$ Then according to the equation (refequal), the estimate of σ is the solution of equation $\begin{matrix} \sum_{j = 1}^{15} σ^{2} exp (- x_{2 t_{j}})^{2} (t_{j + 1} - t_{j})^{2} \\ = & \sum_{j = 1}^{15} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ^{*} exp (- x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$ which implies $\begin{matrix} σ^{*} = \\ {(\frac{\sum_{j = 1}^{15} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ^{*} exp (- x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2}}{\sum_{j = 1}^{15} exp (- x_{2 t_{j}})^{2} (t_{j + 1} - t_{j})^{2}})}^{\frac{1}{2}}, \end{matrix}$ which gives $σ^{*} = 1.9492 .$ Hence, the second-order uncertain differential equation is ${\begin{matrix} \begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} = & 1.7046 exp (- \frac{d X_{t}}{d t}) \\ + & 1.9492 exp (- \frac{d X_{t}}{d t}) \cdot \frac{d C_{t}}{d t} \end{matrix} \\ X_{t} |_{t = 0} = 0, \frac{d X_{t}}{d t} |_{t = 0} = 0 . \end{matrix}$

Example 4. Consider the second-order uncertain differential equation ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} = μ (t - \frac{d X_{t}}{d t}) + σ (t + X_{t}^{2}) \frac{d C_{t}}{d t} \\ X_{t} |_{t = 0} = 1, \frac{d X_{t}}{d t} |_{t = 0} = 1, \end{matrix}$ with two parameters μ and σ > 0 to be estimated. Assume that we have 12 groups of observed data as shown in Table 2.

Table 2

Observed Data in Example 4

i	1	2	3	4	5	6
t _i	0	0.33	0.41	0.66	0.82	0.91
x _{1t_i}	1	1.53	1.73	2.36	2.66	2.19
x _{2t_i}	1	2.71	2.50	2.52	1.88	-5.22
i	7	8	9	10	11	12
t _i	1.13	1.31	1.56	1.71	1.82	2.19
x _{1t_i}	1.88	1.63	1.82	1.55	1.37	1.68
x _{2t_i}	-1.41	-1.39	0.76	-1.80	-1.64	0.84

We write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} . \end{matrix}$ Then we have ${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ d X_{2 t} = μ (t - X_{2 t}) d t + σ (t + X_{1 t}^{2}) d C_{t} . \end{matrix}$ According to the equation (refe5), the estimates μ and σ > 0 solve the following optimization problem $min_{μ} \sum_{j = 1}^{11} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ (t_{j} - x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2},$ we get that the estimate of parameter μ is $μ^{*} = \frac{\sum_{j = 1}^{11} (x_{2 t_{j + 1}} - x_{2 t_{j}}) (t_{j} - x_{2 t_{j}}) (t_{j + 1} - t_{j})}{\sum_{j = 1}^{11} (t_{j} - x_{2 t_{j}})^{2} (t_{j + 1} - t_{j})^{2}},$ which gives $μ^{*} = 2.1133 .$ Then according to the equation (refequal), the estimate of σ is the solution of equation $\begin{matrix} \sum_{j = 1}^{11} σ^{2} (t_{j} + x_{1 t_{j}}^{2})^{2} (t_{j + 1} - t_{j})^{2} \\ = & \sum_{j = 1}^{11} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ^{*} (t_{j} - x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$ which implies $\begin{matrix} σ^{*} = \\ {(\frac{\sum_{j = 1}^{11} {(x_{2 t_{j + 1}} - x_{2 t_{j}} - μ^{*} (t_{j} - x_{2 t_{j}}) (t_{j + 1} - t_{j}))}^{2}}{\sum_{j = 1}^{11} (t_{j} + x_{1 t_{j}}^{2})^{2} (t_{j + 1} - t_{j})^{2}})}^{\frac{1}{2}}, \end{matrix}$ which gives $σ^{*} = 2.7528 .$ Hence, the second-order uncertain differential equation is ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} = 2.1133 (t - \frac{d X_{t}}{d t}) + 2.7528 (t + X_{t}^{2}) \frac{d C_{t}}{d t} \\ X_{t} |_{t = 0} = 1, \frac{d X_{t}}{d t} |_{t = 0} = 1 . \end{matrix}$

5 Application to high-order uncertain spring vibration equations

In this section, applications of the high-order uncertain spring vibration equations verify proposed the least squares estimation method. An high-order uncertain spring vibration equation,

${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t, X_{t}) + σ (t, X_{t}, θ) \frac{d C_{t}}{d t}, \\ qquad t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] \\ X_{t} |_{t = 0} = 0, \frac{d X_{t}}{d t} |_{t = 0} = 0, \end{matrix}$

where δ, ω², θ are three parameters to be estimated. We write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} . \end{matrix}$ Then we have ${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ \begin{matrix} d X_{2 t} = & - 2 δ X_{2 t} d t - ω^{2} X_{1 t} d t \\ + & f (t, X_{1 t}) d t + σ (t, X_{1 t}, θ) d C_{t} . \end{matrix} \end{matrix}$ Assume that x_1t₁, ⋯ , x_{1t_N}, x_2t₁, ⋯ , x_{2t_N} are observations of the high-order uncertain spring vibration equation at the times t₁, ⋯ , t_N with t₁ < ⋯ < t_N. According to the equation (refe5), the estimates δ and ω² solve the following optimization problem

$\begin{matrix} min_{δ^{*}, (ω^{2})^{*}} \sum_{j = 1}^{N - 1} & (x_{2 t_{j + 1}} - x_{2 t_{j}} \\ + 2 δ x_{2 t_{j}} (t_{j + 1} - t_{j}) + ω^{2} x_{1 t_{j}} (t_{j + 1} - t_{j}) \\ {- f (t_{j}, x_{1 t_{j}}) (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$ (7)

we get that the estimates of parameters δ and ω² are $\begin{matrix} [\begin{matrix} δ^{*} \\ (ω^{2})^{*} \end{matrix}] = \\ {[\begin{matrix} 2 \sum_{j = 1}^{N - 1} {(x_{2 t_{j}} h)}^{2} & \sum_{j = 1}^{N - 1} x_{1 t_{j}} x_{2 t_{j}} h^{2} \\ 2 \sum_{j = 1}^{N - 1} x_{1 t_{j}} x_{2 t_{j}} h^{2} & \sum_{j = 1}^{N - 1} (x_{1 t_{j}} h)^{2} \end{matrix}]}^{- 1} \\ \times & [\begin{matrix} - \sum_{j = 1}^{N - 1} (x_{2 t_{j + 1}} - x_{2 t_{j}}) x_{2 t_{j}} h + f (t_{j}, x_{1 t_{j}}) x_{2 t_{j}} h^{2} \\ - \sum_{j = 1}^{N - 1} (x_{2 t_{j + 1}} - x_{2 t_{j}}) x_{1 t_{j}} h + f (t_{j}, x_{1 t_{j}}) x_{1 t_{j}} h^{2} \end{matrix}] \end{matrix}$ where h = t_j+1 - t_j.

Then according to the equation (refequal), the estimate of θ is the solution of equation $\begin{matrix} \sum_{j = 1}^{N - 1} σ (t_{j}, x_{1 t_{j}}, θ)^{2} (t_{j + 1} - t_{j})^{2} \\ = & \sum_{j = 1}^{N - 1} (x_{2 t_{j + 1}} - x_{2 t_{j}} + 2 δ^{*} x_{2 t_{j}} (t_{j + 1} - t_{j}) \\ + (ω^{2})^{*} x_{1 t_{j}} (t_{j + 1} - t_{j}) \\ {- f (t_{j}, x_{1 t_{j}}) (t_{j + 1} - t_{j}))}^{2} . \end{matrix}$ (8)

Example 5. Consider the second-order uncertain spring vibration equation ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = θ \frac{d C_{t}}{d t}, t > 0 \\ X_{t} |_{t = 0} = 0, \frac{d X_{t}}{d t} |_{t = 0} = 0, \end{matrix}$ where δ, ω², θ are three parameters to be estimated. That we have 15 groups of observed data as shown in Table 3.

Table 3

Observed Data in Example 5

i	1	2	3	4	5
t _i	0	0.67	0.93	1.24	2.48
x _{1t_i}	0	5.42	7.32	9.34	13.21
x _{2t_i}	0	8.09	7.31	6.52	3.12
i	6	7	8	9	10
t _i	2.93	3.67	4.95	5.56	6.72
x _{1t_i}	12.69	9.52	0	-5.04	-11.93
x _{2t_i}	-1.16	-4.28	-7.44	-8.26	-5.94
i	11	12	13	14	15
t _i	7.43	8.11	8.52	9.25	9.99
x _{1t_i}	-13.31	-12.24	-10.51	-6.03	0
x _{2t_i}	-1.94	1.57	4.22	6.14	8.15

We write ${\begin{matrix} X_{1 t} = X_{t} \\ X_{2 t} = \frac{d X_{t}}{d t} . \end{matrix}$ Then we have ${\begin{matrix} d X_{1 t} = X_{2 t} d t \\ d X_{2 t} = - 2 δ X_{2 t} d t - ω^{2} X_{1 t} d t + θ d C_{t} . \end{matrix}$ According to the equation (refe7), the estimates δ and ω² solve the following optimization problem

$\begin{matrix} min_{δ^{*}, (ω^{2})^{*}} \sum_{j = 1}^{14} & (x_{2 t_{j + 1}} - x_{2 t_{j}} \\ {+ 2 δ x_{2 t_{j}} (t_{j + 1} - t_{j}) + ω^{2} x_{1 t_{j}} (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$

we get that the estimates of parameters δ and ω² are

$\begin{matrix} [\begin{matrix} δ^{*} \\ (ω^{2})^{*} \end{matrix}] = \\ {[\begin{matrix} 2 \sum_{j = 1}^{14} {(x_{2 t_{j}} (t_{j + 1} - t_{j}))}^{2} & \sum_{j = 1}^{14} x_{1 t_{j}} x_{2 t_{j}} (t_{j + 1} - t_{j})^{2} \\ 2 \sum_{j = 1}^{14} x_{1 t_{j}} x_{2 t_{j}} (t_{j + 1} - t_{j})^{2} & \sum_{j = 1}^{14} {(x_{1 t_{j}} (t_{j + 1} - t_{j}))}^{2} \end{matrix}]}^{- 1} \\ \times & [\begin{matrix} - \sum_{j = 1}^{14} (x_{2 t_{j + 1}} - x_{2 t_{j}}) x_{2 t_{j}} (t_{j + 1} - t_{j}) \\ - \sum_{j = 1}^{14} (x_{2 t_{j + 1}} - x_{2 t_{j}}) x_{1 t_{j}} (t_{j + 1} - t_{j}) \end{matrix}], \end{matrix}$

which gives $δ^{*} = 0.0144, (ω^{2})^{*} = 0.3533 .$ Then according to the equation (refe8), the estimate of θ is the solution of equation

$\begin{matrix} \sum_{j = 1}^{14} & θ^{2} (t_{j + 1} - t_{j})^{2} \\ = \sum_{j = 1}^{14} & (x_{2 t_{j + 1}} - x_{2 t_{j}} \\ {+ 2 δ^{*} x_{2 t_{j}} (t_{j + 1} - t_{j}) + (ω^{2})^{*} x_{1 t_{j}} (t_{j + 1} - t_{j}))}^{2}, \end{matrix}$

which implies $\begin{matrix} θ^{*} = \\ {(\frac{\sum_{j = 1}^{14} (x_{2 t_{j + 1}} - x_{2 t_{j}} + 2 δ^{*} x_{2 t_{j}} h + (ω^{2})^{*} x_{1 t_{j}} h)^{2}}{\sum_{j = 1}^{14} h^{2}})}^{\frac{1}{2}} \end{matrix}$ where h = t_j+1 - t_j, which gives θ^* = 2.9817 .

Hence, the second-order uncertain spring vibration equation is

${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 * 0.0144 \frac{d X_{t}}{d t} + 0.3533 X_{t} = 2.9817 \frac{d C_{t}}{d t}, \\ t > 0 \\ X_{t} |_{t = 0} = 0, \frac{d X_{t}}{d t} |_{t = 0} = 0 . \end{matrix}$

6 Conclusions

In this paper, high-order uncertain differential equations were transformed into system of first-order uncertain differential equations through equivalent substitution. Then based on the least squares estimation method, the parameters in the drift term were obtained by the residual sum of the minimized squares, and the parameters in the diffusion term were obtained by making the noise square sum equal to the residual square sum. Furthermore, some numerical examples were given to illustrate the proposed estimation method of the least squares estimation in high-order uncertain differential equations. Additionally, this method was applied to high-order uncertain spring vibration equation, so as to estimated the unknown parameters in high-order uncertain spring vibration equation.

Footnotes

Acknowledgments

This research was funded by the Natural Science Foundation of Xinjiang (Grant No. 2018D01C036).

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Least squares estimation of high-order uncertain differential equations

Abstract

Keywords

1 Introduction

2 Preliminary

Table 1 Observed Data in Example 4 i 1 2 3 4 5 6 t i 0 0.42 0.53 0.66 0.85 1.16 x 2t i 0 0.82 1.23 1.53 1.68 1.91 i 7 8 9 10 11 12 t i 1.37 1.58 1.99 2.20 2.51 2.68 x 2t i 2.19 2.36 2.46 2.26 1.91 1.88 i 13 14 15 16 t i 2.97 3.12 3.65 3.99 x 2t i 1.73 1.56 1.44 1.31

Footnotes

Acknowledgments

References

Table 1
Observed Data in Example 4

i 1 2 3 4 5 6

t _i 0 0.42 0.53 0.66 0.85 1.16

x _{2t_i} 0 0.82 1.23 1.53 1.68 1.91

i 7 8 9 10 11 12

t _i 1.37 1.58 1.99 2.20 2.51 2.68

x _{2t_i} 2.19 2.36 2.46 2.26 1.91 1.88

i 13 14 15 16

t _i 2.97 3.12 3.65 3.99

x _{2t_i} 1.73 1.56 1.44 1.31