Nonparametric estimation of nonautonomous uncertain differential equations with application to temperature models

Abstract

In recent years, there has been a great development in parameter estimation methods for uncertain differential equations (UDEs). However, the observations we can obtain in real life are limited, in which case the form of function in a UDE is unknown. When dealing with such UDEs, we may use observational data to make nonparametric estimates. There are many nonautonomous systems in real life, and nonautonomous UDEs can simulate some uncertain nonautonomous dynamical systems well. In this paper, a nonparametric estimation method based on the nonautonomous UDEs of the binary Legendre polynomial is proposed. Then, three numerical examples are given to verify the reliability of nonparametric estimation. As an application, a real data example of global average monthly temperatures is used to illustrate the effectiveness of our method.

Keywords

Uncertainty theory uncertain differential equations nonparametric estimation global average monthly temperature

1 Introduction

In practical problems, the existence of uncertainty makes it impossible for us to use general theories to build models. In order to simulate this uncertainty, Liu [1] established uncertainty theory. In 2009, Liu [2] refined it on the basis of normality axiom, duality axiom, subadditivity axiom and product axiom.

In 2008, Liu [3] proposed UDEs driven by Liu processes and applied them to financial markets, and in 2010, Zhu [4] applied UDEs to uncertain optimal control. In 2010, Chen and Liu [5] gave sufficient conditions for the existence and uniqueness of solutions to UDEs, and since in most cases, we cannot get analytical solutions to UDEs, therefore, in 2013, Yao and Chen [6] proposed the Yao-Chen formula to link UDEs with ODEs. On this basis, some scholars have carried out research on numerical solutions of UDEs, in 2013, Yao [7] proposed Euler method, and in 2015, Yang and Shen [8] proposed the Runge-Kutta method.

In the broad application of UDEs, we generally consider how to estimate unknown parameters in UDEs based on observational data. Therefore, how to estimate unknown parameters based on observations of solutions to UDEs is a key problem. To solve this problem, in 2020, Yao and Liu [9] proposed a moment estimation method in the form of a difference based on a UDE, Sheng et al. [10] proposed least squares estimation, and Yang et al. [11] presented the minimum cover estimation. In 2021, Liu [12] proposed a generalized moment estimation method. In addition, in 2022, Liu and Liu [13] proposed a maximum likelihood estimation method and a moment estimation method based on residuals [14]. In addition, Ye and Liu [15] proposed uncertain hypothesis test and applied it to test whether a UDE fits well with the observed data [16].

However, in many cases, the observations we can obtain are limited, in which case the form of the function in UDEs is unknown and cannot be directly estimated using parameters. When dealing with such UDEs, we can use observations to make nonparametric estimates. In 2020, Gu et al. [17] proposed a numerical method for solving optimal control problems via Legendre polynomials, showing that Legendre polynomials have good nonparametric estimation properties for arbitrary approximations of continuous functions. In 2023, He et al. [18] proposed the definition of autonomous UDEs to model uncertain autonomous dynamical systems. A nonparametric estimation method for autonomous UDEs based on Legendre polynomials is proposed. However, there are still nonautonomous uncertain dynamical systems in real life, and in view of this problem, this paper proposes the definition of nonautonomous UDEs and a nonparametric estimation method for such equations based on Legendre polynomials.

The rest of this paper is organized below. In Section 2, we introduce some concepts and theorems about uncertain theory, Weierstrass approximation theorem and binary Legendre polynomials. Then, in Section 3, a nonparametric estimation method for unautonomous UDEs is proposed. In Section 4, three numerical examples are given to illustrate our method. Finally, in Section 5, we use the proposed nonautonomous UDE to simulate the global monthly average temperature change and discuss it by nonparametric estimation method. Finally, some conclusions are made in Section 6.

2 Preliminaries

In this section, we review some concepts and theorems about uncertainty theory, Weierstrass approximation theorem and binary Legendre polynomials.

Definition 1. [2] An uncertain process C_t is called a Liu process if

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

C_t has stationary and independent increments,

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

Theorem 1. [19] Suppose (x₁, x₂, ⋯ , x_p) is a vector of explanatory variables, where x₁, x₂, ⋯ , x_p are uncertain variables with regular uncertainty distributions Φ₁, Φ₂, …, Φ_p, respectively, and y is the response variable with uncertainty distribution Ψ. Suppose the fitted linear regression model is $y = β_{0} + \sum_{j = 1}^{p} β_{j} x_{j} + ɛ,$ the disturbance term ɛ has estimated expected value $\tilde{e}$ and variance ${\tilde{σ}}^{2}$ . Then the forecast value of y $\tilde{y} = β_{0} + \sum_{j = 1}^{p} β_{j} E [x_{j}] + \tilde{e} .$ And the α confidence interval of y is suggested as $[\tilde{y} - b^{*}, \tilde{y} + b^{*}]$ , where $b^{*} = min {b ∣ Ψ (\tilde{y} + b) - Ψ (\tilde{y} - b) \geq α}$ .

The Legendre polynomials l_n (x) are the polynomial solutions of Legendre’s differential equation $\begin{matrix} \frac{d}{dx} ((1 - x^{2}) \frac{d}{dx} l_{n} (x)) + n (n + 1) l_{n} (x) = 0, \\ n = 0, 1, 2, \dots . \end{matrix}$ Then, the binary Legendre polynomials l_i (x, y) are the product of two Legendre polynomials, namely, $\begin{matrix} l_{i} (x, y) = l_{p, q} (x, y) = l_{p} (x) \times l_{q} (y) . \end{matrix}$

Theorem 2. [20] The expression for binary Legendre polynomials is provided by $\begin{matrix} l_{i} (x, y) = & \frac{(\sum_{m = 0}^{\frac{p}{2}} {(- 1)}^{m} C_{p}^{m} C_{2 (p - m)}^{p} x^{p - 2 m})}{2^{p + q}} \times \\ \frac{(\sum_{m = 0}^{\frac{q}{2}} {(- 1)}^{m} C_{q}^{m} C_{2 (q - m)}^{q} y^{q - 2 m})}{2^{p + q}}, \\ i = 1, 2, \dots . \end{matrix}$ (1)

Based on Equation (1), we have $\begin{matrix} l_{1} (x, y) = l_{0} (x) \times l_{0} (y) = 1, \\ l_{2} (x, y) = l_{1} (x) \times l_{0} (y) = x, \\ l_{3} (x, y) = l_{0} (x) \times l_{1} (y) = y, \\ l_{4} (x, y) = l_{2} (x) \times l_{0} (y) = \frac{3 x^{2} - 1}{2}, \\ \dots \end{matrix}$

Theorem 3. [21] Let f (x, y) be a continuous function on the interval [a, b]. Then there exists a sequence of polynomials p_n (x, y) that converges uniformly to f (x, y) on the interval [a, b]. That is, for any given ɛ > 0, there exists a sequence of polynomials p_n (x, y) such that for any x ∈ [a, b] , y ∈ [a, b], $| f (x, y) - p_{n} (x, y) | < ɛ .$ (2)

Theorem 3 suggests that a continuous function f (x, t) (x ∈ [-1, 1] , t ∈ [-1, 1]) may be expanded as $f (x, t) = \sum_{i = 1}^{\infty} c_{i} l_{i} (x, t),$ (3) where $c_{i} \in ℝ$ . In calculation, f (x, t) can be approximated using a sequence of binary Legendre polynomials. $f (x, t) \approx \sum_{i = 1}^{K} c_{i} l_{i} (x, t),$ (4) where K is a properly large positive integer, $c_{i} \in ℝ$ .

3 Nonparametric estimation

In this section, with the help of the approximation ability of binary Legendre polynomials, we give the principle of nonparametric estimation of nonautonomous UDEs based on some known observations.

For some uncertain dynamic systems, changes in their current state are not only affected by their own state, but also related to time. An uncertain dynamical system with this phenomenon is called a nonautonomous uncertain dynamical system. Consider the following nonautonomous UDE ${dX}_{t} = f (X_{t}, t) dt + σ g (X_{t}, t) {dC}_{t},$ (5) where f (X_t, t) and σ are unknown. g (X_t, t) is a known continuous function, C_t is a Liu process. According to Equation (4), f (X_t, t) can be approximated by binary Legendre polynomials. Then the UDE (5) can be written as: ${dX}_{t} \approx \sum_{i = 1}^{K} c_{i} l_{i} {(X}_{t}, t) dt + σ g {(X}_{t}, t) {dC}_{t},$ (6) where l_i (X_t, t) are binary Legendre polynomials, c_i, σ are parameters to be estimated.

Write the Equation (6) in difference form: $\begin{matrix} \frac{X_{t_{j + 1}} - X_{t_{j}}}{g (X_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})} \\ = \sum_{i = 1}^{K} \frac{c_{i} l_{i} {(X}_{t_{j}}, t_{j})}{g (X_{t_{j}}, t_{j})} + σ \frac{C_{t_{j + 1}} - C_{t_{j}}}{t_{j + 1} - t_{j}}, \\ j = 1, 2, \dots, n - 1 . \end{matrix}$ (7)

Bring in n observed values x_{t
₁}, x_{t
₂}, . . . , x_{t
_n} for X_t at time t₁, t₂, ⋯ , t_n. Then Equation (7) can be written as $\begin{matrix} \frac{x_{t_{j + 1}} - x_{t_{j}}}{g (x_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})} \\ = \sum_{i = 1}^{K} \frac{c_{i} l_{i} {(x}_{t_{j}} {, t}_{j})}{g (x_{t_{j}}, t_{j})} + σ \frac{C_{t_{j + 1}} - C_{t_{j}}}{t_{j + 1} - t_{j}} . \end{matrix}$ (8) According to the definition of Liu process C_t, $\begin{matrix} \frac{C_{t_{j + 1}} - C_{t_{j}}}{t_{j + 1} - t_{j}} \sim N (0, 1) . \end{matrix}$ Denote $\begin{matrix} ϵ = σ \frac{C_{t_{j + 1}} - C_{t_{j}}}{t_{j + 1} - t_{j}} \sim N (0, σ) \end{matrix}$ which is the noise.

To estimate the unknown coefficients c_i and σ, we introduce the following uncertain regression model $y = c_{K} η^{T} + ϵ,$ (9) where c _K = (c₁, c₂, …, c_K). The left item y represents the response variable, the right item η = (η₁, η₂, ⋯ , η_K) represent explanatory variables, The observed value of η is denoted as η_j , $\begin{matrix} η_{j} = (\frac{l_{1} {(x}_{t_{j}}, t_{j})}{g {(x}_{t_{j}}, t_{j})}, \frac{l_{2} {(x}_{t_{j}}, t_{j})}{g {(x}_{t_{j}}, t_{j})}, \dots, \frac{l_{K} {(x}_{t_{j}}, t_{j})}{g {(x}_{t_{j}}, t_{j})}), \end{matrix}$ and the observed value of y is denoted as y_j, $\begin{matrix} y_{j} = \frac{x_{t_{j + 1}} - x_{t_{j}}}{g {(x}_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})} . \end{matrix}$ Obviously, the uncertain regression model is equivalent to Equation (8). Thus, the estimation of the unknown function f (X_t, t) in Equation (5) is converted into the estimation of unknown parameter c _K in Equation (9). The estimates for c _K and σ are represented by ${\tilde{c}}_{K} = (c_{1}^{*}, c_{2}^{*}, \dots, c_{K}^{*})$ and σ^*.

According to Lio and Liu [22], in uncertain regression analysis, ${\tilde{c}}_{K}$ is the solution of the problem $min_{{\tilde{c}}_{K}} ⋁_{j = 1}^{n - 1} | y_{j} - {\tilde{c}}_{K} η_{j}^{T} |$ (10) and σ^* is the solution of $max_{σ > 0} \frac{\frac{π}{\sqrt{3} σ} exp (\frac{π}{\sqrt{3} σ} ⋁_{j = 1}^{n - 1} | y_{j} - {\tilde{c}}_{K} η_{j}^{T} |)}{{(1 + exp (\frac{π}{\sqrt{3} σ} ⋁_{j = 1}^{n - 1} | y_{j} - {\tilde{c}}_{K} η_{j}^{T} |))}^{2}} .$ (11) According to [13], we have $σ^{*} = \frac{π}{\sqrt{3} λ} ⋁_{j = 1}^{n - 1} | y_{j} - {\tilde{c}}_{K} η_{j}^{T} |,$ (12) where λ is the root of the transcendental equation 1 + z + exp(z) - z exp(z) =0. In numerical solution, λ takes the value of 1.5434. It is important to find the smallest K, which is denoted by K^*.

Theorem 4. When K→ + ∞, we have $\begin{matrix} lim_{K \to + \infty} ⋁_{j = 1}^{n - 1} | y_{j} - c_{K} η_{j}^{T} | = ⋁_{j = 1}^{n - 1} | y_{j} - f (x_{t_{j}}, t_{j}) | . \end{matrix}$

Proof. According to Eq (4), when K→ + ∞, we can get $\begin{matrix} lim_{K \to + \infty} | f (x_{t_{j}}, t_{j}) - c_{K} η_{j}^{T} | = 0 . \end{matrix}$ Since $\begin{matrix} | y_{j} - c_{K} η_{j}^{T} | \leq & | y_{j} - f (x_{t_{j}}, t_{j}) | + \\ | f (x_{t_{j}}, t_{j}) - c_{K} η_{j}^{T} |, \\ | y_{j} - c_{K} η_{j}^{T} | \geq & | y_{j} - f (x_{t_{j}}, t_{j}) | - \\ | f (x_{t_{j}}, t_{j}) - c_{K} η_{j}^{T} |, \end{matrix}$ based on the Squeeze theorem, we have $\begin{matrix} lim_{K \to + \infty} | y_{j} - c_{K} η_{j}^{T} | = | y_{j} - f (x_{t_{j}}, t_{j}) | . \end{matrix}$ Then, we can get $\begin{matrix} lim_{K \to + \infty} ⋁_{j = 1}^{n - 1} | y_{j} - c_{K} η_{j}^{T} | = ⋁_{j = 1}^{n - 1} | y_{j} - f (x_{t_{j}}, t_{j}) | . \end{matrix}$ The proof is completed.

Denote $\begin{matrix} Φ (K) = ⋁_{j = 1}^{n - 1} | \frac{x_{t_{j + 1}} - x_{t_{j}}}{g {(x}_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})} - \\ \frac{{\tilde{c}}_{K} l_{K}^{T} (x_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})}{g {(x}_{t_{j}}, t_{j})} |, K \in ℕ, \end{matrix}$ (13) where $\begin{matrix} {\tilde{c}}_{K} = (c_{1}, c_{2}, \dots, c_{K}), l_{K}^{T} = (l_{1}, l_{2}, \dots, l_{K}) . \end{matrix}$

To avoid excessive calculations, set an acceptable error denoted as Δ. According to problem (10), we may choose the K^* as the smallest K satisfying $\begin{matrix} | Φ (K) - Φ (K - 1) | < Δ . \end{matrix}$

Now we establish the following algorithm to calculate K^* and ${\tilde{c}}_{K^{*}}$ .

Algorithm[H] 1 Estimation for UDE (5)

Require $n > 0, K \in ℕ$

Set K = 0 ; Δ > 0 ; Opt₀ = 0

whileError ≥ Δ do

Opt = 0

Obtain ${\tilde{c}}_{K}$ from (10)

for j = 1 to n - 1 do

$Opt = Opt ⋁ | \frac{x_{t_{j + 1}} - x_{t_{j}} - {\tilde{c}}_{K} l_{K}^{T} (x_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})}{g (x_{t_{j}}, t_{j}) (t_{j + 1} - t_{j})} |$

end for

Error = |Opt - Opt₀|

Opt₀ = Opt

K^* = K

${\tilde{c}}_{K^{*}} = {\tilde{c}}_{K}$

Obtain σ^* from (12)

K = K + 1

end while

4 Numerical examples

In this section, we give three numerical examples to illustrate our algorithm.

Example 1. Consider the following UDE ${dX}_{t} = f (X_{t}, t) dt + σ {dC}_{t} .$ (14) The observation data of X_t and t are shown in Table 1.

Table 1

Observation data for Example 1

j	1	2	3	4	5	6	7	8	9	10	11	12
t _j	0.04	0.08	0.12	0.16	0.20	0.24	0.28	0.32	0.36	0.4	0.44	0.48
x _{t _j}	0.0065	0.0175	0.0110	0.0237	0.0297	0.0244	0.0250	0.0314	0.0517	0.0745	0.0770	0.1032
j	13	14	15	16	17	18	19	20	21	22	23	24
t _j	0.52	0.56	0.6	0.64	0.68	0.72	0.76	0.80	0.84	0.88	0.92	0.96
x _{t _j}	0.1298	0.1448	0.1678	0.1789	0.1982	0.2315	0.2635	0.3072	0.3446	0.3691	0.4192	0.4800

From (4), we can use the following form to approximate Equation (14), ${dX}_{t} = \sum_{i = 1}^{K} c_{i} l_{i} (X_{t}, t)) dt + σ {dC}_{t} .$

According to Algorithm 1, we obtain K^* = 3 and $\begin{matrix} c_{1}^{*} = 0.0198, c_{2}^{*} = 2.2115, \\ c_{3}^{*} = 0.2313, σ^{*} = 0.4266 . \end{matrix}$ Therefore, the estimated UDE of (14) is $\begin{matrix} {dX}_{t} = & (0.0198 l_{1} (X_{t}, t) + 2.2115 l_{2} (X_{t}, t) + \\ 0.2313 l_{3} (X_{t}, t)) dt + 0.4266 {dC}_{t} . \end{matrix}$ (15)

The residuals of UDE (15) are given in Table 2. According to the uncertain hypothesis test proposed by Ye and Liu [16], for a given significance level α (e.g., α = 0.05), if at most two residuals do not belong to [0.025, 0.975], then UDE (15) is a suitable estimate for the observation data in Table 1. Now it is obvious that all residuals are in [0.025, 0.975]. Thus the UDE (15) is a suitable estimate for the observation data in Table 1.

Table 2

Residuals of UDE (15)

j	2	3	4	5	6	7	8	9	10	11	12	13
ɛ_j	0.7262	0.2652	0.7417	0.5424	0.2442	0.3828	0.5210	0.8115	0.8169	0.2873	0.8240	0.7865
j	14	15	16	17	18	19	20	21	22	23	24
ɛ_j	0.4437	0.6104	0.2567	0.4127	0.7158	0.6084	0.7910	0.5533	0.1761	0.7122	0.8219

Example 2. Consider the following UDE ${dX}_{t} = f (X_{t}, t) dt + σ X_{t} t {dC}_{t} .$ (16) The observation data of X_t and t are shown in Table 3.

Table 3

Observation data from Example 2

j	1	2	3	4	5	6	7	8	9	10	11	12
t _j	0.02	0.04	0.06	0.08	0.10	0.12	0.14	0.16	0.18	0.20	0.22	0.24
x _{t _j}	0.0325	0.0538	0.0759	0.0989	0.1226	0.1476	0.1729	0.1997	0.2274	0.2567	0.2870	0.3194
j	13	14	15	16	17	18	19	20	21	22	23	24
t _j	0.26	0.28	0.30	0.32	0.34	0.36	0.38	0.40	0.42	0.44	0.46	0.48
x _{t _j}	0.3531	0.3856	0.4208	0.4543	0.4853	0.5248	0.5636	0.6033	0.6504	0.6942	0.7395	0.7912

Similar to Example 1, the estimated UDE of (16) is obtained by Algorithm 1 as $\begin{matrix} {dX}_{t} = & (7.0578 l_{1} (X_{t}, t) - 9.9240 l_{2} (X_{t}, t) \\ + 12.0546 l_{3} (X_{t}, t) + 2.3653 l_{4} (X_{t}, t) \\ - 6.1890 l_{5} (X_{t}, t) + 9.4718 l_{6} (X_{t}, t)) dt \\ + 1.8264 X_{t} {tdC}_{t} . \end{matrix}$ (17) The residuals of UDE (17) are given in Table 4. Since there is no residual that does not belong to [0.025, 0.975], the UDE (17) is a suitable estimate for the observations in Table 3.

Table 4

Residuals of UDE (17)

j	2	3	4	5	6	7	8	9	10	11	12	13
ɛ_j	0.6145	0.4134	0.4660	0.2698	0.4863	0.2450	0.4762	0.4426	0.5983	0.6034	0.7563	0.7838
j	14	15	16	17	18	19	20	21	22	23	24
ɛ_j	0.5445	0.6907	0.4491	0.2048	0.6220	0.5208	0.5076	0.7555	0.6022	0.6018	0.7386

Example 2. Consider the following UDE ${dX}_{t} = f (X_{t}, t) dt + σ X_{t} t {dC}_{t} .$ (18) The observation of X_t and t are shown in Table 5.

Table 5

Observation data from Example 3

j	1	2	3	4	5	6	7	8	9	10	11	12
t _j	0.02	0.03	0.05	0.06	0.08	0.09	0.11	0.12	0.14	0.15	0.17	0.18
x _{t _j}	0.4962	0.4989	0.3768	0.5237	0.5701	0.3304	0.2587	0.3228	0.6582	0.5650	0.4257	0.6821
j	13	14	15	16	17	18	19	20	21	22	23	24
t _j	0.20	0.21	0.23	0.24	0.26	0.27	0.29	0.30	0.32	0.33	0.35	0.36
x _{t _j}	0.6156	0.6267	0.2906	0.2103	0.4655	0.5017	0.5381	0.5774	0.6155	0.6507	0.6923	0.7367

Similar to Example 1, the estimated UDE of (18) is obtained by $\begin{matrix} {dX}_{t} = & (12.8806 l_{1} (X_{t}, t) + 193.3850 l_{2} (X_{t}, t) \\ + 692.9641 l_{3} (X_{t}, t) + 14.2562 l_{4} (X_{t}, t) \\ + 83.2884 l_{5} (X_{t}, t) + 5.8680 l_{6} (X_{t}, t) \\ + 158.6208 l_{7} (X_{t}, t) - 884.2083 l_{8} (X_{t}, t) \\ + 221.2000 l_{9} (X_{t}, t) \\ + 615.4459 l_{10} (X_{t}, t)) dt \\ + 1.9252 X_{t} t {dC}_{t} . \end{matrix}$ (19)

The residuals of UDE (19) are given in Table 6. Since there is no residual that does not belong to [0.025, 0.975], the UDE (19) is a suitable estimate for the observations in Table 5.

Table 6

Residuals of UDE (19)

j	2	3	4	5	6	7	8	9	10	11	12	13
ɛ_j	0.4833	0.4732	0.4342	0.5469	0.6140	0.3505	0.2841	0.3374	0.6771	0.5749	0.4386	0.6867
j	14	15	16	17	18	19	20	21	22	23	24
ɛ_j	0.6217	0.6286	0.2925	0.2103	0.2700	0.4303	0.8007	0.7932	0.4646	0.2030	0.6371

5 Application

In this section, we use the proposed nonautonomous UDE to simulate the temperature change and discuss it by nonparametric estimation method. The data of global average monthly temperature from January 2019 to January 2023 are shown in Table 7. In 2019, Lenssen et al. [23] outlined a new and improved uncertainty analysis for the Goddard Institute for Space Studies Surface Temperature product version 4 (GISTEMP v4). The data used in this paper is based on GISS Surface Temperature Analysis (GISTEMP v4) and may be found at National Aeronautics and Space Administration Goddard Institute for Space Studies (https://data.giss.nasa.gov/gistemp/). The unit of these data is Celsius. The date of visiting the website is May 3, 2023.

Table 7
Global average monthly temperature from January 2019 to January 2023

Time 2019.1 2019.2 2019.3 2019.4 2019.5 2019.6 2019.7 2019.8 2019.9 2019.10 2019.11 2019.12

Temperature 0.93 0.95 1.17 1.01 0.85 0.91 0.94 0.95 0.93 1.01 0.99 1.09

Time 2020.1 2020.2 2020.3 2020.4 2020.5 2020.6 2020.7 2020.8 2020.9 2020.10 2020.11 2020.12

Temperature 1.17 1.24 1.17 1.13 1.02 0.92 0.90 0.88 0.99 0.89 1.11 0.81

Time 2021.1 2021.2 2021.3 2021.4 2021.5 2021.6 2021.7 2021.8 2021.9 2021.10 2021.11 2021.12

Temperature 0.82 0.64 0.89 0.76 0.78 0.84 0.92 0.82 0.92 1.00 0.94 0.86

Time 2022.1 2022.2 2022.3 2022.4 2022.5 2022.6 2022.7 2022.8 2022.9 2022.10 2022.11 2022.12

Temperature 0.91 0.89 1.05 0.84 0.84 0.92 0.93 0.95 0.90 0.97 0.73 0.80

Time 2023.1

Temperature 0.87

Time	2019.1	2019.2	2019.3	2019.4	2019.5	2019.6	2019.7	2019.8	2019.9	2019.10	2019.11	2019.12
Temperature	0.93	0.95	1.17	1.01	0.85	0.91	0.94	0.95	0.93	1.01	0.99	1.09
Time	2020.1	2020.2	2020.3	2020.4	2020.5	2020.6	2020.7	2020.8	2020.9	2020.10	2020.11	2020.12
Temperature	1.17	1.24	1.17	1.13	1.02	0.92	0.90	0.88	0.99	0.89	1.11	0.81
Time	2021.1	2021.2	2021.3	2021.4	2021.5	2021.6	2021.7	2021.8	2021.9	2021.10	2021.11	2021.12
Temperature	0.82	0.64	0.89	0.76	0.78	0.84	0.92	0.82	0.92	1.00	0.94	0.86
Time	2022.1	2022.2	2022.3	2022.4	2022.5	2022.6	2022.7	2022.8	2022.9	2022.10	2022.11	2022.12
Temperature	0.91	0.89	1.05	0.84	0.84	0.92	0.93	0.95	0.90	0.97	0.73	0.80
Time	2023.1
Temperature	0.87

We hope to present a UDE to fit the data of global average monthly temperature from January 2019 to December 2022. Then use it to predict the temperature for January 2023. Finally, compare the predicted temperature with the actual temperature to verify whether the predicted result is reasonable. Since binary Legendre polynomials can only approximate a continuous function on interval [-1, 1], we suppose that max m represents the highest global average monthly temperature from January 2019 to December 2022, which is the global average monthly temperature in February 2020, (i.e., 1.24). Multiply all temperatures by $\frac{0.68}{max m}$ , (i.e., 0.5484), denoted as x_{t
₁}, x_{t
₂}, ⋯ , x_{t
₄₈}. Denote the observed time is t₁ = 1, t₂ = 2, ⋯ , t₄₈ = 48. Multiply all time by 0.02. Then the processed data are shown in Table 8.

Table 8

Processed data

j	1	2	3	4	5	6	7	8	9	10	11	12
t _j	0.02	0.04	0.06	0.08	0.10	0.12	0.14	0.16	0.18	0.20	0.22	0.24
x _{t _j}	0.5100	0.5210	0.6416	0.5539	0.4661	0.4990	0.5155	0.5210	0.5100	0.5539	0.5429	0.5977
j	13	14	15	16	17	18	19	20	21	22	23	24
t _j	0.26	0.28	0.30	0.32	0.34	0.36	0.38	0.40	0.42	0.44	0.46	0.48
x _{t _j}	0.6416	0.6800	0.6416	0.6197	0.5594	0.5045	0.4935	0.4826	0.5429	0.4881	0.6087	0.4442
j	25	26	27	28	29	30	31	32	33	34	35	36
t _j	0.50	0.52	0.54	0.56	0.58	0.60	0.62	0.64	0.66	0.68	0.70	0.72
x _{t _j}	0.4497	0.3510	0.4881	0.4168	0.4277	0.4606	0.5045	0.4497	0.5045	0.5484	0.5155	0.4716
j	37	38	39	40	41	42	43	44	45	46	47	48
t _j	0.74	0.76	0.78	0.80	0.82	0.84	0.86	0.88	0.90	0.92	0.94	0.96
x _{t _j}	0.4990	0.4881	0.5758	0.4606	0.4606	0.5045	0.5100	0.5210	0.4935	0.5319	0.4003	0.4387

The temperature change is difficult to measure accurately and is affected by many factors, so we can consider the global monthly average temperature as an uncertain variable, denoted by X_t. First, we adopt the nonparametric estimation method proposed in reference [18] to deal with the data in Table 8. When K gradually increases and the degree of Legendre polynomial used for approximation is already very high, the termination condition is still not reached. That is, we cannot obtain an estimated uncertain differential equations according to the existing nonparametric estimation method. Therefore, the existing method is not suitable for nonautonomous systems. Considering that the temperature change is not only related to the temperature of the previous moment, but also to time, we construct the following global average monthly temperature model based on the processed data in Table 8 by ${dX}_{t} = f (X_{t}, t) dt + σ {dC}_{t},$ (20) where f (X_t, t) and σ are unknown.

From (4), we can use the following form to approximate equation (20), ${dX}_{t} = \sum_{i = 1}^{K} c_{i} l_{i} (X_{t}, t) dt + σ {dC}_{t} .$ (21)

By Algorithm 1, we obtain K^* = 5 and $\begin{matrix} c_{1}^{*} = 10.6057, c_{2}^{*} = - 14.8715, c_{3}^{*} = 16.4626, \\ c_{4}^{*} = 8.2796, c_{5}^{*} = - 42.7276, σ^{*} = 6.8747 . \end{matrix}$ Therefore, the estimated UDE of (20) is $\begin{matrix} {dX}_{t} = & (10.6057 l_{1} (X_{t}, t) - 14.8715 l_{2} (X_{t}, t) \\ + 16.4626 l_{3} (X_{t}, t) + 8.2796 l_{4} (X_{t}, t) \\ - 42.7276 l_{5} (X_{t}, t)) dt + 6.8747 {dC}_{t} . \end{matrix}$ (22) The residuals of UDE (22) are given in Table 9. Since there is no residual that does not belong to [0.025, 0.975], the UDE (22) is a suitable estimate for the observations in Table 8. Namely, the estimated global monthly average temperature model (22) obtained through nonparametric estimation fits observations well.

Table 9

Residuals of UDE (21)

j	2	3	4	5	6	7	8	9	10	11	12	13
ɛ_j	0.4051	0.7505	0.1792	0.1760	0.4842	0.4519	0.4314	0.3891	0.5683	0.4249	0.6394	0.6507
j	14	15	16	17	18	19	20	21	22	23	24	25
ɛ_j	0.6727	0.4677	0.5101	0.3816	0.3560	0.4449	0.4382	0.6590	0.3732	0.8220	0.1761	0.4455
j	26	27	28	29	30	31	32	33	34	35	36	37
ɛ_j	0.1761	0.7057	0.2918	0.4265	0.5211	0.6211	0.3892	0.6419	0.7121	0.5687	0.4737	0.6173
j	38	39	40	41	42	43	44	45	46	47	48
ɛ_j	0.5598	0.8125	0.4068	0.5215	0.6641	0.6549	0.6892	0.6067	0.7383	0.3183	0.5014

According to Theorem 1, we can get the forecast value of X_{t
_j+1} is $\begin{matrix} {\hat{X}}_{t_{j + 1}} = x_{t_{j}} + (\sum_{i = 1}^{K^{*}} c_{i}^{*} l_{i} (x_{t_{j}}, t_{j})) (t_{j + 1} - t_{j}), \end{matrix}$ (23) and the γ-confidence (e.g., γ = 0.95) interval of X_{t
_j+1} is $\begin{matrix} [{\hat{X}}_{t_{j + 1}} - (t_{j + 1} - t_{j}) \frac{σ^{*} \sqrt{3}}{π} \ln \frac{1 + γ}{1 - γ}, \\ {\hat{X}}_{t_{j + 1}} + (t_{j + 1} - t_{j}) \frac{σ^{*} \sqrt{3}}{π} \ln \frac{1 + γ}{1 - γ}] . \end{matrix}$ (24)

With Equation (23), the forecast value of X_{t
₄₉} is 0.4415. Then with (24), the 0.95-confidence interval of X_{t
₄₉} is [0.1638, 0.7192]. Restoring the result to original data, the forecast value of X_{t
₄₉} is 0.8051 and the 0.95-confidence interval of X_{t
₄₉} is [0.2987, 1.3115]. It is clearly that the actual value in January 2023 is 0.87 which is close to forecast value of 0.8051 and is in interval [0.2987, 1.3115]. Therefore, we believe that model (22) has a good predictive effect.

6 Conclusion

In this paper, we generalized nonparametric estimation method to nonautonomous UDEs. An algorithm was presented to do the estimation. Then we gave three numerical examples explain our algorithm. The estimated UDE was proved to be a good estimate of the observed data through residuals and uncertain hypothesis test. Finally, we applied nonparametric estimates to global temperature issues. Based on data of global monthly average temperature from January 2019 to December 2022, we acquired a model based on nonautonomous UDEs using nonparametric estimation methods. Finally, based on data of January 2023, we found the predicted value is close to the actual value, and the actual value is in the confidence interval, namely, the model was proved to be suitable for observational data.

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