Parameter estimation in multifactor uncertain population model

Abstract

This paper extends the traditional multifactor population model from a probabilistic background to an uncertain background, allowing us to better account for the fact that the birth and death rates are sometimes affected by different, independent uncertain noises. The theory of multifactor uncertain differential equation is applied to analyze the novel model. First, a multifactor uncertain population model is developed and the solution of the model is obtained. Secondly, the stability in measure and stability in distribution of the multifactor uncertain population model are studied respectively, and the relationship between the two kinds of stability is further discussed. Then, a method of moment estimation based on residuals is proposed for the unknown parameters in the multifactor uncertain population model. Finally, an example is given to illustrate the validity and rationality of the parameter estimation method.

Keywords

Parameter estimation residual multifactor differential equation stability

1 Introduction

Population model has important application value in the study of population dynamic behavior. Their origin can be traced back to the Malthus growth model [1], which opened the door of deterministic population model described by ordinary differential equation. However, the population system is inevitably affected by all kinds of noises. Then some researchers used random noise described by Wiener process to establish random population models [2]. Next some scholars extended the random model to the multifactor random population model. For example, Hakoyama et al. [3] studied the influence of environmental stochasticity and demographic stochasticity on population extinction. Subsequently, Hill et al. [4] extended the multifactor random population model to the case of two patches.

Although random population models have been widely used in practice, the premise of applying random models is that the probability distribution function of the event is close enough to the actual frequency of the event, that is, the large number theorem must be satisfied. In practice, for economic and technical reasons, sometimes there is no sample data, so only domain experts can be relied on to estimate the probability of an event and give its reliability. Meanwhile, due to the properties of the Wiener process, random population models are inconsistent in describing many time-varying population systems, which was discussed in detail in Zhang and Yang [5]. Therefore, it is not appropriate to use stochastic differential equation to describe the dynamic behavior of population. In order to better describe the various noises in the process of biological population dynamics, new methods need to be considered.

In 2007, Liu proposed uncertainty theory [6] to solve the paradoxes in random models. Uncertainty theory is a branch of mathematics on which reliability models can be built. In order to describe the uncertainty phenomenon changing with time, Liu [7] introduced the uncertain process. Liu process is not only an uncertain process, but also a stationary and independent incremental process with continuous Lipchitz sample path. Unlike random differential equations driven by random processes, uncertain differential equations are driven by uncertain processes. In recent years, uncertain differential equation has become the main tool to deal with dynamic uncertain systems and has achieved fruitful results. Yao [8] studied the first hitting time and extreme value of solution for uncertain differential equations. Barbacioru [9] proposed uncertain delay differential equation. Subsequently, Ge and Zhu [10] used the Banach fixed point theorem to discuss the existence and uniqueness theorem of the solution for uncertain delay differential equations. The backward uncertain differential equation was proposed by Ge and Zhu [11] and it is to study how to set the path for the system to achieve the expected goal under uncertain environment. Yao [12] considered uncertain differential equations with jumps. Zhu [13] explored uncertain fractional differential equations and applied them to the study of interest rates. Li et al. [14] established a class of multifactor uncertain differential equations. Yao [15] discussed higher-order uncertain differential equation. Uncertainty theory was widely used in uncertain planning [16], uncertain reliability analysis [17], uncertain finance [18] and other fields.

Since the factors affecting the population dynamic system in practice usually do not exist separately, the birth rate and death rate are sometimes interfered by different and independent uncertain noises at the same time. Therefore, this paper extends the traditional multifactor population model from a probabilistic background to an uncertain background, and proposes a multifactor uncertain population model which is described by a multifactor uncertain differential equation and in which the birth rate and death rate are disturbed by two independent uncertain variables. The theory of multifactor uncertain differential equation is applied to analyze the novel model.

The rest is arranged as follows. In Section 2, a multifactor uncertain population model is developed, and the solution and α-path of the solution are obtained. In Section 3, it is proved that when t falls on a finite interval, the stability in distribution can be derived from stability in measure for multifactor uncertain population model. In section 4, the moment estimation method based on residuals is used to estimate the unknown parameters in the multifactor uncertain population model. Section 5 shows our parameter estimation method in detail by applying it to real data sets. Finally, a brief summary is given in Section 6.

2 Multifactor uncertain population model

Suppose b and d are the birth and death rates of the population, and they are constants. A simple model of population growth is the Malthus population model

$\frac{{dN}_{t}}{dt} = {bN}_{t} - {dN}_{t},$ (2.1) where N_t is the population at time t. We write b - d as r, where r is defined as the growth rate, and Equation (2.1) simplifies to

$\frac{{dN}_{t}}{dt} = {rN}_{t} .$ (2.2)

Intraspecial competition is a common relationship in biology. Considering density limitation and crowding effect, a density constraint factor $(1 - \frac{N_{t}}{K})$ is added to model (2.2), where K is carrying capacity, and we obtain the Logistic population model

$\frac{{dN}_{t}}{dt} = {rN}_{t} (1 - \frac{N_{t}}{K}),$ (2.3) which was first proposed by Vethurst [19] in 1838.

Next, some scholars have extended the random model to the multifactor random population model. For example, considering the influence of environmental stochasticity and demographic stochasticity on population extinction, Hakoyama et al. [3] studied the multifactor stochastic differential equation

$\begin{matrix} {dN}_{t} = [{rN}_{t} (1 - \frac{N_{t}}{K})] dt + σ_{e} N_{t} {dW}_{e} (t) + \\ \sqrt{N_{t}} {dW}_{d} (t) \end{matrix}$ (2.4) where W_e (t) and W_d (t) are independent white noises for environmental and demographic stochasticities, and σ_e is the intensity of the environmental fluctuations.

Subsequently, Hill et al. [4] extended model (2.4) to the case of two patches and obtained the multifactor random population model is as follows

$\begin{matrix} {dN}_{it} = [{rN}_{it} (1 - \frac{N_{it}}{K_{i}})] dt + D (N_{jt} - N_{it}) + \\ σ_{e} N_{it} {dW}_{e} (t) + σ_{d} \sqrt{N_{it}} {dW}_{d} (t) \end{matrix}$ (2.5) where N_it represent the population size in patch i (i = 1, 2) at time t, and j = 1, 2, i ≠ j, D is the diffusion rate.

These multifactor stochastic differential equation models describe the dynamic behavior of populations disturbed simultaneously by multiple random noises. But in the real world, uncertainty noise is everywhere. Uncertain factors influencing the population dynamic system are usually not alone, and sometimes the birth rate and death rate are affected simultaneously by different and independent uncertain factors. For example, the increase in the death rate due to an unexpected frost or a sudden infectious disease is accompanied by the increase in the birth rate as a result of the implementation of government policies to encourage childbearing.

In this case, it is more accurate to add two independent perturbations to the birth rate and death rate respectively than to add only an uncertain perturbation to the population growth rate. So we add two independent uncertain noises to the birth rate and death rate respectively as follows

$b \to b + σ_{1} \frac{{dC}_{1 t}}{dt}, d \to d + σ_{2} \frac{{dC}_{2 t}}{dt},$ (2.6) where C_1t, C_2t are two independent Liu processes, and σ₁, σ₂ are the intensities of two uncertain noises, respectively. Thus, we establish the multifactor uncertain population model as follows

${dN}_{t} = ({bN}_{t} dt + σ_{1} N_{t} {dC}_{1 t}) - ({dN}_{t} dt + σ_{2} N_{t} {dC}_{2 t}),$ (2.7) for simplicity,

${dN}_{t} = {rN}_{t} dt + σ_{1} N_{t} {dC}_{1 t} - σ_{2} N_{t} {dC}_{2 t} .$ (2.8)

The novel multifactor uncertain population model (2.8) is characterized by a multifactor uncertain differential equation, which extends the traditional multifactor population model from a probability background to an uncertain one, and explains that sometimes the birth and death rates are affected by different and independent uncertain noises. It provides a new way to study the dynamic behavior of populations, and offers a new idea for the exploration of population models in the future.

Next, we will further discuss the distribution function of the solution for the multifactor uncertain population model.

Theorem 2.1. Given the initial population size N₀, the multifactor uncertain population model (2.8) has a solution

$N_{t} = N_{0} exp (rt + σ_{1} C_{1 t} - σ_{2} C_{2 t}) .$ (2.9)

Proof. At first, Equation (2.8) is equivalent to $\frac{{dN}_{t}}{N_{t}} = rdt + σ_{1} {dC}_{1 t} - σ_{2} {dC}_{2 t} .$

According to the fundamental theorem of uncertain calculus, we have $d ln N_{t} = rdt + σ_{1} {dC}_{1 t} - σ_{2} {dC}_{2 t} .$

Then $\int_{0}^{t} \frac{{dN}_{s}}{N_{s}} = rt + \int_{0}^{t} σ_{1} {dC}_{1 s} - \int_{0}^{t} σ_{2} {dC}_{2 s} .$

Therefore, the multifactor uncertain population model (2.8) has a solution (2.9).

Theorem 2.2. The multifactor uncertain population model (2.8) with the initial population size N₀ has an α-path (0 < α < 1)

$N_{t}^{α} = N_{0} exp (rt + | σ_{1} | Φ^{- 1} (α) t + | σ_{2} | Φ^{- 1} (α) t),$ (2.10) where $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}$ is the inverse standard normal uncertain distribution.

Proof. Because $N_{t}^{α}$ is the solution for the ordinary differential equation

$\begin{matrix} {dN}_{t}^{α} = {rN}_{t}^{α} dt + | σ_{1} N_{t}^{α} | Φ^{- 1} (α) dt + \\ | σ_{2} N_{t}^{α} | Φ^{- 1} (α) dt, \end{matrix}$ (2.11) so the multifactor uncertain population model (2.8) has a α-path (2.10) according to the definition of α-path.

Theorem 2.3. Let N_t and $N_{t}^{α}$ be the solution and α-path of the multifactor uncertain population model (2.8), respectively. Then N_t has an inverse uncertainty distribution $Ψ_{t}^{- 1} (α) = N_{t}^{α} .$

Proof. For each s, we have ${N_{t} ⩽ N_{t}^{α}} \supset {N_{s} ⩽ N_{s}^{α}, \forall s},$ ${N_{t} > N_{t}^{α}} \supset {N_{s} > N_{s}^{α}, \forall s} .$

According to the monotonicity of the uncertain measure, there is $M {N_{t} ⩽ N_{t}^{α}} ⩾ M {N_{s} ⩽ N_{s}^{α}, \forall s} = α,$ $M {N_{t} > N_{t}^{α}} ⩾ M {N_{s} > N_{s}^{α}, \forall s} = 1 - α .$

From the duality axiom, we get $M {N_{t} ⩽ N_{t}^{α}} + M {N_{s} > N_{s}^{α}} = 1$

It follows that $\forall s, M {N_{t} ⩽ N_{t}^{α}} = α .$

Thus $Ψ_{t}^{- 1} (α) = N_{t}^{α}$ is the inverse uncertainty distribution of N_t.

3 Stability analysis

The stability of multifactor uncertain population model is important in practical application. If the model is stable, the uncertain dynamic population system is not sensitive to small fluctuations in the initial value and will not cause significant effects. If the model is unstable, a little change in the initial value will result in a significant difference in the behavior of the solution over time.

Zhang [20] focused on the stability in measure and the stability in mean of multifactor uncertain differential equation. Ma [21] discussed the concept of stability in p-th moment for multifactor uncertain differential equation and documents its application. Sheng [22] presented a concept of the almost sure stability of multifactor uncertain differential equation. Ma [23] introduced the stability in distribution for multifactor uncertain differential equation and proved a sufficient condition for a multifactor uncertain differential equation being stable in distribution. This section discusses the stability in measure and the stability in distribution for multifactor uncertain population model, and further explores the relationship between the two kinds of stability.

Zhang [20] gived the definition of stability in measure for multifactor uncertain differential equations when t ⩾ 0. By the definition, it can be proved that the multifactor uncertain population model is unstable. In practice, however, this model is usually used when t is a finite number. Therefore, the definition of stability in measure for multifactor uncertain differential equation when t is on a finite interval is given below.

Definition 3.1. Suppose T is a finite positive number, and t ∈ [0, T]. Let X_t and Y_t be solutions of multifactor uncertain differential equation

${dN}_{t} = f (t, N_{t}) dt + \sum_{i = 1}^{n} g_{i} (t, N_{t}) {dC}_{it},$ (3.1) with initial values X₀ and Y₀, respectively. Then we say that the multifactor uncertain differential equation is stable in measure if for each given number ɛ > 0,

$lim_{| X_{0} - Y_{0} | \to 0} M {sup_{0 ⩽ t ⩽ T} | X_{t} - Y_{t} | ⩽ ɛ} = 1 .$ (3.2)

Theorem 3.1. Suppose T is a finite positive number, and t ∈ [0, T]. Let X_t and Y_t be solutions of multifactor uncertain population model with initial values X₀ and Y₀, respectively. Then the multifactor uncertain population model is stable in measure.

Proof. The solutions with initial values X₀ and Y₀ are $X_{t} = X_{0} exp (rt + σ_{1} C_{1 t} - σ_{2} C_{2 t}),$ and $Y_{t} = Y_{0} exp (rt + σ_{1} C_{1 t} - σ_{2} C_{2 t}),$ respectively. Since

$\begin{matrix} | X_{t} - Y_{t} | \\ = | X_{0} - Y_{0} | exp (rt + σ_{1} C_{1 t} - σ_{2} C_{2 t}) \to 0, \end{matrix}$ (3.3) when |X₀ - Y₀| → 0 and 0 ⩽ t ⩽ T, we can get $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{0 ⩽ t ⩽ T} | X_{t} - Y_{t} | ⩽ ɛ} = 1 .$

According to Definition 3.1, the multifactor uncertain population model is stable in measure.

Ma [23] gave the definition of stability in distribution for multifactor uncertain differential equation when t ⩾ 0. According to the definition, it can be proved that the multifactor uncertain population model is unstable in distribution. Considering that the model is usually used when t is a finite number, the following is the definition of stability in distribution for multifactor uncertain differential equations when t is on a finite interval.

Definition 3.2. Let T be a finite positive number, and t ∈ [0, T]. Suppose X_t and Y_t are two solutions of multifactor uncertain differential equation ${dN}_{t} = f (t, N_{t}) dt + \sum_{i = 1}^{n} g_{i} (t, N_{t}) {dC}_{it},$ with different initial values X₀ and Y₀, respectively. Assume the uncertainty distributions of X_t and Y_t are Φ_t (x) and Ψ_t (x), respectively. Then we say that the multifactor uncertain differential equation is stable in distribution if

$lim_{| X_{0} - Y_{0} | \to 0} | Φ_{t} (x) - Ψ_{t} (x) | = 0 \forall t \in [0, T],$ (3.4) for all x at which Φ and Ψ are continuous.

Theorem 3.2. Let T be a finite positive number, and t ∈ [0, T]. Suppose X_t and Y_t are solutions of multifactor uncertain population model with initial values X₀ and Y₀, respectively. Then multifactor uncertain population model is stable in distribution.

Proof. The solutions with initial values X₀ and Y₀ are $X_{t} = X_{0} exp (rt + σ_{1} C_{1 t} - σ_{2} C_{2 t})$

and $Y_{t} = Y_{0} exp (rt + σ_{1} C_{1 t} - σ_{2} C_{2 t}),$

whose inverse uncertainty distributions are

$\begin{matrix} Φ_{t}^{- 1} (α) = X_{0} \cdot exp (rt + | σ_{1} | t \frac{\sqrt{3}}{π} ln \frac{α}{1 - α} + \\ | σ_{2} | t \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}), \end{matrix}$ (3.5) and

$\begin{matrix} Ψ_{t}^{- 1} (α) = Y_{0} \cdot exp (rt + | σ_{1} | t \frac{\sqrt{3}}{π} ln \frac{α}{1 - α} + \\ | σ_{2} | t \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}), \end{matrix}$ (3.6) respectively. Since

$\begin{matrix} | Φ_{t}^{- 1} (α) - Ψ_{t}^{- 1} (α) | = exp (rt + | σ_{1} | t \frac{\sqrt{3}}{π} ln \\ \frac{α}{1 - α} + | σ_{2} | t \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}) | X_{0} - Y_{0} | . \end{matrix}$ (3.7)

When |X₀ - Y₀| → 0 and 0 ⩽ t ⩽ T, there is $lim_{| X_{0} - Y_{0} | \to 0} | Φ_{t} (x) - Ψ_{t} (x) | = 0 \forall t \in [0, T] .$

According to Definition 3.2, the multifactor uncertain population model is stable in distribution.

Theorem 3.3. Let T be a finite positive number, and t ∈ [0, T]. If the multifactor uncertain population model is stable in measure, then we say that it is stable in distribution.

Proof. By Definition 3.1, for solutions X_t and Y_t with initial values X₀ and Y₀, respectively, we obtain

$lim_{| X_{0} - Y_{0} | \to 0} M {sup_{0 ⩽ t ⩽ T} | X_{t} - Y_{t} | ⩽ ɛ} = 1,$ (3.8)

for each given real number ɛ > 0. Suppose x is any given real number, for any y > x. Then $\begin{matrix} {X_{t} ⩽ x} = {X_{t} ⩽ x, Y_{t} ⩽ y} \cup {X_{t} ⩽ x, Y_{t} > y} \\ \subset {Y_{t} ⩽ y} \cup {| X_{t} - Y_{t} | ⩾ y - x} \\ \subset {Y_{t} ⩽ y} \cup {sup_{0 < t < T} | X_{t} - Y_{t} | > y - x} . \end{matrix}$

According to the monotonicity theorem and subadditivity axiom, it can be obtained $Φ_{t} (x) - Ψ_{t} (x) ⩽ M {sup_{0 < t < T} | X_{t} - Y_{t} | ⩾ y - x} .$

Since $M {sup_{0 < t < T} | X_{t} - Y_{t} | ⩾ y - x} \to 0$ as $| X_{0} - Y_{0} | \to 0,$ for any y > x, we get $lim_{| X_{0} - Y_{0} | \to 0} Φ_{t} (x) - Ψ_{t} (x) ⩽ 0 .$

Let y → x⁺, we obtain

$lim_{| X_{0} - Y_{0} | \to 0} Φ_{t} (x) - Ψ_{t} (x) ⩽ 0 .$ (3.9)

For any z < x, we get $\begin{matrix} {Y_{t} ⩽ z} = {X_{t} ⩽ x, Y_{t} ⩽ z} \cup {X_{t} > x, Y_{t} ⩽ z} \\ \subset {X_{t} ⩽ x} \cup {sup_{0 < t < T} | X_{t} - Y_{t} | ⩾ x - z} . \end{matrix}$

From the monotonicity theorem and subadditivity axiom, we can get $Ψ_{t} (z) - Φ_{t} (x) ⩽ M {sup_{0 < t < T} | X_{t} - Y_{t} | ⩾ x - z} .$

Since $M {sup_{0 < t < T} | X_{t} - Y_{t} | ⩾ x - z} \to 0$

as $| X_{0} - Y_{0} | \to 0,$

we have $lim_{| X_{0} - Y_{0} | \to 0} Ψ_{t} (x) - Φ_{t} (x) ⩽ 0$

for any z < x.

Let z → x^-, we obtain

$lim_{| X_{0} - Y_{0} | \to 0} Ψ_{t} (x) - Φ_{t} (x) ⩽ 0 .$ (3.10)

By in equation (3.9) and (3.10), we get $lim_{| X_{0} - Y_{0} | \to 0} Φ_{t} (x) - Ψ_{t} (x) = 0 .$

It follows that $lim_{| X_{0} - Y_{0} | \to 0} | Φ_{t} (x) - Ψ_{t} (x) | = 0 .$

From Definition 3.2, the multifactor uncertain population model is stable in distribution.

4 Parameter estimation

How to estimate unknown parameters based on some observed data is a key problem in the practical application of the multifactor uncertain population model. Liu and Liu [24] proposed the concept of residual of uncertain differential equations. The parameter estimation methods based on residuals are different from the parameter estimation methods based on difference scheme, and they are suitable when the observation time intervals are not short enough. In multifactor uncertain population model, r, σ₁ and σ₂ are unknown parameters. In this section, we define the residual of the multifactor uncertain differential equations, and use a method of moments based on residuals to estimate the unknown parameters.

Assume

$n_{t_{1}}, n_{t_{2}}, \dots, n_{t_{n}}$ (4.1) are the observed values of the uncertain process N_t at the times t₁, t₂, ⋯ , t_n with t₁ < t₂ < ⋯ < t_n, respectively. For any given index i with 2 ⩽ i ⩽ n, the updated multifactor uncertain differential equation is

$\begin{matrix} {dN}_{t} = {rN}_{t} dt + σ_{1} N_{t} {dC}_{1 t} - σ_{2} N_{t} {dC}_{2 t} \\ N_{t_{i - 1}} = n_{t_{i - 1}}, \end{matrix}$ (4.2) where N_{t
_i-1} is the new initial value of the new initial time t_i-1. The uncertainty distribution of N_{t
_i} is thus obtained and represented by Φ_{t
_i}. Thus, the uncertainty distribution of N_{t
_i} is obtained, denoted by Φ_{t
_i}. For any x and 0 < x < 1, we have

$\begin{matrix} M {Φ_{t_{i}} (N_{t_{i}}) ⩽ x} = M {N_{t_{i}} ⩽ Φ_{t_{i}}^{- 1} (x)} \\ = Φ_{t_{i}} (Φ_{t_{i}}^{- 1} (x)) \\ = x . \end{matrix}$ (4.3)

So Φ_{t
_i} (N_{t
_i}) is always going to be a linear uncertain variable. Substitute N_{t
_i} with the observed value n_{t
_i}, and write

$ɛ_{i} = Φ_{t_{i}} (n_{t_{i}}) .$ (4.4)

Then ɛ_i is the sample of linear uncertainty distribution L (0, 1).

Definition 4.1. For each index i and 2 ⩽ i ⩽ n, the term ɛ_i defined by (4.3) is called the ith residual of multifactor uncertain population model (2.8) corresponding to the observed data (4.1).

We obtain the uncertainty distribution of N_{t
_i} as follows,

$\begin{matrix} Φ_{t_{i}} (x) = \\ {(1 + exp (\frac{π (ln x_{t_{i - 1}} + r (t_{i} - t_{i - 1}) - ln x)}{\sqrt{3} (σ_{1} + σ_{2}) (t_{i} - t_{i - 1})}))}^{- 1} . \end{matrix}$ (4.5)

It follows from definition 4.1, that the ith residual is

$\begin{matrix} ɛ_{i} = \\ {(1 + exp (\frac{π (ln n_{t_{i - 1}} + r (t_{i} - t_{i - 1}) - ln n_{t_{i}})}{\sqrt{3} (σ_{1} + σ_{2}) (t_{i} - t_{i - 1})}))}^{- 1} . \end{matrix}$ (4.6)

In order to calculate unknown parameters more conveniently, we can let ν₁ be the weight of unknown parameter σ₁ and ν₂ be the weight of unknown parameter σ₂ satisfying ν₁ + ν₂ = 1. Then we replace every unknown parameter σ₁ and σ₂ with ν₁σ₁ + ν₂σ₂. So the multifactor uncertain population model can be approximately written as

$\begin{matrix} {dN}_{t} = {rN}_{t} dt + (ν_{1} σ_{1} + ν_{2} σ_{2}) N_{t} {dC}_{1 t} + \\ (ν_{1} σ_{1} + ν_{2} σ_{2}) N_{t} {dC}_{2 t} . \end{matrix}$ (4.7)

We solve the updated multifactor uncertain population model and obtain the uncertainty distribution of N_{t
_i} as follows,

$\begin{matrix} Φ_{t_{i}} (x) = \\ {(1 + exp (\frac{π (ln x_{t_{i - 1}} + r (t_{i} - t_{i - 1}) - ln x}{\sqrt{3} (ν_{1} σ_{1} + ν_{2} σ_{2}) (t_{i} - t_{i - 1})}))}^{- 1} . \end{matrix}$ (4.8)

For i = 2, 3, ⋯ , n, we obtain n - 1 residuals of

$\begin{matrix} ɛ_{i} = \\ {(1 + exp (\frac{π (ln n_{t_{i - 1}} + r (t_{i} - t_{i - 1}) - ln n_{t_{i}}}{\sqrt{3} (ν_{1} σ_{1} + ν_{2} σ_{2}) (t_{i} - t_{i - 1})}))}^{- 1}, \end{matrix}$ (4.9) the n - 1 residuals can be regarded as samples of the linear uncertainty distribution L (0, 1), i.e.,

$ɛ_{2}, ɛ_{3}, \dots, ɛ_{n} \sim L (0, 1) .$ (4.10)

For each positive integer k, the kth sample moment of the n - 1 residuals ɛ₂, ɛ₃, ⋯ , ɛ_n is

$\frac{1}{n - 1} \sum_{i = 2}^{n} ɛ_{i}^{k} (r, σ_{1}, σ_{2}),$ (4.11) and the kth population moment of the linear uncertainty distribution L (0, 1) is

$\frac{1}{k + 1} .$ (4.12)

Since the number of unknown parameters is 3 and the first three moments of the linear uncertainty distribution L (0, 1) are 1/ - 2, 1/ - 3, and 1/ - 4, according to the work of Liu and Liu [24], the moment estimation of (r, σ₁, σ₂) is the solution of the system of equations (4.13),

${\begin{matrix} \frac{1}{n - 1} \sum_{i = 2}^{n} ɛ_{i} (r, σ_{1}, σ_{2}) = \frac{1}{2}, \\ \frac{1}{n - 1} \sum_{i = 2}^{n} ɛ_{i}^{2} (r, σ_{1}, σ_{2}) = \frac{1}{3} \\ \frac{1}{n - 1} \sum_{i = 2}^{n} ɛ_{i}^{3} (r, σ_{1}, σ_{2}) = \frac{1}{4} . \end{matrix},$ (4.13)

In order to test whether the multifactor uncertain population model (4.7) fits the observed data (4.1), we need to test whether the linear uncertainty distribution L (0, 1) fits the n - 1 residuals. Therefore, Ye and Liu [25] suggested uncertain hypothesis testing. Given a significance level α, the test is

$\begin{matrix} W = {(z_{2}, z_{3}, \dots, z_{n}) : there are more than α of \\ indexes s with i^{,} 2 ⩽ i ⩽ n such that z_{i} < \frac{α}{2} or z_{i} \\ > 1 - \frac{α}{2}} . \end{matrix}$ (4.14)

If the vector of the n - 1 residuals ɛ₂, ɛ₃, ⋯ , ɛ_n belongs to the test W, i.e., $ɛ_{2}, ɛ_{3}, \dots, ɛ_{n} \in W,$

then the multifactor uncertain population model (4.7) is not a good fit to the observed data (4.1). If $ɛ_{2}, ɛ_{3}, \dots, ɛ_{n} \notin W,$

then the multifactor uncertain population model fits the observed data (4.1) well.

5 An example: the American population

In order to further stimulate the application of the multifactor uncertain population model, this section models the American population by multifactor uncertain population model. Table 1 shows the demographic data of the United States from 1790 to 2000. Denote the year from 1790 to 2000 by i = 1, 2, ⋯ , 22, and let N_{t
_i} represent the American population in Table 1 and Fig. 1.

Table 1
American populations (millions) from 1790 to 2000

i t _i N _{t
_i} i t _i N _{t
_i} i t _i N _{t
_i}

1 0 3.900 9 80 38.558 17 160 151.33

2 10 5.308 10 90 50.189 18 170 167.32

3 20 7.240 11 100 62.980 19 180 203.3

4 30 9.638 12 110 76.212 20 190 226.54

5 40 12.861 13 120 92.228 21 200 248.71

6 50 17.064 14 130 106.02 22 210 281.42

7 60 23.192 15 140 123.20

8 70 31.443 16 150 132.16

i	t _i	N _{t _i}	i	t _i	N _{t _i}	i	t _i	N _{t _i}
1	0	3.900	9	80	38.558	17	160	151.33
2	10	5.308	10	90	50.189	18	170	167.32
3	20	7.240	11	100	62.980	19	180	203.3
4	30	9.638	12	110	76.212	20	190	226.54
5	40	12.861	13	120	92.228	21	200	248.71
6	50	17.064	14	130	106.02	22	210	281.42
7	60	23.192	15	140	123.20
8	70	31.443	16	150	132.16

Fig. 1

The American populations from 1790 to 2000.

Assume N_t is an uncertain process and follows the multifactor uncertain population model $\begin{matrix} \begin{matrix} {dN}_{t} = {rN}_{t} dt + (ν_{1} σ_{1} + ν_{2} σ_{2}) N_{t} {dC}_{1 t} + \\ (ν_{1} σ_{1} + ν_{2} σ_{2}) N_{t} {dC}_{2 t} . \end{matrix} \end{matrix}$

There are three parameters r, σ₁ and σ₂ to be estimated. Let $ν_{1} = \frac{1}{3}, ν_{2} = \frac{2}{3}$ , we obtain 21 residuals

$\begin{matrix} ɛ_{i} = \\ {(1 + exp (\frac{π (ln n_{t_{i - 1}} + r (t_{i} - t_{i - 1}) - ln n_{t_{i}}}{\sqrt{3} (\frac{1}{3} σ_{1} + \frac{2}{3} σ_{2}) (t_{i} - t_{i - 1})}))}^{- 1} \\ (i = 2, \dots, 22) . \end{matrix}$ (5.1)

Use the equation (4.13), we get the following equations

${\begin{matrix} \frac{1}{21} \sum_{i = 2}^{22} ɛ_{i} (r, σ_{1}, σ_{2}) = \frac{1}{2}, \\ \frac{1}{21} \sum_{i = 2}^{22} ɛ_{i}^{2} (r, σ_{1}, σ_{2}) = \frac{1}{3}, \\ \frac{1}{21} \sum_{i = 2}^{22} ɛ_{i}^{3} (r, σ_{1}, σ_{2}) = \frac{1}{4} . \end{matrix}$ (5.2)

Solving the above system of equations, we obtain

$(r, σ_{1}, σ_{2}) = (0.2041, 0.0586, - 0.1818)$ (5.3)

Thus we obtain the multifactor uncertain population model

${dN}_{t} = 0.2041 N_{t} dt + 0.0586 N_{t} {dC}_{1 t} + 0.1818 N_{t} {dC}_{2 t} .$ (5.4)

Substituting r, σ₁ and σ₂ into (4.9), we can get 21 residuals ɛ₂, ɛ₃, ⋯, ɛ₂₂ of the multifactor uncertain population model corresponding to the American populations. See Fig. 2.

We divide the 21 residuals into two parts, i.e.,

Fig. 2

Residual plot corresponding to American populations.

$ɛ_{2}, ɛ_{3}, \dots, ɛ_{11} and ɛ_{12}, ɛ_{13}, \dots, ɛ_{22} .$

By the two-sample Kolmogorov-Smirnov test, it is shown that the two parts in the above residuals do not come from the same population. So the residuals ɛ₂, ɛ₃, ⋯ , ɛ₂₂ are not white noise in the sense of probability theory. This is why we use uncertain differential equation instead of stochastic differential equation.

Finally, let us test whether the multifactor uncertain population model fits American populations n_{t
₁}, n_{t
₂}, ⋯ , n_{t
₂₂}. Given a significance level α = 0.05, it follows from α × 21 = 1.05 and the test is $\begin{matrix} W = {(z_{2}, z_{3}, \dots, z_{22}) : there are at least 1 of \\ indexes i^{,} s with 2 ⩽ i ⩽ 22 such that z_{i} < 0.025 \\ or z_{i} > 0.975} . \end{matrix}$

Since all residuals are in [0.025, 0.975], we have ɛ₂, ɛ₃, ⋯ , ɛ₂₂ ∉ W. Thus the multifactor uncertain population model is a good fit to the American populations.

6 Conclusion

Noting the paradox of stochastic population models, this paper develops a multifactor uncertain population model based on a multifactor uncertain differential equation to better describe the fact that the birth and death rates are sometimes affected by different, independent uncertain noises. It broadens the application of uncertain differential equations in the field of population. First, the inverse uncertainty distribution of the solution for the multifactor uncertain population model is obtained. Secondly, applying the uncertainty analysis method, the stability in measure and stability in distribution of the multifactor uncertain population model are studied respectively, and the relationship between the two kinds of stability is further discussed. Then, a method of moment estimation based on residuals for unknown parameters in the multifactor uncertain population model is derived. Finally, a numerical example related to American population from 1790 to 2000 is given to verify the validity and rationality of our parameter estimation method.

Footnotes

Acknowledgments

This research is supported by Fundamental Research Program of Shanxi Province Scientific No. 20210302124310 and Basic Research Foundation of Shanxi Datong University No. 2022K24.

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