Application of non-equidistant GM(1,1) model based on the fractional-order accumulation in building settlement monitoring

Abstract

Non-equidistant GM(1,1) (abbreviated as NEGM) model is widely used in building settlement prediction because of its high accuracy and outstanding adaptability. To improve the building settlement prediction accuracy of the NEGM model, the fractional-order non-equidistant GM(1,1) model (abbreviated as FNEGM) is established in this study. In the modeling process of the FNEGM model, the fractional-order accumulated generating sequence is extended based on the first-order accumulated generating sequence, and the optimal parameters that increase the prediction precision of the model are obtained by using the whale optimization algorithm. The FNEGM model and the other two grey prediction models are applied to three cases, and five prediction performance indexes are used to evaluate the prediction precision of the three models. The results show that the FNEGM model is more suitable for predicting the settlement of buildings than the other two grey prediction models.

Keywords

Non-equidistant GM(1 1) model fractional-order accumulation grey prediction model

1 Introduction

In the construction and service period, buildings will have different degrees of settlement due to load increase or self-weight consolidation of foundation soil layers. The settlement of structures often leads to floor subsidence, cracking, pipeline deformation, and even fracture. When the settlement is too large and uneven, buildings will incline, which will affect the service safety of buildings. Scientific and reasonable monitoring means and prediction models are important to building safety assessments. Many methods are being used for predicting settlement, such as multi-point network settlement monitoring system and multi-point scanning settlement monitoring system [1], machine learning approach [2], the three-point hyperbolic combination model [3], the iterative prediction methodology [4] and so on.

In fact, the monitoring data are often incomplete due to the influence of hydrogeology, climate conditions, monitoring means, and other factors, which shows that it is suitable for monitoring data to be dealt with by grey system theory [5]. In recent years, GM(1,1) model in grey system theory has become a significant method for building settlement prediction. Generally, the traditional GM(1,1) model is based on the equidistant sequence. This means that the settlement data need to be equidistant. In fact, most of the settlement data do not meet this condition. Thus, non-equidistant GM(1,1) (abbreviated as NEGM) model is needed to deal with settlement data.

Now, the NEGM model plays an important role in settlement prediction [6 –8]. However, as a nascent model, it has some defects. In recent years, many scholars have put forward some improved NEGM models, such as the model with appropriate morphing parameters [9], the NEGM model with improved initial conditions, and the optimized whitening equation [10], robust weighted non-equidistant GM(1,1) model [11] and so on. They all have been applied to the settlement prediction successfully.

These studies greatly promote the application of the grey prediction model in building settlement. In fact, most of the optimization measures are very limited, and they can not be widely used in various cases. The fractional-order accumulation idea arises at the historic moment [12], whose purpose is to extend the accumulated order of the grey prediction model. The idea makes the accumulated order no longer limited to positive integers and enhances the generalizability of the model [13]. Thus, it is very effective to improve the accuracy of the grey prediction model.

At present, GM(1,1) model based on the non-equidistant idea mostly stays in the cases of positive integer order accumulation, the study of fractional-order accumulation GM(1,1) model has not received considerable attention. Aiming at the non-equidistant characteristics of building settlement data and the superiority of fractional-order accumulated GM(1,1) model, the fractional-order non-equidistant GM(1,1) (abbreviated as FNEGM) model with good generalizability and adaptability is proposed in the present paper.

The rest of the present paper is arranged as follows. Section 2 is literature review on grey prediction model. Section 3 specifically studies the proposed FNEGM model. Section 4 takes three cases to verify the effectiveness of the proposed FNEGM model. Section 5 comes to conclusions.

2 Literature review

Owing to both the existence of internal and external disturbances of information system and the understanding limitation of people, the available information tends to contain various kinds of uncertainty and noises [14, 15]. To describe and deal with the problems involving small samples and poor information in uncertain information systems, Chinese scholar Professor Julong Deng [5] proposed the grey system theory in 1982. In grey system theory, the dynamics behaviors and evolution laws of uncertain systems characterized by incomplete and inadequate information can be correctly described and effectively monitored through generating, excavating and extracting available information. The main components of grey systems theory contain grey number theory, sequence operator, grey incidence model, grey cluster evaluation model, grey prediction, grey decision model, grey game models, grey input and output and combined grey models, etc [16, 17].

In the grey system theory, the theory and application of grey prediction model is one of the most active research branches. With the advantage of a simple modeling process and low demands regarding the distribution pattern of the data sequence, the grey prediction model has been widely applied in many fields, such as energy [18, 19], economy [20], society [21, 22] and sustainability [23]. The fundamental principle of grey prediction model is to generate a new data sequence through the first-order accumulated generating operator of uncertain (no obvious trends) observation data and to then establish the corresponding grey differential equation to solve [5]. The general form of the grey prediction models is called the GM(h,n) model, where the “h” represents the order of the grey model and “n” represents the number of variables. According to the modelling techniques, the existing grey prediction models can be simply categorized in two classes: the continuous grey models and the discrete grey models.

The GM(1,1) is the simplest continuous grey model. Due to its effectiveness of time series forecasting with small samples, the model is very popular in the applications. Meanwhile, the GM(1,1) also represents the main methodology of the traditional grey models, and many variants have been developed using the similar modelling procedures of the GM(1,1) by researchers to promote model forecasting accuracy. For the initial condition of model, Xiong et al. [24] proposed a novel GM(1,1) model based on modifying the initial value. For original data processing, Zhang et al. [25] proposed improved GM(1,1) model based on function transformation criterion to reduce stepwise ratio variance. For oscillatory sequence, Zeng et al. [26] deduces a novel grey prediction model by using a smoothing algorithm which can compress the amplitude of the oscillation sequence. For background value building, Zhang et al. [27] established an improved GM(1,1) model by using variable weight to construct the background value. On the problem of fractional order, Wu et al. [12, 28] first proposed fractional accumulation GM(1,1) model(the FAGM(1,1) model). Xia et al. [29] analyzed the error of traditional FAGM(1,1) model, improved the background value of FAGM (1,1) model by using Simpson integral formula, established SFAGM(1,1) model and determined the optimal order of the model by genetic algorithm. Zeng [30] proposed a fractional order opposite-direction accumulative grey prediction model with time-power. Aiming at the unequal interval data sequence existed, Zhang et al. [31] used the integral median theorem to optimize the background value of the non-equidistant GM(1,1) model, and used the optimized model to predict the settlement of the building.

3 Mathematical model

3.1 The basic theory of fractional-order accumulated generation

Supposing there is a non-negative original sequence X⁽⁰⁾ = {x⁽⁰⁾ (1) , x⁽⁰⁾ (2) , ⋯ , x⁽⁰⁾ (n)} , n ⩾ 4, if assumed

${\begin{matrix} x^{(1)} (1) = x^{(0)} (1) \\ x^{(1)} (k) = x^{(0)} (k) + x^{(1)} (k - 1), k = 2, 3, \dots, n \end{matrix}$ (1) then sequence X⁽¹⁾ = {x⁽¹⁾ (1) , x⁽¹⁾ (2) , ⋯ , x⁽¹⁾ (n)} is called as the first order accumulated generating sequence of X⁽⁰⁾.

Expression (1) can be expressed in matrix form as $\begin{matrix} X^{(1)} = X^{(0)} A, A = [\begin{matrix} 1 & 1 & \dots & 1 \\ 0 & 1 & \dots & 1 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}] \end{matrix}$ where X⁽¹⁾ is called as the first order accumulated generating sequence of X⁽⁰⁾, and A is called as the first order accumulated generating matrix.

And then, we can get $\begin{matrix} X^{(0)} = X^{(1)} A^{- 1} \end{matrix}$ where the k term of X⁽¹⁾ can be expressed as $\begin{matrix} x^{(1)} (k) = \sum_{i = 1}^{k} x^{(0)} (i), k = 1, 2, \dots, n \end{matrix}$

As an analogy, the first order accumulated generating sequence of X⁽¹⁾ is called as the second order accumulated generating sequence of X⁽⁰⁾, and A² is called as the second order accumulated generating matrix. That is $\begin{matrix} X^{(2)} & = X^{(1)} A = X^{(0)} A^{2}, A^{2} \\ = [\begin{matrix} 1 & 2 & \dots & C_{2 + n - 2}^{n - 1} \\ 0 & 1 & \dots & C_{2 + n - 3}^{n - 2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}] \end{matrix}$

And then, we can get $\begin{matrix} X^{(0)} = X^{(2)} (A^{2})^{- 1} \end{matrix}$

Where the k term of X⁽²⁾ can be expressed as $\begin{matrix} x^{(2)} (k) = \sum_{i = 1}^{k} C_{k - i + 1}^{k - i} x^{(0)} (i), k = 1, 2, \dots, n \end{matrix}$

By analogy, the first order accumulated generating sequence of X^(N-1) is called as the n-th order accumulated generating sequence of X⁽⁰⁾, and A^N is called as the n-th order accumulated generating matrix. That is $\begin{matrix} X^{(N)} & = X^{(N - 1)} A = X^{(0)} A^{N}, A^{N} \\ = [\begin{matrix} 1 & C_{N}^{1} & C_{N + 1}^{2} & \dots & C_{N + n - 2}^{n - 1} \\ 0 & 1 & C_{N}^{1} & \dots & C_{N + n - 3}^{n - 2} \\ 0 & 0 & 1 & \dots & C_{N + n - 4}^{n - 3} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{matrix}] \end{matrix}$

And then, we can get $\begin{matrix} X^{(0)} = X^{(N)} (A^{N})^{- 1} \end{matrix}$

Where the k term of X^(N) can be expressed as $\begin{matrix} x^{(N)} (k) = \sum_{i = 1}^{k} C_{k - i + N - 1}^{k - i} x^{(0)} (i), k = 1, 2, \dots, n . \end{matrix}$

According to the generalized definition of combinatorial number [32], we generalize A^N to obtain fractional-order accumulated generating matrix A^r (r > 0). A^r is called as the r-th order accumulated generating matrix, and the first order accumulated generating sequence of X^(r-1) is called as the r-th order accumulated generating sequence of X⁽⁰⁾. That is $\begin{matrix} X^{(r)} & = X^{(0)} A^{r}, A^{r} \\ = [\begin{matrix} 1 & C_{r}^{1} & C_{r + 1}^{2} & \dots & C_{r + n - 2}^{n - 1} \\ 0 & 1 & C_{r}^{1} & \dots & C_{r + n - 3}^{n - 2} \\ 0 & 0 & 1 & \dots & C_{r + n - 4}^{n - 3} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \end{matrix}] \end{matrix}$

And then, we can get $\begin{matrix} X^{(0)} = X^{(r)} (A^{r})^{- 1} \end{matrix}$

Where the k term of X^(r) can be expressed as

$x^{(r)} (k) = \sum_{i = 1}^{k} C_{k - i + r - 1}^{k - i} x^{(0)} (i), k = 1, 2, \dots, n$ (2)

For example, if r = 1.4, then $\begin{matrix} A^{1.4} = [\begin{matrix} 1 & 1.4 & 1.68 & 1.904 \\ 0 & 1 & 1.4 & 1.68 \\ 0 & 0 & 1 & 1.4 \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}$

3.2 Non-equidistant GM (1,1) model (NEGM)

If X⁽⁰⁾ = {x⁽⁰⁾ (k₁) , x⁽⁰⁾ (k₂) , ⋯ , x⁽⁰⁾ (k_n)} is the original sequence, where Δk_i = k_i - k_i-1 ≠ const, i = 2, 3, ⋯ , n, then X⁽⁰⁾ is called non-equidistant sequence.

Let X⁽¹⁾ = {x⁽¹⁾ (k₁) , x⁽¹⁾ (k₂) , ⋯ , x⁽¹⁾ (k_n)}, where $x^{(1)} (k_{i}) = \sum_{j = 1}^{i} x^{(0)} (k_{j}) Δ k_{j}$ , Δk_i = k_i - k_i-1 ≠ const, i = 2, 3, ⋯ , n, then X⁽¹⁾ is called the first order accumulated generating sequence of X⁽⁰⁾.

If X⁽⁰⁾ and X⁽¹⁾ are mentioned above,

$\frac{{dx}^{(1)} (t)}{dt} + {ax}^{(1)} (t) = b$ (3) is called whitening equation of NEGM model.

$x^{(0)} (k_{i}) + {az}^{(1)} (k_{i}) = b$ (4) is called grey differential equation of NEGM model, where $\begin{matrix} z^{(1)} (k_{i}) = \frac{x^{(1)} (k_{i - 1}) + x^{(1)} (k_{i})}{2}, i = 2, 3, \dots, n \end{matrix}$ is called the background value of the NEGM model. Similar to the traditional regression prediction model, the least squares method is used to estimate the parameters of the model, namely $\begin{matrix} (\hat{a}, \hat{b})^{T} = (B^{T} B)^{- 1} B^{T} Y \end{matrix}$

And

$\begin{matrix} {\hat{x}}^{(1)} (k_{i}) \\ = [x^{(1)} (k_{1}) - \frac{\hat{b}}{\hat{a}}] e^{- \hat{a} (k_{i} - k_{1})} + \frac{\hat{b}}{\hat{a}}, i = 1, 2, \dots, n \end{matrix}$ (5) is called the time response function of NEGM model.

Since NEGM model is based on the first-order accumulated generation sequence, it is necessary to carry out first order inverse accumulated generation to the first-order prediction sequence. That is,

${\begin{matrix} {\hat{x}}^{(0)} (k_{1}) = x^{(0)} (k_{1}); \\ {\hat{x}}^{(0)} (k_{i}) = \frac{{\hat{x}}^{(1)} (k_{i}) - {\hat{x}}^{(1)} (k_{i - 1})}{Δ k_{i}}, i = 2, 3, \dots, n \end{matrix}$ (6)

3.3 Fractional-order non-equidistant GM (1,1) model (FNEGM)

Definition 1. if the original sequence is X⁽⁰⁾ = {x⁽⁰⁾ (k₁) , x⁽⁰⁾ (k₂) , ⋯ , x⁽⁰⁾ (k_n)}, where Δk_i = k_i - k_i-1 ≠ const, i = 2, 3, ⋯ , n, then X^(r) = {x^(r) (k₁) , x^(r) (k₂) , ⋯ , x^(r) (k_n)} is called r-order accumulated generating sequence of X⁽⁰⁾, where

$\begin{matrix} x^{(r)} (k_{i}) = \sum_{j = 1}^{i} C_{i - j + r - 1}^{i - j} x^{(0)} (k_{j}) Δ k_{j}, i = 1, 2, \dots, n \end{matrix}$ $\begin{matrix} {\begin{matrix} C_{k - i + r - 1}^{k - i} = \frac{(r + k - i - 1) (r + k - i - 2) \dots (r + 1) r}{(k - i)!} \\ C_{r - 1}^{0} = 1, C_{k - 1}^{k} = 0, k = 1, 2, \dots, n \end{matrix} \end{matrix}$

The nearest mean generating sequence of X^(r) is Z^(r) = {z^(r) (k₁) , z^(r) (k₂) , ⋯ , z^(r) (k_n)} , where

$z^{(r)} (k_{i}) = \frac{x^{(r)} (k_{i - 1}) + x^{(r)} (k_{i})}{2}, i = 2, 3, \dots, n$ (7)

The following differential equations are constructed for the sequence X^(r)

$\frac{{dx}^{(r)} (t)}{dt} + {ax}^{(r)} (t) = b$ (8)

By integrating both sides of Eq. (8) in the interval [k_i-1, k_i] , i = 2, 3, ⋯ , n, we get $\begin{matrix} \int_{k_{i - 1}}^{k_{i}} \frac{{dx}^{(r)} (t)}{dt} + \int_{k_{i - 1}}^{k_{i}} {ax}^{(r)} (t) dt = \int_{k_{i - 1}}^{k_{i}} bdt \end{matrix}$

That is $\begin{matrix} x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1}) + a \int_{k_{i - 1}}^{k_{i}} x^{(r)} (t) dt = b Δ k_{i} \end{matrix}$

In the interval [k_i-1, k_i], trapezoid area z^(r) (k_i) Δk_i is used to replace curved edge trapezoid area $\int_{k_{i - 1}}^{k_{i}} x^{(r)} (t) dt$ , and then we get $\begin{matrix} x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1}) + {az}^{(r)} (k_{i}) Δ k_{i} = b Δ k_{i} \end{matrix}$

That is

$\frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}} + {az}^{(r)} (k_{i}) = b, i = 2, 3, \dots, n$ (9)

Definition 2. The Eq. (9) is called fractional-order non-equidistant GM (1,1) model, abbreviated as FNEGM, and Eq. (8) is the albino form of FNEGM model.

Theorem 1. If X^(r), Z^(r) are mentioned above, then the matrix form of the least squares estimation of FNEGM model parameters a and b is $\hat{Φ} = (\hat{a}, \hat{b})^{T} = (B^{T} B)^{- 1} B^{T} Y$ , where $\begin{matrix} B = [\begin{matrix} - z^{(r)} (k_{2}) & 1 \\ - z^{(r)} (k_{3}) & 1 \\ ⋮ & ⋮ \\ - z^{(r)} (k_{n}) & 1 \end{matrix}], \\ Y = [\begin{matrix} [x^{(r)} (k_{2}) - x^{(r)} (k_{1})] / Δ k_{2} \\ [x^{(r)} (k_{3}) - x^{(r)} (k_{2})] / Δ k_{3} \\ ⋮ \\ [x^{(r)} (k_{n}) - x^{(r)} (k_{n - 1})] / Δ k_{n} \end{matrix}] \end{matrix}$

Then the least square estimation of the identification parameters in the FNEGM model is $\begin{matrix} {\begin{matrix} \hat{a} = \frac{(n - 1) \cdot [- \sum_{i = 2}^{n} z^{(r)} (k_{i}) Q (i)] + \sum_{i = 2}^{n} z^{(r)} (k_{i}) \sum_{i = 2}^{n} Q (i)}{(n - 1) \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} - [\sum_{i = 2}^{n} z^{(r)} (k_{i})]^{2}}, Q (i) = \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}} \\ \hat{b} = \frac{\sum_{i = 2}^{n} z^{(r)} (k_{i}) \cdot [- \sum_{i = 2}^{n} z^{(r)} (k_{i}) Q (i)] + \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} \sum_{i = 2}^{n} Q (i)}{(n - 1) \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} - [\sum_{i = 2}^{n} z^{(r)} (k_{i})]^{2}}, Q (i) = \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}} \end{matrix} \end{matrix}$

Proof. It can be obtained from the formula of least square method that

$\begin{matrix} (\hat{a}, \hat{b})^{T} = (B^{T} B)^{- 1} B^{T} Y \end{matrix}$ because

$\begin{matrix} B^{T} B = [\begin{matrix} \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} & - \sum_{i = 2}^{n} z^{(r)} (k_{i}) \\ - \sum_{i = 2}^{n} z^{(r)} (k_{i}) & n - 1 \end{matrix}] \end{matrix}$ when |B^TB|≠0,

$(B^{T} B)^{- 1} = \frac{(B^{T} B)^{*}}{| B^{T} B |} = \frac{[\begin{matrix} n - 1 & \sum_{i = 2}^{n} z^{(r)} (k_{i}) \\ \sum_{i = 2}^{n} z^{(r)} (k_{i}) & \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} \end{matrix}]}{(n - 1) \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} - [\sum_{i = 2}^{n} z^{(r)} (k_{i})]^{2}}$ and $B^{T} Y = [\begin{matrix} - \sum_{i = 2}^{n} z^{(r)} (k_{i}) \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}} \\ \sum_{i = 2}^{n} \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}} \end{matrix}]$ then $\begin{matrix} (\hat{a}, \hat{b})^{T} = (B^{T} B)^{- 1} B^{T} Y = [\begin{matrix} \frac{(n - 1) \cdot [- \sum_{i = 2}^{n} z^{(r)} (k_{i}) \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}} + \sum_{i = 2}^{n} z^{(r)} (k_{i}) \sum_{i = 2}^{n} \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}}}{(n - 1) \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} - [\sum_{i = 2}^{n} z^{(r)} (k_{i})]^{2}} \\ \frac{\sum_{i = 2}^{n} z^{(r)} (k_{i}) \cdot [- \sum_{i = 2}^{n} z^{(r)} (k_{i}) \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}}] + \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} \sum_{i = 2}^{n} \frac{x^{(r)} (k_{i}) - x^{(r)} (k_{i - 1})}{Δ k_{i}}}{(n - 1) \sum_{i = 2}^{n} [z^{(r)} (k_{i})]^{2} - [\sum_{i = 2}^{n} z^{(r)} (k_{i})]^{2}} \end{matrix}] \end{matrix}$

Theorem 2. If $\hat{a}, \hat{b}$ are described in Theorem 1, then the time response function of FNEGM model is

$\begin{matrix} {\hat{x}}^{(r)} (k_{i}) = [x^{(0)} (k_{1}) - \frac{\hat{b}}{\hat{a}}] e^{- \hat{a} (k_{i} - k_{1})} + \frac{\hat{b}}{\hat{a}}, \\ i = 1, 2, \dots, n \end{matrix}$ (10)

The proposed FNEGM model is constructed based on the fractional-order accumulation sequence, so it is necessary to carry out first order inverse accumulated generating operation for prediction sequence.

That is

${\hat{X}}^{(0)} = {\hat{X}}^{(r)} (A^{r})^{- 1}$ (11)

The proof process is similar to the traditional NEGM model.

3.4 Solution of FNEGM model

The proposed FNEGM model is based on the fractional-order r, which can be established for effective prediction only when the fractional-order r is determined. The best fractional-order r should make the model have the highest accuracy under a given sample. Therefore, we only need to establish an optimization problem whose goal is to minimize the error of the model by changing the value of fractional-order r, and the constraint conditions follow the modeling steps of the proposed model. In this paper, the mean absolute percentage error (MAPE) is chosen as the objective function to evaluate the validity of the model

$\begin{matrix} \underset{0 ⩽ r ⩽ 1}{\arg \min} f (r) \\ = \frac{1}{n - 1} \sum_{i = 2}^{n} | \frac{{\hat{x}}^{(0)} (k_{i}) - x^{(0)} (k_{i})}{x^{(0)} (k_{i})} | \times 100 % \end{matrix}$ (12)

$\begin{matrix} s . t . {\begin{matrix} {\hat{x}}^{(r)} (k_{i}) = [x^{(0)} (k_{1}) - \frac{\hat{b}}{\hat{a}}] e^{- \hat{a} (k_{i} - k_{1})} + \frac{\hat{b}}{\hat{a}}, i = 1, 2, \dots, n \\ Φ = (\hat{a}, \hat{b})^{T} = (B^{T} B)^{- 1} B^{T} Y \end{matrix} \end{matrix}$

The above nonlinear programming problems could be solved by intelligent algorithms. In this paper, we use the whale optimization algorithm [33] to solve it, and then directly get the prediction results and the best parameter r. To clearly understand the process of optimizing parameters for the grey model, the modeling flowchart of the proposed WOA-based FENGM model is presented in Fig. 1 in the study.

Fig. 1

Modeling flowchart of the WOA-based FENGM model.

3.5 Whale optimization algorithm

Inspired by the social behavior of whale groups, Mirijalli and Lewis proposed a new intelligent optimization algorithm called whale optimization algorithm (WOA) in 2016 [33]. In recent years, WOA has been widely used in classification [34], bioinformatics [35], feature selection [36], and image processing [37]. At the same time, WOA is also effective for training complex nonlinear optimization problems such as multilayer perceptron neural networks [38]. In this paper, WOA algorithm is used to solve nonlinear programming problem (12). The main idea and model are described as follows.

The main idea of WOA is to simulate the predatory behavior of humpback whales. When humpback whales fish, they usually surround the fish, and the location of the schools is considered to be the best candidate target at present. The whales then update their positions based on the candidate targets. Mathematically, the behavior of encirclement is expressed as follows. $\begin{matrix} \begin{matrix} \vec{D} = | 2 \vec{r} \cdot {\vec{P}}^{*} (i) - \vec{P} (i) |; \vec{C} = 2 f (i) \cdot \vec{r} - f (i); \\ f (i) = 2 - \frac{2 i}{T}; \vec{P} (i + 1) = {\vec{P}}^{*} (i) - \vec{C} \cdot \vec{D} \end{matrix} \end{matrix}$ where $\vec{P} (i)$ is the current position of humpback whale, ${\vec{P}}^{*} (i)$ is the best current position of humpback whale, the vector $\vec{r}$ is randomly generated in the interval [0, 1], and T is the maximum number of iterations. In addition, humpback whales move spirally when they catch prey. In order to simulate the spiral motion, the spiral updated position is expressed as follows: $\begin{matrix} \vec{P} (i + 1) = | {\vec{P}}^{*} (i) - \vec{P} (i) | \cdot e^{β l} \cdot cos (2 π l) + {\vec{P}}^{*} (i) \end{matrix}$ where the coefficient l is a random number in the interval [-1, 1], and β is an arbitrary constant that determines the shape of the spiral motion. However, in the real world, encirclement and spiral motion occur simultaneously. In order to keep it simple in this model, the entire predatory movement of humpback whales is mathematically expressed as follows: $\begin{matrix} \vec{P} (i + 1) \\ = {\begin{matrix} {\vec{P}}^{*} (i) - [2 f (i) \cdot \vec{r} - f (i)] \cdot \vec{D}, if ξ < 0.5 \\ | {\vec{P}}^{*} (i) - \vec{P} (i) | \cdot e^{β l} \cdot cos (2 π l) + {\vec{P}}^{*} (i), if ξ ⩾ 0.5 \end{matrix} \end{matrix}$

In the above formula, ξ is the probability of choosing the motion strategy from the encirclement and spiral motion behaviors. When the norm of $\vec{C}$ is greater than 1, the positions of all whales are updated according to the randomly selected position of the whale rather than the best one. Mathematically, this model can be expressed as follows $\begin{matrix} \vec{P} (i + 1) = {\vec{P}}_{r} (i) - \vec{C} \cdot | 2 {\vec{P}}_{r} (i) \cdot \vec{r} - \vec{P} (i) | \end{matrix}$ where ${\vec{P}}_{r} (i)$ is the location of the whales randomly selected from the humpback herd. According to the principle of humpback whale predation behavior, the position of each whale is updated iteratively until the stop criteria are met.

3.6 Error measure indexes for model prediction performance

In this section, several error measures indexes widely used in the prediction model are considered to illustrate the effectiveness and applicability of the grey model. They are defined as follows (Table 1).

Table 1
Error measure indexes for testing model prediction performance

Index Formula

Absolute percentage error (APE) $APE (k) = | \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} | \times$ 100

Mean absolute error (MAE) $MAE = \frac{1}{n} \sum_{k = 1}^{n} | x^{(0)} (k) - {\hat{x}}^{(0)} (k) |$

Mean squares error (MSE) $MSE = \frac{1}{n} \sum_{k = 1}^{n} [x^{(0)} (k) - {\hat{x}}^{(0)} (k)]^{2}$

Mean absolute percentage error (MAPE) $MAPE = \frac{1}{n} \sum_{k = 1}^{n} | \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} | \times$ 100

Root mean squares percentage error (RMSPE) $RMSPE = \sqrt{\frac{1}{n} \sum_{k = 1}^{n} {(\frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)})}^{2}}$

Index	Formula
Absolute percentage error (APE)	$APE (k) = \| \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} \| \times$ 100
Mean absolute error (MAE)	$MAE = \frac{1}{n} \sum_{k = 1}^{n} \| x^{(0)} (k) - {\hat{x}}^{(0)} (k) \|$
Mean squares error (MSE)	$MSE = \frac{1}{n} \sum_{k = 1}^{n} [x^{(0)} (k) - {\hat{x}}^{(0)} (k)]^{2}$
Mean absolute percentage error (MAPE)	$MAPE = \frac{1}{n} \sum_{k = 1}^{n} \| \frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)} \| \times$ 100
Root mean squares percentage error (RMSPE)	$RMSPE = \sqrt{\frac{1}{n} \sum_{k = 1}^{n} {(\frac{x^{(0)} (k) - {\hat{x}}^{(0)} (k)}{x^{(0)} (k)})}^{2}}$

4 Empirical applications

In this section, three cases will be used to verify the feasibility and effectiveness of the proposed FNEGM model.

4.1 Case 1

The settlement data of the teaching and experimental building of Huaihai Institute of technology is taken as an example, which is presented in literature [39]. The first eight data are selected to establish NEGM model, the improved NEGM model in literature [40] (short for INEGM) and the proposed FNEGM model in this study respectively. The last two data are used to test the prediction performance of the three prediction models.

The time response function of the NEGM model for data in Case 1 $\begin{matrix} \begin{matrix} {\hat{x}}^{(0)} (k_{i}) = 10339.15 [e^{0 . 00104247 (k_{i + 1} - k_{1})} \\ - e^{0 . 00104247 (k_{i} - k_{1})}] / Δ k_{i} i = 1, 2, \dots, n \end{matrix} \end{matrix}$

The time response function of the INEGM model in literature [40] for data in Case 1 $\begin{matrix} \begin{matrix} {\hat{x}}^{(0)} (k_{i}) = 9645.558 [e^{0.00113 (k_{i + 1} - k_{1})} \\ - e^{0.00113 (k_{i} - k_{1})}] / Δ k_{i} i = 1, 2, \dots, n \end{matrix} \end{matrix}$

The time response function of the proposed FNEGM model for data in Case 1 $\begin{matrix} {\hat{x}}^{(0 . 980019)} (k_{i}) = 11530 e^{0 . 00092243 (k_{i} - k_{1})} - 11521, \\ i = 1, 2, \dots, n \end{matrix}$

The fractional-order r = 0.980019 is obtained by the whale algorithm and the prediction results of the model are shown in Table 2 and Fig. 2. The prediction performance of the three prediction models is shown in Table 3 and Fig. 2.

Table 2
Comparative results by three grey models in settlement observation points of teaching and experimental building of Huaihai Institute of Technology in Case 1

No. Time Δk Actual values NEGM INEGM FNEGM

Fitting values APE Fitting values APE Fitting values APE

1 1 – 9.28 9.280 0.000% 9.280 0.000% 9.280 0.000%

2 25 24 10.71 10.914 1.907% 10.942 2.167% 10.762 0.488%

3 53 28 11.31 11.214 0.847% 11.250 0.531% 11.203 0.944%

4 83 30 11.64 11.558 0.701% 11.603 0.314% 11.610 0.261%

5 116 33 12.00 11.944 0.463% 12.000 0.000% 12.004 0.032%

6 147 31 12.23 12.350 0.977% 12.417 1.526% 12.469 1.953%

7 177 30 13.05 12.748 2.311% 12.827 1.708% 12.885 1.262%

8 237 60 13.16 13.362 1.538% 13.460 2.276% 13.160 0.000%

No. Time Δk Actual values Predicting values APE Predicting values APE Predicting values APE

9 269 32 13.61 14.017 2.991% 14.135 3.855% 14.264 4.807%

10 355 86 13.94 14.911 6.963% 15.057 8.015% 14.468 3.787%

No.	Time	Δk	Actual values	NEGM	INEGM	FNEGM
1	1	–	9.28	9.280	0.000%	9.280	0.000%	9.280	0.000%
2	25	24	10.71	10.914	1.907%	10.942	2.167%	10.762	0.488%
3	53	28	11.31	11.214	0.847%	11.250	0.531%	11.203	0.944%
4	83	30	11.64	11.558	0.701%	11.603	0.314%	11.610	0.261%
5	116	33	12.00	11.944	0.463%	12.000	0.000%	12.004	0.032%
6	147	31	12.23	12.350	0.977%	12.417	1.526%	12.469	1.953%
7	177	30	13.05	12.748	2.311%	12.827	1.708%	12.885	1.262%
8	237	60	13.16	13.362	1.538%	13.460	2.276%	13.160	0.000%
No.	Time	Δk	Actual values	Predicting values	APE	Predicting values	APE	Predicting values	APE
9	269	32	13.61	14.017	2.991%	14.135	3.855%	14.264	4.807%
10	355	86	13.94	14.911	6.963%	15.057	8.015%	14.468	3.787%

Table 3

Fitting and predicting performance of the three grey models in Case 1

Fitting	NEGM	INEGM	FNEGM	Predicting	NEGM	INEGM	FNEGM
MAPE	1.2490	1.2190	0.7060	MAPE	4.9770	5.9350	4.2960
RMSPE	0.0200	0.0220	0.0090	RMSPE	0.2870	0.3950	0.1870
MAE	0.1515	0.1484	0.0853	MAE	0.6889	0.8210	0.5910
MSE	0.0295	0.0334	0.0142	MSE	0.5539	0.7617	0.3533

Fig. 2

Fitting and prediction results for the three grey models in Case 1.

Fig. 3

Fitting and prediction results for the three grey models in Case 2.

From the fitting results shown in Table 1 and Fig. 2, the fitting results of the three prediction models have little difference. From the prediction results, the three prediction models overestimate the settlement. From the fitting indicators and the prediction indicators shown in Table 2 and Fig. 2, the FNEGM model is better than other two grey models. To sum up, the FNEGM model proposed in this study shows better prediction performance than other two grey models in this case.

4.2 Case 2

The settlement data of D15 point on the south side of foundation pit of Zhuque building in Xi’an City proposed in literature [41] is taken as an example. The first seven data are selected to establish NEGM model, the improved NEGM model in literature [40] (short for INEGM) and the proposed FNEGM model in this study respectively. The last three data are used to test the prediction performance of the three prediction models.

The time response function of NEGM model for data in Case 2 $\begin{matrix} \begin{matrix} {\hat{x}}^{(0)} (k_{i}) = 135.5315 [e^{0.01061 (k_{i + 1} - k_{1})} \\ - e^{0.01061 (k_{i} - k_{1})}] / Δ k_{i} i = 1, 2, \dots, n \end{matrix} \end{matrix}$

The time response function of improved NEGM model in literature [40] (short for INEGM) for data in Case 2 $\begin{matrix} \begin{matrix} {\hat{x}}^{(0)} (k_{i}) = 135.3695 [e^{0.01058 (k_{i + 1} - k_{1})} \\ - e^{0.01058 (k_{i} - k_{1})}] / Δ k_{i} i = 1, 2, \dots, n \end{matrix} \end{matrix}$

The time response function of the proposed FNEGM model for data in Case 2 $\begin{matrix} {\hat{x}}^{(0.852608)} (k_{i}) = 181.26 e^{0.0075 (k_{i} - k_{1})} - 180.15, \\ i = 1, 2, \dots, n \end{matrix}$

The fractional-order r = 0.852608 obtained by the whale algorithm and the prediction results of the model are shown in Table 4 and Fig. 3. The prediction performances of the three grey models are shown in Table 5 and Fig. 3.

Table 4
Comparative results by three grey models in D15 settlement point on the south side of foundation pit of Zhuque building in Xi’an City in Case 2

No. Time Δk Actual values NEGM INEGM FNEGM

Fitting values APE Fitting values APE Fitting values APE

1 1 – 1.11 1.110 0.000% 1.110 0.000% 1.110 0.000%

2 16 15 1.45 1.558 7.454% 1.535 5.841% 1.450 0.000%

3 32 16 1.93 1.837 4.832% 1.806 6.413% 1.821 5.629%

4 44 12 2.08 2.130 2.386% 2.091 0.539% 2.271 9.164%

5 58 14 2.52 2.445 2.974% 2.398 4.854% 2.459 2.404%

6 72 14 2.92 2.836 2.862% 2.777 4.884% 2.809 3.811%

7 85 13 3.20 3.273 2.270% 3.200 0.000% 3.225 0.792%

No. Time Δk Actual values Predicting values APE Predicting values APE Predicting values APE

8 128 43 3.96 4.439 12.095% 4.327 9.270% 3.286 17.012%

9 141 13 4.38 5.927 35.321% 5.762 31.544% 5.807 32.570%

10 155 14 4.68 6.840 46.161% 6.640 41.880% 5.856 25.135%

No.	Time	Δk	Actual values	NEGM	INEGM	FNEGM
1	1	–	1.11	1.110	0.000%	1.110	0.000%	1.110	0.000%
2	16	15	1.45	1.558	7.454%	1.535	5.841%	1.450	0.000%
3	32	16	1.93	1.837	4.832%	1.806	6.413%	1.821	5.629%
4	44	12	2.08	2.130	2.386%	2.091	0.539%	2.271	9.164%
5	58	14	2.52	2.445	2.974%	2.398	4.854%	2.459	2.404%
6	72	14	2.92	2.836	2.862%	2.777	4.884%	2.809	3.811%
7	85	13	3.20	3.273	2.270%	3.200	0.000%	3.225	0.792%
No.	Time	Δk	Actual values	Predicting values	APE	Predicting values	APE	Predicting values	APE
8	128	43	3.96	4.439	12.095%	4.327	9.270%	3.286	17.012%
9	141	13	4.38	5.927	35.321%	5.762	31.544%	5.807	32.570%
10	155	14	4.68	6.840	46.161%	6.640	41.880%	5.856	25.135%

Table 5

Fitting and predicting performance of the three prediction models in Case 2

Fitting	NEGM	INEGM	FNEGM	Predicting	NEGM	INEGM	FNEGM
MAPE	3.8010	3.7590	3.6390	MAPE	31.1900	27.567	24.909
RMSPE	0.1780	0.2060	0.2290	RMSPE	11.7470	9.4510	6.6090
MAE	0.0805	0.0808	0.0828	MAE	1.3953	1.2363	1.0923
MSE	0.0068	0.0097	0.0108	MSE	2.4294	1.9621	1.2912

From the fitting and predicting results shown in Table 3 and Fig. 3, the fitting results of the three prediction models have little difference; from the prediction results, the three prediction models overestimate the settlement. From the prediction indicators shown in Table 4 and Fig. 3, the FNEGM is better than other two models. To sum up, the FNEGM model proposed in this study is more suitable than other two grey models in this case.

4.3 Case 3

The settlement monitoring data of buildings around the foundation pit of China (Jiangmen) international green light source Expo and trading center proposed in literature [42] is taken as an example. The first nine data are selected to establish NEGM model, the improved NEGM model in literature [40] (short for INEGM) and the proposed FNEGM model in this study respectively. The last four data are used to test the prediction performance of the three prediction models.

The time response function of NEGM model for data in Case 3 $\begin{matrix} \begin{matrix} {\hat{x}}^{(0)} (k_{i}) = 26853.62 [e^{0.00294 (k_{i + 1} - k_{1})} \\ - e^{0.00294 (k_{i} - k_{1})}] / Δ k_{i} i = 1, 2, \dots, n \end{matrix} \end{matrix}$

The time response function of improved NEGM model in literature [40] (short for INEGM) for data in Case 3 $\begin{matrix} \begin{matrix} {\hat{x}}^{(0)} (k_{i}) = 26909.96 [e^{0.00294 (k_{i + 1} - k_{1})} \\ - e^{0.00294 (k_{i} - k_{1})}] / Δ k_{i} i = 1, 2, \dots, n \end{matrix} \end{matrix}$

The time response function of the proposed FNEGM model for data in Case 3 $\begin{matrix} {\hat{x}}^{(0.996945)} (k_{i}) = 31524 e^{0.0025 (k_{i} - k_{1})} - 31446, \\ i = 1, 2, \dots, n \end{matrix}$

The fractional-order r = 0.996945 is solved by the whale algorithm and the prediction results of the model are shown in Table 6 and Fig. 4. The prediction performances of the three grey models are shown in Table 7 and Fig. 4.

Table 6
Comparative results by three grey models in the settlement point of buildings around the foundation pit of the international green light source Expo Trading Center in Case 3

No. Time Δk Actual values NEGM INEGM FNEGM

Fitting values APE Fitting values APE Fitting values APE

1 1 – 78.6 78.600 0.000% 78.600 0.000% 78.600 0.000%

2 3 2 78.4 79.287 1.131% 79.393 1.266% 79.221 1.047%

3 5 2 80.3 79.755 0.678% 79.861 0.546% 79.799 0.623%

4 8 3 80.7 80.345 0.440% 80.451 0.309% 80.266 0.538%

5 10 2 81.1 80.938 0.200% 81.045 0.068% 81.098 0.002%

6 12 2 81.5 81.416 0.103% 81.523 0.028% 81.499 0.002%

7 15 3 82.1 82.017 0.101% 82.125 0.030% 81.826 0.334%

8 17 2 82.6 82.623 0.028% 82.731 0.158% 82.700 0.121%

9 19 2 82.8 83.111 0.376% 83.219 0.506% 83.079 0.337%

No. Time Δk Actual values Predicting values APE Predicting values APE Predicting values APE

10 22 3 83.7 83.725 0.030% 83.833 0.159% 83.352 0.416%

11 25 3 83.8 84.468 0.797% 84.576 0.926% 84.072 0.324%

12 29 4 84.0 85.343 1.599% 85.452 1.728% 84.699 0.832%

13 31 2 84.7 86.099 1.652% 86.209 1.781% 86.017 1.555%

No.	Time	Δk	Actual values	NEGM	INEGM	FNEGM
1	1	–	78.6	78.600	0.000%	78.600	0.000%	78.600	0.000%
2	3	2	78.4	79.287	1.131%	79.393	1.266%	79.221	1.047%
3	5	2	80.3	79.755	0.678%	79.861	0.546%	79.799	0.623%
4	8	3	80.7	80.345	0.440%	80.451	0.309%	80.266	0.538%
5	10	2	81.1	80.938	0.200%	81.045	0.068%	81.098	0.002%
6	12	2	81.5	81.416	0.103%	81.523	0.028%	81.499	0.002%
7	15	3	82.1	82.017	0.101%	82.125	0.030%	81.826	0.334%
8	17	2	82.6	82.623	0.028%	82.731	0.158%	82.700	0.121%
9	19	2	82.8	83.111	0.376%	83.219	0.506%	83.079	0.337%
No.	Time	Δk	Actual values	Predicting values	APE	Predicting values	APE	Predicting values	APE
10	22	3	83.7	83.725	0.030%	83.833	0.159%	83.352	0.416%
11	25	3	83.8	84.468	0.797%	84.576	0.926%	84.072	0.324%
12	29	4	84.0	85.343	1.599%	85.452	1.728%	84.699	0.832%
13	31	2	84.7	86.099	1.652%	86.209	1.781%	86.017	1.555%

Fig. 4

Fitting and prediction results for the three grey models in Case 3.

Table 7

Fitting and predicting performance of the three prediction models in Case 3

Fitting	NEGM	INEGM	FNEGM	Predicting	NEGM	INEGM	FNEGM
MAPE	0.3820	0.3640	0.3760	MAPE	1.0190	1.1490	0.7820
RMSPE	0.0030	0.0030	0.0030	RMSPE	0.0150	0.0180	0.0080
MAE	0.3063	0.2918	0.3015	MAE	0.8588	0.9675	0.6590
MSE	0.1684	0.1797	0.1595	MSE	1.0520	1.2513	0.6045

From the fitting and predicting results shown in Table 5 and Fig. 4, the fitting results of the three prediction models have little difference; from the prediction results, the three prediction models overestimate the settlement value. From the fitting indexes shown in Table 6 and Fig. 4, the FNEGM is better than other two models. To sum up, the FNEGM model proposed in this study is more suitable than other two grey models in this case.

5 Conclusion

Fractional-order non-equidistant GM(1,1) model is established based on the theory introduced in this paper, which improves the prediction accuracy by expanding the range of the accumulated order of the model. The improved model is compared with the other two grey prediction models to verify the feasibility and effectiveness of FNEGM. The results show that the prediction accuracy and fitting accuracy of FNEGM are improved compared with those of NEGM. Therefore, the model proposed in this paper is more suitable for predicting building settlement than other grey prediction models.

The accumulated order of the traditional grey prediction model is 1, while the accumulated order of FNEGM is more extensive. Thus, the advantage of fractional-order non-equidistant GM(1,1) model is its good generalizability and adaptability and high accuracy. The author believes that with the continuous research and efforts of many experts and scholars, an improved model can be established, which can be widely generalized with higher accuracy.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Authorship contribution statement

Jun Zhang: Writing-original draft, Methodology; Yanping Qin: Software, Visualization; Xinyu Zhang: Modification; Gen Che: Writing-review; Xuan Sun: Validation; Huaqiong Duo: Updating draft.

Footnotes

Acknowledgments

The authors are grateful to anonymous for their helpful and constructive comments on this paper. The relevant works done are supported by National Natural Science Foundation of China (No. 32160332), Inner Mongolia Agricultural University High-level Talents Scientific Research Project (No. NDYB2019-35), Natural Science Foundation of Inner Mongolia Autonomous Region, China (No. 2018MS03047).

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